Engineering Statistics Handbook

1.
Exploratory Data Analysis

http://www.itl.nist.gov/div898/handbook/eda/eda.htm[6/27/2012 2:04:03 PM]

1. Exploratory Data Analysis
This chapter presents the assumptions, principles, and techniques necessary
to gain insight into data via EDA--exploratory data analysis.
1. EDA Introduction
1. What is EDA?
2. EDA vs Classical &
Bayesian
3. EDA vs Summary
4. EDA Goals
5. The Role of Graphics
6. An EDA/Graphics Example
7. General Problem Categories
2. EDA Assumptions
1. Underlying Assumptions
2. Importance
3. Techniques for Testing
Assumptions
4. Interpretation of 4-Plot
5. Consequences
3. EDA Techniques
1. Introduction
2. Analysis Questions
3. Graphical Techniques:
Alphabetical
4. Graphical Techniques: By
Problem Category
5. Quantitative Techniques
6. Probability Distributions
4. EDA Case Studies
1. Introduction
2. By Problem Category
Detailed Chapter Table of Contents
References
Dataplot Commands for EDA Techniques
http://www.itl.nist.gov/div898/handbook/eda/eda_d.htm[6/27/2012 2:00:18 PM]

1. Exploratory Data Analysis - Detailed Table of Contents [1.]
This chapter presents the assumptions, principles, and techniques necessary to gain insight into data via EDA--
exploratory data analysis.
1. EDA Introduction [1.1.]
1. What is EDA? [1.1.1.]
2. How Does Exploratory Data Analysis differ from Classical Data Analysis? [1.1.2.]
1. Model [1.1.2.1.]
2. Focus [1.1.2.2.]
3. Techniques [1.1.2.3.]
4. Rigor [1.1.2.4.]
5. Data Treatment [1.1.2.5.]
6. Assumptions [1.1.2.6.]
3. How Does Exploratory Data Analysis Differ from Summary Analysis? [1.1.3.]
4. What are the EDA Goals? [1.1.4.]
5. The Role of Graphics [1.1.5.]
6. An EDA/Graphics Example [1.1.6.]
7. General Problem Categories [1.1.7.]
2. EDA Assumptions [1.2.]
1. Underlying Assumptions [1.2.1.]
2. Importance [1.2.2.]
3. Techniques for Testing Assumptions [1.2.3.]
4. Interpretation of 4-Plot [1.2.4.]
5. Consequences [1.2.5.]
1. Consequences of Non-Randomness [1.2.5.1.]
2. Consequences of Non-Fixed Location Parameter [1.2.5.2.]
3. Consequences of Non-Fixed Variation Parameter [1.2.5.3.]
4. Consequences Related to Distributional Assumptions [1.2.5.4.]
3. EDA Techniques [1.3.]
1. Introduction [1.3.1.]
2. Analysis Questions [1.3.2.]
3. Graphical Techniques: Alphabetic [1.3.3.]
1. Autocorrelation Plot [1.3.3.1.]
1. Autocorrelation Plot: Random Data [1.3.3.1.1.]
2. Autocorrelation Plot: Moderate Autocorrelation [1.3.3.1.2.]
3. Autocorrelation Plot: Strong Autocorrelation and Autoregressive Model [1.3.3.1.3.]
4. Autocorrelation Plot: Sinusoidal Model [1.3.3.1.4.]
2. Bihistogram [1.3.3.2.]
3. Block Plot [1.3.3.3.]
4. Bootstrap Plot [1.3.3.4.]
5. Box-Cox Linearity Plot [1.3.3.5.]
6. Box-Cox Normality Plot [1.3.3.6.]
7. Box Plot [1.3.3.7.]
8. Complex Demodulation Amplitude Plot [1.3.3.8.]
9. Complex Demodulation Phase Plot [1.3.3.9.]
10. Contour Plot [1.3.3.10.]
1. DOE Contour Plot [1.3.3.10.1.]
11. DOE Scatter Plot [1.3.3.11.]
12. DOE Mean Plot [1.3.3.12.]
13. DOE Standard Deviation Plot [1.3.3.13.]
14. Histogram [1.3.3.14.]
1. Histogram Interpretation: Normal [1.3.3.14.1.]
2. Histogram Interpretation: Symmetric, Non-Normal, Short-Tailed [1.3.3.14.2.]
3. Histogram Interpretation: Symmetric, Non-Normal, Long-Tailed [1.3.3.14.3.]
4. Histogram Interpretation: Symmetric and Bimodal [1.3.3.14.4.]
5. Histogram Interpretation: Bimodal Mixture of 2 Normals [1.3.3.14.5.]
6. Histogram Interpretation: Skewed (Non-Normal) Right [1.3.3.14.6.]
7. Histogram Interpretation: Skewed (Non-Symmetric) Left [1.3.3.14.7.]
8. Histogram Interpretation: Symmetric with Outlier [1.3.3.14.8.]
15. Lag Plot [1.3.3.15.]
1. Lag Plot: Random Data [1.3.3.15.1.]
2. Lag Plot: Moderate Autocorrelation [1.3.3.15.2.]
3. Lag Plot: Strong Autocorrelation and Autoregressive Model [1.3.3.15.3.]
4. Lag Plot: Sinusoidal Models and Outliers [1.3.3.15.4.]
16. Linear Correlation Plot [1.3.3.16.]
17. Linear Intercept Plot [1.3.3.17.]
18. Linear Slope Plot [1.3.3.18.]
19. Linear Residual Standard Deviation Plot [1.3.3.19.]
20. Mean Plot [1.3.3.20.]
21. Normal Probability Plot [1.3.3.21.]
1. Normal Probability Plot: Normally Distributed Data [1.3.3.21.1.]
2. Normal Probability Plot: Data Have Short Tails [1.3.3.21.2.]
3. Normal Probability Plot: Data Have Long Tails [1.3.3.21.3.]
4. Normal Probability Plot: Data are Skewed Right [1.3.3.21.4.]
22. Probability Plot [1.3.3.22.]
23. Probability Plot Correlation Coefficient Plot [1.3.3.23.]
24. Quantile-Quantile Plot [1.3.3.24.]
25. Run-Sequence Plot [1.3.3.25.]
26. Scatter Plot [1.3.3.26.]
1. Scatter Plot: No Relationship [1.3.3.26.1.]
2. Scatter Plot: Strong Linear (positive correlation) Relationship [1.3.3.26.2.]
3. Scatter Plot: Strong Linear (negative correlation) Relationship [1.3.3.26.3.]
4. Scatter Plot: Exact Linear (positive correlation) Relationship [1.3.3.26.4.]
5. Scatter Plot: Quadratic Relationship [1.3.3.26.5.]
6. Scatter Plot: Exponential Relationship [1.3.3.26.6.]
7. Scatter Plot: Sinusoidal Relationship (damped) [1.3.3.26.7.]
8. Scatter Plot: Variation of Y Does Not Depend on X (homoscedastic) [1.3.3.26.8.]
9. Scatter Plot: Variation of Y Does Depend on X (heteroscedastic) [1.3.3.26.9.]
10. Scatter Plot: Outlier [1.3.3.26.10.]
11. Scatterplot Matrix [1.3.3.26.11.]
12. Conditioning Plot [1.3.3.26.12.]
27. Spectral Plot [1.3.3.27.]
1. Spectral Plot: Random Data [1.3.3.27.1.]
2. Spectral Plot: Strong Autocorrelation and Autoregressive Model [1.3.3.27.2.]
3. Spectral Plot: Sinusoidal Model [1.3.3.27.3.]
28. Standard Deviation Plot [1.3.3.28.]
29. Star Plot [1.3.3.29.]
30. Weibull Plot [1.3.3.30.]
31. Youden Plot [1.3.3.31.]
1. DOE Youden Plot [1.3.3.31.1.]
32. 4-Plot [1.3.3.32.]
33. 6-Plot [1.3.3.33.]
4. Graphical Techniques: By Problem Category [1.3.4.]
5. Quantitative Techniques [1.3.5.]
1. Measures of Location [1.3.5.1.]
2. Confidence Limits for the Mean [1.3.5.2.]
3. Two-Sample t-Test for Equal Means [1.3.5.3.]
1. Data Used for Two-Sample t-Test [1.3.5.3.1.]
4. One-Factor ANOVA [1.3.5.4.]
5. Multi-factor Analysis of Variance [1.3.5.5.]
6. Measures of Scale [1.3.5.6.]
7. Bartlett's Test [1.3.5.7.]
8. Chi-Square Test for the Standard Deviation [1.3.5.8.]
1. Data Used for Chi-Square Test for the Standard Deviation [1.3.5.8.1.]
9. F-Test for Equality of Two Standard Deviations [1.3.5.9.]
10. Levene Test for Equality of Variances [1.3.5.10.]
11. Measures of Skewness and Kurtosis [1.3.5.11.]
12. Autocorrelation [1.3.5.12.]
13. Runs Test for Detecting Non-randomness [1.3.5.13.]
14. Anderson-Darling Test [1.3.5.14.]
15. Chi-Square Goodness-of-Fit Test [1.3.5.15.]
16. Kolmogorov-Smirnov Goodness-of-Fit Test [1.3.5.16.]
17. Grubbs' Test for Outliers [1.3.5.17.]
18. Yates Analysis [1.3.5.18.]
1. Defining Models and Prediction Equations [1.3.5.18.1.]
2. Important Factors [1.3.5.18.2.]
6. Probability Distributions [1.3.6.]
1. What is a Probability Distribution [1.3.6.1.]
2. Related Distributions [1.3.6.2.]
3. Families of Distributions [1.3.6.3.]
4. Location and Scale Parameters [1.3.6.4.]
5. Estimating the Parameters of a Distribution [1.3.6.5.]
1. Method of Moments [1.3.6.5.1.]
2. Maximum Likelihood [1.3.6.5.2.]
3. Least Squares [1.3.6.5.3.]
4. PPCC and Probability Plots [1.3.6.5.4.]
6. Gallery of Distributions [1.3.6.6.]
1. Normal Distribution [1.3.6.6.1.]
2. Uniform Distribution [1.3.6.6.2.]
3. Cauchy Distribution [1.3.6.6.3.]
4. t Distribution [1.3.6.6.4.]
5. F Distribution [1.3.6.6.5.]
6. Chi-Square Distribution [1.3.6.6.6.]
7. Exponential Distribution [1.3.6.6.7.]
8. Weibull Distribution [1.3.6.6.8.]
9. Lognormal Distribution [1.3.6.6.9.]
10. Fatigue Life Distribution [1.3.6.6.10.]
11. Gamma Distribution [1.3.6.6.11.]
12. Double Exponential Distribution [1.3.6.6.12.]
13. Power Normal Distribution [1.3.6.6.13.]
14. Power Lognormal Distribution [1.3.6.6.14.]
15. Tukey-Lambda Distribution [1.3.6.6.15.]
16. Extreme Value Type I Distribution [1.3.6.6.16.]
17. Beta Distribution [1.3.6.6.17.]
18. Binomial Distribution [1.3.6.6.18.]
19. Poisson Distribution [1.3.6.6.19.]
7. Tables for Probability Distributions [1.3.6.7.]
1. Cumulative Distribution Function of the Standard Normal Distribution [1.3.6.7.1.]
2. Upper Critical Values of the Student's-t Distribution [1.3.6.7.2.]
3. Upper Critical Values of the F Distribution [1.3.6.7.3.]
4. Critical Values of the Chi-Square Distribution [1.3.6.7.4.]
5. Critical Values of the t
*
Distribution [1.3.6.7.5.]
6. Critical Values of the Normal PPCC Distribution [1.3.6.7.6.]
4. EDA Case Studies [1.4.]
1. Case Studies Introduction [1.4.1.]
2. Case Studies [1.4.2.]
1. Normal Random Numbers [1.4.2.1.]
1. Background and Data [1.4.2.1.1.]
2. Graphical Output and Interpretation [1.4.2.1.2.]
3. Quantitative Output and Interpretation [1.4.2.1.3.]
4. Work This Example Yourself [1.4.2.1.4.]
2. Uniform Random Numbers [1.4.2.2.]
3. Random Walk [1.4.2.3.]
2. Test Underlying Assumptions [1.4.2.3.2.]
3. Develop A Better Model [1.4.2.3.3.]
4. Validate New Model [1.4.2.3.4.]
4. J osephson J unction Cryothermometry [1.4.2.4.]
5. Beam Deflections [1.4.2.5.]
2. Test Underlying Assumptions [1.4.2.5.2.]
3. Develop a Better Model [1.4.2.5.3.]
4. Validate New Model [1.4.2.5.4.]
6. Filter Transmittance [1.4.2.6.]
7. Standard Resistor [1.4.2.7.]
8. Heat Flow Meter 1 [1.4.2.8.]
9. Fatigue Life of Aluminum Alloy Specimens [1.4.2.9.]
10. Ceramic Strength [1.4.2.10.]
2. Analysis of the Response Variable [1.4.2.10.2.]
3. Analysis of the Batch Effect [1.4.2.10.3.]
4. Analysis of the Lab Effect [1.4.2.10.4.]
5. Analysis of Primary Factors [1.4.2.10.5.]
3. References For Chapter 1: Exploratory Data Analysis [1.4.3.]
1.1. EDA Introduction
http://www.itl.nist.gov/div898/handbook/eda/section1/eda1.htm[6/27/2012 2:00:24 PM]

Summary What is exploratory data analysis? How did it begin? How
and where did it originate? How is it differentiated from other
data analysis approaches, such as classical and Bayesian? Is
EDA the same as statistical graphics? What role does
statistical graphics play in EDA? Is statistical graphics
identical to EDA?
These questions and related questions are dealt with in this
section. This section answers these questions and provides the
necessary frame of reference for EDA assumptions, principles,
and techniques.
Table of
Contents
for Section
1
1. What is EDA?
2. EDA versus Classical and Bayesian
1. Models
2. Focus
3. Techniques
4. Rigor
5. Data Treatment
6. Assumptions
3. EDA vs Summary
4. EDA Goals
5. The Role of Graphics
6. An EDA/Graphics Example
7. General Problem Categories
1.1.1. What is EDA?

1.1.1. What is EDA?
Approach Exploratory Data Analysis (EDA) is an approach/philosophy
for data analysis that employs a variety of techniques (mostly
graphical) to
1. maximize insight into a data set;
2. uncover underlying structure;
3. extract important variables;
4. detect outliers and anomalies;
5. test underlying assumptions;
6. develop parsimonious models; and
7. determine optimal factor settings.
Focus The EDA approach is precisely that--an approach--not a set of
techniques, but an attitude/philosophy about how a data
analysis should be carried out.
Philosophy EDA is not identical to statistical graphics although the two
terms are used almost interchangeably. Statistical graphics is a
collection of techniques--all graphically based and all
focusing on one data characterization aspect. EDA
encompasses a larger venue; EDA is an approach to data
analysis that postpones the usual assumptions about what kind
of model the data follow with the more direct approach of
allowing the data itself to reveal its underlying structure and
model. EDA is not a mere collection of techniques; EDA is a
philosophy as to how we dissect a data set; what we look for;
how we look; and how we interpret. It is true that EDA
heavily uses the collection of techniques that we call
"statistical graphics", but it is not identical to statistical
graphics per se.
History The seminal work in EDA is Exploratory Data Analysis,
Tukey, (1977). Over the years it has benefitted from other
noteworthy publications such as Data Analysis and
Regression, Mosteller and Tukey (1977), Interactive Data
Analysis, Hoaglin (1977), The ABC's of EDA, Velleman and
Hoaglin (1981) and has gained a large following as "the" way
to analyze a data set.
Techniques Most EDA techniques are graphical in nature with a few
quantitative techniques. The reason for the heavy reliance on
1.1.1. What is EDA?
graphics is that by its very nature the main role of EDA is to
open-mindedly explore, and graphics gives the analysts
unparalleled power to do so, enticing the data to reveal its
structural secrets, and being always ready to gain some new,
often unsuspected, insight into the data. In combination with
the natural pattern-recognition capabilities that we all possess,
graphics provides, of course, unparalleled power to carry this
out.
The particular graphical techniques employed in EDA are
often quite simple, consisting of various techniques of:
1. Plotting the raw data (such as data traces, histograms,
bihistograms, probability plots, lag plots, block plots,
and Youden plots.
2. Plotting simple statistics such as mean plots, standard
deviation plots, box plots, and main effects plots of the
raw data.
3. Positioning such plots so as to maximize our natural
pattern-recognition abilities, such as using multiple
plots per page.
1.1.2. How Does Exploratory Data Analysis differ from Classical Data Analysis?

1.1.2. How Does Exploratory Data Analysis
differ from Classical Data Analysis?
Data
Analysis
Approaches
EDA is a data analysis approach. What other data analysis
approaches exist and how does EDA differ from these other
approaches? Three popular data analysis approaches are:
1. Classical
2. Exploratory (EDA)
3. Bayesian
Paradigms
for Analysis
Techniques
These three approaches are similar in that they all start with
a general science/engineering problem and all yield
science/engineering conclusions. The difference is the
sequence and focus of the intermediate steps.
For classical analysis, the sequence is
Problem => Data => Model => Analysis =>
Conclusions
For EDA, the sequence is
Problem => Data => Analysis => Model =>
Conclusions
For Bayesian, the sequence is
Problem => Data => Model => Prior Distribution =>
Analysis => Conclusions
Method of
dealing with
underlying
model for
the data
distinguishes
the 3
approaches
Thus for classical analysis, the data collection is followed by
the imposition of a model (normality, linearity, etc.) and the
analysis, estimation, and testing that follows are focused on
the parameters of that model. For EDA, the data collection is
not followed by a model imposition; rather it is followed
immediately by analysis with a goal of inferring what model
would be appropriate. Finally, for a Bayesian analysis, the
analyst attempts to incorporate scientific/engineering
knowledge/expertise into the analysis by imposing a data-
independent distribution on the parameters of the selected
model; the analysis thus consists of formally combining both
the prior distribution on the parameters and the collected
data to jointly make inferences and/or test assumptions about
the model parameters.
In the real world, data analysts freely mix elements of all of
the above three approaches (and other approaches). The
above distinctions were made to emphasize the major
differences among the three approaches.
Further
discussion of
the
distinction
between the
classical and
EDA
approaches
Focusing on EDA versus classical, these two approaches
differ as follows:
1. Models
2. Focus
3. Techniques
4. Rigor
5. Data Treatment
6. Assumptions
1.1.2.1. Model

1.1.2.1. Model
Classical The classical approach imposes models (both deterministic
and probabilistic) on the data. Deterministic models include,
for example, regression models and analysis of variance
(ANOVA) models. The most common probabilistic model
assumes that the errors about the deterministic model are
normally distributed--this assumption affects the validity of
the ANOVA F tests.
Exploratory The Exploratory Data Analysis approach does not impose
deterministic or probabilistic models on the data. On the
contrary, the EDA approach allows the data to suggest
admissible models that best fit the data.
1.1.2.2. Focus

1.1.2.2. Focus
Classical The two approaches differ substantially in focus. For classical
analysis, the focus is on the model--estimating parameters of
the model and generating predicted values from the model.
Exploratory For exploratory data analysis, the focus is on the data--its
structure, outliers, and models suggested by the data.
1.1.2.3. Techniques

1.1.2.3. Techniques
Classical Classical techniques are generally quantitative in nature. They
include ANOVA, t tests, chi-squared tests, and F tests.
Exploratory EDA techniques are generally graphical. They include scatter
plots, character plots, box plots, histograms, bihistograms,
probability plots, residual plots, and mean plots.
1.1.2.4. Rigor

1.1.2.4. Rigor
Classical Classical techniques serve as the probabilistic foundation of
science and engineering; the most important characteristic of
classical techniques is that they are rigorous, formal, and
"objective".
Exploratory EDA techniques do not share in that rigor or formality. EDA
techniques make up for that lack of rigor by being very
suggestive, indicative, and insightful about what the
appropriate model should be.
EDA techniques are subjective and depend on interpretation
which may differ from analyst to analyst, although
experienced analysts commonly arrive at identical
conclusions.
1.1.2.5. Data Treatment

1.1.2.5. Data Treatment
Classical Classical estimation techniques have the characteristic of
taking all of the data and mapping the data into a few
numbers ("estimates"). This is both a virtue and a vice. The
virtue is that these few numbers focus on important
characteristics (location, variation, etc.) of the population. The
vice is that concentrating on these few characteristics can
filter out other characteristics (skewness, tail length,
autocorrelation, etc.) of the same population. In this sense
there is a loss of information due to this "filtering" process.
Exploratory The EDA approach, on the other hand, often makes use of
(and shows) all of the available data. In this sense there is no
corresponding loss of information.
1.1.2.6. Assumptions

Classical The "good news" of the classical approach is that tests based
on classical techniques are usually very sensitive--that is, if a
true shift in location, say, has occurred, such tests frequently
have the power to detect such a shift and to conclude that
such a shift is "statistically significant". The "bad news" is
that classical tests depend on underlying assumptions (e.g.,
normality), and hence the validity of the test conclusions
becomes dependent on the validity of the underlying
assumptions. Worse yet, the exact underlying assumptions
may be unknown to the analyst, or if known, untested. Thus
the validity of the scientific conclusions becomes intrinsically
linked to the validity of the underlying assumptions. In
practice, if such assumptions are unknown or untested, the
validity of the scientific conclusions becomes suspect.
Exploratory Many EDA techniques make little or no assumptions--they
present and show the data--all of the data--as is, with fewer
encumbering assumptions.
1.1.3. How Does Exploratory Data Analysis Differ from Summary Analysis?

1.1.3. How Does Exploratory Data Analysis
Differ from Summary Analysis?
Summary A summary analysis is simply a numeric reduction of a
historical data set. It is quite passive. Its focus is in the past.
Quite commonly, its purpose is to simply arrive at a few key
statistics (for example, mean and standard deviation) which
may then either replace the data set or be added to the data
set in the form of a summary table.
Exploratory In contrast, EDA has as its broadest goal the desire to gain
insight into the engineering/scientific process behind the data.
Whereas summary statistics are passive and historical, EDA
is active and futuristic. In an attempt to "understand" the
process and improve it in the future, EDA uses the data as a
"window" to peer into the heart of the process that generated
the data. There is an archival role in the research and
manufacturing world for summary statistics, but there is an
enormously larger role for the EDA approach.
1.1.4. What are the EDA Goals?

1.1.4. What are the EDA Goals?
Primary
and
Secondary
Goals
The primary goal of EDA is to maximize the analyst's insight
into a data set and into the underlying structure of a data set,
while providing all of the specific items that an analyst would
want to extract from a data set, such as:
1. a good-fitting, parsimonious model
2. a list of outliers
3. a sense of robustness of conclusions
4. estimates for parameters
5. uncertainties for those estimates
6. a ranked list of important factors
7. conclusions as to whether individual factors are
statistically significant
8. optimal settings
Insight
into the
Data
Insight implies detecting and uncovering underlying structure
in the data. Such underlying structure may not be encapsulated
in the list of items above; such items serve as the specific
targets of an analysis, but the real insight and "feel" for a data
set comes as the analyst judiciously probes and explores the
various subtleties of the data. The "feel" for the data comes
almost exclusively from the application of various graphical
techniques, the collection of which serves as the window into
the essence of the data. Graphics are irreplaceable--there are
no quantitative analogues that will give the same insight as
well-chosen graphics.
To get a "feel" for the data, it is not enough for the analyst to
know what is in the data; the analyst also must know what is
not in the data, and the only way to do that is to draw on our
own human pattern-recognition and comparative abilities in
the context of a series of judicious graphical techniques
applied to the data.
1.1.5. The Role of Graphics

Quantitative/
Graphical
Statistics and data analysis procedures can broadly be split
into two parts:
quantitative
graphical
Quantitative Quantitative techniques are the set of statistical procedures
that yield numeric or tabular output. Examples of
quantitative techniques include:
hypothesis testing
analysis of variance
point estimates and confidence intervals
least squares regression
These and similar techniques are all valuable and are
mainstream in terms of classical analysis.
Graphical On the other hand, there is a large collection of statistical
tools that we generally refer to as graphical techniques.
These include:
scatter plots
histograms
probability plots
residual plots
box plots
block plots
EDA
Approach
Relies
Heavily on
Graphical
Techniques
The EDA approach relies heavily on these and similar
graphical techniques. Graphical procedures are not just tools
that we could use in an EDA context, they are tools that we
must use. Such graphical tools are the shortest path to
gaining insight into a data set in terms of
testing assumptions
model selection
model validation
estimator selection
relationship identification
factor effect determination
outlier detection
If one is not using statistical graphics, then one is forfeiting
insight into one or more aspects of the underlying structure
of the data.
1.1.6. An EDA/Graphics Example

Anscombe
Example
A simple, classic (Anscombe) example of the central role
that graphics play in terms of providing insight into a data
set starts with the following data set:
Data
X Y
10. 00 8. 04
8. 00 6. 95
13. 00 7. 58
9. 00 8. 81
11. 00 8. 33
14. 00 9. 96
6. 00 7. 24
4. 00 4. 26
12. 00 10. 84
7. 00 4. 82
5. 00 5. 68
Summary
Statistics
If the goal of the analysis is to compute summary statistics
plus determine the best linear fit for Y as a function of X,
the results might be given as:
N = 11
Mean of X = 9.0
Mean of Y = 7.5
Intercept = 3
Slope = 0.5
Residual standard deviation = 1.237
Correlation = 0.816
The above quantitative analysis, although valuable, gives us
only limited insight into the data.
Scatter Plot In contrast, the following simple scatter plot of the data
suggests the following:
1. The data set "behaves like" a linear curve with some
scatter;
2. there is no justification for a more complicated model
(e.g., quadratic);
3. there are no outliers;
4. the vertical spread of the data appears to be of equal
height irrespective of the X-value; this indicates that
the data are equally-precise throughout and so a
"regular" (that is, equi-weighted) fit is appropriate.
Three
Additional
Data Sets
This kind of characterization for the data serves as the core
for getting insight/feel for the data. Such insight/feel does
not come from the quantitative statistics; on the contrary,
calculations of quantitative statistics such as intercept and
slope should be subsequent to the characterization and will
make sense only if the characterization is true. To illustrate
the loss of information that results when the graphics
insight step is skipped, consider the following three data
sets [Anscombe data sets 2, 3, and 4]:
X2 Y2 X3 Y3 X4 Y4
10. 00 9. 14 10. 00 7. 46 8. 00 6. 58
8. 00 8. 14 8. 00 6. 77 8. 00 5. 76
13. 00 8. 74 13. 00 12. 74 8. 00 7. 71
9. 00 8. 77 9. 00 7. 11 8. 00 8. 84
11. 00 9. 26 11. 00 7. 81 8. 00 8. 47
14. 00 8. 10 14. 00 8. 84 8. 00 7. 04
6. 00 6. 13 6. 00 6. 08 8. 00 5. 25
4. 00 3. 10 4. 00 5. 39 19. 00 12. 50
12. 00 9. 13 12. 00 8. 15 8. 00 5. 56
7. 00 7. 26 7. 00 6. 42 8. 00 7. 91
5. 00 4. 74 5. 00 5. 73 8. 00 6. 89
Quantitative
Statistics for
Data Set 2
A quantitative analysis on data set 2 yields
N = 11
Mean of X = 9.0
Mean of Y = 7.5
Intercept = 3
Slope = 0.5
Correlation = 0.816
which is identical to the analysis for data set 1. One might
naively assume that the two data sets are "equivalent" since
that is what the statistics tell us; but what do the statistics
not tell us?
Quantitative
Statistics for
Data Sets 3
and 4
Remarkably, a quantitative analysis on data sets 3 and 4
also yields
N = 11
Mean of X = 9.0
Mean of Y = 7.5
Intercept = 3
Slope = 0.5
Correlation = 0.816 (0.817 for data set 4)
which implies that in some quantitative sense, all four of
the data sets are "equivalent". In fact, the four data sets are
far from "equivalent" and a scatter plot of each data set,
which would be step 1 of any EDA approach, would tell us
that immediately.
Scatter Plots
Interpretation
of Scatter
Plots
Conclusions from the scatter plots are:
1. data set 1 is clearly linear with some scatter.
2. data set 2 is clearly quadratic.
3. data set 3 clearly has an outlier.
4. data set 4 is obviously the victim of a poor
experimental design with a single point far removed
from the bulk of the data "wagging the dog".
Importance These points are exactly the substance that provide and
of
Exploratory
Analysis
define "insight" and "feel" for a data set. They are the goals
and the fruits of an open exploratory data analysis (EDA)
approach to the data. Quantitative statistics are not wrong
per se, but they are incomplete. They are incomplete
because they are numeric summaries which in the
summarization operation do a good job of focusing on a
particular aspect of the data (e.g., location, intercept, slope,
degree of relatedness, etc.) by judiciously reducing the data
to a few numbers. Doing so also filters the data, necessarily
omitting and screening out other sometimes crucial
information in the focusing operation. Quantitative statistics
focus but also filter; and filtering is exactly what makes the
quantitative approach incomplete at best and misleading at
worst.
The estimated intercepts (= 3) and slopes (= 0.5) for data
sets 2, 3, and 4 are misleading because the estimation is
done in the context of an assumed linear model and that
linearity assumption is the fatal flaw in this analysis.
The EDA approach of deliberately postponing the model
selection until further along in the analysis has many
rewards, not the least of which is the ultimate convergence
to a much-improved model and the formulation of valid
and supportable scientific and engineering conclusions.
1.1.7. General Problem Categories

Problem
Classification
The following table is a convenient way to classify EDA
problems.
Univariate
and Control
UNIVARIATE
Data:
A single column of
numbers, Y.
Model:
y = constant + error
Output:
1. A number (the
estimated constant in
the model).
2. An estimate of
uncertainty for the
constant.
3. An estimate of the
distribution for the
error.
Techniques:
4-Plot
Probability Plot
PPCC Plot
CONTROL
Data:
A single column of
numbers, Y.
Model:
y = constant + error
Output:
A "yes" or "no" to the
question "Is the
system out of control?
".
Techniques:
Control Charts
Comparative
and
Screening
COMPARATIVE
Data:
A single response
variable and k
independent variables
(Y, X
1
, X
2
, ... , X
k
),
primary focus is on
SCREENING
Data:
A single response
variable and k
(Y, X
1
, X
2
, ... , X
k
).
one (the primary
factor) of these
independent variables.
Model:
y = f(x
1
, x
2
, ..., x
k
) +
error
Output:
A "yes" or "no" to the
question "Is the
primary factor
significant?".
Techniques:
Block Plot
Scatter Plot
Box Plot
Model:
y = f(x
1
, x
2
, ..., x
k
) +
error
Output:
1. A ranked list (from
most important to
least important) of
factors.
2. Best settings for the
factors.
3. A good
model/prediction
equation relating Y to
the factors.
Techniques:
Block Plot
Probability Plot
Bihistogram
Optimization
and
Regression
OPTIMIZATION
Data:
A single response
variable and k
(Y, X
1
, X
2
, ... , X
k
).
Model:
y = f(x
1
, x
2
, ..., x
k
) +
error
Output:
Best settings for the
factor variables.
Techniques:
Block Plot
Least Squares Fitting
Contour Plot
REGRESSION
Data:
A single response
variable and k
(Y, X
1
, X
2
, ... , X
k
).
The independent
variables can be
continuous.
Model:
y = f(x
1
, x
2
, ..., x
k
) +
error
Output:
A good
model/prediction
equation relating Y to
the factors.
Techniques:
Least Squares Fitting
Scatter Plot
6-Plot
Time Series
and
Multivariate
TIME SERIES
Data:
A column of
time dependent
numbers, Y. In
addition, time is
an indpendent
variable. The
time variable
can be either
explicit or
implied. If the
data are not
equi-spaced, the
time variable
should be
explicitly
provided.
Model:
y
t
= f(t) + error
The model can
be either a time
domain based or
frequency
domain based.
Output:
A good
model/prediction
equation relating
Y to previous
values of Y.
Techniques:
Autocorrelation
Plot
Spectrum
Complex
Demodulation
Amplitude Plot
Complex
Demodulation
Phase Plot
ARIMA Models
MULTIVARIATE
Data:
k factor variables (X
1
, X
2
, ...
, X
k
).
Model:
The model is not explicit.
Output:
Identify underlying
correlation structure in the
data.
Techniques:
Star Plot
Scatter Plot Matrix
Conditioning Plot
Profile Plot
Principal Components
Clustering
Discrimination/Classification
Note that multivarate analysis is
only covered lightly in this
Handbook.
1.2. EDA Assumptions

Summary The gamut of scientific and engineering experimentation is
virtually limitless. In this sea of diversity is there any common
basis that allows the analyst to systematically and validly
arrive at supportable, repeatable research conclusions?
Fortunately, there is such a basis and it is rooted in the fact
that every measurement process, however complicated, has
certain underlying assumptions. This section deals with what
those assumptions are, why they are important, how to go
about testing them, and what the consequences are if the
assumptions do not hold.
Table of
Contents
for Section
2
1. Underlying Assumptions
2. Importance
3. Testing Assumptions
4. Importance of Plots
5. Consequences
1.2.1. Underlying Assumptions

Assumptions
Underlying a
Measurement
Process
There are four assumptions that typically underlie all
measurement processes; namely, that the data from the
process at hand "behave like":
1. random drawings;
2. from a fixed distribution;
3. with the distribution having fixed location; and
4. with the distribution having fixed variation.
Univariate or
Single
Response
Variable
The "fixed location" referred to in item 3 above differs for
different problem types. The simplest problem type is
univariate; that is, a single variable. For the univariate
problem, the general model
response = deterministic component + random
component
becomes
response = constant + error
Assumptions
for
Univariate
Model
For this case, the "fixed location" is simply the unknown
constant. We can thus imagine the process at hand to be
operating under constant conditions that produce a single
column of data with the properties that
the data are uncorrelated with one another;
the random component has a fixed distribution;
the deterministic component consists of only a
constant; and
the random component has fixed variation.
Extrapolation
to a Function
of Many
Variables
The universal power and importance of the univariate model
is that it can easily be extended to the more general case
where the deterministic component is not just a constant,
but is in fact a function of many variables, and the
engineering objective is to characterize and model the
function.
Residuals The key point is that regardless of how many factors there
Will Behave
According to
Univariate
Assumptions
are, and regardless of how complicated the function is, if
the engineer succeeds in choosing a good model, then the
differences (residuals) between the raw response data and
the predicted values from the fitted model should
themselves behave like a univariate process. Furthermore,
the residuals from this univariate process fit will behave
like:
random drawings;
from a fixed distribution;
with fixed location (namely, 0 in this case); and
with fixed variation.
Validation of
Model
Thus if the residuals from the fitted model do in fact behave
like the ideal, then testing of underlying assumptions
becomes a tool for the validation and quality of fit of the
chosen model. On the other hand, if the residuals from the
chosen fitted model violate one or more of the above
univariate assumptions, then the chosen fitted model is
inadequate and an opportunity exists for arriving at an
improved model.
1.2.2. Importance

1.2.2. Importance
Predictability
and
Statistical
Control
Predictability is an all-important goal in science and
engineering. If the four underlying assumptions hold, then
we have achieved probabilistic predictability--the ability to
make probability statements not only about the process in
the past, but also about the process in the future. In short,
such processes are said to be "in statistical control".
Validity of
Engineering
Conclusions
Moreover, if the four assumptions are valid, then the
process is amenable to the generation of valid scientific and
engineering conclusions. If the four assumptions are not
valid, then the process is drifting (with respect to location,
variation, or distribution), unpredictable, and out of control.
A simple characterization of such processes by a location
estimate, a variation estimate, or a distribution "estimate"
inevitably leads to engineering conclusions that are not
valid, are not supportable (scientifically or legally), and
which are not repeatable in the laboratory.
1.2.3. Techniques for Testing Assumptions

Testing
Underlying
Assumptions
Helps Assure the
Validity of
Scientific and
Engineering
Conclusions
Because the validity of the final scientific/engineering
conclusions is inextricably linked to the validity of the
underlying univariate assumptions, it naturally follows that
there is a real necessity that each and every one of the
above four assumptions be routinely tested.
Four Techniques
to Test
Underlying
Assumptions
The following EDA techniques are simple, efficient, and
powerful for the routine testing of underlying
assumptions:
1. run sequence plot (Y
i
versus i)
2. lag plot (Y
i
versus Y
i-1
)
3. histogram (counts versus subgroups of Y)
4. normal probability plot (ordered Y versus theoretical
ordered Y)
Plot on a Single
Page for a
Quick
Characterization
of the Data
The four EDA plots can be juxtaposed for a quick look at
the characteristics of the data. The plots below are ordered
as follows:
1. Run sequence plot - upper left
2. Lag plot - upper right
3. Histogram - lower left
4. Normal probability plot - lower right
Sample Plot:
Assumptions
Hold
This 4-plot reveals a process that has fixed location, fixed
variation, is random, apparently has a fixed approximately
normal distribution, and has no outliers.
Sample Plot:
Assumptions Do
Not Hold
If one or more of the four underlying assumptions do not
hold, then it will show up in the various plots as
demonstrated in the following example.
This 4-plot reveals a process that has fixed location, fixed
variation, is non-random (oscillatory), has a non-normal,
U-shaped distribution, and has several outliers.
1.2.4. Interpretation of 4-Plot

Interpretation
of EDA
Plots:
Flat and
Equi-Banded,
Random,
Bell-Shaped,
and Linear
The four EDA plots discussed on the previous page are
used to test the underlying assumptions:
1. Fixed Location:
If the fixed location assumption holds, then the run
sequence plot will be flat and non-drifting.
2. Fixed Variation:
If the fixed variation assumption holds, then the
vertical spread in the run sequence plot will be the
approximately the same over the entire horizontal
axis.
3. Randomness:
If the randomness assumption holds, then the lag plot
will be structureless and random.
4. Fixed Distribution:
If the fixed distribution assumption holds, in
particular if the fixed normal distribution holds, then
1. the histogram will be bell-shaped, and
2. the normal probability plot will be linear.
Plots Utilized
to Test the
Assumptions
Conversely, the underlying assumptions are tested using the
EDA plots:
Run Sequence Plot:
If the run sequence plot is flat and non-drifting, the
fixed-location assumption holds. If the run sequence
plot has a vertical spread that is about the same over
the entire plot, then the fixed-variation assumption
holds.
Lag Plot:
If the lag plot is structureless, then the randomness
assumption holds.
Histogram:
If the histogram is bell-shaped, the underlying
distribution is symmetric and perhaps approximately
normal.
Normal Probability Plot:
If the normal probability plot is linear, the underlying
distribution is approximately normal.
If all four of the assumptions hold, then the process is said
definitionally to be "in statistical control".
1.2.5. Consequences

1.2.5. Consequences
What If
Assumptions
Do Not Hold?
If some of the underlying assumptions do not hold, what
can be done about it? What corrective actions can be
taken? The positive way of approaching this is to view the
testing of underlying assumptions as a framework for
learning about the process. Assumption-testing promotes
insight into important aspects of the process that may not
have surfaced otherwise.
Primary Goal
is Correct
and Valid
Scientific
Conclusions
The primary goal is to have correct, validated, and
complete scientific/engineering conclusions flowing from
the analysis. This usually includes intermediate goals such
as the derivation of a good-fitting model and the
computation of realistic parameter estimates. It should
always include the ultimate goal of an understanding and a
"feel" for "what makes the process tick". There is no more
powerful catalyst for discovery than the bringing together
of an experienced/expert scientist/engineer and a data set
ripe with intriguing "anomalies" and characteristics.
Consequences
of Invalid
Assumptions
The following sections discuss in more detail the
consequences of invalid assumptions:
1. Consequences of non-randomness
2. Consequences of non-fixed location parameter
3. Consequences of non-fixed variation
4. Consequences related to distributional assumptions
1.2.5.1. Consequences of Non-Randomness

1.2.5. Consequences
Randomness
Assumption
There are four underlying assumptions:
1. randomness;
2. fixed location;
3. fixed variation; and
4. fixed distribution.
The randomness assumption is the most critical but the
least tested.
Consequeces of
Non-
Randomness
If the randomness assumption does not hold, then
1. All of the usual statistical tests are invalid.
2. The calculated uncertainties for commonly used
statistics become meaningless.
3. The calculated minimal sample size required for a
pre-specified tolerance becomes meaningless.
4. The simple model: y = constant + error becomes
invalid.
5. The parameter estimates become suspect and non-
supportable.
Non-
Randomness
Due to
Autocorrelation
One specific and common type of non-randomness is
autocorrelation. Autocorrelation is the correlation
between Y
t
and Y
t-k
, where k is an integer that defines the
lag for the autocorrelation. That is, autocorrelation is a
time dependent non-randomness. This means that the
value of the current point is highly dependent on the
previous point if k = 1 (or k points ago if k is not 1).
Autocorrelation is typically detected via an
autocorrelation plot or a lag plot.
If the data are not random due to autocorrelation, then
1. Adjacent data values may be related.
2. There may not be n independent snapshots of the
phenomenon under study.
3. There may be undetected "junk"-outliers.
4. There may be undetected "information-rich"-
outliers.
1.2.5.2. Consequences of Non-Fixed Location Parameter

1.2.5. Consequences
1.2.5.2. Consequences of Non-Fixed Location
Parameter
Location
Estimate
The usual estimate of location is the mean
from N measurements Y
1
, Y
2
, ... , Y
N
.
Consequences
of Non-Fixed
Location
If the run sequence plot does not support the assumption of
fixed location, then
1. The location may be drifting.
2. The single location estimate may be meaningless (if
the process is drifting).
3. The choice of location estimator (e.g., the sample
mean) may be sub-optimal.
4. The usual formula for the uncertainty of the mean:
may be invalid and the numerical value optimistically
small.
5. The location estimate may be poor.
6. The location estimate may be biased.
1.2.5.3. Consequences of Non-Fixed Variation Parameter

1.2.5. Consequences
1.2.5.3. Consequences of Non-Fixed Variation
Parameter
Variation
Estimate
The usual estimate of variation is the standard deviation
from N measurements Y
1
, Y
2
, ... , Y
N
.
Consequences
of Non-Fixed
Variation
If the run sequence plot does not support the assumption of
fixed variation, then
1. The variation may be drifting.
2. The single variation estimate may be meaningless (if
the process variation is drifting).
3. The variation estimate may be poor.
4. The variation estimate may be biased.
1.2.5.4. Consequences Related to Distributional Assumptions

1.2.5. Consequences
1.2.5.4. Consequences Related to Distributional
Assumptions
Distributional
Analysis
Scientists and engineers routinely use the mean (average) to
estimate the "middle" of a distribution. It is not so well
known that the variability and the noisiness of the mean as
a location estimator are intrinsically linked with the
underlying distribution of the data. For certain distributions,
the mean is a poor choice. For any given distribution, there
exists an optimal choice-- that is, the estimator with
minimum variability/noisiness. This optimal choice may be,
for example, the median, the midrange, the midmean, the
mean, or something else. The implication of this is to
"estimate" the distribution first, and then--based on the
distribution--choose the optimal estimator. The resulting
engineering parameter estimators will have less variability
than if this approach is not followed.
Case Studies The airplane glass failure case study gives an example of
determining an appropriate distribution and estimating the
parameters of that distribution. The uniform random
numbers case study gives an example of determining a
more appropriate centrality parameter for a non-normal
distribution.
Other consequences that flow from problems with
distributional assumptions are:
Distribution 1. The distribution may be changing.
2. The single distribution estimate may be meaningless
(if the process distribution is changing).
3. The distribution may be markedly non-normal.
4. The distribution may be unknown.
5. The true probability distribution for the error may
remain unknown.
Model 1. The model may be changing.
2. The single model estimate may be meaningless.
3. The default model
Y = constant + error
may be invalid.
4. If the default model is insufficient, information about
1.2.5.4. Consequences Related to Distributional Assumptions
a better model may remain undetected.
5. A poor deterministic model may be fit.
6. Information about an improved model may go
undetected.
Process 1. The process may be out-of-control.
2. The process may be unpredictable.
3. The process may be un-modelable.
1.3. EDA Techniques

1.3. EDA Techniques
Summary After you have collected a set of data, how do you do an
exploratory data analysis? What techniques do you employ?
What do the various techniques focus on? What conclusions
can you expect to reach?
This section provides answers to these kinds of questions via a
gallery of EDA techniques and a detailed description of each
technique. The techniques are divided into graphical and
quantitative techniques. For exploratory data analysis, the
emphasis is primarily on the graphical techniques.
Table of
Contents
for Section
3
1. Introduction
2. Analysis Questions
3. Graphical Techniques: Alphabetical
4. Graphical Techniques: By Problem Category
5. Quantitative Techniques: Alphabetical
6. Probability Distributions
1.3.1. Introduction

1.3. EDA Techniques
1.3.1. Introduction
Graphical
and
Quantitative
Techniques
This section describes many techniques that are commonly
used in exploratory and classical data analysis. This list is by
no means meant to be exhaustive. Additional techniques
(both graphical and quantitative) are discussed in the other
chapters. Specifically, the product comparisons chapter has a
much more detailed description of many classical statistical
techniques.
EDA emphasizes graphical techniques while classical
techniques emphasize quantitative techniques. In practice, an
analyst typically uses a mixture of graphical and quantitative
techniques. In this section, we have divided the descriptions
into graphical and quantitative techniques. This is for
organizational clarity and is not meant to discourage the use
of both graphical and quantitiative techniques when
analyzing data.
Use of
Techniques
Shown in
Case
Studies
This section emphasizes the techniques themselves; how the
graph or test is defined, published references, and sample
output. The use of the techniques to answer engineering
questions is demonstrated in the case studies section. The
case studies do not demonstrate all of the techniques.
Availability
in Software
The sample plots and output in this section were generated
with the Dataplot software program. Other general purpose
statistical data analysis programs can generate most of the
plots, intervals, and tests discussed here, or macros can be
written to acheive the same result.
1.3.2. Analysis Questions

1.3. EDA Techniques
EDA
Questions
Some common questions that exploratory data analysis is
used to answer are:
1. What is a typical value?
2. What is the uncertainty for a typical value?
3. What is a good distributional fit for a set of numbers?
4. What is a percentile?
5. Does an engineering modification have an effect?
6. Does a factor have an effect?
7. What are the most important factors?
8. Are measurements coming from different laboratories
equivalent?
9. What is the best function for relating a response
variable to a set of factor variables?
10. What are the best settings for factors?
11. Can we separate signal from noise in time dependent
data?
12. Can we extract any structure from multivariate data?
13. Does the data have outliers?
Analyst
Should
Identify
Relevant
Questions
for his
Engineering
Problem
A critical early step in any analysis is to identify (for the
engineering problem at hand) which of the above questions
are relevant. That is, we need to identify which questions we
want answered and which questions have no bearing on the
problem at hand. After collecting such a set of questions, an
equally important step, which is invaluable for maintaining
focus, is to prioritize those questions in decreasing order of
importance. EDA techniques are tied in with each of the
questions. There are some EDA techniques (e.g., the scatter
plot) that are broad-brushed and apply almost universally. On
the other hand, there are a large number of EDA techniques
that are specific and whose specificity is tied in with one of
the above questions. Clearly if one chooses not to explicitly
identify relevant questions, then one cannot take advantage of
these question-specific EDA technqiues.
EDA
Approach
Emphasizes
Graphics
Most of these questions can be addressed by techniques
discussed in this chapter. The process modeling and process
improvement chapters also address many of the questions
above. These questions are also relevant for the classical
approach to statistics. What distinguishes the EDA approach
is an emphasis on graphical techniques to gain insight as
opposed to the classical approach of quantitative tests. Most
data analysts will use a mix of graphical and classical
quantitative techniques to address these problems.
1.3.3. Graphical Techniques: Alphabetic

1.3. EDA Techniques
This section provides a gallery of some useful graphical
techniques. The techniques are ordered alphabetically, so this
section is not intended to be read in a sequential fashion. The
use of most of these graphical techniques is demonstrated in
the case studies in this chapter. A few of these graphical
techniques are demonstrated in later chapters.
Autocorrelation
Plot: 1.3.3.1
Bihistogram:
1.3.3.2
Block Plot:
1.3.3.3
Bootstrap Plot:
1.3.3.4
Box-Cox
Linearity Plot:
1.3.3.5
Box-Cox
Normality Plot:
1.3.3.6
Box Plot: 1.3.3.7 Complex
Demodulation
Amplitude Plot:
1.3.3.8
Complex
Demodulation
Phase Plot:
1.3.3.9
Contour Plot:
1.3.3.10
DOE Scatter
Plot: 1.3.3.11
DOE Mean Plot:
1.3.3.12
DOE Standard
Deviation Plot:
1.3.3.13
Histogram:
1.3.3.14
Lag Plot:
1.3.3.15
Linear
Correlation Plot:
1.3.3.16
Linear Intercept
Plot: 1.3.3.17
Linear Slope
Plot: 1.3.3.18
Linear Residual
Standard
Deviation Plot:
1.3.3.19
Mean Plot:
1.3.3.20
Normal
Probability Plot:
1.3.3.21
Probability Plot:
1.3.3.22
Probability Plot
Correlation
Coefficient Plot:
1.3.3.23
Quantile-
Quantile Plot:
1.3.3.24
Run Sequence
Plot: 1.3.3.25
Scatter Plot:
1.3.3.26
Spectrum:
1.3.3.27
Standard
Deviation Plot:
1.3.3.28
Star Plot:
1.3.3.29
Weibull Plot:
1.3.3.30
Youden Plot:
1.3.3.31
4-Plot: 1.3.3.32
6-Plot: 1.3.3.33
1.3.3.1. Autocorrelation Plot

1.3. EDA Techniques
Purpose:
Check
Randomness
Autocorrelation plots (Box and Jenkins, pp. 28-32) are a
commonly-used tool for checking randomness in a data
set. This randomness is ascertained by computing
autocorrelations for data values at varying time lags. If
random, such autocorrelations should be near zero for any
and all time-lag separations. If non-random, then one or
more of the autocorrelations will be significantly non-
zero.
In addition, autocorrelation plots are used in the model
identification stage for Box-Jenkins autoregressive,
moving average time series models.
Sample Plot:
Autocorrelations
should be near-
zero for
randomness.
Such is not the
case in this
example and
thus the
randomness
assumption fails
This sample autocorrelation plot shows that the time series
is not random, but rather has a high degree of
autocorrelation between adjacent and near-adjacent
observations.
Definition:
r(h) versus h
Autocorrelation plots are formed by
Vertical axis: Autocorrelation coefficient
where C
h
is the autocovariance function
and C
0
is the variance function
Note--R
h
is between -1 and +1.
Note--Some sources may use the following formula
for the autocovariance function
Although this definition has less bias, the (1/N)
formulation has some desirable statistical properties
and is the form most commonly used in the
statistics literature. See pages 20 and 49-50 in
Chatfield for details.
Horizontal axis: Time lag h (h = 1, 2, 3, ...)
The above line also contains several horizontal
reference lines. The middle line is at zero. The other
four lines are 95 % and 99 % confidence bands.
Note that there are two distinct formulas for
generating the confidence bands.
1. If the autocorrelation plot is being used to test
for randomness (i.e., there is no time
dependence in the data), the following
formula is recommended:
where N is the sample size, z is the
cumulative distribution function of the
standard normal distribution and is the
significance level. In this case, the confidence
bands have fixed width that depends on the
sample size. This is the formula that was used
to generate the confidence bands in the above
plot.
2. Autocorrelation plots are also used in the
model identification stage for fitting ARIMA
models. In this case, a moving average model
is assumed for the data and the following
confidence bands should be generated:
where k is the lag, N is the sample size, z is
the cumulative distribution function of the
standard normal distribution and is the
significance level. In this case, the confidence
bands increase as the lag increases.
Questions The autocorrelation plot can provide answers to the
following questions:
1. Are the data random?
2. Is an observation related to an adjacent
observation?
3. Is an observation related to an observation twice-
removed? (etc.)
4. Is the observed time series white noise?
5. Is the observed time series sinusoidal?
6. Is the observed time series autoregressive?
7. What is an appropriate model for the observed time
series?
8. Is the model
valid and sufficient?
9. Is the formula valid?
Importance:
Ensure validity
of engineering
conclusions
Randomness (along with fixed model, fixed variation, and
fixed distribution) is one of the four assumptions that
typically underlie all measurement processes. The
randomness assumption is critically important for the
following three reasons:
1. Most standard statistical tests depend on
randomness. The validity of the test conclusions is
directly linked to the validity of the randomness
assumption.
2. Many commonly-used statistical formulae depend
on the randomness assumption, the most common
formula being the formula for determining the
standard deviation of the sample mean:
where is the standard deviation of the data.
Although heavily used, the results from using this
formula are of no value unless the randomness
assumption holds.
3. For univariate data, the default model is
If the data are not random, this model is incorrect
and invalid, and the estimates for the parameters
(such as the constant) become nonsensical and
invalid.
In short, if the analyst does not check for randomness,
then the validity of many of the statistical conclusions
becomes suspect. The autocorrelation plot is an excellent
way of checking for such randomness.
Examples Examples of the autocorrelation plot for several common
situations are given in the following pages.
1. Random (= White Noise)
2. Weak autocorrelation
3. Strong autocorrelation and autoregressive model
4. Sinusoidal model
Related
Techniques
Partial Autocorrelation Plot
Lag Plot
Spectral Plot
Seasonal Subseries Plot
Case Study The autocorrelation plot is demonstrated in the beam
deflection data case study.
Software Autocorrelation plots are available in most general
purpose statistical software programs.
1.3.3.1.1. Autocorrelation Plot: Random Data

1.3. EDA Techniques
Autocorrelation
Plot
The following is a sample autocorrelation plot.
Conclusions We can make the following conclusions from this plot.
1. There are no significant autocorrelations.
2. The data are random.
Discussion Note that with the exception of lag 0, which is always 1 by
definition, almost all of the autocorrelations fall within the
95% confidence limits. In addition, there is no apparent
pattern (such as the first twenty-five being positive and the
second twenty-five being negative). This is the abscence
of a pattern we expect to see if the data are in fact random.
A few lags slightly outside the 95% and 99% confidence
limits do not neccessarily indicate non-randomness. For a
95% confidence interval, we might expect about one out
of twenty lags to be statistically significant due to random
fluctuations.
There is no associative ability to infer from a current value
Y
i
as to what the next value Y
i+1
will be. Such non-
association is the essense of randomness. In short, adjacent
observations do not "co-relate", so we call this the "no
autocorrelation" case.
1.3.3.1.2. Autocorrelation Plot: Moderate Autocorrelation

1.3. EDA Techniques
1.3.3.1.2. Autocorrelation Plot: Moderate
Autocorrelation
Autocorrelation
Plot
Conclusions We can make the following conclusions from this plot.
1. The data come from an underlying autoregressive
model with moderate positive autocorrelation.
Discussion The plot starts with a moderately high autocorrelation at
lag 1 (approximately 0.75) that gradually decreases. The
decreasing autocorrelation is generally linear, but with
significant noise. Such a pattern is the autocorrelation plot
signature of "moderate autocorrelation", which in turn
provides moderate predictability if modeled properly.
Recommended
Next Step
The next step would be to estimate the parameters for the
autoregressive model:
Such estimation can be performed by using least squares
linear regression or by fitting a Box-Jenkins autoregressive
(AR) model.
1.3.3.1.2. Autocorrelation Plot: Moderate Autocorrelation
The randomness assumption for least squares fitting
applies to the residuals of the model. That is, even though
the original data exhibit non-randomness, the residuals
after fitting Y
i
against Y
i-1
should result in random
residuals. Assessing whether or not the proposed model in
fact sufficiently removed the randomness is discussed in
detail in the Process Modeling chapter.
The residual standard deviation for this autoregressive
model will be much smaller than the residual standard
deviation for the default model
1.3.3.1.3. Autocorrelation Plot: Strong Autocorrelation and Autoregressive Model

1.3. EDA Techniques
1.3.3.1.3. Autocorrelation Plot: Strong
Autocorrelation and Autoregressive
Model
Autocorrelation
Plot for Strong
Autocorrelation
Conclusions We can make the following conclusions from the above
plot.
model with strong positive autocorrelation.
Discussion The plot starts with a high autocorrelation at lag 1 (only
slightly less than 1) that slowly declines. It continues
decreasing until it becomes negative and starts showing an
incresing negative autocorrelation. The decreasing
autocorrelation is generally linear with little noise. Such a
pattern is the autocorrelation plot signature of "strong
autocorrelation", which in turn provides high
predictability if modeled properly.
Recommended
Next Step
The next step would be to estimate the parameters for the
1.3.3.1.3. Autocorrelation Plot: Strong Autocorrelation and Autoregressive Model
Such estimation can be performed by using least squares
linear regression or by fitting a Box-Jenkins
autoregressive (AR) model.
The randomness assumption for least squares fitting
applies to the residuals of the model. That is, even though
the original data exhibit non-randomness, the residuals
after fitting Y
i
against Y
i-1
should result in random
residuals. Assessing whether or not the proposed model in
fact sufficiently removed the randomness is discussed in
detail in the Process Modeling chapter.
1.3.3.1.4. Autocorrelation Plot: Sinusoidal Model

1.3. EDA Techniques
1.3.3.1.4. Autocorrelation Plot: Sinusoidal
Model
Autocorrelation
Plot for
Sinusoidal
Model
plot.
1. The data come from an underlying sinusoidal
model.
Discussion The plot exhibits an alternating sequence of positive and
negative spikes. These spikes are not decaying to zero.
Such a pattern is the autocorrelation plot signature of a
sinusoidal model.
Recommended
Next Step
The beam deflection case study gives an example of
modeling a sinusoidal model.
1.3.3.2. Bihistogram

1.3. EDA Techniques
Purpose:
Check for a
change in
location,
variation, or
distribution
The bihistogram is an EDA tool for assessing whether a
before-versus-after engineering modification has caused a
change in
location;
variation; or
distribution.
It is a graphical alternative to the two-sample t-test. The
bihistogram can be more powerful than the t-test in that all
of the distributional features (location, scale, skewness,
outliers) are evident on a single plot. It is also based on the
common and well-understood histogram.
Sample Plot:
This
bihistogram
reveals that
there is a
significant
difference in
ceramic
breaking
strength
between
batch 1
(above) and
batch 2
(below)
From the above bihistogram, we can see that batch 1 is
centered at a ceramic strength value of approximately 725
while batch 2 is centered at a ceramic strength value of
approximately 625. That indicates that these batches are
displaced by about 100 strength units. Thus the batch factor
has a significant effect on the location (typical value) for
strength and hence batch is said to be "significant" or to
"have an effect". We thus see graphically and convincingly
what a t-test or analysis of variance would indicate
quantitatively.
With respect to variation, note that the spread (variation) of
the above-axis batch 1 histogram does not appear to be that
much different from the below-axis batch 2 histogram. With
respect to distributional shape, note that the batch 1
histogram is skewed left while the batch 2 histogram is more
symmetric with even a hint of a slight skewness to the right.
Thus the bihistogram reveals that there is a clear difference
between the batches with respect to location and
distribution, but not in regard to variation. Comparing batch
1 and batch 2, we also note that batch 1 is the "better batch"
due to its 100-unit higher average strength (around 725).
Definition:
Two
adjoined
histograms
Bihistograms are formed by vertically juxtaposing two
histograms:
Above the axis: Histogram of the response variable
for condition 1
Below the axis: Histogram of the response variable for
condition 2
Questions The bihistogram can provide answers to the following
questions:
1. Is a (2-level) factor significant?
2. Does a (2-level) factor have an effect?
3. Does the location change between the 2 subgroups?
4. Does the variation change between the 2 subgroups?
5. Does the distributional shape change between
subgroups?
6. Are there any outliers?
Importance:
Checks 3 out
of the 4
underlying
assumptions
of a
measurement
process
The bihistogram is an important EDA tool for determining if
a factor "has an effect". Since the bihistogram provides
insight into the validity of three (location, variation, and
distribution) out of the four (missing only randomness)
underlying assumptions in a measurement process, it is an
especially valuable tool. Because of the dual (above/below)
nature of the plot, the bihistogram is restricted to assessing
factors that have only two levels. However, this is very
common in the before-versus-after character of many
scientific and engineering experiments.
Related
Techniques
t test (for shift in location)
F test (for shift in variation)
Kolmogorov-Smirnov test (for shift in distribution)
Quantile-quantile plot (for shift in location and distribution)
Case Study The bihistogram is demonstrated in the ceramic strength
data case study.
Software The bihistogram is not widely available in general purpose
statistical software programs. Bihistograms can be generated
using Dataplot and R software.
1.3.3.3. Block Plot

1.3. EDA Techniques
1.3.3.3. Block Plot
Purpose:
Check to
determine if
a factor of
interest has
an effect
robust over
all other
factors
The block plot (Filliben 1993) is an EDA tool for assessing
whether the factor of interest (the primary factor) has a
statistically significant effect on the response, and whether
that conclusion about the primary factor effect is valid
robustly over all other nuisance or secondary factors in the
experiment.
It replaces the analysis of variance test with a less
assumption-dependent binomial test and should be routinely
used whenever we are trying to robustly decide whether a
primary factor has an effect.
Sample
Plot:
Weld
method 2 is
lower
(better)
than weld
method 1 in
10 of 12
cases
This block plot reveals that in 10 of the 12 cases (bars), weld
method 2 is lower (better) than weld method 1. From a
binomial point of view, weld method is statistically
significant.
Definition Block Plots are formed as follows:
Vertical axis: Response variable Y
Horizontal axis: All combinations of all levels of all
nuisance (secondary) factors X1, X2, ...
Plot Character: Levels of the primary factor XP
1.3.3.3. Block Plot
Discussion:
Primary
factor is
denoted by
plot
character:
within-bar
plot
character.
Average number of defective lead wires per hour from a
study with four factors,
1. weld strength (2 levels)
2. plant (2 levels)
3. speed (2 levels)
4. shift (3 levels)
are shown in the plot above. Weld strength is the primary
factor and the other three factors are nuisance factors. The 12
distinct positions along the horizontal axis correspond to all
possible combinations of the three nuisance factors, i.e., 12 =
2 plants x 2 speeds x 3 shifts. These 12 conditions provide the
framework for assessing whether any conclusions about the 2
levels of the primary factor (weld method) can truly be
called "general conclusions". If we find that one weld method
setting does better (smaller average defects per hour) than the
other weld method setting for all or most of these 12 nuisance
factor combinations, then the conclusion is in fact general
and robust.
Ordering
along the
horizontal
axis
In the above chart, the ordering along the horizontal axis is as
follows:
The left 6 bars are from plant 1 and the right 6 bars are
from plant 2.
The first 3 bars are from speed 1, the next 3 bars are
from speed 2, the next 3 bars are from speed 1, and the
last 3 bars are from speed 2.
Bars 1, 4, 7, and 10 are from the first shift, bars 2, 5, 8,
and 11 are from the second shift, and bars 3, 6, 9, and
12 are from the third shift.
Setting 2 is
better than
setting 1 in
10 out of 12
cases
In the block plot for the first bar (plant 1, speed 1, shift 1),
weld method 1 yields about 28 defects per hour while weld
method 2 yields about 22 defects per hour--hence the
difference for this combination is about 6 defects per hour
and weld method 2 is seen to be better (smaller number of
defects per hour).
Is "weld method 2 is better than weld method 1" a general
conclusion?
For the second bar (plant 1, speed 1, shift 2), weld method 1
is about 37 while weld method 2 is only about 18. Thus weld
method 2 is again seen to be better than weld method 1.
Similarly for bar 3 (plant 1, speed 1, shift 3), we see weld
method 2 is smaller than weld method 1. Scanning over all of
the 12 bars, we see that weld method 2 is smaller than weld
method 1 in 10 of the 12 cases, which is highly suggestive of
a robust weld method effect.
An event What is the chance of 10 out of 12 happening by chance?
1.3.3.3. Block Plot
with chance
probability
of only 2%
This is probabilistically equivalent to testing whether a coin
is fair by flipping it and getting 10 heads in 12 tosses. The
chance (from the binomial distribution) of getting 10 (or
more extreme: 11, 12) heads in 12 flips of a fair coin is about
2%. Such low-probability events are usually rejected as
untenable and in practice we would conclude that there is a
difference in weld methods.
Advantage:
Graphical
and
binomial
The advantages of the block plot are as follows:
A quantitative procedure (analysis of variance) is
replaced by a graphical procedure.
An F-test (analysis of variance) is replaced with a
binomial test, which requires fewer assumptions.
Questions The block plot can provide answers to the following
questions:
1. Is the factor of interest significant?
2. Does the factor of interest have an effect?
3. Does the location change between levels of the primary
factor?
4. Has the process improved?
5. What is the best setting (= level) of the primary factor?
6. How much of an average improvement can we expect
with this best setting of the primary factor?
7. Is there an interaction between the primary factor and
one or more nuisance factors?
8. Does the effect of the primary factor change depending
on the setting of some nuisance factor?
Importance:
Robustly
checks the
significance
of the factor
of interest
The block plot is a graphical technique that pointedly focuses
on whether or not the primary factor conclusions are in fact
robustly general. This question is fundamentally different
from the generic multi-factor experiment question where the
analyst asks, "What factors are important and what factors
are not" (a screening problem)? Global data analysis
techniques, such as analysis of variance, can potentially be
improved by local, focused data analysis techniques that take
advantage of this difference.
Related
Techniques
t test (for shift in location for exactly 2 levels)
ANOVA (for shift in location for 2 or more levels)
Bihistogram (for shift in location, variation, and distribution
for exactly 2 levels).
Case Study The block plot is demonstrated in the ceramic strength data
case study.
Software Block plots are not currently available in most general
1.3.3.3. Block Plot
purpose statistical software programs. However they can be
generated using Dataplot and, with some programming, R
software.
1.3.3.4. Bootstrap Plot

1.3. EDA Techniques
Purpose:
Estimate
uncertainty
The bootstrap (Efron and Gong) plot is used to estimate the
uncertainty of a statistic.
Generate
subsamples
with
replacement
To generate a bootstrap uncertainty estimate for a given
statistic from a set of data, a subsample of a size less than or
equal to the size of the data set is generated from the data,
and the statistic is calculated. This subsample is generated
with replacement so that any data point can be sampled
multiple times or not sampled at all. This process is repeated
for many subsamples, typically between 500 and 1000. The
computed values for the statistic form an estimate of the
sampling distribution of the statistic.
For example, to estimate the uncertainty of the median from
a dataset with 50 elements, we generate a subsample of 50
elements and calculate the median. This is repeated at least
500 times so that we have at least 500 values for the median.
Although the number of bootstrap samples to use is
somewhat arbitrary, 500 subsamples is usually sufficient. To
calculate a 90% confidence interval for the median, the
sample medians are sorted into ascending order and the value
of the 25th median (assuming exactly 500 subsamples were
taken) is the lower confidence limit while the value of the
475th median (assuming exactly 500 subsamples were taken)
is the upper confidence limit.
Sample
Plot:
This bootstrap plot was generated from 500 uniform random
numbers. Bootstrap plots and corresponding histograms were
generated for the mean, median, and mid-range. The
histograms for the corresponding statistics clearly show that
for uniform random numbers the mid-range has the smallest
variance and is, therefore, a superior location estimator to the
mean or the median.
Definition The bootstrap plot is formed by:
Vertical axis: Computed value of the desired statistic
for a given subsample.
Horizontal axis: Subsample number.
The bootstrap plot is simply the computed value of the
statistic versus the subsample number. That is, the bootstrap
plot generates the values for the desired statistic. This is
usually immediately followed by a histogram or some other
distributional plot to show the location and variation of the
sampling distribution of the statistic.
Questions The bootstrap plot is used to answer the following questions:
What does the sampling distribution for the statistic
look like?
What is a 95% confidence interval for the statistic?
Which statistic has a sampling distribution with the
smallest variance? That is, which statistic generates the
narrowest confidence interval?
Importance The most common uncertainty calculation is generating a
confidence interval for the mean. In this case, the uncertainty
formula can be derived mathematically. However, there are
many situations in which the uncertainty formulas are
mathematically intractable. The bootstrap provides a method
for calculating the uncertainty in these cases.
Cautuion on
use of the
bootstrap
The bootstrap is not appropriate for all distributions and
statistics (Efron and Tibrashani). For example, because of
the shape of the uniform distribution, the bootstrap is not
appropriate for estimating the distribution of statistics that are
heavily dependent on the tails, such as the range.
Related
Techniques
Histogram
Jackknife
The jacknife is a technique that is closely related to the
bootstrap. The jackknife is beyond the scope of this
handbook. See the Efron and Gong article for a discussion of
the jackknife.
Case Study The bootstrap plot is demonstrated in the uniform random
numbers case study.
Software The bootstrap is becoming more common in general purpose
statistical software programs. However, it is still not
supported in many of these programs. Both R software and
Dataplot support a bootstrap capability.
1.3.3.5. Box-Cox Linearity Plot

1.3. EDA Techniques
Purpose:
Find the
transformation
of the X
variable that
maximizes the
correlation
between a Y
and an X
variable
When performing a linear fit of Y against X, an
appropriate transformation of X can often significantly
improve the fit. The Box-Cox transformation (Box and
Cox, 1964) is a particularly useful family of
transformations. It is defined as:
where X is the variable being transformed and is the
transformation parameter. For = 0, the natural log of the
data is taken instead of using the above formula.
The Box-Cox linearity plot is a plot of the correlation
between Y and the transformed X for given values of .
That is, is the coordinate for the horizontal axis variable
and the value of the correlation between Y and the
transformed X is the coordinate for the vertical axis of the
plot. The value of corresponding to the maximum
correlation (or minimum for negative correlation) on the
plot is then the optimal choice for .
Transforming X is used to improve the fit. The Box-Cox
transformation applied to Y can be used as the basis for
meeting the error assumptions. That case is not covered
here. See page 225 of (Draper and Smith, 1981) or page
77 of (Ryan, 1997) for a discussion of this case.
Sample Plot
The plot of the original data with the predicted values from
a linear fit indicate that a quadratic fit might be preferable.
The Box-Cox linearity plot shows a value of = 2.0. The
plot of the transformed data with the predicted values from
a linear fit with the transformed data shows a better fit
(verified by the significant reduction in the residual
standard deviation).
Definition Box-Cox linearity plots are formed by
Vertical axis: Correlation coefficient from the
transformed X and Y
Horizontal axis: Value for
Questions The Box-Cox linearity plot can provide answers to the
1. Would a suitable transformation improve my fit?
2. What is the optimal value of the transformation
parameter?
Importance:
Find a
suitable
transformation
Transformations can often significantly improve a fit. The
Box-Cox linearity plot provides a convenient way to find
a suitable transformation without engaging in a lot of trial
and error fitting.
Related
Techniques
Linear Regression
Box-Cox Normality Plot
Case Study The Box-Cox linearity plot is demonstrated in the Alaska
pipeline data case study.
Software Box-Cox linearity plots are not a standard part of most
general purpose statistical software programs. However,
the underlying technique is based on a transformation and
computing a correlation coefficient. So if a statistical
program supports these capabilities, writing a macro for a
Box-Cox linearity plot should be feasible.
1.3.3.6. Box-Cox Normality Plot

1.3. EDA Techniques
Purpose:
Find
transformation
to normalize
data
Many statistical tests and intervals are based on the
assumption of normality. The assumption of normality
often leads to tests that are simple, mathematically
tractable, and powerful compared to tests that do not make
the normality assumption. Unfortunately, many real data
sets are in fact not approximately normal. However, an
appropriate transformation of a data set can often yield a
data set that does follow approximately a normal
distribution. This increases the applicability and usefulness
of statistical techniques based on the normality
assumption.
The Box-Cox transformation is a particulary useful family
of transformations. It is defined as:
where Y is the response variable and is the
transformation parameter. For = 0, the natural log of the
data is taken instead of using the above formula.
Given a particular transformation such as the Box-Cox
transformation defined above, it is helpful to define a
measure of the normality of the resulting transformation.
One measure is to compute the correlation coefficient of a
normal probability plot. The correlation is computed
between the vertical and horizontal axis variables of the
probability plot and is a convenient measure of the
linearity of the probability plot (the more linear the
probability plot, the better a normal distribution fits the
data).
The Box-Cox normality plot is a plot of these correlation
coefficients for various values of the parameter. The
value of corresponding to the maximum correlation on
the plot is then the optimal choice for .
Sample Plot
The histogram in the upper left-hand corner shows a data
set that has significant right skewness (and so does not
follow a normal distribution). The Box-Cox normality plot
shows that the maximum value of the correlation
coefficient is at = -0.3. The histogram of the data after
applying the Box-Cox transformation with = -0.3 shows
a data set for which the normality assumption is
reasonable. This is verified with a normal probability plot
of the transformed data.
Definition Box-Cox normality plots are formed by:
Vertical axis: Correlation coefficient from the
normal probability plot after applying Box-Cox
transformation
Horizontal axis: Value for
Questions The Box-Cox normality plot can provide answers to the
1. Is there a transformation that will normalize my
data?
2. What is the optimal value of the transformation
parameter?
Importance:
Normalization
Improves
Validity of
Tests
Normality assumptions are critical for many univariate
intervals and hypothesis tests. It is important to test the
normality assumption. If the data are in fact clearly not
normal, the Box-Cox normality plot can often be used to
find a transformation that will approximately normalize the
data.
Related
Techniques
Normal Probability Plot
Box-Cox Linearity Plot
Software Box-Cox normality plots are not a standard part of most
general purpose statistical software programs. However,
the underlying technique is based on a normal probability
plot and computing a correlation coefficient. So if a
statistical program supports these capabilities, writing a
macro for a Box-Cox normality plot should be feasible.
1.3.3.7. Box Plot

1.3. EDA Techniques
1.3.3.7. Box Plot
Purpose:
Check
location
and
variation
shifts
Box plots (Chambers 1983) are an excellent tool for
conveying location and variation information in data sets,
particularly for detecting and illustrating location and
variation changes between different groups of data.
Sample
Plot:
This box
plot reveals
that
machine
has a
significant
effect on
energy with
respect to
location
and
possibly
variation
This box plot, comparing four machines for energy output,
shows that machine has a significant effect on energy with
respect to both location and variation. Machine 3 has the
highest energy response (about 72.5); machine 4 has the least
variable energy response with about 50% of its readings
being within 1 energy unit.
Definition Box plots are formed by
Vertical axis: Response variable
Horizontal axis: The factor of interest
More specifically, we
1. Calculate the median and the quartiles (the lower
quartile is the 25th percentile and the upper quartile is
the 75th percentile).
1.3.3.7. Box Plot
2. Plot a symbol at the median (or draw a line) and draw
a box (hence the name--box plot) between the lower
and upper quartiles; this box represents the middle
50% of the data--the "body" of the data.
3. Draw a line from the lower quartile to the minimum
point and another line from the upper quartile to the
maximum point. Typically a symbol is drawn at these
minimum and maximum points, although this is
optional.
Thus the box plot identifies the middle 50% of the data, the
median, and the extreme points.
Single or
multiple
box plots
can be
drawn
A single box plot can be drawn for one batch of data with no
distinct groups. Alternatively, multiple box plots can be
drawn together to compare multiple data sets or to compare
groups in a single data set. For a single box plot, the width of
the box is arbitrary. For multiple box plots, the width of the
box plot can be set proportional to the number of points in
the given group or sample (some software implementations
of the box plot simply set all the boxes to the same width).
Box plots
with fences
There is a useful variation of the box plot that more
specifically identifies outliers. To create this variation:
1. Calculate the median and the lower and upper
quartiles.
2. Plot a symbol at the median and draw a box between
the lower and upper quartiles.
3. Calculate the interquartile range (the difference
between the upper and lower quartile) and call it IQ.
4. Calculate the following points:
L1 = lower quartile - 1.5*IQ
L2 = lower quartile - 3.0*IQ
U1 = upper quartile + 1.5*IQ
U2 = upper quartile + 3.0*IQ
5. The line from the lower quartile to the minimum is
now drawn from the lower quartile to the smallest
point that is greater than L1. Likewise, the line from
the upper quartile to the maximum is now drawn to the
largest point smaller than U1.
6. Points between L1 and L2 or between U1 and U2 are
drawn as small circles. Points less than L2 or greater
than U2 are drawn as large circles.
Questions The box plot can provide answers to the following questions:
1.3.3.7. Box Plot
1. Is a factor significant?
2. Does the location differ between subgroups?
3. Does the variation differ between subgroups?
Importance:
Check the
significance
of a factor
The box plot is an important EDA tool for determining if a
factor has a significant effect on the response with respect to
either location or variation.
The box plot is also an effective tool for summarizing large
quantities of information.
Related
Techniques
Mean Plot
Analysis of Variance
Case Study The box plot is demonstrated in the ceramic strength data
case study.
Software Box plots are available in most general purpose statistical
software programs.
1.3.3.8. Complex Demodulation Amplitude Plot

1.3. EDA Techniques
Purpose:
Detect
Changing
Amplitude
in
Sinusoidal
Models
In the frequency analysis of time series models, a common
model is the sinusoidal model:
In this equation, is the amplitude, is the phase shift, and
is the dominant frequency. In the above model, and are
constant, that is they do not vary with time, t
i
.
The complex demodulation amplitude plot (Granger, 1964) is
used to determine if the assumption of constant amplitude is
justifiable. If the slope of the complex demodulation
amplitude plot is not zero, then the above model is typically
replaced with the model:
where is some type of linear model fit with standard least
squares. The most common case is a linear fit, that is the
model becomes
Quadratic models are sometimes used. Higher order models
are relatively rare.
Sample
Plot:
This complex demodulation amplitude plot shows that:
the amplitude is fixed at approximately 390;
there is a start-up effect; and
there is a change in amplitude at around x = 160 that
should be investigated for an outlier.
Definition: The complex demodulation amplitude plot is formed by:
Vertical axis: Amplitude
Horizontal axis: Time
The mathematical computations for determining the
amplitude are beyond the scope of the Handbook. Consult
Granger (Granger, 1964) for details.
Questions The complex demodulation amplitude plot answers the
1. Does the amplitude change over time?
2. Are there any outliers that need to be investigated?
3. Is the amplitude different at the beginning of the series
(i.e., is there a start-up effect)?
Importance:
Assumption
Checking
As stated previously, in the frequency analysis of time series
models, a common model is the sinusoidal model:
In this equation, is assumed to be constant, that is it does
not vary with time. It is important to check whether or not
this assumption is reasonable.
The complex demodulation amplitude plot can be used to
verify this assumption. If the slope of this plot is essentially
zero, then the assumption of constant amplitude is justified. If
it is not, should be replaced with some type of time-
varying model. The most common cases are linear (B
0
+
B
1
*t) and quadratic (B
0
+ B
1
*t + B
2
*t
2
).
Related
Techniques
Spectral Plot
Complex Demodulation Phase Plot
Non-Linear Fitting
Case Study The complex demodulation amplitude plot is demonstrated in
the beam deflection data case study.
Software Complex demodulation amplitude plots are available in some,
but not most, general purpose statistical software programs.
1.3.3.9. Complex Demodulation Phase Plot

1.3. EDA Techniques
Purpose:
Improve
the
estimate of
frequency
in
sinusoidal
time series
models
As stated previously, in the frequency analysis of time series
models, a common model is the sinusoidal model:
In this equation, is the amplitude, is the phase shift, and
is the dominant frequency. In the above model, and are
constant, that is they do not vary with time t
i
.
The complex demodulation phase plot (Granger, 1964) is used
to improve the estimate of the frequency (i.e., ) in this
model.
If the complex demodulation phase plot shows lines sloping
from left to right, then the estimate of the frequency should be
increased. If it shows lines sloping right to left, then the
frequency should be decreased. If there is essentially zero
slope, then the frequency estimate does not need to be
modified.
Sample
Plot:
This complex demodulation phase plot shows that:
the specified demodulation frequency is incorrect;
the demodulation frequency should be increased.
Definition The complex demodulation phase plot is formed by:
Vertical axis: Phase
Horizontal axis: Time
The mathematical computations for the phase plot are beyond
the scope of the Handbook. Consult Granger (Granger, 1964)
for details.
Questions The complex demodulation phase plot answers the following
question:
Is the specified demodulation frequency correct?
Importance
of a Good
Initial
Estimate
for the
Frequency
The non-linear fitting for the sinusoidal model:
is usually quite sensitive to the choice of good starting values.
The initial estimate of the frequency, , is obtained from a
spectral plot. The complex demodulation phase plot is used to
assess whether this estimate is adequate, and if it is not,
whether it should be increased or decreased. Using the
complex demodulation phase plot with the spectral plot can
significantly improve the quality of the non-linear fits
obtained.
Related
Techniques
Spectral Plot
Non-Linear Fitting
Case Study The complex demodulation amplitude plot is demonstrated in
the beam deflection data case study.
Software Complex demodulation phase plots are available in some, but
not most, general purpose statistical software programs.
1.3.3.10. Contour Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33a.htm[6/27/2012 2:00:55 PM]

1.3. EDA Techniques
Purpose:
Display 3-d
surface on
2-d plot
A contour plot is a graphical technique for representing a 3-
dimensional surface by plotting constant z slices, called
contours, on a 2-dimensional format. That is, given a value
for z, lines are drawn for connecting the (x,y) coordinates
where that z value occurs.
The contour plot is an alternative to a 3-D surface plot.
Sample
Plot:
This contour plot shows that the surface is symmetric and
peaks in the center.
Definition The contour plot is formed by:
Vertical axis: Independent variable 2
Horizontal axis: Independent variable 1
Lines: iso-response values
The independent variables are usually restricted to a regular
grid. The actual techniques for determining the correct iso-
response values are rather complex and are almost always
computer generated.
An additional variable may be required to specify the Z
values for drawing the iso-lines. Some software packages
require explicit values. Other software packages will
determine them automatically.
If the data (or function) do not form a regular grid, you
typically need to perform a 2-D interpolation to form a
regular grid.
Questions The contour plot is used to answer the question
How does Z change as a function of X and Y?
Importance:
Visualizing
3-
dimensional
data
For univariate data, a run sequence plot and a histogram are
considered necessary first steps in understanding the data.
For 2-dimensional data, a scatter plot is a necessary first step
in understanding the data.
In a similar manner, 3-dimensional data should be plotted.
Small data sets, such as result from designed experiments,
can typically be represented by block plots, DOE mean plots,
and the like ("DOE" stands for "Design of Experiments").
For large data sets, a contour plot or a 3-D surface plot
should be considered a necessary first step in understanding
the data.
DOE
Contour
Plot
The DOE contour plot is a specialized contour plot used in
the design of experiments. In particular, it is useful for full
and fractional designs.
Related
Techniques
3-D Plot
Software Contour plots are available in most general purpose statistical
software programs. They are also available in many general
purpose graphics and mathematics programs. These programs
vary widely in the capabilities for the contour plots they
generate. Many provide just a basic contour plot over a
rectangular grid while others permit color filled or shaded
contours.
Most statistical software programs that support design of
experiments will provide a DOE contour plot capability.
1.3.3.10.1. DOE Contour Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33a1.htm[6/27/2012 2:00:55 PM]

1.3. EDA Techniques
DOE
Contour Plot:
Introduction
The DOE contour plot is a specialized contour plot used in the analysis of full and fractional
experimental designs. These designs often have a low level, coded as "-1" or "-", and a high
level, coded as "+1" or "+" for each factor. In addition, there can optionally be one or more
center points. Center points are at the mid-point between the low and high level for each
factor and are coded as "0".
The DOE contour plot is generated for two factors. Typically, this would be the two most
important factors as determined by previous analyses (e.g., through the use of the DOE
mean plots and an analysis of variance). If more than two factors are important, you may
want to generate a series of DOE contour plots, each of which is drawn for two of these
factors. You can also generate a matrix of all pairwise DOE contour plots for a number of
important factors (similar to the scatter plot matrix for scatter plots).
The typical application of the DOE contour plot is in determining settings that will
maximize (or minimize) the response variable. It can also be helpful in determining settings
that result in the response variable hitting a pre-determined target value. The DOE contour
plot plays a useful role in determining the settings for the next iteration of the experiment.
That is, the initial experiment is typically a fractional factorial design with a fairly large
number of factors. After the most important factors are determined, the DOE contour plot
can be used to help define settings for a full factorial or response surface design based on a
smaller number of factors.
Construction
of DOE
Contour Plot
The following are the primary steps in the construction of the DOE contour plot.
1. The x and y axes of the plot represent the values of the first and second factor
(independent) variables.
2. The four vertex points are drawn. The vertex points are (-1,-1), (-1,1), (1,1), (1,-1). At
each vertex point, the average of all the response values at that vertex point is printed.
3. Similarly, if there are center points, a point is drawn at (0,0) and the average of the
response values at the center points is printed.
4. The linear DOE contour plot assumes the model:
where is the overall mean of the response variable. The values of , , , and
are estimated from the vertex points using least squares estimation.
In order to generate a single contour line, we need a value for Y, say Y . Next, we
0
solve for U
2
in terms of U
1
and, after doing the algebra, we have the equation:
We generate a sequence of points for U
1
in the range -2 to 2 and compute the
corresponding values of U
2
. These points constitute a single contour line
corresponding to Y = Y
0
.
The user specifies the target values for which contour lines will be generated.
The above algorithm assumes a linear model for the design. DOE contour plots can also be
generated for the case in which we assume a quadratic model for the design. The algebra for
solving for U
2
in terms of U
1
becomes more complicated, but the fundamental idea is the
same. Quadratic models are needed for the case when the average for the center points does
not fall in the range defined by the vertex point (i.e., there is curvature).
Sample DOE
Contour Plot
The following is a DOE contour plot for the data used in the Eddy current case study. The
analysis in that case study demonstrated that X1 and X2 were the most important factors.
Interpretation
of the Sample
DOE
Contour Plot
From the above DOE contour plot we can derive the following information.
1. Interaction significance;
2. Best (data) setting for these two dominant factors;
Interaction
Significance
Note the appearance of the contour plot. If the contour curves are linear, then that implies
that the interaction term is not significant; if the contour curves have considerable curvature,
then that implies that the interaction term is large and important. In our case, the contour
curves do not have considerable curvature, and so we conclude that the X1*X2 term is not
significant.
Best Settings To determine the best factor settings for the already-run experiment, we first must define
what "best" means. For the Eddy current data set used to generate this DOE contour plot,
"best" means to maximize (rather than minimize or hit a target) the response. Hence from
the contour plot we determine the best settings for the two dominant factors by simply
scanning the four vertices and choosing the vertex with the largest value (= average
response). In this case, it is (X1 = +1, X2 = +1).
As for factor X3, the contour plot provides no best setting information, and so we would
resort to other tools: the main effects plot, the interaction effects matrix, or the ordered data
to determine optimal X3 settings.
Case Study The Eddy current case study demonstrates the use of the DOE contour plot in the context of
the analysis of a full factorial design.
Software DOE Contour plots are available in many statistical software programs that analyze data
from designed experiments.
1.3.3.11. DOE Scatter Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33b.htm[6/27/2012 2:00:56 PM]

1.3. EDA Techniques
Purpose:
Determine
Important
Factors with
Respect to
Location and
Scale
The DOE scatter plot shows the response values for each level of each factor (i.e.,
independent) variable. This graphically shows how the location and scale vary for both
within a factor variable and between different factor variables. This graphically shows
which are the important factors and can help provide a ranked list of important factors from
a designed experiment. The DOE scatter plot is a complement to the traditional analyis of
variance of designed experiments.
DOE scatter plots are typically used in conjunction with the DOE mean plot and the DOE
standard deviation plot. The DOE mean plot replaces the raw response values with mean
response values while the DOE standard deviation plot replaces the raw response values
with the standard deviation of the response values. There is value in generating all 3 of these
plots. The DOE mean and standard deviation plots are useful in that the summary measures
of location and spread stand out (they can sometimes get lost with the raw plot). However,
the raw data points can reveal subtleties, such as the presence of outliers, that might get lost
with the summary statistics.
Sample Plot:
Factors 4, 2,
3, and 7 are
the Important
Factors.
Description
of the Plot
For this sample plot, there are seven factors and each factor has two levels. For each factor,
we define a distinct x coordinate for each level of the factor. For example, for factor 1, level
1 is coded as 0.8 and level 2 is coded as 1.2. The y coordinate is simply the value of the
response variable. The solid horizontal line is drawn at the overall mean of the response
variable. The vertical dotted lines are added for clarity.
Although the plot can be drawn with an arbitrary number of levels for a factor, it is really
only useful when there are two or three levels for a factor.
Conclusions This sample DOE scatter plot shows that:
1. there does not appear to be any outliers;
2. the levels of factors 2 and 4 show distinct location differences; and
3. the levels of factor 1 show distinct scale differences.
Definition:
Response
Values
Versus
Factor
Variables
DOE scatter plots are formed by:
Vertical axis: Value of the response variable
Horizontal axis: Factor variable (with each level of the factor coded with a slightly
offset x coordinate)
Questions The DOE scatter plot can be used to answer the following questions:
1. Which factors are important with respect to location and scale?
2. Are there outliers?
Importance:
Identify
Important
Factors with
Respect to
Location and
Scale
The goal of many designed experiments is to determine which factors are important with
respect to location and scale. A ranked list of the important factors is also often of interest.
DOE scatter, mean, and standard deviation plots show this graphically. The DOE scatter plot
additionally shows if outliers may potentially be distorting the results.
DOE scatter plots were designed primarily for analyzing designed experiments. However,
they are useful for any type of multi-factor data (i.e., a response variable with two or more
factor variables having a small number of distinct levels) whether or not the data were
generated from a designed experiment.
Extension for
Interaction
Effects
Using the concept of the scatterplot matrix, the DOE scatter plot can be extended to display
first order interaction effects.
Specifically, if there are k factors, we create a matrix of plots with k rows and k columns.
On the diagonal, the plot is simply a DOE scatter plot with a single factor. For the off-
diagonal plots, we multiply the values of X
i
and X
j
. For the common 2-level designs (i.e.,
each factor has two levels) the values are typically coded as -1 and 1, so the multiplied
values are also -1 and 1. We then generate a DOE scatter plot for this interaction variable.
This plot is called a DOE interaction effects plot and an example is shown below.
Interpretation
of the DOE
Interaction
Effects Plot
We can first examine the diagonal elements for the main effects. These diagonal plots show
a great deal of overlap between the levels for all three factors. This indicates that location
and scale effects will be relatively small.
We can then examine the off-diagonal plots for the first order interaction effects. For
example, the plot in the first row and second column is the interaction between factors X1
and X2. As with the main effect plots, no clear patterns are evident.
Related
Techniques
DOE mean plot
DOE standard deviation plot
Block plot
Box plot
Analysis of variance
Case Study The DOE scatter plot is demonstrated in the ceramic strength data case study.
Software DOE scatter plots are available in some general purpose statistical software programs,
although the format may vary somewhat between these programs. They are essentially just
scatter plots with the X variable defined in a particular way, so it should be feasible to write
macros for DOE scatter plots in most statistical software programs.
1.3.3.12. DOE Mean Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33c.htm[6/27/2012 2:00:58 PM]

1.3. EDA Techniques
Purpose:
Detect
Important
Factors
With
Respect to
Location
The DOE mean plot is appropriate for analyzing data from a designed
experiment, with respect to important factors, where the factors are at two or
more levels. The plot shows mean values for the two or more levels of each
factor plotted by factor. The means for a single factor are connected by a
straight line. The DOE mean plot is a complement to the traditional analysis
of variance of designed experiments.
This plot is typically generated for the mean. However, it can be generated
for other location statistics such as the median.
Sample
Plot:
Factors 4,
2, and 1 Are
the Most
Important
Factors
This sample DOE mean plot shows that:
1. factor 4 is the most important;
2. factor 2 is the second most important;
3. factor 1 is the third most important;
4. factor 7 is the fourth most important;
5. factor 6 is the fifth most important;
6. factors 3 and 5 are relatively unimportant.
In summary, factors 4, 2, and 1 seem to be clearly important, factors 3 and 5
seem to be clearly unimportant, and factors 6 and 7 are borderline factors
whose inclusion in any subsequent models will be determined by further
analyses.
Definition:
Mean
Response
Versus
Factor
Variables
DOE mean plots are formed by:
Vertical axis: Mean of the response variable for each level of the
factor
Horizontal axis: Factor variable
Questions The DOE mean plot can be used to answer the following questions:
1. Which factors are important? The DOE mean plot does not provide a
definitive answer to this question, but it does help categorize factors as
"clearly important", "clearly not important", and "borderline
importance".
2. What is the ranking list of the important factors?
Importance:
Determine
Significant
Factors
The goal of many designed experiments is to determine which factors are
significant. A ranked order listing of the important factors is also often of
interest. The DOE mean plot is ideally suited for answering these types of
questions and we recommend its routine use in analyzing designed
experiments.
Extension
for
Interaction
Effects
Using the concept of the scatter plot matrix, the DOE mean plot can be
extended to display first-order interaction effects.
Specifically, if there are k factors, we create a matrix of plots with k rows
and k columns. On the diagonal, the plot is simply a DOE mean plot with a
single factor. For the off-diagonal plots, measurements at each level of the
interaction are plotted versus level, where level is X
i
times X
j
and X
i
is the
code for the ith main effect level and X
j
is the code for the jth main effect.
For the common 2-level designs (i.e., each factor has two levels) the values
are typically coded as -1 and 1, so the multiplied values are also -1 and 1.
We then generate a DOE mean plot for this interaction variable. This plot is
called a DOE interaction effects plot and an example is shown below.
DOE
Interaction
Effects Plot
This plot shows that the most significant factor is X1 and the most
significant interaction is between X1 and X3.
Related
Techniques
DOE scatter plot
Block plot
Box plot
Case Study The DOE mean plot and the DOE interaction effects plot are demonstrated in
the ceramic strength data case study.
Software DOE mean plots are available in some general purpose statistical software
programs, although the format may vary somewhat between these programs.
It may be feasible to write macros for DOE mean plots in some statistical
software programs that do not support this plot directly.
1.3.3.13. DOE Standard Deviation Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33d.htm[6/27/2012 2:00:59 PM]

1.3. EDA Techniques
Purpose:
Detect
Important
Factors
With
Respect to
Scale
The DOE standard deviation plot is appropriate for analyzing data from a
designed experiment, with respect to important factors, where the factors are
at two or more levels and there are repeated values at each level. The plot
shows standard deviation values for the two or more levels of each factor
plotted by factor. The standard deviations for a single factor are connected
by a straight line. The DOE standard deviation plot is a complement to the
traditional analysis of variance of designed experiments.
This plot is typically generated for the standard deviation. However, it can
also be generated for other scale statistics such as the range, the median
absolute deviation, or the average absolute deviation.
Sample Plot
This sample DOE standard deviation plot shows that:
1. factor 1 has the greatest difference in standard deviations between
factor levels;
2. factor 4 has a significantly lower average standard deviation than the
average standard deviations of other factors (but the level 1 standard
deviation for factor 1 is about the same as the level 1 standard
deviation for factor 4);
3. for all factors, the level 1 standard deviation is smaller than the level 2
standard deviation.
Definition:
Response
Standard
Deviations
Versus
Factor
Variables
DOE standard deviation plots are formed by:
Vertical axis: Standard deviation of the response variable for each
level of the factor
Horizontal axis: Factor variable
Questions The DOE standard deviation plot can be used to answer the following
questions:
1. How do the standard deviations vary across factors?
2. How do the standard deviations vary within a factor?
3. Which are the most important factors with respect to scale?
4. What is the ranked list of the important factors with respect to scale?
Importance:
Assess
Variability
The goal with many designed experiments is to determine which factors are
significant. This is usually determined from the means of the factor levels
(which can be conveniently shown with a DOE mean plot). A secondary
goal is to assess the variability of the responses both within a factor and
between factors. The DOE standard deviation plot is a convenient way to do
this.
Related
Techniques
DOE scatter plot
DOE mean plot
Block plot
Box plot
Case Study The DOE standard deviation plot is demonstrated in the ceramic strength
data case study.
Software DOE standard deviation plots are not available in most general purpose
statistical software programs. It may be feasible to write macros for DOE
standard deviation plots in some statistical software programs that do not
support them directly.
1.3.3.14. Histogram
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33e.htm[6/27/2012 2:00:59 PM]

1.3. EDA Techniques
1.3.3.14. Histogram
Purpose:
Summarize
a
Univariate
Data Set
The purpose of a histogram (Chambers) is to graphically
summarize the distribution of a univariate data set.
The histogram graphically shows the following:
1. center (i.e., the location) of the data;
2. spread (i.e., the scale) of the data;
3. skewness of the data;
4. presence of outliers; and
5. presence of multiple modes in the data.
These features provide strong indications of the proper
distributional model for the data. The probability plot or a
goodness-of-fit test can be used to verify the distributional
model.
The examples section shows the appearance of a number of
common features revealed by histograms.
Sample
Plot
Definition The most common form of the histogram is obtained by
splitting the range of the data into equal-sized bins (called
classes). Then for each bin, the number of points from the data
set that fall into each bin are counted. That is
1.3.3.14. Histogram
Vertical axis: Frequency (i.e., counts for each bin)
Horizontal axis: Response variable
The classes can either be defined arbitrarily by the user or via
some systematic rule. A number of theoretically derived rules
have been proposed by Scott (Scott 1992).
The cumulative histogram is a variation of the histogram in
which the vertical axis gives not just the counts for a single
bin, but rather gives the counts for that bin plus all bins for
smaller values of the response variable.
Both the histogram and cumulative histogram have an
additional variant whereby the counts are replaced by the
normalized counts. The names for these variants are the
relative histogram and the relative cumulative histogram.
There are two common ways to normalize the counts.
1. The normalized count is the count in a class divided by
the total number of observations. In this case the relative
counts are normalized to sum to one (or 100 if a
percentage scale is used). This is the intuitive case
where the height of the histogram bar represents the
proportion of the data in each class.
2. The normalized count is the count in the class divided
by the number of observations times the class width. For
this normalization, the area (or integral) under the
histogram is equal to one. From a probabilistic point of
view, this normalization results in a relative histogram
that is most akin to the probability density function and
a relative cumulative histogram that is most akin to the
cumulative distribution function. If you want to overlay
a probability density or cumulative distribution function
on top of the histogram, use this normalization.
Although this normalization is less intuitive (relative
frequencies greater than 1 are quite permissible), it is the
appropriate normalization if you are using the histogram
to model a probability density function.
Questions The histogram can be used to answer the following questions:
1. What kind of population distribution do the data come
from?
2. Where are the data located?
3. How spread out are the data?
4. Are the data symmetric or skewed?
5. Are there outliers in the data?
Examples 1. Normal
2. Symmetric, Non-Normal, Short-Tailed
1.3.3.14. Histogram
3. Symmetric, Non-Normal, Long-Tailed
4. Symmetric and Bimodal
5. Bimodal Mixture of 2 Normals
6. Skewed (Non-Symmetric) Right
7. Skewed (Non-Symmetric) Left
8. Symmetric with Outlier
Related
Techniques
Box plot
Probability plot
The techniques below are not discussed in the Handbook.
However, they are similar in purpose to the histogram.
Additional information on them is contained in the Chambers
and Scott references.
Frequency Plot
Stem and Leaf Plot
Density Trace
Case Study The histogram is demonstrated in the heat flow meter data
case study.
Software Histograms are available in most general purpose statistical
software programs. They are also supported in most general
purpose charting, spreadsheet, and business graphics
programs.
1.3.3.14.1. Histogram Interpretation: Normal
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33e1.htm[6/27/2012 2:01:00 PM]

1.3. EDA Techniques
1.3.3.14. Histogram
1.3.3.14.1. Histogram Interpretation: Normal
Symmetric,
Moderate-
Tailed
Histogram
Note the classical bell-shaped, symmetric histogram with
most of the frequency counts bunched in the middle and
with the counts dying off out in the tails. From a physical
science/engineering point of view, the normal distribution
is that distribution which occurs most often in nature (due
in part to the central limit theorem).
Recommended
Next Step
If the histogram indicates a symmetric, moderate tailed
distribution, then the recommended next step is to do a
normal probability plot to confirm approximate normality.
If the normal probability plot is linear, then the normal
distribution is a good model for the data.
1.3.3.14.2. Histogram Interpretation: Symmetric, Non-Normal, Short-Tailed

1.3. EDA Techniques
1.3.3.14. Histogram
1.3.3.14.2. Histogram Interpretation:
Symmetric, Non-Normal, Short-
Tailed
Symmetric,
Short-Tailed
Histogram
Description of
What Short-
Tailed Means
For a symmetric distribution, the "body" of a distribution
refers to the "center" of the distribution--commonly that
region of the distribution where most of the probability
resides--the "fat" part of the distribution. The "tail" of a
distribution refers to the extreme regions of the
distribution--both left and right. The "tail length" of a
distribution is a term that indicates how fast these extremes
approach zero.
For a short-tailed distribution, the tails approach zero very
fast. Such distributions commonly have a truncated
("sawed-off") look. The classical short-tailed distribution is
the uniform (rectangular) distribution in which the
probability is constant over a given range and then drops to
zero everywhere else--we would speak of this as having no
tails, or extremely short tails.
For a moderate-tailed distribution, the tails decline to zero
in a moderate fashion. The classical moderate-tailed
distribution is the normal (Gaussian) distribution.
1.3.3.14.2. Histogram Interpretation: Symmetric, Non-Normal, Short-Tailed
For a long-tailed distribution, the tails decline to zero very
slowly--and hence one is apt to see probability a long way
from the body of the distribution. The classical long-tailed
distribution is the Cauchy distribution.
In terms of tail length, the histogram shown above would
be characteristic of a "short-tailed" distribution.
The optimal (unbiased and most precise) estimator for
location for the center of a distribution is heavily
dependent on the tail length of the distribution. The
common choice of taking N observations and using the
calculated sample mean as the best estimate for the center
of the distribution is a good choice for the normal
distribution (moderate tailed), a poor choice for the
uniform distribution (short tailed), and a horrible choice for
the Cauchy distribution (long tailed). Although for the
normal distribution the sample mean is as precise an
estimator as we can get, for the uniform and Cauchy
distributions, the sample mean is not the best estimator.
For the uniform distribution, the midrange
midrange = (smallest + largest) / 2
is the best estimator of location. For a Cauchy distribution,
the median is the best estimator of location.
Recommended
Next Step
If the histogram indicates a symmetric, short-tailed
distribution, the recommended next step is to generate a
uniform probability plot. If the uniform probability plot is
linear, then the uniform distribution is an appropriate
model for the data.
1.3.3.14.3. Histogram Interpretation: Symmetric, Non-Normal, Long-Tailed

1.3. EDA Techniques
1.3.3.14. Histogram
Symmetric, Non-Normal, Long-
Tailed
Symmetric,
Long-Tailed
Histogram
Description of
Long-Tailed
The previous example contains a discussion of the
distinction between short-tailed, moderate-tailed, and long-
tailed distributions.
In terms of tail length, the histogram shown above would
be characteristic of a "long-tailed" distribution.
Recommended
Next Step
If the histogram indicates a symmetric, long tailed
distribution, the recommended next step is to do a Cauchy
probability plot. If the Cauchy probability plot is linear,
then the Cauchy distribution is an appropriate model for the
data. Alternatively, a Tukey Lambda PPCC plot may
provide insight into a suitable distributional model for the
data.
1.3.3.14.4. Histogram Interpretation: Symmetric and Bimodal

1.3. EDA Techniques
1.3.3.14. Histogram
Symmetric and Bimodal
Symmetric,
Bimodal
Histogram
Description of
Bimodal
The mode of a distribution is that value which is most
frequently occurring or has the largest probability of
occurrence. The sample mode occurs at the peak of the
histogram.
For many phenomena, it is quite common for the
distribution of the response values to cluster around a
single mode (unimodal) and then distribute themselves
with lesser frequency out into the tails. The normal
distribution is the classic example of a unimodal
distribution.
The histogram shown above illustrates data from a bimodal
(2 peak) distribution. The histogram serves as a tool for
diagnosing problems such as bimodality. Questioning the
underlying reason for distributional non-unimodality
frequently leads to greater insight and improved
deterministic modeling of the phenomenon under study.
For example, for the data presented above, the bimodal
histogram is caused by sinusoidality in the data.
Recommended
Next Step
If the histogram indicates a symmetric, bimodal
distribution, the recommended next steps are to:
1.3.3.14.4. Histogram Interpretation: Symmetric and Bimodal
1. Do a run sequence plot or a scatter plot to check for
sinusoidality.
2. Do a lag plot to check for sinusoidality. If the lag
plot is elliptical, then the data are sinusoidal.
3. If the data are sinusoidal, then a spectral plot is used
to graphically estimate the underlying sinusoidal
frequency.
4. If the data are not sinusoidal, then a Tukey Lambda
PPCC plot may determine the best-fit symmetric
distribution for the data.
5. The data may be fit with a mixture of two
distributions. A common approach to this case is to
fit a mixture of 2 normal or lognormal distributions.
Further discussion of fitting mixtures of distributions
is beyond the scope of this Handbook.
1.3.3.14.5. Histogram Interpretation: Bimodal Mixture of 2 Normals

1.3. EDA Techniques
1.3.3.14. Histogram
1.3.3.14.5. Histogram Interpretation: Bimodal
Mixture of 2 Normals
Histogram
from Mixture
of 2 Normal
Distributions
Discussion of
Unimodal and
Bimodal
The histogram shown above illustrates data from a bimodal
(2 peak) distribution.
In contrast to the previous example, this example
illustrates bimodality due not to an underlying
deterministic model, but bimodality due to a mixture of
probability models. In this case, each of the modes appears
to have a rough bell-shaped component. One could easily
imagine the above histogram being generated by a process
consisting of two normal distributions with the same
standard deviation but with two different locations (one
centered at approximately 9.17 and the other centered at
approximately 9.26). If this is the case, then the research
challenge is to determine physically why there are two
similar but separate sub-processes.
Recommended
Next Steps
If the histogram indicates that the data might be
appropriately fit with a mixture of two normal
distributions, the recommended next step is:
Fit the normal mixture model using either least squares or
maximum likelihood. The general normal mixing model is
1.3.3.14.5. Histogram Interpretation: Bimodal Mixture of 2 Normals
where p is the mixing proportion (between 0 and 1) and
and are normal probability density functions with
location and scale parameters , , , and ,
respectively. That is, there are 5 parameters to estimate in
the fit.
Whether maximum likelihood or least squares is used, the
quality of the fit is sensitive to good starting values. For the
mixture of two normals, the histogram can be used to
provide initial estimates for the location and scale
parameters of the two normal distributions.
Both Dataplot code and R code can be used to fit a
mixture of two normals.
1.3.3.14.6. Histogram Interpretation: Skewed (Non-Normal) Right

1.3. EDA Techniques
1.3.3.14. Histogram
1.3.3.14.6. Histogram Interpretation: Skewed
(Non-Normal) Right
Right-Skewed
Histogram
Discussion of
Skewness
A symmetric distribution is one in which the 2 "halves" of
the histogram appear as mirror-images of one another. A
skewed (non-symmetric) distribution is a distribution in
which there is no such mirror-imaging.
For skewed distributions, it is quite common to have one
tail of the distribution considerably longer or drawn out
relative to the other tail. A "skewed right" distribution is
one in which the tail is on the right side. A "skewed left"
distribution is one in which the tail is on the left side. The
above histogram is for a distribution that is skewed right.
Skewed distributions bring a certain philosophical
complexity to the very process of estimating a "typical
value" for the distribution. To be specific, suppose that the
analyst has a collection of 100 values randomly drawn
from a distribution, and wishes to summarize these 100
observations by a "typical value". What does typical value
mean? If the distribution is symmetric, the typical value is
unambiguous-- it is a well-defined center of the
distribution. For example, for a bell-shaped symmetric
distribution, a center point is identical to that value at the
peak of the distribution.
For a skewed distribution, however, there is no "center" in
the usual sense of the word. Be that as it may, several
"typical value" metrics are often used for skewed
distributions. The first metric is the mode of the
distribution. Unfortunately, for severely-skewed
distributions, the mode may be at or near the left or right
tail of the data and so it seems not to be a good
representative of the center of the distribution. As a second
choice, one could conceptually argue that the mean (the
point on the horizontal axis where the distributiuon would
balance) would serve well as the typical value. As a third
choice, others may argue that the median (that value on the
horizontal axis which has exactly 50% of the data to the
left (and also to the right) would serve as a good typical
value.
For symmetric distributions, the conceptual problem
disappears because at the population level the mode, mean,
and median are identical. For skewed distributions,
however, these 3 metrics are markedly different. In
practice, for skewed distributions the most commonly
reported typical value is the mean; the next most common
is the median; the least common is the mode. Because each
of these 3 metrics reflects a different aspect of
"centerness", it is recommended that the analyst report at
least 2 (mean and median), and preferably all 3 (mean,
median, and mode) in summarizing and characterizing a
data set.
Some Causes
for Skewed
Data
Skewed data often occur due to lower or upper bounds on
the data. That is, data that have a lower bound are often
skewed right while data that have an upper bound are often
skewed left. Skewness can also result from start-up effects.
For example, in reliability applications some processes
may have a large number of initial failures that could cause
left skewness. On the other hand, a reliability process
could have a long start-up period where failures are rare
resulting in right-skewed data.
Data collected in scientific and engineering applications
often have a lower bound of zero. For example, failure data
must be non-negative. Many measurement processes
generate only positive data. Time to occurence and size are
common measurements that cannot be less than zero.
Recommended
Next Steps
If the histogram indicates a right-skewed data set, the
recommended next steps are to:
1. Quantitatively summarize the data by computing and
reporting the sample mean, the sample median, and
the sample mode.
2. Determine the best-fit distribution (skewed-right)
from the
Weibull family (for the maximum)
Gamma family
Chi-square family
Lognormal family
Power lognormal family
3. Consider a normalizing transformation such as the
Box-Cox transformation.
1.3.3.14.7. Histogram Interpretation: Skewed (Non-Symmetric) Left

1.3. EDA Techniques
1.3.3.14. Histogram
1.3.3.14.7. Histogram Interpretation: Skewed
(Non-Symmetric) Left
Skewed
Left
Histogram
The issues for skewed left data are similar to those for skewed
right data.
1.3.3.14.8. Histogram Interpretation: Symmetric with Outlier

1.3. EDA Techniques
1.3.3.14. Histogram
Symmetric with Outlier
Symmetric
Histogram
with Outlier
Discussion of
Outliers
A symmetric distribution is one in which the 2 "halves" of
the histogram appear as mirror-images of one another. The
above example is symmetric with the exception of outlying
data near Y = 4.5.
An outlier is a data point that comes from a distribution
different (in location, scale, or distributional form) from
the bulk of the data. In the real world, outliers have a range
of causes, from as simple as
1. operator blunders
2. equipment failures
3. day-to-day effects
4. batch-to-batch differences
5. anomalous input conditions
6. warm-up effects
to more subtle causes such as
1. A change in settings of factors that (knowingly or
unknowingly) affect the response.
2. Nature is trying to tell us something.
1.3.3.14.8. Histogram Interpretation: Symmetric with Outlier
Outliers
Should be
Investigated
All outliers should be taken seriously and should be
investigated thoroughly for explanations. Automatic
outlier-rejection schemes (such as throw out all data
beyond 4 sample standard deviations from the sample
mean) are particularly dangerous.
The classic case of automatic outlier rejection becoming
automatic information rejection was the South Pole ozone
depletion problem. Ozone depletion over the South Pole
would have been detected years earlier except for the fact
that the satellite data recording the low ozone readings had
outlier-rejection code that automatically screened out the
"outliers" (that is, the low ozone readings) before the
analysis was conducted. Such inadvertent (and incorrect)
purging went on for years. It was not until ground-based
South Pole readings started detecting low ozone readings
that someone decided to double-check as to why the
satellite had not picked up this fact--it had, but it had
gotten thrown out!
The best attitude is that outliers are our "friends", outliers
are trying to tell us something, and we should not stop
until we are comfortable in the explanation for each outlier.
Recommended
Next Steps
If the histogram shows the presence of outliers, the
recommended next steps are:
1. Graphically check for outliers (in the commonly
encountered normal case) by generating a box plot.
In general, box plots are a much better graphical tool
for detecting outliers than are histograms.
2. Quantitatively check for outliers (in the commonly
encountered normal case) by carrying out Grubbs
test which indicates how many sample standard
deviations away from the sample mean are the data
in question. Large values indicate outliers.
1.3.3.15. Lag Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33f.htm[6/27/2012 2:01:06 PM]

1.3. EDA Techniques
1.3.3.15. Lag Plot
Purpose:
Check for
randomness
A lag plot checks whether a data set or time series is random
or not. Random data should not exhibit any identifiable
structure in the lag plot. Non-random structure in the lag plot
indicates that the underlying data are not random. Several
common patterns for lag plots are shown in the examples
below.
Sample
Plot
This sample lag plot exhibits a linear pattern. This shows that
the data are strongly non-random and further suggests that an
autoregressive model might be appropriate.
Definition A lag is a fixed time displacement. For example, given a data
set Y
1
, Y
2
..., Y
n
, Y
2
and Y
7
have lag 5 since 7 - 2 = 5. Lag
plots can be generated for any arbitrary lag, although the
most commonly used lag is 1.
A plot of lag 1 is a plot of the values of Y
i
versus Y
i-1
Vertical axis: Y
i
for all i
Horizontal axis: Y
i-1
for all i
Questions Lag plots can provide answers to the following questions:
1.3.3.15. Lag Plot
2. Is there serial correlation in the data?
3. What is a suitable model for the data?
Importance Inasmuch as randomness is an underlying assumption for
most statistical estimation and testing techniques, the lag plot
should be a routine tool for researchers.
Examples Random (White Noise)
Weak autocorrelation
Strong autocorrelation and autoregressive model
Sinusoidal model and outliers
Related
Techniques
Autocorrelation Plot
Spectrum
Runs Test
Case Study The lag plot is demonstrated in the beam deflection data case
study.
Software Lag plots are not directly available in most general purpose
statistical software programs. Since the lag plot is essentially
a scatter plot with the 2 variables properly lagged, it should
be feasible to write a macro for the lag plot in most statistical
programs.
1.3.3.15.1. Lag Plot: Random Data
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33f1.htm[6/27/2012 2:01:07 PM]

1.3. EDA Techniques
1.3.3.15. Lag Plot
1.3.3.15.1. Lag Plot: Random Data
Lag Plot
Conclusions We can make the following conclusions based on the above
plot.
2. The data exhibit no autocorrelation.
3. The data contain no outliers.
Discussion The lag plot shown above is for lag = 1. Note the absence of
structure. One cannot infer, from a current value Y
i-1
, the
next value Y
i
. Thus for a known value Y
i-1
on the horizontal
axis (say, Y
i-1
= +0.5), the Y
i
-th value could be virtually
anything (from Y
i
= -2.5 to Y
i
= +1.5). Such non-association
is the essence of randomness.
1.3.3.15.2. Lag Plot: Moderate Autocorrelation

1.3. EDA Techniques
1.3.3.15. Lag Plot
Lag Plot
Conclusions We can make the conclusions based on the above plot.
1. The data are from an underlying autoregressive
model with moderate positive autocorrelation
Discussion In the plot above for lag = 1, note how the points tend to
cluster (albeit noisily) along the diagonal. Such clustering
is the lag plot signature of moderate autocorrelation.
If the process were completely random, knowledge of a
current observation (say Y
i-1
= 0) would yield virtually no
knowledge about the next observation Y
i
. If the process has
moderate autocorrelation, as above, and if Y
i-1
= 0, then
the range of possible values for Y
i
is seen to be restricted
to a smaller range (.01 to +.01). This suggests prediction is
possible using an autoregressive model.
Recommended
Next Step
Estimate the parameters for the autoregressive model:
Since Y and Y are precisely the axes of the lag plot,
i i-1
such estimation is a linear regression straight from the lag
plot.
The residual standard deviation for the autoregressive
1.3.3.15.3. Lag Plot: Strong Autocorrelation and Autoregressive Model

1.3. EDA Techniques
1.3.3.15. Lag Plot
1.3.3.15.3. Lag Plot: Strong Autocorrelation and
Autoregressive Model
Lag Plot
Conclusions We can make the following conclusions based on the
above plot.
model with strong positive autocorrelation
Discussion Note the tight clustering of points along the diagonal. This
is the lag plot signature of a process with strong positive
autocorrelation. Such processes are highly non-random--
there is strong association between an observation and a
succeeding observation. In short, if you know Y
i-1
you can
make a strong guess as to what Y
i
will be.
If the above process were completely random, the plot
would have a shotgun pattern, and knowledge of a current
observation (say Y
i-1
= 3) would yield virtually no
knowledge about the next observation Y
i
(it could here be
anywhere from -2 to +8). On the other hand, if the process
had strong autocorrelation, as seen above, and if Y
i-1
= 3,
then the range of possible values for Y is seen to be
1.3.3.15.3. Lag Plot: Strong Autocorrelation and Autoregressive Model
i
restricted to a smaller range (2 to 4)--still wide, but an
improvement nonetheless (relative to -2 to +8) in
predictive power.
Recommended
Next Step
When the lag plot shows a strongly autoregressive pattern
and only successive observations appear to be correlated,
the next steps are to:
1. Extimate the parameters for the autoregressive
model:
Since Y
i
and Y
i-1
are precisely the axes of the lag
plot, such estimation is a linear regression straight
from the lag plot.
The residual standard deviation for this
autoregressive model will be much smaller than the
residual standard deviation for the default model
2. Reexamine the system to arrive at an explanation for
the strong autocorrelation. Is it due to the
1. phenomenon under study; or
2. drifting in the environment; or
3. contamination from the data acquisition
system?
Sometimes the source of the problem is
contamination and carry-over from the data
acquisition system where the system does not have
time to electronically recover before collecting the
next data point. If this is the case, then consider
slowing down the sampling rate to achieve
randomness.
1.3.3.15.4. Lag Plot: Sinusoidal Models and Outliers

1.3. EDA Techniques
1.3.3.15. Lag Plot
1.3.3.15.4. Lag Plot: Sinusoidal Models and
Outliers
Lag Plot
Conclusions We can make the following conclusions based on the
above plot.
1. The data come from an underlying single-cycle
sinusoidal model.
2. The data contain three outliers.
Discussion In the plot above for lag = 1, note the tight elliptical
clustering of points. Processes with a single-cycle
sinusoidal model will have such elliptical lag plots.
Consequences
of Ignoring
Cyclical
Pattern
If one were to naively assume that the above process came
from the null model
and then estimate the constant by the sample mean, then
the analysis would suffer because
1. the sample mean would be biased and meaningless;
2. the confidence limits would be meaningless and
optimistically small.
The proper model
(where is the amplitude, is the frequency--between 0
and .5 cycles per observation--, and is the phase) can be
fit by standard non-linear least squares, to estimate the
coefficients and their uncertainties.
The lag plot is also of value in outlier detection. Note in
the above plot that there appears to be 4 points lying off the
ellipse. However, in a lag plot, each point in the original
data set Y shows up twice in the lag plot--once as Y
i
and
once as Y
i-1
. Hence the outlier in the upper left at Y
i
= 300
is the same raw data value that appears on the far right at
Y
i-1
= 300. Thus (-500,300) and (300,200) are due to the
same outlier, namely the 158th data point: 300. The correct
value for this 158th point should be approximately -300
and so it appears that a sign got dropped in the data
collection. The other two points lying off the ellipse, at
roughly (100,100) and at (0,-50), are caused by two faulty
data values: the third data point of -15 should be about
+125 and the fourth data point of +141 should be about -
50, respectively. Hence the 4 apparent lag plot outliers are
traceable to 3 actual outliers in the original run sequence:
at points 4 (-15), 5 (141) and 158 (300). In retrospect, only
one of these (point 158 (= 300)) is an obvious outlier in the
run sequence plot.
Unexpected
Value of EDA
Frequently a technique (e.g., the lag plot) is constructed to
check one aspect (e.g., randomness) which it does well.
Along the way, the technique also highlights some other
anomaly of the data (namely, that there are 3 outliers).
Such outlier identification and removal is extremely
important for detecting irregularities in the data collection
system, and also for arriving at a "purified" data set for
modeling. The lag plot plays an important role in such
outlier identification.
Recommended
Next Step
When the lag plot indicates a sinusoidal model with
possible outliers, the recommended next steps are:
1. Do a spectral plot to obtain an initial estimate of the
frequency of the underlying cycle. This will be
helpful as a starting value for the subsequent non-
linear fitting.
2. Omit the outliers.
3. Carry out a non-linear fit of the model to the 197
points.
1.3.3.16. Linear Correlation Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33g.htm[6/27/2012 2:01:10 PM]

1.3. EDA Techniques
Purpose:
Detect
changes in
correlation
between
groups
Linear correlation plots are used to assess whether or not
correlations are consistent across groups. That is, if your
data is in groups, you may want to know if a single
correlation can be used across all the groups or whether
separate correlations are required for each group.
Linear correlation plots are often used in conjunction with
linear slope, linear intercept, and linear residual standard
deviation plots. A linear correlation plot could be generated
intially to see if linear fitting would be a fruitful direction. If
the correlations are high, this implies it is worthwhile to
continue with the linear slope, intercept, and residual
standard deviation plots. If the correlations are weak, a
different model needs to be pursued.
In some cases, you might not have groups. Instead you may
have different data sets and you want to know if the same
correlation can be adequately applied to each of the data
sets. In this case, simply think of each distinct data set as a
group and apply the linear slope plot as for groups.
Sample Plot
This linear correlation plot shows that the correlations are
high for all groups. This implies that linear fits could
provide a good model for each of these groups.
Definition:
Group
Correlations
Versus
Group ID
Linear correlation plots are formed by:
Vertical axis: Group correlations
Horizontal axis: Group identifier
A reference line is plotted at the correlation between the full
data sets.
Questions The linear correlation plot can be used to answer the
following questions.
1. Are there linear relationships across groups?
2. Are the strength of the linear relationships relatively
constant across the groups?
Importance:
Checking
Group
Homogeneity
For grouped data, it may be important to know whether the
different groups are homogeneous (i.e., similar) or
heterogeneous (i.e., different). Linear correlation plots help
answer this question in the context of linear fitting.
Related
Techniques
Linear Intercept Plot
Linear Slope Plot
Linear Residual Standard Deviation Plot
Linear Fitting
Case Study The linear correlation plot is demonstrated in the Alaska
Software Most general purpose statistical software programs do not
support a linear correlation plot. However, if the statistical
program can generate correlations over a group, it should be
feasible to write a macro to generate this plot.
1.3.3.17. Linear Intercept Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33h.htm[6/27/2012 2:01:11 PM]

1.3. EDA Techniques
Purpose:
Detect
changes in
linear
intercepts
between
groups
Linear intercept plots are used to graphically assess whether
or not linear fits are consistent across groups. That is, if your
data have groups, you may want to know if a single fit can
be used across all the groups or whether separate fits are
required for each group.
Linear intercept plots are typically used in conjunction with
linear slope and linear residual standard deviation plots.
In some cases you might not have groups. Instead, you have
different data sets and you want to know if the same fit can
be adequately applied to each of the data sets. In this case,
simply think of each distinct data set as a group and apply
the linear intercept plot as for groups.
Sample Plot
This linear intercept plot shows that there is a shift in
intercepts. Specifically, the first three intercepts are lower
than the intercepts for the other groups. Note that these are
small differences in the intercepts.
Definition:
Group
Intercepts
Versus
Linear intercept plots are formed by:
Vertical axis: Group intercepts from linear fits
Group ID
A reference line is plotted at the intercept from a linear fit
using all the data.
Questions The linear intercept plot can be used to answer the following
questions.
1. Is the intercept from linear fits relatively constant
across groups?
2. If the intercepts vary across groups, is there a
discernible pattern?
Importance:
Checking
Group
Homogeneity
heterogeneous (i.e., different). Linear intercept plots help
Related
Techniques
Linear Correlation Plot
Linear Slope Plot
Linear Fitting
Case Study The linear intercept plot is demonstrated in the Alaska
support a linear intercept plot. However, if the statistical
program can generate linear fits over a group, it should be
1.3.3.18. Linear Slope Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33i.htm[6/27/2012 2:01:12 PM]

1.3. EDA Techniques
Purpose:
Detect
changes in
linear slopes
between
groups
Linear slope plots are used to graphically assess whether or
not linear fits are consistent across groups. That is, if your
data have groups, you may want to know if a single fit can
be used across all the groups or whether separate fits are
required for each group.
Linear slope plots are typically used in conjunction with
linear intercept and linear residual standard deviation plots.
the linear slope plot as for groups.
Sample Plot
This linear slope plot shows that the slopes are about 0.174
(plus or minus 0.002) for all groups. There does not appear
to be a pattern in the variation of the slopes. This implies
that a single fit may be adequate.
Definition:
Group
Slopes
Versus
Group ID
Linear slope plots are formed by:
Vertical axis: Group slopes from linear fits
A reference line is plotted at the slope from a linear fit using
all the data.
Questions The linear slope plot can be used to answer the following
questions.
1. Do you get the same slope across groups for linear
fits?
2. If the slopes differ, is there a discernible pattern in the
slopes?
Importance:
Checking
Group
Homogeneity
heterogeneous (i.e., different). Linear slope plots help
Related
Techniques
Linear Fitting
Case Study The linear slope plot is demonstrated in the Alaska pipeline
data case study.
support a linear slope plot. However, if the statistical
program can generate linear fits over a group, it should be
1.3.3.19. Linear Residual Standard Deviation Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33j.htm[6/27/2012 2:01:13 PM]

1.3. EDA Techniques
1.3.3.19. Linear Residual Standard Deviation
Plot
Purpose:
Detect
Changes in
Linear
Residual
Standard
Deviation
Between
Groups
Linear residual standard deviation (RESSD) plots are used
to graphically assess whether or not linear fits are consistent
across groups. That is, if your data have groups, you may
want to know if a single fit can be used across all the groups
or whether separate fits are required for each group.
The residual standard deviation is a goodness-of-fit
measure. That is, the smaller the residual standard deviation,
the closer is the fit to the data.
Linear RESSD plots are typically used in conjunction with
linear intercept and linear slope plots. The linear intercept
and slope plots convey whether or not the fits are consistent
across groups while the linear RESSD plot conveys whether
the adequacy of the fit is consistent across groups.
the linear RESSD plot as for groups.
Sample Plot
1.3.3.19. Linear Residual Standard Deviation Plot
This linear RESSD plot shows that the residual standard
deviations from a linear fit are about 0.0025 for all the
groups.
Definition:
Group
Residual
Standard
Deviation
Versus
Group ID
Linear RESSD plots are formed by:
Vertical axis: Group residual standard deviations from
linear fits
A reference line is plotted at the residual standard deviation
from a linear fit using all the data. This reference line will
typically be much greater than any of the individual residual
standard deviations.
Questions The linear RESSD plot can be used to answer the following
questions.
1. Is the residual standard deviation from a linear fit
constant across groups?
2. If the residual standard deviations vary, is there a
discernible pattern across the groups?
Importance:
Checking
Group
Homogeneity
heterogeneous (i.e., different). Linear RESSD plots help
Related
Techniques
Linear Slope Plot
Linear Fitting
Case Study The linear residual standard deviation plot is demonstrated
in the Alaska pipeline data case study.
support a linear residual standard deviation plot. However, if
the statistical program can generate linear fits over a group,
it should be feasible to write a macro to generate this plot.
1.3.3.20. Mean Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33k.htm[6/27/2012 2:01:13 PM]

1.3. EDA Techniques
1.3.3.20. Mean Plot
Purpose:
Detect
changes in
location
between
groups
Mean plots are used to see if the mean varies between
different groups of the data. The grouping is determined by
the analyst. In most cases, the data set contains a specific
grouping variable. For example, the groups may be the levels
of a factor variable. In the sample plot below, the months of
the year provide the grouping.
Mean plots can be used with ungrouped data to determine if
the mean is changing over time. In this case, the data are
split into an arbitrary number of equal-sized groups. For
example, a data series with 400 points can be divided into 10
groups of 40 points each. A mean plot can then be generated
with these groups to see if the mean is increasing or
decreasing over time.
Although the mean is the most commonly used measure of
location, the same concept applies to other measures of
location. For example, instead of plotting the mean of each
group, the median or the trimmed mean might be plotted
instead. This might be done if there were significant outliers
in the data and a more robust measure of location than the
mean was desired.
Mean plots are typically used in conjunction with standard
deviation plots. The mean plot checks for shifts in location
while the standard deviation plot checks for shifts in scale.
Sample Plot
1.3.3.20. Mean Plot
This sample mean plot shows a shift of location after the 6th
month.
Definition:
Group
Means
Versus
Group ID
Mean plots are formed by:
Vertical axis: Group mean
A reference line is plotted at the overall mean.
Questions The mean plot can be used to answer the following
questions.
1. Are there any shifts in location?
2. What is the magnitude of the shifts in location?
3. Is there a distinct pattern in the shifts in location?
Importance:
Checking
Assumptions
A common assumption in 1-factor analyses is that of
constant location. That is, the location is the same for
different levels of the factor variable. The mean plot provides
a graphical check for that assumption. A common assumption
for univariate data is that the location is constant. By
grouping the data into equal intervals, the mean plot can
provide a graphical test of this assumption.
Related
Techniques
Standard Deviation Plot
DOE Mean Plot
Box Plot
support a mean plot. However, if the statistical program can
generate the mean over a group, it should be feasible to write
a macro to generate this plot.
1.3.3.20. Mean Plot
1.3.3.21. Normal Probability Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33l.htm[6/27/2012 2:01:14 PM]

1.3. EDA Techniques
Purpose:
Check If Data
Are
Approximately
Normally
Distributed
The normal probability plot (Chambers 1983) is a
graphical technique for assessing whether or not a data set
is approximately normally distributed.
The data are plotted against a theoretical normal
distribution in such a way that the points should form an
approximate straight line. Departures from this straight line
indicate departures from normality.
The normal probability plot is a special case of the
probability plot. We cover the normal probability plot
separately due to its importance in many applications.
Sample Plot
The points on this plot form a nearly linear pattern, which
indicates that the normal distribution is a good model for
this data set.
Definition:
Ordered
Response
Values Versus
Normal Order
Statistic
Medians
The normal probability plot is formed by:
Vertical axis: Ordered response values
Horizontal axis: Normal order statistic medians
The observations are plotted as a function of the
corresponding normal order statistic medians which are
defined as:
N(i) = G(U(i))
where U(i) are the uniform order statistic medians (defined
below) and G is the percent point function of the normal
distribution. The percent point function is the inverse of
the cumulative distribution function (probability that x is
less than or equal to some value). That is, given a
probability, we want the corresponding x of the cumulative
distribution function.
The uniform order statistic medians are defined as:
U(i) = 1 - U(n) for i = 1
U(i) = (i - 0.3175)/(n + 0.365) for i = 2, 3, ..., n-1
U(i) = 0.5
(1/n)
for i = n
In addition, a straight line can be fit to the points and
added as a reference line. The further the points vary from
this line, the greater the indication of departures from
normality.
Probability plots for distributions other than the normal are
computed in exactly the same way. The normal percent
point function (the G) is simply replaced by the percent
point function of the desired distribution. That is, a
probability plot can easily be generated for any distribution
for which you have the percent point function.
One advantage of this method of computing probability
plots is that the intercept and slope estimates of the fitted
line are in fact estimates for the location and scale
parameters of the distribution. Although this is not too
important for the normal distribution since the location and
scale are estimated by the mean and standard deviation,
respectively, it can be useful for many other distributions.
The correlation coefficient of the points on the normal
probability plot can be compared to a table of critical
values to provide a formal test of the hypothesis that the
data come from a normal distribution.
Questions The normal probability plot is used to answer the
1. Are the data normally distributed?
2. What is the nature of the departure from normality
(data skewed, shorter than expected tails, longer than
expected tails)?
Importance:
Check
The underlying assumptions for a measurement process are
that the data should behave like:
Normality
Assumption
1. random drawings;
3. with fixed location;
4. with fixed scale.
Probability plots are used to assess the assumption of a
fixed distribution. In particular, most statistical models are
of the form:
response = deterministic + random
where the deterministic part is the fit and the random part
is error. This error component in most common statistical
models is specifically assumed to be normally distributed
with fixed location and scale. This is the most frequent
application of normal probability plots. That is, a model is
fit and a normal probability plot is generated for the
residuals from the fitted model. If the residuals from the
fitted model are not normally distributed, then one of the
major assumptions of the model has been violated.
Examples 1. Data are normally distributed
2. Data have short tails
3. Data have fat tails
4. Data are skewed right
Related
Techniques
Histogram
Probability plots for other distributions (e.g., Weibull)
Probability plot correlation coefficient plot (PPCC plot)
Anderson-Darling Goodness-of-Fit Test
Chi-Square Goodness-of-Fit Test
Kolmogorov-Smirnov Goodness-of-Fit Test
Case Study The normal probability plot is demonstrated in the heat
flow meter data case study.
Software Most general purpose statistical software programs can
generate a normal probability plot.
1.3.3.21.1. Normal Probability Plot: Normally Distributed Data
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33l1.htm[6/27/2012 2:01:15 PM]

1.3. EDA Techniques
1.3.3.21.1. Normal Probability Plot: Normally
Distributed Data
Normal
Probability
Plot
The following normal probability plot is from the heat flow
meter data.
Conclusions We can make the following conclusions from the above plot.
1. The normal probability plot shows a strongly linear
pattern. There are only minor deviations from the line
fit to the points on the probability plot.
2. The normal distribution appears to be a good model for
these data.
Discussion Visually, the probability plot shows a strongly linear pattern.
This is verified by the correlation coefficient of 0.9989 of the
line fit to the probability plot. The fact that the points in the
lower and upper extremes of the plot do not deviate
significantly from the straight-line pattern indicates that there
are not any significant outliers (relative to a normal
distribution).
In this case, we can quite reasonably conclude that the
normal distribution provides an excellent model for the data.
The intercept and slope of the fitted line give estimates of
1.3.3.21.1. Normal Probability Plot: Normally Distributed Data
9.26 and 0.023 for the location and scale parameters of the
fitted normal distribution.
1.3.3.21.2. Normal Probability Plot: Data Have Short Tails

1.3. EDA Techniques
1.3.3.21.2. Normal Probability Plot: Data Have
Short Tails
Normal
Probability
Plot for
Data with
Short Tails
The following is a normal probability plot for 500 random
numbers generated from a Tukey-Lambda distribution with
the parameter equal to 1.1.
1. The normal probability plot shows a non-linear pattern.
2. The normal distribution is not a good model for these
data.
Discussion For data with short tails relative to the normal distribution,
the non-linearity of the normal probability plot shows up in
two ways. First, the middle of the data shows an S-like
pattern. This is common for both short and long tails.
Second, the first few and the last few points show a marked
departure from the reference fitted line. In comparing this
plot to the long tail example in the next section, the
important difference is the direction of the departure from the
fitted line for the first few and last few points. For short tails,
the first few points show increasing departure from the fitted
line above the line and last few points show increasing
departure from the fitted line below the line. For long tails,
1.3.3.21.2. Normal Probability Plot: Data Have Short Tails
this pattern is reversed.
In this case, we can reasonably conclude that the normal
distribution does not provide an adequate fit for this data set.
For probability plots that indicate short-tailed distributions,
the next step might be to generate a Tukey Lambda PPCC
plot. The Tukey Lambda PPCC plot can often be helpful in
identifying an appropriate distributional family.
1.3.3.21.3. Normal Probability Plot: Data Have Long Tails

1.3. EDA Techniques
1.3.3.21.3. Normal Probability Plot: Data Have
Long Tails
Normal
Probability
Plot for
Data with
Long Tails
The following is a normal probability plot of 500 numbers
generated from a double exponential distribution. The double
exponential distribution is symmetric, but relative to the
normal it declines rapidly and has longer tails.
1. The normal probability plot shows a reasonably linear
pattern in the center of the data. However, the tails,
particularly the lower tail, show departures from the
fitted line.
2. A distribution other than the normal distribution would
be a good model for these data.
Discussion For data with long tails relative to the normal distribution, the
non-linearity of the normal probability plot can show up in
two ways. First, the middle of the data may show an S-like
pattern. This is common for both short and long tails. In this
particular case, the S pattern in the middle is fairly mild.
Second, the first few and the last few points show marked
departure from the reference fitted line. In the plot above,
this is most noticeable for the first few data points. In
1.3.3.21.3. Normal Probability Plot: Data Have Long Tails
comparing this plot to the short-tail example in the previous
section, the important difference is the direction of the
departure from the fitted line for the first few and the last few
points. For long tails, the first few points show increasing
departure from the fitted line below the line and last few
points show increasing departure from the fitted line above
the line. For short tails, this pattern is reversed.
In this case we can reasonably conclude that the normal
distribution can be improved upon as a model for these data.
For probability plots that indicate long-tailed distributions,
the next step might be to generate a Tukey Lambda PPCC
plot. The Tukey Lambda PPCC plot can often be helpful in
identifying an appropriate distributional family.
1.3.3.21.4. Normal Probability Plot: Data are Skewed Right

1.3. EDA Techniques
1.3.3.21.4. Normal Probability Plot: Data are
Skewed Right
Normal
Probability
Plot for
Data that
are Skewed
Right
1. The normal probability plot shows a strongly non-
linear pattern. Specifically, it shows a quadratic pattern
in which all the points are below a reference line
drawn between the first and last points.
2. The normal distribution is not a good model for these
data.
Discussion This quadratic pattern in the normal probability plot is the
signature of a significantly right-skewed data set. Similarly,
if all the points on the normal probability plot fell above the
reference line connecting the first and last points, that would
be the signature pattern for a significantly left-skewed data
set.
In this case we can quite reasonably conclude that we need to
model these data with a right skewed distribution such as the
Weibull or lognormal.
1.3.3.21.4. Normal Probability Plot: Data are Skewed Right
1.3.3.22. Probability Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33m.htm[6/27/2012 2:01:18 PM]

1.3. EDA Techniques
Purpose:
Check If
Data Follow
a Given
Distribution
The probability plot (Chambers 1983) is a graphical
technique for assessing whether or not a data set follows a
given distribution such as the normal or Weibull.
The data are plotted against a theoretical distribution in such
a way that the points should form approximately a straight
line. Departures from this straight line indicate departures
from the specified distribution.
The correlation coefficient associated with the linear fit to
the data in the probability plot is a measure of the goodness
of the fit. Estimates of the location and scale parameters of
the distribution are given by the intercept and slope.
Probability plots can be generated for several competing
distributions to see which provides the best fit, and the
probability plot generating the highest correlation
coefficient is the best choice since it generates the
straightest probability plot.
For distributions with shape parameters (not counting
location and scale parameters), the shape parameters must
be known in order to generate the probability plot. For
distributions with a single shape parameter, the probability
plot correlation coefficient (PPCC) plot provides an
excellent method for estimating the shape parameter.
We cover the special case of the normal probability plot
separately due to its importance in many statistical
applications.
Sample Plot
This data is a set of 500 Weibull random numbers with a
shape parameter = 2, location parameter = 0, and scale
parameter = 1. The Weibull probability plot indicates that
the Weibull distribution does in fact fit these data well.
Definition:
Ordered
Response
Values
Versus Order
Statistic
Medians for
the Given
Distribution
The probability plot is formed by:
Vertical axis: Ordered response values
Horizontal axis: Order statistic medians for the given
distribution
The order statistic medians are defined as:
N(i) = G(U(i))
where the U(i) are the uniform order statistic medians
(defined below) and G is the percent point function for the
desired distribution. The percent point function is the
inverse of the cumulative distribution function (probability
that x is less than or equal to some value). That is, given a
probability, we want the corresponding x of the cumulative
The uniform order statistic medians are defined as:
m(i) = 1 - m(n) for i = 1
m(i) = (i - 0.3175)/(n + 0.365) for i = 2, 3, ..., n-1
m(i) = 0.5**(1/n) for i = n
In addition, a straight line can be fit to the points and added
as a reference line. The further the points vary from this
line, the greater the indication of a departure from the
specified distribution.
This definition implies that a probability plot can be easily
generated for any distribution for which the percent point
function can be computed.
One advantage of this method of computing proability plots
is that the intercept and slope estimates of the fitted line are
in fact estimates for the location and scale parameters of the
distribution. Although this is not too important for the
normal distribution (the location and scale are estimated by
the mean and standard deviation, respectively), it can be
useful for many other distributions.
Questions The probability plot is used to answer the following
questions:
Does a given distribution, such as the Weibull,
provide a good fit to my data?
What distribution best fits my data?
What are good estimates for the location and scale
parameters of the chosen distribution?
Importance:
Check
distributional
assumption
The discussion for the normal probability plot covers the
use of probability plots for checking the fixed distribution
assumption.
Some statistical models assume data have come from a
population with a specific type of distribution. For example,
in reliability applications, the Weibull, lognormal, and
exponential are commonly used distributional models.
Probability plots can be useful for checking this
distributional assumption.
Related
Techniques
Histogram
Probability Plot Correlation Coefficient (PPCC) Plot
Hazard Plot
Quantile-Quantile Plot
Anderson-Darling Goodness of Fit
Chi-Square Goodness of Fit
Kolmogorov-Smirnov Goodness of Fit
Case Study The probability plot is demonstrated in the uniform random
numbers case study.
Software Most general purpose statistical software programs support
probability plots for at least a few common distributions.
1.3.3.23. Probability Plot Correlation Coefficient Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33n.htm[6/27/2012 2:01:19 PM]

1.3. EDA Techniques
1.3.3.23. Probability Plot Correlation
Coefficient Plot
Purpose:
Graphical
Technique for
Finding the
Shape
Parameter of
a
Distributional
Family that
Best Fits a
Data Set
The probability plot correlation coefficient (PPCC) plot
(Filliben 1975) is a graphical technique for identifying the
shape parameter for a distributional family that best
describes the data set. This technique is appropriate for
families, such as the Weibull, that are defined by a single
shape parameter and location and scale parameters, and it is
not appropriate for distributions, such as the normal, that
are defined only by location and scale parameters.
The PPCC plot is generated as follows. For a series of
values for the shape parameter, the correlation coefficient is
computed for the probability plot associated with a given
value of the shape parameter. These correlation coefficients
are plotted against their corresponding shape parameters.
The maximum correlation coefficient corresponds to the
optimal value of the shape parameter. For better precision,
two iterations of the PPCC plot can be generated; the first
is for finding the right neighborhood and the second is for
fine tuning the estimate.
The PPCC plot is used first to find a good value of the
shape parameter. The probability plot is then generated to
find estimates of the location and scale parameters and in
addition to provide a graphical assessment of the adequacy
of the distributional fit.
Compare
Distributions
In addition to finding a good choice for estimating the
shape parameter of a given distribution, the PPCC plot can
be useful in deciding which distributional family is most
appropriate. For example, given a set of reliabilty data, you
might generate PPCC plots for a Weibull, lognormal,
gamma, and inverse Gaussian distributions, and possibly
others, on a single page. This one page would show the
best value for the shape parameter for several distributions
and would additionally indicate which of these
distributional families provides the best fit (as measured by
the maximum probability plot correlation coefficient). That
is, if the maximum PPCC value for the Weibull is 0.99 and
only 0.94 for the lognormal, then we could reasonably
conclude that the Weibull family is the better choice.
Tukey-
Lambda
PPCC Plot
for Symmetric
Distributions
The Tukey Lambda PPCC plot, with shape parameter , is
particularly useful for symmetric distributions. It indicates
whether a distribution is short or long tailed and it can
further indicate several common distributions. Specifically,
1. = -1: distribution is approximately Cauchy
2. = 0: distribution is exactly logistic
3. = 0.14: distribution is approximately normal
4. = 0.5: distribution is U-shaped
5. = 1: distribution is exactly uniform
If the Tukey Lambda PPCC plot gives a maximum value
near 0.14, we can reasonably conclude that the normal
distribution is a good model for the data. If the maximum
value is less than 0.14, a long-tailed distribution such as the
double exponential or logistic would be a better choice. If
the maximum value is near -1, this implies the selection of
very long-tailed distribution, such as the Cauchy. If the
maximum value is greater than 0.14, this implies a short-
tailed distribution such as the Beta or uniform.
The Tukey-Lambda PPCC plot is used to suggest an
appropriate distribution. You should follow-up with PPCC
and probability plots of the appropriate alternatives.
Use
Judgement
When
Selecting An
Appropriate
Distributional
Family
When comparing distributional models, do not simply
choose the one with the maximum PPCC value. In many
cases, several distributional fits provide comparable PPCC
values. For example, a lognormal and Weibull may both fit
a given set of reliability data quite well. Typically, we
would consider the complexity of the distribution. That is, a
simpler distribution with a marginally smaller PPCC value
may be preferred over a more complex distribution.
Likewise, there may be theoretical justification in terms of
the underlying scientific model for preferring a distribution
with a marginally smaller PPCC value in some cases. In
other cases, we may not need to know if the distributional
model is optimal, only that it is adequate for our purposes.
That is, we may be able to use techniques designed for
normally distributed data even if other distributions fit the
data somewhat better.
Sample Plot The following is a PPCC plot of 100 normal random
numbers. The maximum value of the correlation coefficient
= 0.997 at = 0.099.
This PPCC plot shows that:
1. the best-fit symmetric distribution is nearly normal;
2. the data are not long tailed;
3. the sample mean would be an appropriate estimator
of location.
We can follow-up this PPCC plot with a normal probability
plot to verify the normality model for the data.
Definition: The PPCC plot is formed by:
Vertical axis: Probability plot correlation coefficient;
Horizontal axis: Value of shape parameter.
Questions The PPCC plot answers the following questions:
1. What is the best-fit member within a distributional
family?
2. Does the best-fit member provide a good fit (in terms
of generating a probability plot with a high
correlation coefficient)?
3. Does this distributional family provide a good fit
compared to other distributions?
4. How sensitive is the choice of the shape parameter?
Importance Many statistical analyses are based on distributional
assumptions about the population from which the data have
been obtained. However, distributional families can have
radically different shapes depending on the value of the
shape parameter. Therefore, finding a reasonable choice for
the shape parameter is a necessary step in the analysis. In
many analyses, finding a good distributional model for the
data is the primary focus of the analysis. In both of these
cases, the PPCC plot is a valuable tool.
Related
Techniques
Probability Plot
Maximum Likelihood Estimation
Least Squares Estimation
Method of Moments Estimation
Software PPCC plots are currently not available in most common
general purpose statistical software programs. However, the
underlying technique is based on probability plots and
correlation coefficients, so it should be possible to write
macros for PPCC plots in statistical programs that support
these capabilities. Dataplot supports PPCC plots.
1.3.3.24. Quantile-Quantile Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33o.htm[6/27/2012 2:01:20 PM]

1.3. EDA Techniques
Purpose:
Check If
Two Data
Sets Can Be
Fit With the
Same
Distribution
The quantile-quantile (q-q) plot is a graphical technique for
determining if two data sets come from populations with a
common distribution.
A q-q plot is a plot of the quantiles of the first data set
against the quantiles of the second data set. By a quantile, we
mean the fraction (or percent) of points below the given
value. That is, the 0.3 (or 30%) quantile is the point at which
30% percent of the data fall below and 70% fall above that
value.
A 45-degree reference line is also plotted. If the two sets
come from a population with the same distribution, the points
should fall approximately along this reference line. The
greater the departure from this reference line, the greater the
evidence for the conclusion that the two data sets have come
from populations with different distributions.
The advantages of the q-q plot are:
1. The sample sizes do not need to be equal.
2. Many distributional aspects can be simultaneously
tested. For example, shifts in location, shifts in scale,
changes in symmetry, and the presence of outliers can
all be detected from this plot. For example, if the two
data sets come from populations whose distributions
differ only by a shift in location, the points should lie
along a straight line that is displaced either up or down
from the 45-degree reference line.
The q-q plot is similar to a probability plot. For a probability
plot, the quantiles for one of the data samples are replaced
with the quantiles of a theoretical distribution.
Sample Plot
This q-q plot shows that
1. These 2 batches do not appear to have come from
populations with a common distribution.
2. The batch 1 values are significantly higher than the
corresponding batch 2 values.
3. The differences are increasing from values 525 to 625.
Then the values for the 2 batches get closer again.
Definition:
Quantiles
for Data Set
1 Versus
Quantiles of
Data Set 2
The q-q plot is formed by:
Vertical axis: Estimated quantiles from data set 1
Horizontal axis: Estimated quantiles from data set 2
Both axes are in units of their respective data sets. That is,
the actual quantile level is not plotted. For a given point on
the q-q plot, we know that the quantile level is the same for
both points, but not what that quantile level actually is.
If the data sets have the same size, the q-q plot is essentially
a plot of sorted data set 1 against sorted data set 2. If the data
sets are not of equal size, the quantiles are usually picked to
correspond to the sorted values from the smaller data set and
then the quantiles for the larger data set are interpolated.
Questions The q-q plot is used to answer the following questions:
Do two data sets come from populations with a
common distribution?
Do two data sets have common location and scale?
Do two data sets have similar distributional shapes?
Do two data sets have similar tail behavior?
Importance:
Check for
Common
When there are two data samples, it is often desirable to
know if the assumption of a common distribution is justified.
If so, then location and scale estimators can pool both data
Distribution sets to obtain estimates of the common location and scale. If
two samples do differ, it is also useful to gain some
understanding of the differences. The q-q plot can provide
more insight into the nature of the difference than analytical
methods such as the chi-square and Kolmogorov-Smirnov 2-
sample tests.
Related
Techniques
Bihistogram
T Test
F Test
2-Sample Chi-Square Test
2-Sample Kolmogorov-Smirnov Test
Case Study The quantile-quantile plot is demonstrated in the ceramic
strength data case study.
Software Q-Q plots are available in some general purpose statistical
software programs. If the number of data points in the two
samples are equal, it should be relatively easy to write a
macro in statistical programs that do not support the q-q plot.
If the number of points are not equal, writing a macro for a
q-q plot may be difficult.
1.3.3.25. Run-Sequence Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33p.htm[6/27/2012 2:01:21 PM]

1.3. EDA Techniques
Purpose:
Check for
Shifts in
Location
and Scale
and Outliers
Run sequence plots (Chambers 1983) are an easy way to
graphically summarize a univariate data set. A common
assumption of univariate data sets is that they behave like:
1. random drawings;
3. with a common location; and
4. with a common scale.
With run sequence plots, shifts in location and scale are
typically quite evident. Also, outliers can easily be detected.
Sample
Plot:
Last Third
of Data
Shows a
Shift of
Location
This sample run sequence plot shows that the location shifts
up for the last third of the data.
Definition:
y(i) Versus i
Run sequence plots are formed by:
Vertical axis: Response variable Y(i)
Horizontal axis: Index i (i = 1, 2, 3, ... )
Questions The run sequence plot can be used to answer the following
questions
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33p.htm[6/27/2012 2:01:21 PM]
1. Are there any shifts in location?
2. Are there any shifts in variation?
The run sequence plot can also give the analyst an excellent
feel for the data.
Importance:
Check
Univariate
Assumptions
For univariate data, the default model is
where the error is assumed to be random, from a fixed
distribution, and with constant location and scale. The
validity of this model depends on the validity of these
assumptions. The run sequence plot is useful for checking for
constant location and scale.
Even for more complex models, the assumptions on the error
term are still often the same. That is, a run sequence plot of
the residuals (even from very complex models) is still vital
for checking for outliers and for detecting shifts in location
and scale.
Related
Techniques
Scatter Plot
Histogram
Lag Plot
Case Study The run sequence plot is demonstrated in the Filter
transmittance data case study.
Software Run sequence plots are available in most general purpose
statistical software programs.
1.3.3.26. Scatter Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33q.htm[6/27/2012 2:01:21 PM]

1.3. EDA Techniques
Purpose:
Check for
Relationship
A scatter plot (Chambers 1983) reveals relationships or
association between two variables. Such relationships
manifest themselves by any non-random structure in the plot.
Various common types of patterns are demonstrated in the
examples.
Sample
Plot:
Linear
Relationship
Between
Variables Y
and X
This sample plot reveals a linear relationship between the
two variables indicating that a linear regression model might
be appropriate.
Definition:
Y Versus X
A scatter plot is a plot of the values of Y versus the
corresponding values of X:
Vertical axis: variable Y--usually the response variable
Horizontal axis: variable X--usually some variable we
suspect may ber related to the response
Questions Scatter plots can provide answers to the following questions:
1. Are variables X and Y related?
2. Are variables X and Y linearly related?
3. Are variables X and Y non-linearly related?
4. Does the variation in Y change depending on X?
Examples 1. No relationship
2. Strong linear (positive correlation)
3. Strong linear (negative correlation)
4. Exact linear (positive correlation)
5. Quadratic relationship
6. Exponential relationship
7. Sinusoidal relationship (damped)
8. Variation of Y doesn't depend on X (homoscedastic)
9. Variation of Y does depend on X (heteroscedastic)
10. Outlier
Combining
Scatter
Plots
Scatter plots can also be combined in multiple plots per page
to help understand higher-level structure in data sets with
more than two variables.
The scatterplot matrix generates all pairwise scatter plots on
a single page. The conditioning plot, also called a co-plot or
subset plot, generates scatter plots of Y versus X dependent
on the value of a third variable.
Causality Is
Not Proved
By
Association
The scatter plot uncovers relationships in data.
"Relationships" means that there is some structured
association (linear, quadratic, etc.) between X and Y. Note,
however, that even though
causality implies association
association does NOT imply causality.
Scatter plots are a useful diagnostic tool for determining
association, but if such association exists, the plot may or
may not suggest an underlying cause-and-effect mechanism.
A scatter plot can never "prove" cause and effect--it is
ultimately only the researcher (relying on the underlying
science/engineering) who can conclude that causality actually
exists.
Appearance The most popular rendition of a scatter plot is
1. some plot character (e.g., X) at the data points, and
2. no line connecting data points.
Other scatter plot format variants include
1. an optional plot character (e.g, X) at the data points,
but
2. a solid line connecting data points.
In both cases, the resulting plot is referred to as a scatter plot,
although the former (discrete and disconnected) is the
author's personal preference since nothing makes it onto the
screen except the data--there are no interpolative artifacts to
bias the interpretation.
Related
Techniques
Run Sequence Plot
Box Plot
Block Plot
Case Study The scatter plot is demonstrated in the load cell calibration
data case study.
Software Scatter plots are a fundamental technique that should be
available in any general purpose statistical software program.
Scatter plots are also available in most graphics and
spreadsheet programs as well.
1.3.3.26.1. Scatter Plot: No Relationship
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33q1.htm[6/27/2012 2:01:22 PM]

1.3. EDA Techniques
1.3.3.26.1. Scatter Plot: No Relationship
Scatter Plot
with No
Relationship
Discussion Note in the plot above how for a given value of X (say X =
0.5), the corresponding values of Y range all over the place
from Y = -2 to Y = +2. The same is true for other values of X.
This lack of predictablility in determining Y from a given
value of X, and the associated amorphous, non-structured
appearance of the scatter plot leads to the summary
conclusion: no relationship.
1.3.3.26.2. Scatter Plot: Strong Linear (positive correlation) Relationship

1.3. EDA Techniques
1.3.3.26.2. Scatter Plot: Strong Linear (positive
correlation) Relationship
Scatter
Plot
Showing
Strong
Positive
Linear
Correlation
Discussion Note in the plot above how a straight line comfortably fits
through the data; hence a linear relationship exists. The
scatter about the line is quite small, so there is a strong linear
relationship. The slope of the line is positive (small values of
X correspond to small values of Y; large values of X
correspond to large values of Y), so there is a positive co-
relation (that is, a positive correlation) between X and Y.
1.3.3.26.3. Scatter Plot: Strong Linear (negative correlation) Relationship

1.3. EDA Techniques
1.3.3.26.3. Scatter Plot: Strong Linear (negative
Scatter
Plot
Showing a
Strong
Negative
Correlation
through the data; hence there is a linear relationship. The
scatter about the line is quite small, so there is a strong linear
relationship. The slope of the line is negative (small values of
X correspond to large values of Y; large values of X
correspond to small values of Y), so there is a negative co-
relation (that is, a negative correlation) between X and Y.
1.3.3.26.4. Scatter Plot: Exact Linear (positive correlation) Relationship

1.3. EDA Techniques
1.3.3.26.4. Scatter Plot: Exact Linear (positive
Scatter Plot
Showing an
Exact
Linear
Relationship
through the data; hence there is a linear relationship. The
scatter about the line is zero--there is perfect predictability
between X and Y), so there is an exact linear relationship.
The slope of the line is positive (small values of X
correspond to small values of Y; large values of X correspond
to large values of Y), so there is a positive co-relation (that
is, a positive correlation) between X and Y.
1.3.3.26.5. Scatter Plot: Quadratic Relationship

1.3. EDA Techniques
1.3.3.26.5. Scatter Plot: Quadratic Relationship
Scatter Plot
Showing
Quadratic
Relationship
Discussion Note in the plot above how no imaginable simple straight
line could ever adequately describe the relationship between
X and Y--a curved (or curvilinear, or non-linear) function is
needed. The simplest such curvilinear function is a quadratic
model
for some A, B, and C. Many other curvilinear functions are
possible, but the data analysis principle of parsimony
suggests that we try fitting a quadratic function first.
1.3.3.26.6. Scatter Plot: Exponential Relationship

1.3. EDA Techniques
1.3.3.26.6. Scatter Plot: Exponential
Relationship
Scatter Plot
Showing
Exponential
Relationship
Discussion Note that a simple straight line is grossly inadequate in
describing the relationship between X and Y. A quadratic
model would prove lacking, especially for large values of X.
In this example, the large values of X correspond to nearly
constant values of Y, and so a non-linear function beyond the
quadratic is needed. Among the many other non-linear
functions available, one of the simpler ones is the
exponential model
for some A, B, and C. In this case, an exponential function
would, in fact, fit well, and so one is led to the summary
conclusion of an exponential relationship.
1.3.3.26.7. Scatter Plot: Sinusoidal Relationship (damped)

1.3. EDA Techniques
1.3.3.26.7. Scatter Plot: Sinusoidal Relationship
(damped)
Scatter Plot
Showing a
Sinusoidal
Relationship
Discussion The complex relationship between X and Y appears to be
basically oscillatory, and so one is naturally drawn to the
trigonometric sinusoidal model:
Closer inspection of the scatter plot reveals that the amount
of swing (the amplitude in the model) does not appear to
be constant but rather is decreasing (damping) as X gets
large. We thus would be led to the conclusion: damped
sinusoidal relationship, with the simplest corresponding
model being
1.3.3.26.8. Scatter Plot: Variation of Y Does Not Depend on X (homoscedastic)

1.3. EDA Techniques
1.3.3.26.8. Scatter Plot: Variation of Y Does Not
Depend on X (homoscedastic)
Scatter Plot
Showing
Homoscedastic
Variability
Discussion This scatter plot reveals a linear relationship between X
and Y: for a given value of X, the predicted value of Y will
fall on a line. The plot further reveals that the variation in
Y about the predicted value is about the same (+- 10 units),
regardless of the value of X. Statistically, this is referred to
as homoscedasticity. Such homoscedasticity is very
important as it is an underlying assumption for regression,
and its violation leads to parameter estimates with inflated
variances. If the data are homoscedastic, then the usual
regression estimates can be used. If the data are not
homoscedastic, then the estimates can be improved using
weighting procedures as shown in the next example.
1.3.3.26.9. Scatter Plot: Variation of Y Does Depend on X (heteroscedastic)

1.3. EDA Techniques
1.3.3.26.9. Scatter Plot: Variation of Y Does
Depend on X (heteroscedastic)
Scatter Plot
Showing
Heteroscedastic
Variability
Discussion This scatter plot reveals an approximate linear relationship
between X and Y, but more importantly, it reveals a
statistical condition referred to as heteroscedasticity (that
is, nonconstant variation in Y over the values of X). For a
heteroscedastic data set, the variation in Y differs
depending on the value of X. In this example, small values
of X yield small scatter in Y while large values of X result
in large scatter in Y.
Heteroscedasticity complicates the analysis somewhat, but
its effects can be overcome by:
1. proper weighting of the data with noisier data being
weighted less, or by
2. performing a Y variable transformation to achieve
homoscedasticity. The Box-Cox normality plot can
help determine a suitable transformation.
Impact of
Ignoring
Unequal
Fortunately, unweighted regression analyses on
heteroscedastic data produce estimates of the coefficients
that are unbiased. However, the coefficients will not be as
1.3.3.26.9. Scatter Plot: Variation of Y Does Depend on X (heteroscedastic)
Variability in
the Data
precise as they would be with proper weighting.
Note further that if heteroscedasticity does exist, it is
frequently useful to plot and model the local variation
as a function of X, as in
. This modeling has two
advantages:
1. it provides additional insight and understanding as
to how the response Y relates to X; and
2. it provides a convenient means of forming weights
for a weighted regression by simply using
The topic of non-constant variation is discussed in some
detail in the process modeling chapter.
1.3.3.26.10. Scatter Plot: Outlier
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33qa.htm[6/27/2012 2:01:29 PM]

1.3. EDA Techniques
1.3.3.26.10. Scatter Plot: Outlier
Scatter
Plot
Showing
Outliers
Discussion The scatter plot here reveals
1. a basic linear relationship between X and Y for most of
the data, and
2. a single outlier (at X = 375).
An outlier is defined as a data point that emanates from a
different model than do the rest of the data. The data here
appear to come from a linear model with a given slope and
variation except for the outlier which appears to have been
generated from some other model.
Outlier detection is important for effective modeling. Outliers
should be excluded from such model fitting. If all the data
here are included in a linear regression, then the fitted model
will be poor virtually everywhere. If the outlier is omitted
from the fitting process, then the resulting fit will be excellent
almost everywhere (for all points except the outlying point).
1.3.3.26.11. Scatterplot Matrix
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33qb.htm[6/27/2012 2:01:30 PM]

1.3. EDA Techniques
Purpose:
Check
Pairwise
Relationships
Between
Variables
Given a set of variables X
1
, X
2
, ... , X
k
, the scatterplot
matrix contains all the pairwise scatter plots of the variables
on a single page in a matrix format. That is, if there are k
variables, the scatterplot matrix will have k rows and k
columns and the ith row and jth column of this matrix is a
plot of X
i
versus X
j
.
Although the basic concept of the scatterplot matrix is
simple, there are numerous alternatives in the details of the
plots.
1. The diagonal plot is simply a 45-degree line since we
are plotting X
i
versus X
i
. Although this has some
usefulness in terms of showing the univariate
distribution of the variable, other alternatives are
common. Some users prefer to use the diagonal to
print the variable label. Another alternative is to plot
the univariate histogram on the diagonal.
Alternatively, we could simply leave the diagonal
blank.
2. Since X
i
versus X
j
is equivalent to X
j
versus X
i
with
the axes reversed, some prefer to omit the plots below
the diagonal.
3. It can be helpful to overlay some type of fitted curve
on the scatter plot. Although a linear or quadratic fit
can be used, the most common alternative is to
overlay a lowess curve.
4. Due to the potentially large number of plots, it can be
somewhat tricky to provide the axes labels in a way
that is both informative and visually pleasing. One
alternative that seems to work well is to provide axis
labels on alternating rows and columns. That is, row
one will have tic marks and axis labels on the left
vertical axis for the first plot only while row two will
have the tic marks and axis labels for the right
vertical axis for the last plot in the row only. This
alternating pattern continues for the remaining rows.
A similar pattern is used for the columns and the
horizontal axes labels. Another alternative is to put
the minimum and maximum scale value in the
diagonal plot with the variable name.
5. Some analysts prefer to connect the scatter plots.
Others prefer to leave a little gap between each plot.
6. Although this plot type is most commonly used for
scatter plots, the basic concept is both simple and
powerful and extends easily to other plot formats that
involve pairwise plots such as the quantile-quantile
plot and the bihistogram.
Sample Plot
This sample plot was generated from pollution data
collected by NIST chemist Lloyd Currie.
There are a number of ways to view this plot. If we are
primarily interested in a particular variable, we can scan the
row and column for that variable. If we are interested in
finding the strongest relationship, we can scan all the plots
and then determine which variables are related.
Definition Given k variables, scatter plot matrices are formed by
creating k rows and k columns. Each row and column
defines a single scatter plot
The individual plot for row i and column j is defined as
Vertical axis: Variable X
i
Horizontal axis: Variable X
j
Questions The scatterplot matrix can provide answers to the following
questions:
1. Are there pairwise relationships between the
variables?
2. If there are relationships, what is the nature of these
relationships?
4. Is there clustering by groups in the data?
Linking and
Brushing
The scatterplot matrix serves as the foundation for the
concepts of linking and brushing.
By linking, we mean showing how a point, or set of points,
behaves in each of the plots. This is accomplished by
highlighting these points in some fashion. For example, the
highlighted points could be drawn as a filled circle while
the remaining points could be drawn as unfilled circles. A
typical application of this would be to show how an outlier
shows up in each of the individual pairwise plots. Brushing
extends this concept a bit further. In brushing, the points to
be highlighted are interactively selected by a mouse and the
scatterplot matrix is dynamically updated (ideally in real
time). That is, we can select a rectangular region of points
in one plot and see how those points are reflected in the
other plots. Brushing is discussed in detail by Becker,
Cleveland, and Wilks in the paper "Dynamic Graphics for
Data Analysis" (Cleveland and McGill, 1988).
Related
Techniques
Star plot
Scatter plot
Conditioning plot
Locally weighted least squares
Software Scatterplot matrices are becoming increasingly common in
general purpose statistical software programs. If a software
program does not generate scatterplot matrices, but it does
provide multiple plots per page and scatter plots, it should
be possible to write a macro to generate a scatterplot matrix.
Brushing is available in a few of the general purpose
statistical software programs that emphasize graphical
approaches.
1.3.3.26.12. Conditioning Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33qc.htm[6/27/2012 2:01:31 PM]

1.3. EDA Techniques
Purpose:
Check
pairwise
relationship
between
two
variables
conditional
on a third
variable
A conditioning plot, also known as a coplot or subset plot, is
a plot of two variables conditional on the value of a third
variable (called the conditioning variable). The conditioning
variable may be either a variable that takes on only a few
discrete values or a continuous variable that is divided into a
limited number of subsets.
One limitation of the scatterplot matrix is that it cannot show
interaction effects with another variable. This is the strength
of the conditioning plot. It is also useful for displaying scatter
plots for groups in the data. Although these groups can also
be plotted on a single plot with different plot symbols, it can
often be visually easier to distinguish the groups using the
conditioning plot.
Although the basic concept of the conditioning plot matrix is
simple, there are numerous alternatives in the details of the
plots.
1. It can be helpful to overlay some type of fitted curve on
the scatter plot. Although a linear or quadratic fit can
be used, the most common alternative is to overlay a
lowess curve.
2. Due to the potentially large number of plots, it can be
somewhat tricky to provide the axis labels in a way that
is both informative and visually pleasing. One
alternative that seems to work well is to provide axis
labels on alternating rows and columns. That is, row
one will have tic marks and axis labels on the left
vertical axis for the first plot only while row two will
have the tic marks and axis labels for the right vertical
axis for the last plot in the row only. This alternating
pattern continues for the remaining rows. A similar
pattern is used for the columns and the horizontal axis
labels. Note that this approach only works if the axes
limits are fixed to common values for all of the plots.
3. Some analysts prefer to connect the scatter plots.
Others prefer to leave a little gap between each plot.
Alternatively, each plot can have its own labeling with
the plots not connected.
4. Although this plot type is most commonly used for
scatter plots, the basic concept is both simple and
powerful and extends easily to other plot formats.
Sample
Plot
In this case, temperature has six distinct values. We plot
torque versus time for each of these temperatures. This
example is discussed in more detail in the process modeling
chapter.
Definition Given the variables X, Y, and Z, the conditioning plot is
formed by dividing the values of Z into k groups. There are
several ways that these groups may be formed. There may be
a natural grouping of the data, the data may be divided into
several equal sized groups, the grouping may be determined
by clusters in the data, and so on. The page will be divided
into n rows and c columns where . Each row and
column defines a single scatter plot.
The individual plot for row i and column j is defined as
Vertical axis: Variable Y
Horizontal axis: Variable X
where only the points in the group corresponding to the ith
row and jth column are used.
Questions The conditioning plot can provide answers to the following
questions:
1. Is there a relationship between two variables?
2. If there is a relationship, does the nature of the
relationship depend on the value of a third variable?
3. Are groups in the data similar?
Related
Techniques
Scatter plot
Scatterplot matrix
Locally weighted least squares
Software Scatter plot matrices are becoming increasingly common in
general purpose statistical software programs, including. If a
software program does not generate conditioning plots, but it
does provide multiple plots per page and scatter plots, it
should be possible to write a macro to generate a
conditioning plot.
1.3.3.27. Spectral Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33r.htm[6/27/2012 2:01:32 PM]

1.3. EDA Techniques
Purpose:
Examine
Cyclic
Structure
A spectral plot ( Jenkins and Watts 1968 or Bloomfield 1976)
is a graphical technique for examining cyclic structure in the
frequency domain. It is a smoothed Fourier transform of the
autocovariance function.
The frequency is measured in cycles per unit time where unit
time is defined to be the distance between 2 points. A
frequency of 0 corresponds to an infinite cycle while a
frequency of 0.5 corresponds to a cycle of 2 data points. Equi-
spaced time series are inherently limited to detecting
frequencies between 0 and 0.5.
Trends should typically be removed from the time series
before applying the spectral plot. Trends can be detected from
a run sequence plot. Trends are typically removed by
differencing the series or by fitting a straight line (or some
other polynomial curve) and applying the spectral analysis to
the residuals.
Spectral plots are often used to find a starting value for the
frequency, , in the sinusoidal model
See the beam deflection case study for an example of this.
Sample
Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33r.htm[6/27/2012 2:01:32 PM]
This spectral plot shows one dominant frequency of
approximately 0.3 cycles per observation.
Definition:
Variance
Versus
Frequency
The spectral plot is formed by:
Vertical axis: Smoothed variance (power)
Horizontal axis: Frequency (cycles per observation)
The computations for generating the smoothed variances can
be involved and are not discussed further here. The details can
be found in the Jenkins and Bloomfield references and in
most texts that discuss the frequency analysis of time series.
Questions The spectral plot can be used to answer the following
questions:
1. How many cyclic components are there?
2. Is there a dominant cyclic frequency?
3. If there is a dominant cyclic frequency, what is it?
Importance
Check
Cyclic
Behavior
of Time
Series
The spectral plot is the primary technique for assessing the
cyclic nature of univariate time series in the frequency
domain. It is almost always the second plot (after a run
sequence plot) generated in a frequency domain analysis of a
time series.
Examples 1. Random (= White Noise)
2. Strong autocorrelation and autoregressive model
3. Sinusoidal model
Related
Techniques
Complex Demodulation Amplitude Plot
Case Study The spectral plot is demonstrated in the beam deflection data
case study.
Software Spectral plots are a fundamental technique in the frequency
analysis of time series. They are available in many general
1.3.3.27.1. Spectral Plot: Random Data
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33r1.htm[6/27/2012 2:01:33 PM]

1.3. EDA Techniques
1.3.3.27.1. Spectral Plot: Random Data
Spectral
Plot of 200
Normal
Random
Numbers
1. There are no dominant peaks.
2. There is no identifiable pattern in the spectrum.
Discussion For random data, the spectral plot should show no dominant
peaks or distinct pattern in the spectrum. For the sample plot
above, there are no clearly dominant peaks and the peaks
seem to fluctuate at random. This type of appearance of the
spectral plot indicates that there are no significant cyclic
patterns in the data.
1.3.3.27.2. Spectral Plot: Strong Autocorrelation and Autoregressive Model

1.3. EDA Techniques
1.3.3.27.2. Spectral Plot: Strong Autocorrelation
and Autoregressive Model
Spectral Plot
for Random
Walk Data
plot.
1. Strong dominant peak near zero.
2. Peak decays rapidly towards zero.
3. An autoregressive model is an appropriate model.
Discussion This spectral plot starts with a dominant peak near zero
and rapidly decays to zero. This is the spectral plot
signature of a process with strong positive autocorrelation.
Such processes are highly non-random in that there is high
association between an observation and a succeeding
observation. In short, if you know Y
i
you can make a strong
guess as to what Y
i+1
will be.
Recommended
Next Step
The next step would be to determine the parameters for the
Such estimation can be done by linear regression or by
1.3.3.27.2. Spectral Plot: Strong Autocorrelation and Autoregressive Model
fitting a Box-Jenkins autoregressive (AR) model.
Then the system should be reexamined to find an
explanation for the strong autocorrelation. Is it due to the
1. phenomenon under study; or
2. drifting in the environment; or
3. contamination from the data acquisition system
(DAS)?
Oftentimes the source of the problem is item (3) above
where contamination and carry-over from the data
acquisition system result because the DAS does not have
time to electronically recover before collecting the next
data point. If this is the case, then consider slowing down
the sampling rate to re-achieve randomness.
1.3.3.27.3. Spectral Plot: Sinusoidal Model

1.3. EDA Techniques
Spectral Plot
for Sinusoidal
Model
plot.
1. There is a single dominant peak at approximately
0.3.
2. There is an underlying single-cycle sinusoidal
model.
Discussion This spectral plot shows a single dominant frequency. This
indicates that a single-cycle sinusoidal model might be
appropriate.
If one were to naively assume that the data represented by
the graph could be fit by the model
and then estimate the constant by the sample mean, the
analysis would be incorrect because
the sample mean is biased;
the confidence interval for the mean, which is valid
only for random data, is meaningless and too small.
On the other hand, the choice of the proper model
where is the amplitude, is the frequency (between 0
and .5 cycles per observation), and is the phase can be fit
by non-linear least squares. The beam deflection data case
study demonstrates fitting this type of model.
Recommended
Next Steps
The recommended next steps are to:
1. Estimate the frequency from the spectral plot. This
will be helpful as a starting value for the subsequent
non-linear fitting. A complex demodulation phase
plot can be used to fine tune the estimate of the
frequency before performing the non-linear fit.
2. Do a complex demodulation amplitude plot to obtain
an initial estimate of the amplitude and to determine
if a constant amplitude is justified.
3. Carry out a non-linear fit of the model
1.3.3.28. Standard Deviation Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33s.htm[6/27/2012 2:01:35 PM]

1.3. EDA Techniques
Purpose:
Detect
Changes in
Scale
Between
Groups
Standard deviation plots are used to see if the standard
deviation varies between different groups of the data. The
grouping is determined by the analyst. In most cases, the data
provide a specific grouping variable. For example, the groups
may be the levels of a factor variable. In the sample plot
below, the months of the year provide the grouping.
Standard deviation plots can be used with ungrouped data to
determine if the standard deviation is changing over time. In
this case, the data are broken into an arbitrary number of
equal-sized groups. For example, a data series with 400
points can be divided into 10 groups of 40 points each. A
standard deviation plot can then be generated with these
groups to see if the standard deviation is increasing or
decreasing over time.
Although the standard deviation is the most commonly used
measure of scale, the same concept applies to other measures
of scale. For example, instead of plotting the standard
deviation of each group, the median absolute deviation or the
average absolute deviation might be plotted instead. This
might be done if there were significant outliers in the data
and a more robust measure of scale than the standard
deviation was desired.
Standard deviation plots are typically used in conjunction
with mean plots. The mean plot would be used to check for
shifts in location while the standard deviation plot would be
used to check for shifts in scale.
Sample Plot
This sample standard deviation plot shows
1. there is a shift in variation;
2. greatest variation is during the summer months.
Definition:
Group
Standard
Deviations
Versus
Group ID
Standard deviation plots are formed by:
Vertical axis: Group standard deviations
A reference line is plotted at the overall standard deviation.
Questions The standard deviation plot can be used to answer the
1. Are there any shifts in variation?
2. What is the magnitude of the shifts in variation?
3. Is there a distinct pattern in the shifts in variation?
Importance:
Checking
Assumptions
A common assumption in 1-factor analyses is that of equal
variances. That is, the variance is the same for different
levels of the factor variable. The standard deviation plot
provides a graphical check for that assumption. A common
assumption for univariate data is that the variance is constant.
By grouping the data into equi-sized intervals, the standard
deviation plot can provide a graphical test of this assumption.
Related
Techniques
Mean Plot
DOE Standard Deviation Plot
support a standard deviation plot. However, if the statistical
program can generate the standard deviation for a group, it
should be feasible to write a macro to generate this plot.
1.3.3.29. Star Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33t.htm[6/27/2012 2:01:35 PM]

1.3. EDA Techniques
1.3.3.29. Star Plot
Purpose:
Display
Multivariate
Data
The star plot (Chambers 1983) is a method of displaying
multivariate data. Each star represents a single observation.
Typically, star plots are generated in a multi-plot format with
many stars on each page and each star representing one
observation.
Star plots are used to examine the relative values for a single
data point (e.g., point 3 is large for variables 2 and 4, small
for variables 1, 3, 5, and 6) and to locate similar points or
dissimilar points.
Sample Plot The plot below contains the star plots of 16 cars. The data
file actually contains 74 cars, but we restrict the plot to what
can reasonably be shown on one page. The variable list for
the sample star plot is
1 Price
2 Mileage (MPG)
3 1978 Repair Record (1 = Worst, 5 = Best)
4 1977 Repair Record (1 = Worst, 5 = Best)
5 Headroom
6 Rear Seat Room
7 Trunk Space
8 Weight
9 Length
1.3.3.29. Star Plot
We can look at these plots individually or we can use them
to identify clusters of cars with similar features. For example,
we can look at the star plot of the Cadillac Seville and see
that it is one of the most expensive cars, gets below average
(but not among the worst) gas mileage, has an average repair
record, and has average-to-above-average roominess and
size. We can then compare the Cadillac models (the last three
plots) with the AMC models (the first three plots). This
comparison shows distinct patterns. The AMC models tend
to be inexpensive, have below average gas mileage, and are
small in both height and weight and in roominess. The
Cadillac models are expensive, have poor gas mileage, and
are large in both size and roominess.
Definition The star plot consists of a sequence of equi-angular spokes,
called radii, with each spoke representing one of the
variables. The data length of a spoke is proportional to the
magnitude of the variable for the data point relative to the
maximum magnitude of the variable across all data points. A
line is drawn connecting the data values for each spoke. This
gives the plot a star-like appearance and the origin of the
name of this plot.
Questions The star plot can be used to answer the following questions:
1. What variables are dominant for a given observation?
2. Which observations are most similar, i.e., are there
clusters of observations?
Weakness
in
Technique
Star plots are helpful for small-to-moderate-sized
multivariate data sets. Their primary weakness is that their
effectiveness is limited to data sets with less than a few
hundred points. After that, they tend to be overwhelming.
Graphical techniques suited for large data sets are discussed
1.3.3.29. Star Plot
by Scott.
Related
Techniques
Alternative ways to plot multivariate data are discussed in
Chambers, du Toit, and Everitt.
Software Star plots are available in some general purpose statistical
software progams.
1.3.3.30. Weibull Plot
http://www.itl.nist.gov/div898/handbook/eda/section3/eda33u.htm[6/27/2012 2:01:36 PM]

1.3. EDA Techniques
Purpose:
Graphical
Check To See
If Data Come
From a
Population
That Would
Be Fit by a
Weibull
Distribution
The Weibull plot (Nelson 1982) is a graphical technique for
determining if a data set comes from a population that
would logically be fit by a 2-parameter Weibull distribution
(the location is assumed to be zero).
The Weibull plot has special scales that are designed so that
if the data do in fact follow a Weibull distribution, the
points will be linear (or nearly linear). The least squares fit
of this line yields estimates for the shape and scale
parameters of the Weibull distribution (the location is
assumed to be zero).
Specifically, the shape parameter is the reciprocal of the
slope of the fitted line and the scale parameter is the
exponent of the intercept of the fitted line.
The Weibull distribution also has the property that the scale
parameter falls at the 63.2% point irrespective of the value
of the shape parameter. The plot shows a horizontal line at
this 63.2% point and a vertical line where the horizontal
line intersects the least squares fitted line. This vertical line
shows the value of scale parameter.
Sample Plot
This Weibull plot shows that:
1. the assumption of a Weibull distribution is
reasonable;
2. the scale parameter estimate is computed to be 33.32;
3. the shape parameter estimate is computed to be 5.28;
and
4. there are no outliers.
Note that the values on the x-axis ("0", "1", and "2") are the
exponents. These actually denote the value 10
0
= 1, 10
1
=
10, and 10
2
= 100.
Definition:
Weibull
Cumulative
Probability
Versus
LN(Ordered
Response)
The Weibull plot is formed by:
Vertical axis: Weibull cumulative probability
expressed as a percentage
Horizontal axis: ordered failure times (in a LOG10
scale)
The vertical scale is ln(-ln(1-p)) where p=(i-0.3)/(n+0.4)
and i is the rank of the observation. This scale is chosen in
order to linearize the resulting plot for Weibull data.
Questions The Weibull plot can be used to answer the following
questions:
1. Do the data follow a 2-parameter Weibull
distribution?
2. What is the best estimate of the shape parameter for
the 2-parameter Weibull distribution?
3. What is the best estimate of the scale (= variation)
parameter for the 2-parameter Weibull distribution?
Importance:
Check
Distributional
Assumptions
Many statistical analyses, particularly in the field of
reliability, are based on the assumption that the data follow
a Weibull distribution. If the analysis assumes the data
follow a Weibull distribution, it is important to verify this
assumption and, if verified, find good estimates of the
Weibull parameters.
Related
Techniques
Weibull Probability Plot
Weibull PPCC Plot
Weibull Hazard Plot
The Weibull probability plot (in conjunction with the
Weibull PPCC plot), the Weibull hazard plot, and the
Weibull plot are all similar techniques that can be used for
assessing the adequacy of the Weibull distribution as a
model for the data, and additionally providing estimation
for the shape, scale, or location parameters.
The Weibull hazard plot and Weibull plot are designed to
handle censored data (which the Weibull probability plot
does not).
Case Study The Weibull plot is demonstrated in the fatigue life of
aluminum alloy specimens case study.
Software Weibull plots are generally available in statistical software
programs that are designed to analyze reliability data.
1.3.3.31. Youden Plot

1.3. EDA Techniques
Purpose:
Interlab
Comparisons
Youden plots are a graphical technique for analyzing
interlab data when each lab has made two runs on the same
product or one run on two different products.
The Youden plot is a simple but effective method for
comparing both the within-laboratory variability and the
between-laboratory variability.
Sample Plot
This plot shows:
1. Not all labs are equivalent.
2. Lab 4 is biased low.
3. Lab 3 has within-lab variability problems.
4. Lab 5 has an outlying run.
Definition:
Response 1
Versus
Response 2
Coded by
Lab
Youden plots are formed by:
1. Vertical axis: Response variable 1 (i.e., run 1 or
product 1 response value)
2. Horizontal axis: Response variable 2 (i.e., run 2 or
product 2 response value)
In addition, the plot symbol is the lab id (typically an
integer from 1 to k where k is the number of labs).
Sometimes a 45-degree reference line is drawn. Ideally, a
lab generating two runs of the same product should produce
reasonably similar results. Departures from this reference
line indicate inconsistency from the lab. If two different
products are being tested, then a 45-degree line may not be
appropriate. However, if the labs are consistent, the points
should lie near some fitted straight line.
Questions The Youden plot can be used to answer the following
questions:
1. Are all labs equivalent?
2. What labs have between-lab problems
(reproducibility)?
3. What labs have within-lab problems (repeatability)?
4. What labs are outliers?
Importance In interlaboratory studies or in comparing two runs from the
same lab, it is useful to know if consistent results are
generated. Youden plots should be a routine plot for
analyzing this type of data.
DOE Youden
Plot
The DOE Youden plot is a specialized Youden plot used in
the design of experiments. In particular, it is useful for full
and fractional designs.
Related
Techniques
Scatter Plot
Software The Youden plot is essentially a scatter plot, so it should be
feasible to write a macro for a Youden plot in any general
purpose statistical program that supports scatter plots.
1.3.3.31.1. DOE Youden Plot

1.3. EDA Techniques
DOE Youden
Plot:
Introduction
The DOE (Design of Experiments) Youden plot is a specialized Youden plot used in the
analysis of full and fractional experiment designs. In particular, it is used in conjunction
with the Yates algorithm. These designs may have a low level, coded as "-1" or "-", and a
high level, coded as "+1" or "+", for each factor. In addition, there can optionally be one or
more center points. Center points are at the midpoint between the low and high levels for
each factor and are coded as "0".
The Yates agorithm and the the DOE Youden plot only use the "-1" and "+1" points. The
Yates agorithm is used to estimate factor effects. The DOE Youden plot can be used to help
determine the approriate model to based on the effect estimates from the Yates algorithm.
Construction
of DOE
Youden Plot
The following are the primary steps in the construction of the DOE Youden plot.
1. For a given factor or interaction term, compute the mean of the response variable for
the low level of the factor and for the high level of the factor. Any center points are
omitted from the computation.
2. Plot the point where the y-coordinate is the mean for the high level of the factor and
the x-coordinate is the mean for the low level of the factor. The character used for the
plot point should identify the factor or interaction term (e.g., "1" for factor 1, "13" for
the interaction between factors 1 and 3).
3. Repeat steps 1 and 2 for each factor and interaction term of the data.
The high and low values of the interaction terms are obtained by multiplying the
corresponding values of the main level factors. For example, the interaction term X
13
is
obtained by multiplying the values for X
1
with the corresponding values of X
3
. Since the
values for X
1
and X
3
are either "-1" or "+1", the resulting values for X
13
are also either "-1"
or "+1".
In summary, the DOE Youden plot is a plot of the mean of the response variable for the
high level of a factor or interaction term against the mean of the response variable for the
low level of that factor or interaction term.
For unimportant factors and interaction terms, these mean values should be nearly the same.
For important factors and interaction terms, these mean values should be quite different. So
the interpretation of the plot is that unimportant factors should be clustered together near the
grand mean. Points that stand apart from this cluster identify important factors that should
be included in the model.
Sample DOE The following is a DOE Youden plot for the data used in the Eddy current case study. The
Youden Plot analysis in that case study demonstrated that X1 and X2 were the most important factors.
Interpretation
of the Sample
DOE Youden
Plot
From the above DOE Youden plot, we see that factors 1 and 2 stand out from the others.
That is, the mean response values for the low and high levels of factor 1 and factor 2 are
quite different. For factor 3 and the 2 and 3-term interactions, the mean response values for
the low and high levels are similar.
We would conclude from this plot that factors 1 and 2 are important and should be included
in our final model while the remaining factors and interactions should be omitted from the
final model.
Case Study The Eddy current case study demonstrates the use of the DOE Youden plot in the context of
the analysis of a full factorial design.
Software DOE Youden plots are not typically available as built-in plots in statistical software
programs. However, it should be relatively straightforward to write a macro to generate this
plot in most general purpose statistical software programs.
1.3.3.32. 4-Plot

1.3. EDA Techniques
1.3.3.32. 4-Plot
Purpose:
Check
Underlying
Statistical
Assumptions
The 4-plot is a collection of 4 specific EDA graphical
techniques whose purpose is to test the assumptions that
underlie most measurement processes. A 4-plot consists of
a
1. run sequence plot;
2. lag plot;
3. histogram;
4. normal probability plot.
If the 4 underlying assumptions of a typical measurement
process hold, then the above 4 plots will have a
characteristic appearance (see the normal random numbers
case study below); if any of the underlying assumptions
fail to hold, then it will be revealed by an anomalous
appearance in one or more of the plots. Several commonly
encountered situations are demonstrated in the case studies
below.
Although the 4-plot has an obvious use for univariate and
time series data, its usefulness extends far beyond that.
Many statistical models of the form
have the same underlying assumptions for the error term.
That is, no matter how complicated the functional fit, the
assumptions on the underlying error term are still the
same. The 4-plot can and should be routinely applied to
the residuals when fitting models regardless of whether the
model is simple or complicated.
Sample Plot:
Process Has
Fixed
Location,
Fixed
Variation,
Non-Random
(Oscillatory),
Non-Normal
U-Shaped
1.3.3.32. 4-Plot
Distribution,
and Has 3
Outliers.
This 4-plot reveals the following:
1. the fixed location assumption is justified as shown
by the run sequence plot in the upper left corner.
2. the fixed variation assumption is justified as shown
by the run sequence plot in the upper left corner.
3. the randomness assumption is violated as shown by
the non-random (oscillatory) lag plot in the upper
right corner.
4. the assumption of a common, normal distribution is
violated as shown by the histogram in the lower left
corner and the normal probability plot in the lower
right corner. The distribution is non-normal and is a
U-shaped distribution.
5. there are several outliers apparent in the lag plot in
the upper right corner.
Definition:
1. Run
Sequence
Plot;
2. Lag Plot;
3. Histogram;
4. Normal
Probability
Plot
The 4-plot consists of the following:
1. Run sequence plot to test fixed location and
variation.
Vertically: Y
i
Horizontally: i
2. Lag Plot to test randomness.
Vertically: Y
i
Horizontally: Y
i-1
3. Histogram to test (normal) distribution.
Vertically: Counts
Horizontally: Y
4. Normal probability plot to test normal distribution.
Vertically: Ordered Y
i
Horizontally: Theoretical values from a
normal N(0,1) distribution for ordered Y
i
1.3.3.32. 4-Plot
Questions 4-plots can provide answers to many questions:
1. Is the process in-control, stable, and predictable?
2. Is the process drifting with respect to location?
3. Is the process drifting with respect to variation?
5. Is an observation related to an adjacent observation?
6. If the data are a time series, is is white noise?
7. If the data are a time series and not white noise, is it
sinusoidal, autoregressive, etc.?
8. If the data are non-random, what is a better model?
9. Does the process follow a normal distribution?
10. If non-normal, what distribution does the process
follow?
11. Is the model
valid and sufficient?
12. If the default model is insufficient, what is a better
model?
13. Is the formula valid?
14. Is the sample mean a good estimator of the process
location?
15. If not, what would be a better estimator?
Importance:
Testing
Underlying
Assumptions
Helps Ensure
the Validity of
the Final
Scientific and
Engineering
Conclusions
There are 4 assumptions that typically underlie all
measurement processes; namely, that the data from the
process at hand "behave like":
1. random drawings;
3. with that distribution having a fixed location; and
4. with that distribution having fixed variation.
Predictability is an all-important goal in science and
engineering. If the above 4 assumptions hold, then we
have achieved probabilistic predictability--the ability to
make probability statements not only about the process in
the past, but also about the process in the future. In short,
such processes are said to be "statistically in control". If
the 4 assumptions do not hold, then we have a process that
is drifting (with respect to location, variation, or
distribution), is unpredictable, and is out of control. A
simple characterization of such processes by a location
estimate, a variation estimate, or a distribution "estimate"
inevitably leads to optimistic and grossly invalid
engineering conclusions.
Inasmuch as the validity of the final scientific and
engineering conclusions is inextricably linked to the
1.3.3.32. 4-Plot
validity of these same 4 underlying assumptions, it
naturally follows that there is a real necessity for all 4
assumptions to be routinely tested. The 4-plot (run
sequence plot, lag plot, histogram, and normal probability
plot) is seen as a simple, efficient, and powerful way of
carrying out this routine checking.
Interpretation:
Flat, Equi-
Banded,
Random, Bell-
Shaped, and
Linear
Of the 4 underlying assumptions:
1. If the fixed location assumption holds, then the run
sequence plot will be flat and non-drifting.
2. If the fixed variation assumption holds, then the
vertical spread in the run sequence plot will be
approximately the same over the entire horizontal
axis.
3. If the randomness assumption holds, then the lag
plot will be structureless and random.
4. If the fixed distribution assumption holds (in
particular, if the fixed normal distribution
assumption holds), then the histogram will be bell-
shaped and the normal probability plot will be
approximatelylinear.
If all 4 of the assumptions hold, then the process is
"statistically in control". In practice, many processes fall
short of achieving this ideal.
Related
Techniques
Run Sequence Plot
Lag Plot
Histogram
Spectral Plot
PPCC Plot
Case Studies The 4-plot is used in most of the case studies in this
chapter:
1. Normal random numbers (the ideal)
2. Uniform random numbers
3. Random walk
4. Josephson junction cryothermometry
5. Beam deflections
6. Filter transmittance
7. Standard resistor
8. Heat flow meter 1
Software It should be feasible to write a macro for the 4-plot in any
general purpose statistical software program that supports
the capability for multiple plots per page and supports the
underlying plot techniques.
1.3.3.32. 4-Plot
1.3.3.33. 6-Plot

1.3. EDA Techniques
1.3.3.33. 6-Plot
Purpose:
Graphical
Model
Validation
The 6-plot is a collection of 6 specific graphical techniques
whose purpose is to assess the validity of a Y versus X fit.
The fit can be a linear fit, a non-linear fit, a LOWESS
(locally weighted least squares) fit, a spline fit, or any other
fit utilizing a single independent variable.
The 6 plots are:
1. Scatter plot of the response and predicted values versus
the independent variable;
2. Scatter plot of the residuals versus the independent
variable;
3. Scatter plot of the residuals versus the predicted values;
4. Lag plot of the residuals;
5. Histogram of the residuals;
6. Normal probability plot of the residuals.
Sample Plot
This 6-plot, which followed a linear fit, shows that the linear
model is not adequate. It suggests that a quadratic model
would be a better model.
Definition:
6
The 6-plot consists of the following:
1.3.3.33. 6-Plot
Component
Plots
1. Response and predicted values
Vertical axis: Response variable, predicted
values
Horizontal axis: Independent variable
2. Residuals versus independent variable
Vertical axis: Residuals
Horizontal axis: Independent variable
3. Residuals versus predicted values
Vertical axis: Residuals
Horizontal axis: Predicted values
4. Lag plot of residuals
Vertical axis: RES(I)
Horizontal axis: RES(I-1)
5. Histogram of residuals
Vertical axis: Counts
Horizontal axis: Residual values
6. Normal probability plot of residuals
Vertical axis: Ordered residuals
Horizontal axis: Theoretical values from a
normal N(0,1) distribution for ordered residuals
Questions The 6-plot can be used to answer the following questions:
1. Are the residuals approximately normally distributed
with a fixed location and scale?
3. Is the fit adequate?
4. Do the residuals suggest a better fit?
Importance:
Validating
Model
A model involving a response variable and a single
independent variable has the form:
where Y is the response variable, X is the independent
variable, f is the linear or non-linear fit function, and E is the
random component. For a good model, the error component
should behave like:
1. random drawings (i.e., independent);
3. with fixed location; and
4. with fixed variation.
In addition, for fitting models it is usually further assumed
that the fixed distribution is normal and the fixed location is
zero. For a good model the fixed variation should be as small
as possible. A necessary component of fitting models is to
verify these assumptions for the error component and to
assess whether the variation for the error component is
sufficiently small. The histogram, lag plot, and normal
probability plot are used to verify the fixed distribution,
1.3.3.33. 6-Plot
location, and variation assumptions on the error component.
The plot of the response variable and the predicted values
versus the independent variable is used to assess whether the
variation is sufficiently small. The plots of the residuals
versus the independent variable and the predicted values is
used to assess the independence assumption.
Assessing the validity and quality of the fit in terms of the
above assumptions is an absolutely vital part of the model-
fitting process. No fit should be considered complete without
an adequate model validation step.
Related
Techniques
Linear Least Squares
Non-Linear Least Squares
Scatter Plot
Run Sequence Plot
Lag Plot
Histogram
Case Study The 6-plot is used in the Alaska pipeline data case study.
Software It should be feasible to write a macro for the 6-plot in any
general purpose statistical software program that supports the
capability for multiple plots per page and supports the
underlying plot techniques.
1.3.4. Graphical Techniques: By Problem Category

1.3. EDA Techniques
1.3.4. Graphical
Techniques: By
Problem Category
Univariate
y = c + e
Run Sequence
Plot: 1.3.3.25
Lag Plot:
1.3.3.15
Histogram:
1.3.3.14

Normal
Probability Plot:
1.3.3.21
4-Plot:
1.3.3.32
PPCC Plot:
1.3.3.23

Weibull Plot:
1.3.3.30
Probability
Plot: 1.3.3.22
Box-Cox
Linearity Plot:
1.3.3.5

Box-Cox
Normality Plot:
1.3.3.6
Bootstrap
Plot: 1.3.3.4
Time Series
y = f(t) + e
Run Sequence
Plot: 1.3.3.25
Spectral Plot:
1.3.3.27
Autocorrelation
Plot: 1.3.3.1

Complex
Demodulation
Amplitude Plot:
1.3.3.8
Complex
Demodulation
Phase Plot:
1.3.3.9
1 Factor
y = f(x) + e
Scatter Plot:
1.3.3.26
Box Plot:
1.3.3.7
Bihistogram:
1.3.3.2

Quantile-
Quantile Plot:
1.3.3.24
Mean Plot:
1.3.3.20
Standard
Deviation Plot:
1.3.3.28
Multi-
Factor/Comparative
y =
f(xp, x1,x2,...,xk) +
e
Block Plot:
1.3.3.3
Multi-
Factor/Screening
y =
f(x1,x2,x3,...,xk) +
e DOE Scatter
Plot: 1.3.3.11
DOE Mean
Plot: 1.3.3.12
DOE Standard
Deviation Plot:
1.3.3.13
Contour Plot:
1.3.3.10
Regression
y =
f(x1,x2,x3,...,xk) +
e
Scatter Plot:
1.3.3.26
6-Plot:
1.3.3.33
Linear
Correlation
Plot: 1.3.3.16

Linear Intercept
Plot: 1.3.3.17
Linear Slope
Plot: 1.3.3.18
Linear Residual
Standard
Deviation
Plot:1.3.3.19
Interlab
(y1,y2) = f(x) + e
Youden Plot:
1.3.3.31
Multivariate
(y1,y2,...,yp)
Star Plot:
1.3.3.29
1.3.5. Quantitative Techniques

1.3. EDA Techniques
Confirmatory
Statistics
The techniques discussed in this section are classical
statistical methods as opposed to EDA techniques. EDA and
classical techniques are not mutually exclusive and can be
used in a complementary fashion. For example, the analysis
can start with some simple graphical techniques such as the
4-plot followed by the classical confirmatory methods
discussed herein to provide more rigorous statements about
the conclusions. If the classical methods yield different
conclusions than the graphical analysis, then some effort
should be invested to explain why. Often this is an
indication that some of the assumptions of the classical
techniques are violated.
Many of the quantitative techniques fall into two broad
categories:
1. Interval estimation
2. Hypothesis tests
Interval
Estimates
It is common in statistics to estimate a parameter from a
sample of data. The value of the parameter using all of the
possible data, not just the sample data, is called the
population parameter or true value of the parameter. An
estimate of the true parameter value is made using the
sample data. This is called a point estimate or a sample
estimate.
For example, the most commonly used measure of location
is the mean. The population, or true, mean is the sum of all
the members of the given population divided by the number
of members in the population. As it is typically impractical
to measure every member of the population, a random
sample is drawn from the population. The sample mean is
calculated by summing the values in the sample and
dividing by the number of values in the sample. This sample
mean is then used as the point estimate of the population
mean.
Interval estimates expand on point estimates by
incorporating the uncertainty of the point estimate. In the
example for the mean above, different samples from the
same population will generate different values for the
sample mean. An interval estimate quantifies this
uncertainty in the sample estimate by computing lower and
upper values of an interval which will, with a given level of
confidence (i.e., probability), contain the population
parameter.
Hypothesis
Tests
Hypothesis tests also address the uncertainty of the sample
estimate. However, instead of providing an interval, a
hypothesis test attempts to refute a specific claim about a
population parameter based on the sample data. For
example, the hypothesis might be one of the following:
the population mean is equal to 10
the population standard deviation is equal to 5
the means from two populations are equal
the standard deviations from 5 populations are equal
To reject a hypothesis is to conclude that it is false.
However, to accept a hypothesis does not mean that it is
true, only that we do not have evidence to believe
otherwise. Thus hypothesis tests are usually stated in terms
of both a condition that is doubted (null hypothesis) and a
condition that is believed (alternative hypothesis).
A common format for a hypothesis test is:
H
0
: A statement of the null hypothesis, e.g., two
population means are equal.
H
a
: A statement of the alternative hypothesis, e.g.,
two population means are not equal.
Test
Statistic:
The test statistic is based on the specific
hypothesis test.
Significance
Level:
The significance level, , defines the
sensitivity of the test. A value of = 0.05
means that we inadvertently reject the null
hypothesis 5% of the time when it is in fact
true. This is also called the type I error. The
choice of is somewhat arbitrary, although in
practice values of 0.1, 0.05, and 0.01 are
commonly used.
The probability of rejecting the null
hypothesis when it is in fact false is called the
power of the test and is denoted by 1 - . Its
complement, the probability of accepting the
null hypothesis when the alternative
hypothesis is, in fact, true (type II error), is
called and can only be computed for a
specific alternative hypothesis.
Critical
Region:
The critical region encompasses those values
of the test statistic that lead to a rejection of
the null hypothesis. Based on the distribution
of the test statistic and the significance level,
a cut-off value for the test statistic is
computed. Values either above or below or
both (depending on the direction of the test)
this cut-off define the critical region.
Practical
Versus
Statistical
Significance
It is important to distinguish between statistical significance
and practical significance. Statistical significance simply
means that we reject the null hypothesis. The ability of the
test to detect differences that lead to rejection of the null
hypothesis depends on the sample size. For example, for a
particularly large sample, the test may reject the null
hypothesis that two process means are equivalent. However,
in practice the difference between the two means may be
relatively small to the point of having no real engineering
significance. Similarly, if the sample size is small, a
difference that is large in engineering terms may not lead to
rejection of the null hypothesis. The analyst should not just
blindly apply the tests, but should combine engineering
judgement with statistical analysis.
Bootstrap
Uncertainty
Estimates
In some cases, it is possible to mathematically derive
appropriate uncertainty intervals. This is particularly true for
intervals based on the assumption of a normal distribution.
However, there are many cases in which it is not possible to
mathematically derive the uncertainty. In these cases, the
bootstrap provides a method for empirically determining an
appropriate interval.
Table of
Contents
Some of the more common classical quantitative techniques
are listed below. This list of quantitative techniques is by no
means meant to be exhaustive. Additional discussions of
classical statistical techniques are contained in the product
comparisons chapter.
Location
1. Measures of Location
2. Confidence Limits for the Mean and One
Sample t-Test
3. Two Sample t-Test for Equal Means
4. One Factor Analysis of Variance
5. Multi-Factor Analysis of Variance
Scale (or variability or spread)
1. Measures of Scale
2. Bartlett's Test
3. Chi-Square Test
4. F-Test
5. Levene Test
Skewness and Kurtosis
1. Measures of Skewness and Kurtosis
Randomness
1. Autocorrelation
2. Runs Test
Distributional Measures
1. Anderson-Darling Test
2. Chi-Square Goodness-of-Fit Test
3. Kolmogorov-Smirnov Test
Outliers
1. Detection of Outliers
2. Grubbs Test
3. Tietjen-Moore Test
4. Generalized Extreme Deviate Test
2-Level Factorial Designs
1. Yates Algorithm
1.3.5.1. Measures of Location

1.3. EDA Techniques
Location A fundamental task in many statistical analyses is to estimate
a location parameter for the distribution; i.e., to find a typical
or central value that best describes the data.
Definition
of Location
The first step is to define what we mean by a typical value.
For univariate data, there are three common definitions:
1. mean - the mean is the sum of the data points divided
by the number of data points. That is,
The mean is that value that is most commonly referred
to as the average. We will use the term average as a
synonym for the mean and the term typical value to
refer generically to measures of location.
2. median - the median is the value of the point which has
half the data smaller than that point and half the data
larger than that point. That is, if X
1
, X
2
, ... ,X
N
is a
random sample sorted from smallest value to largest
value, then the median is defined as:
3. mode - the mode is the value of the random sample that
occurs with the greatest frequency. It is not necessarily
unique. The mode is typically used in a qualitative
fashion. For example, there may be a single dominant
hump in the data perhaps two or more smaller humps
in the data. This is usually evident from a histogram of
the data.
When taking samples from continuous populations, we
need to be somewhat careful in how we define the
mode. That is, any specific value may not occur more
than once if the data are continuous. What may be a
more meaningful, if less exact measure, is the midpoint
of the class interval of the histogram with the highest
peak.
Why
Different
Measures
A natural question is why we have more than one measure of
the typical value. The following example helps to explain
why these alternative definitions are useful and necessary.
This plot shows histograms for 10,000 random numbers
generated from a normal, an exponential, a Cauchy, and a
lognormal distribution.
Normal
Distribution
The first histogram is a sample from a normal distribution.
The mean is 0.005, the median is -0.010, and the mode is -
0.144 (the mode is computed as the midpoint of the
histogram interval with the highest peak).
The normal distribution is a symmetric distribution with well-
behaved tails and a single peak at the center of the
distribution. By symmetric, we mean that the distribution can
be folded about an axis so that the 2 sides coincide. That is, it
behaves the same to the left and right of some center point.
For a normal distribution, the mean, median, and mode are
actually equivalent. The histogram above generates similar
estimates for the mean, median, and mode. Therefore, if a
histogram or normal probability plot indicates that your data
are approximated well by a normal distribution, then it is
reasonable to use the mean as the location estimator.
Exponential
Distribution
The second histogram is a sample from an exponential
distribution. The mean is 1.001, the median is 0.684, and the
mode is 0.254 (the mode is computed as the midpoint of the
The exponential distribution is a skewed, i. e., not symmetric,
distribution. For skewed distributions, the mean and median
are not the same. The mean will be pulled in the direction of
the skewness. That is, if the right tail is heavier than the left
tail, the mean will be greater than the median. Likewise, if
the left tail is heavier than the right tail, the mean will be less
than the median.
For skewed distributions, it is not at all obvious whether the
mean, the median, or the mode is the more meaningful
measure of the typical value. In this case, all three measures
are useful.
Cauchy
Distribution
The third histogram is a sample from a Cauchy distribution.
The mean is 3.70, the median is -0.016, and the mode is -
0.362 (the mode is computed as the midpoint of the
For better visual comparison with the other data sets, we
restricted the histogram of the Cauchy distribution to values
between -10 and 10. The full Cauchy data set in fact has a
minimum of approximately -29,000 and a maximum of
approximately 89,000.
The Cauchy distribution is a symmetric distribution with
heavy tails and a single peak at the center of the distribution.
The Cauchy distribution has the interesting property that
collecting more data does not provide a more accurate
estimate of the mean. That is, the sampling distribution of the
mean is equivalent to the sampling distribution of the original
data. This means that for the Cauchy distribution the mean is
useless as a measure of the typical value. For this histogram,
the mean of 3.7 is well above the vast majority of the data.
This is caused by a few very extreme values in the tail.
However, the median does provide a useful measure for the
typical value.
Although the Cauchy distribution is an extreme case, it does
illustrate the importance of heavy tails in measuring the
mean. Extreme values in the tails distort the mean. However,
these extreme values do not distort the median since the
median is based on ranks. In general, for data with extreme
values in the tails, the median provides a better estimate of
location than does the mean.
Lognormal
Distribution
The fourth histogram is a sample from a lognormal
distribution. The mean is 1.677, the median is 0.989, and the
mode is 0.680 (the mode is computed as the midpoint of the
The lognormal is also a skewed distribution. Therefore the
mean and median do not provide similar estimates for the
location. As with the exponential distribution, there is no
obvious answer to the question of which is the more
meaningful measure of location.
Robustness There are various alternatives to the mean and median for
measuring location. These alternatives were developed to
address non-normal data since the mean is an optimal
estimator if in fact your data are normal.
Tukey and Mosteller defined two types of robustness where
robustness is a lack of susceptibility to the effects of
nonnormality.
1. Robustness of validity means that the confidence
intervals for the population location have a 95%
chance of covering the population location regardless
of what the underlying distribution is.
2. Robustness of efficiency refers to high effectiveness in
the face of non-normal tails. That is, confidence
intervals for the population location tend to be almost
as narrow as the best that could be done if we knew the
true shape of the distributuion.
The mean is an example of an estimator that is the best we
can do if the underlying distribution is normal. However, it
lacks robustness of validity. That is, confidence intervals
based on the mean tend not to be precise if the underlying
distribution is in fact not normal.
The median is an example of a an estimator that tends to
have robustness of validity but not robustness of efficiency.
The alternative measures of location try to balance these two
concepts of robustness. That is, the confidence intervals for
the case when the data are normal should be almost as
narrow as the confidence intervals based on the mean.
However, they should maintain their validity even if the
underlying data are not normal. In particular, these
alternatives address the problem of heavy-tailed distributions.
Alternative
Measures
of Location
A few of the more common alternative location measures are:
1. Mid-Mean - computes a mean using the data between
the 25th and 75th percentiles.
2. Trimmed Mean - similar to the mid-mean except
different percentile values are used. A common choice
is to trim 5% of the points in both the lower and upper
tails, i.e., calculate the mean for data between the 5th
and 95th percentiles.
3. Winsorized Mean - similar to the trimmed mean.
However, instead of trimming the points, they are set
to the lowest (or highest) value. For example, all data
below the 5th percentile are set equal to the value of
the 5th percentile and all data greater than the 95th
percentile are set equal to the 95th percentile.
4. Mid-range = (smallest + largest)/2.
The first three alternative location estimators defined above
have the advantage of the median in the sense that they are
not unduly affected by extremes in the tails. However, they
generate estimates that are closer to the mean for data that are
normal (or nearly so).
The mid-range, since it is based on the two most extreme
points, is not robust. Its use is typically restricted to situations
in which the behavior at the extreme points is relevant.
Case Study The uniform random numbers case study compares the
performance of several different location estimators for a
particular non-normal distribution.
compute at least some of the measures of location discussed
above.
1.3.5.2. Confidence Limits for the Mean

1.3. EDA Techniques
Purpose:
Interval
Estimate
for Mean
Confidence limits for the mean (Snedecor and Cochran, 1989) are an interval estimate for the
mean. Interval estimates are often desirable because the estimate of the mean varies from
sample to sample. Instead of a single estimate for the mean, a confidence interval generates a
lower and upper limit for the mean. The interval estimate gives an indication of how much
uncertainty there is in our estimate of the true mean. The narrower the interval, the more
precise is our estimate.
Confidence limits are expressed in terms of a confidence coefficient. Although the choice of
confidence coefficient is somewhat arbitrary, in practice 90 %, 95 %, and 99 % intervals are
often used, with 95 % being the most commonly used.
As a technical note, a 95 % confidence interval does not mean that there is a 95 % probability
that the interval contains the true mean. The interval computed from a given sample either
contains the true mean or it does not. Instead, the level of confidence is associated with the
method of calculating the interval. The confidence coefficient is simply the proportion of
samples of a given size that may be expected to contain the true mean. That is, for a 95 %
confidence interval, if many samples are collected and the confidence interval computed, in
the long run about 95 % of these intervals would contain the true mean.
Definition:
Confidence
Interval
Confidence limits are defined as:
where is the sample mean, s is the sample standard deviation, N is the sample size, is the
desired significance level, and t
1-/2, N-1
is the 100(1-/2) percentile of the t distribution with
N - 1 degrees of freedom. Note that the confidence coefficient is 1 - .
From the formula, it is clear that the width of the interval is controlled by two factors:
1. As N increases, the interval gets narrower from the term.
That is, one way to obtain more precise estimates for the mean is to increase the sample
size.
2. The larger the sample standard deviation, the larger the confidence interval. This simply
means that noisy data, i.e., data with a large standard deviation, are going to generate
wider intervals than data with a smaller standard deviation.
Definition:
Hypothesis
Test
To test whether the population mean has a specific value, , against the two-sided alternative
that it does not have a value , the confidence interval is converted to hypothesis-test form.
The test is a one-sample t-test, and it is defined as:
H
0
:
H
a
:
Test Statistic:

where , N, and are defined as above.
Significance Level: . The most commonly used value for is 0.05.
Critical Region: Reject the null hypothesis that the mean is a specified value, , if
or
Confidence
Interval
Example
We generated a 95 %, two-sided confidence interval for the ZARR13.DAT data set based on
the following information.
N = 195
MEAN = 9.261460
STANDARD DEVIATION = 0.022789
t
1-0.025,N-1
= 1.9723
LOWER LIMIT = 9.261460 - 1.9723*0.022789/195
UPPER LIMIT = 9.261460 + 1.9723*0.022789/195
Thus, a 95 % confidence interval for the mean is (9.258242, 9.264679).
t-Test
Example
We performed a two-sided, one-sample t-test using the ZARR13.DAT data set to test the null
hypothesis that the population mean is equal to 5.
H
0
: = 5
H
a
: 5
Test statistic: T = 2611.284
Degrees of freedom: = 194
Significance level: = 0.05
Critical value: t
1-/2,
= 1.9723
Critical region: Reject H
0
if |T| > 1.9723
We reject the null hypotheses for our two-tailed t-test because the absolute value of the test
statistic is greater than the critical value. If we were to perform an upper, one-tailed test, the
critical value would be t
1-,
= 1.6527, and we would still reject the null hypothesis.
The confidence interval provides an alternative to the hypothesis test. If the confidence
interval contains 5, then H
0
cannot be rejected. In our example, the confidence interval
(9.258242, 9.264679) does not contain 5, indicating that the population mean does not equal 5
at the 0.05 level of significance.
In general, there are three possible alternative hypotheses and rejection regions for the one-
sample t-test:
Alternative Hypothesis Rejection Region
H
a
:
0
|T| > t
1-/2,
H
a
: >
0
T > t
1-,
H
a
: <
0
T < t
,
The rejection regions for three posssible alternative hypotheses using our example data are
shown in the following graphs.
Questions Confidence limits for the mean can be used to answer the following questions:
1. What is a reasonable estimate for the mean?
2. How much variability is there in the estimate of the mean?
3. Does a given target value fall within the confidence limits?
Related
Techniques
Two-Sample t-Test
Confidence intervals for other location estimators such as the median or mid-mean tend to be
mathematically difficult or intractable. For these cases, confidence intervals can be obtained
using the bootstrap.
Case Study Heat flow meter data.
Software Confidence limits for the mean and one-sample t-tests are available in just about all general
purpose statistical software programs. Both Dataplot code and R code can be used to generate
the analyses in this section.
1.3.5.3. Two-Sample t-Test for Equal Means

1.3. EDA Techniques
1.3.5.3. Two-Sample t-Test for Equal Means
Purpose:
Test if two
population
means are
equal
The two-sample t-test (Snedecor and Cochran, 1989) is used to determine if two population
means are equal. A common application is to test if a new process or treatment is superior to a
current process or treatment.
There are several variations on this test.
1. The data may either be paired or not paired. By paired, we mean that there is a one-to-
one correspondence between the values in the two samples. That is, if X
1
, X
2
, ..., X
n
and
Y
1
, Y
2
, ... , Y
n
are the two samples, then X
i
corresponds to Y
i
. For paired samples, the
difference X
i
- Y
i
is usually calculated. For unpaired samples, the sample sizes for the
two samples may or may not be equal. The formulas for paired data are somewhat
simpler than the formulas for unpaired data.
2. The variances of the two samples may be assumed to be equal or unequal. Equal
variances yields somewhat simpler formulas, although with computers this is no longer
a significant issue.
3. In some applications, you may want to adopt a new process or treatment only if it
exceeds the current treatment by some threshold. In this case, we can state the null
hypothesis in the form that the difference between the two populations means is equal to
some constant ( ) where the constant is the desired threshold.
Definition The two-sample t-test for unpaired data is defined as:
H
0
:
H
a
:
Test Statistic:
where N
1
and N
2
are the sample sizes, and are the sample means, and
and are the sample variances.
If equal variances are assumed, then the formula reduces to:
where
Significance
Level:
.
Critical
Region:
Reject the null hypothesis that the two means are equal if
|T| > t
1-/2,
where t
1-/2,
is the critical value of the t distribution with degrees of
freedom where
If equal variances are assumed, then
Two-
Sample t-
Test
Example
The following two-sample t-test was generated for the AUTO83B.DAT data set. The data set
contains miles per gallon for U.S. cars (sample 1) and for Japanese cars (sample 2); the
summary statistics for each sample are shown below.
SAMPLE 1:
NUMBER OF OBSERVATIONS = 249
MEAN = 20.14458
STANDARD ERROR OF THE MEAN = 0.40652

SAMPLE 2:
MEAN = 30.48101
STANDARD ERROR OF THE MEAN = 0.68717
We are testing the hypothesis that the population means are equal for the two samples. We
assume that the variances for the two samples are equal.
H
0
:
1
=
2
H
a
:
1

2
Test statistic: T = -12.62059
Pooled standard deviation: s
p
= 6.34260
Critical value (upper tail): t
1-/2,
= 1.9673
0
if |T| > 1.9673
The absolute value of the test statistic for our example, 12.62059, is greater than the critical
value of 1.9673, so we reject the null hypothesis and conclude that the two population means
are different at the 0.05 significance level.
In general, there are three possible alternative hypotheses and rejection regions for the one-
sample t-test:
Alternative Hypothesis Rejection Region
H
a
:
1

2
|T| > t
1-/2,
H
a
:
1
>
2
T > t
1-,
H
a
:
1
<
2
T < t
,
For our two-tailed t-test, the critical value is t
1-/2,
= 1.9673, where = 0.05 and = 326. If
we were to perform an upper, one-tailed test, the critical value would be t
1-,
= 1.6495. The
rejection regions for three posssible alternative hypotheses using our example data are shown
below.
Questions Two-sample t-tests can be used to answer the following questions:
1. Is process 1 equivalent to process 2?
2. Is the new process better than the current process?
3. Is the new process better than the current process by at least some pre-determined
threshold amount?
Related
Techniques
Confidence Limits for the Mean
Case Study Ceramic strength data.
Software Two-sample t-tests are available in just about all general purpose statistical software
programs. Both Dataplot code and R code can be used to generate the analyses in this section.
1.3.5.3.1. Data Used for Two-Sample t-Test

1.3. EDA Techniques
1.3.5.3. Two-Sample t-Test for Equal Means
1.3.5.3.1. Data Used for Two-Sample t-Test
Data Used
for Two-
Sample t-
Test
Example
The following is the data used for the two-sample t-test
example. The first column is miles per gallon for U.S. cars and
the second column is miles per gallon for Japanese cars. For
the t-test example, rows with the second column equal to -999
were deleted.
18 24
15 27
18 27
16 25
17 31
15 35
14 24
14 19
14 28
15 23
15 27
14 20
15 22
14 18
22 20
18 31
21 32
21 31
10 32
10 24
11 26
9 29
28 24
25 24
19 33
16 33
17 32
19 28
18 19
14 32
14 34
14 26
14 30
12 22
13 22
13 33
18 39
22 36
19 28
18 27
23 21
26 24
25 30
20 34
21 32
13 38
14 37
15 30
14 31
17 37
11 32
13 47
12 41
13 45
15 34
13 33
13 24
14 32
22 39
28 35
13 32
14 37
13 38
14 34
15 34
12 32
13 33
13 32
14 25
13 24
12 37
13 31
18 36
16 36
18 34
18 38
23 32
11 38
12 32
13 -999
12 -999
18 -999
21 -999
19 -999
21 -999
15 -999
16 -999
15 -999
11 -999
20 -999
21 -999
19 -999
15 -999
26 -999
25 -999
16 -999
16 -999
18 -999
16 -999
13 -999
14 -999
14 -999
14 -999
28 -999
19 -999
18 -999
15 -999
15 -999
16 -999
15 -999
16 -999
14 -999
17 -999
16 -999
15 -999
18 -999
21 -999
20 -999
13 -999
23 -999
20 -999
23 -999
18 -999
19 -999
25 -999
26 -999
18 -999
16 -999
16 -999
15 -999
22 -999
22 -999
24 -999
23 -999
29 -999
25 -999
20 -999
18 -999
19 -999
18 -999
27 -999
13 -999
17 -999
13 -999
13 -999
13 -999
30 -999
26 -999
18 -999
17 -999
16 -999
15 -999
18 -999
21 -999
19 -999
19 -999
16 -999
16 -999
16 -999
16 -999
25 -999
26 -999
31 -999
34 -999
36 -999
20 -999
19 -999
20 -999
19 -999
21 -999
20 -999
25 -999
21 -999
19 -999
21 -999
21 -999
19 -999
18 -999
19 -999
18 -999
18 -999
18 -999
30 -999
31 -999
23 -999
24 -999
22 -999
20 -999
22 -999
20 -999
21 -999
17 -999
18 -999
17 -999
18 -999
17 -999
16 -999
19 -999
19 -999
36 -999
27 -999
23 -999
24 -999
34 -999
35 -999
28 -999
29 -999
27 -999
34 -999
32 -999
28 -999
26 -999
24 -999
19 -999
28 -999
24 -999
27 -999
27 -999
26 -999
24 -999
30 -999
39 -999
35 -999
34 -999
30 -999
22 -999
27 -999
20 -999
18 -999
28 -999
27 -999
34 -999
31 -999
29 -999
27 -999
24 -999
23 -999
38 -999
36 -999
25 -999
38 -999
26 -999
22 -999
36 -999
27 -999
27 -999
32 -999
28 -999
31 -999
1.3.5.4. One-Factor ANOVA

1.3. EDA Techniques
Purpose:
Test for
Equal
Means
Across
Groups
One factor analysis of variance (Snedecor and Cochran, 1989)
is a special case of analysis of variance (ANOVA), for one
factor of interest, and a generalization of the two-sample t-
test. The two-sample t-test is used to decide whether two
groups (levels) of a factor have the same mean. One-way
analysis of variance generalizes this to levels where k, the
number of levels, is greater than or equal to 2.
For example, data collected on, say, five instruments have one
factor (instruments) at five levels. The ANOVA tests whether
instruments have a significant effect on the results.
Definition The Product and Process Comparisons chapter (chapter 7)
contains a more extensive discussion of one-factor ANOVA,
including the details for the mathematical computations of
one-way analysis of variance.
The model for the analysis of variance can be stated in two
mathematically equivalent ways. In the following discussion,
each level of each factor is called a cell. For the one-way
case, a cell and a level are equivalent since there is only one
factor. In the following, the subscript i refers to the level and
the subscript j refers to the observation within a level. For
example, Y
23
refers to the third observation in the second
level.
The first model is
This model decomposes the response into a mean for each cell
and an error term. The analysis of variance provides estimates
for each cell mean. These estimated cell means are the
predicted values of the model and the differences between the
response variable and the estimated cell means are the
residuals. That is
The second model is
This model decomposes the response into an overall (grand)
mean, the effect of the ith factor level, and an error term. The
analysis of variance provides estimates of the grand mean and
the effect of the ith factor level. The predicted values and the
residuals of the model are
The distinction between these models is that the second model
divides the cell mean into an overall mean and the effect of
the ith factor level. This second model makes the factor effect
more explicit, so we will emphasize this approach.
Model
Validation
Note that the ANOVA model assumes that the error term, E
ij
,
should follow the assumptions for a univariate measurement
process. That is, after performing an analysis of variance, the
model should be validated by analyzing the residuals.
One-Way
ANOVA
Example
A one-way analysis of variance was generated for the
GEAR.DAT data set. The data set contains 10 measurements
of gear diameter for ten different batches for a total of 100
measurements.
DEGREES OF SUM OF MEAN
SOURCE FREEDOM SQUARES SQUARE
F STATISTIC
---------------- ---------- -------- ------
-- -----------
BATCH 9 0.000729
0.000081 2.2969
RESIDUAL 90 0.003174
0.000035
TOTAL (CORRECTED) 99 0.003903
0.000039

RESIDUAL STANDARD DEVIATION = 0.00594

BATCH N MEAN SD(MEAN)
---------------------------------
1 10 0.99800 0.00178
2 10 0.99910 0.00178
3 10 0.99540 0.00178
4 10 0.99820 0.00178
5 10 0.99190 0.00178
6 10 0.99880 0.00178
7 10 1.00150 0.00178
8 10 1.00040 0.00178
9 10 0.99830 0.00178
10 10 0.99480 0.00178
The ANOVA table decomposes the variance into the
following component sum of squares:
Total sum of squares. The degrees of freedom for this
entry is the number of observations minus one.
Sum of squares for the factor. The degrees of freedom
for this entry is the number of levels minus one. The
mean square is the sum of squares divided by the
number of degrees of freedom.
Residual sum of squares. The degrees of freedom is the
total degrees of freedom minus the factor degrees of
freedom. The mean square is the sum of squares divided
by the number of degrees of freedom.
The sums of squares summarize how much of the variance in
the data (total sum of squares) is accounted for by the factor
effect (batch sum of squares) and how much is random error
(residual sum of squares). Ideally, we would like most of the
variance to be explained by the factor effect.
The ANOVA table provides a formal F test for the factor
effect. For our example, we are testing the following
hypothesis.
H
0
: All individual batch means are equal.
H
a
: At least one batch mean is not equal to the others.
The F statistic is the batch mean square divided by the residual
mean square. This statistic follows an F distribution with (k-1)
and (N-k) degrees of freedom. For our example, the critical F
value (upper tail) for = 0.05, (k-1) = 10, and (N-k) = 90 is
1.9376. Since the F statistic, 2.2969, is greater than the critical
value, we conclude that there is a significant batch effect at
the 0.05 level of significance.
Once we have determined that there is a significant batch
effect, we might be interested in comparing individual batch
means. The batch means and the standard errors of the batch
means provide some information about the individual batches,
however, we may want to employ multiple comparison
methods for a more formal analysis. (See Box, Hunter, and
Hunter for more information.)
In addition to the quantitative ANOVA output, it is
recommended that any analysis of variance be complemented
with model validation. At a minimum, this should include:
1. a run sequence plot of the residuals,
2. a normal probability plot of the residuals, and
3. a scatter plot of the predicted values against the
residuals.
Question The analysis of variance can be used to answer the following
question
Are means the same across groups in the data?
Importance The analysis of uncertainty depends on whether the factor
significantly affects the outcome.
Related
Techniques
Two-sample t-test
Multi-factor analysis of variance
Regression
Box plot
generate an analysis of variance. Both Dataplot code and R
code can be used to generate the analyses in this section.
1.3.5.5. Multi-factor Analysis of Variance

1.3. EDA Techniques
Purpose:
Detect
significant
factors
The analysis of variance (ANOVA) (Neter, Wasserman, and
Kunter, 1990) is used to detect significant factors in a multi-
factor model. In the multi-factor model, there is a response
(dependent) variable and one or more factor (independent)
variables. This is a common model in designed experiments
where the experimenter sets the values for each of the factor
variables and then measures the response variable.
Each factor can take on a certain number of values. These are
referred to as the levels of a factor. The number of levels can
vary betweeen factors. For designed experiments, the number
of levels for a given factor tends to be small. Each factor and
level combination is a cell. Balanced designs are those in
which the cells have an equal number of observations and
unbalanced designs are those in which the number of
observations varies among cells. It is customary to use
balanced designs in designed experiments.
Definition The Product and Process Comparisons chapter (chapter 7)
contains a more extensive discussion of two-factor ANOVA,
including the details for the mathematical computations.
The model for the analysis of variance can be stated in two
mathematically equivalent ways. We explain the model for a
two-way ANOVA (the concepts are the same for additional
factors). In the following discussion, each combination of
factors and levels is called a cell. In the following, the
subscript i refers to the level of factor 1, j refers to the level of
factor 2, and the subscript k refers to the kth observation
within the (i,j)th cell. For example, Y
235
refers to the fifth
observation in the second level of factor 1 and the third level
of factor 2.
The first model is
This model decomposes the response into a mean for each cell
and an error term. The analysis of variance provides estimates
for each cell mean. These cell means are the predicted values
of the model and the differences between the response
variable and the estimated cell means are the residuals. That is
The second model is
This model decomposes the response into an overall (grand)
mean, factor effects ( and represent the effects of the ith
level of the first factor and the jth level of the second factor,
respectively), and an error term. The analysis of variance
provides estimates of the grand mean and the factor effects.
The predicted values and the residuals of the model are
The distinction between these models is that the second model
divides the cell mean into an overall mean and factor effects.
This second model makes the factor effect more explicit, so
we will emphasize this approach.
Model
Validation
Note that the ANOVA model assumes that the error term, E
ijk
,
should follow the assumptions for a univariate measurement
process. That is, after performing an analysis of variance, the
model should be validated by analyzing the residuals.
Multi-
Factor
ANOVA
Example
An analysis of variance was performed for the
JAHANMI2.DAT data set. The data contains four, two-level
factors: table speed, down feed rate, wheel grit size, and batch.
There are 30 measurements of ceramic strength for each factor
combination for a total of 480 measurements.
SOURCE DF SUM OF SQUARES MEAN
SQUARE F STATISTIC
-------------------------------------------------
-----------------
TABLE SPEED 1 26672.726562
26672.726562 6.7080
DOWN FEED RATE 1 11524.053711
11524.053711 2.8982
WHEEL GRIT SIZE 1 14380.633789
14380.633789 3.6166
BATCH 1 727143.125000
727143.125000 182.8703
RESIDUAL 475 1888731.500000
3976.276855
TOTAL (CORRECTED) 479 2668446.000000
5570.868652

RESIDUAL STANDARD DEVIATION = 63.05772781

FACTOR LEVEL N MEAN SD(MEAN)
------------------------------------------------
TABLE SPEED -1 240 657.53168 2.87818
1 240 642.62286 2.87818
DOWN FEED RATE -1 240 645.17755 2.87818
1 240 654.97723 2.87818
WHEEL GRIT SIZE -1 240 655.55084 2.87818
1 240 644.60376 2.87818
BATCH 1 240 688.99890 2.87818
2 240 611.15594 2.87818
The ANOVA decomposes the variance into the following
component sum of squares:
Total sum of squares. The degrees of freedom for this
entry is the number of observations minus one.
Sum of squares for each of the factors. The degrees of
freedom for these entries are the number of levels for
the factor minus one. The mean square is the sum of
squares divided by the number of degrees of freedom.
Residual sum of squares. The degrees of freedom is the
total degrees of freedom minus the sum of the factor
degrees of freedom. The mean square is the sum of
squares divided by the number of degrees of freedom.
The analysis of variance summarizes how much of the
variance in the data (total sum of squares) is accounted for by
the factor effects (factor sum of squares) and how much is due
to random error (residual sum of squares). Ideally, we would
like most of the variance to be explained by the factor effects.
The ANOVA table provides a formal F test for the factor
effects. To test the overall batch effect in our example we use
the following hypotheses.
H
0
: All individual batch means are equal.
H
a
: At least one batch mean is not equal to the others.
The F statistic is the mean square for the factor divided by the
residual mean square. This statistic follows an F distribution
with (k-1) and (N-k) degrees of freedom where k is the
number of levels for the given factor. Here, we see that the
size of the "direction" effect dominates the size of the other
effects. For our example, the critical F value (upper tail) for
= 0.05, (k-1) = 1, and (N-k) = 475 is 3.86111. Thus, "table
speed" and "batch" are significant at the 5 % level while
"down feed rate" and "wheel grit size" are not significant at
the 5 % level.
In addition to the quantitative ANOVA output, it is
recommended that any analysis of variance be complemented
with model validation. At a minimum, this should include
1. A run sequence plot of the residuals.
2. A normal probability plot of the residuals.
3. A scatter plot of the predicted values against the
residuals.
Questions The analysis of variance can be used to answer the following
questions:
1. Do any of the factors have a significant effect?
2. Which is the most important factor?
3. Can we account for most of the variability in the data?
Related
Techniques
One-factor analysis of variance
Two-sample t-test
Box plot
Block plot
DOE mean plot
Case Study The quantitative ANOVA approach can be contrasted with the
more graphical EDA approach in the ceramic strength case
study.
perform multi-factor analysis of variance. Both Dataplot code
and R code can be used to generate the analyses in this
section.
1.3.5.6. Measures of Scale

1.3. EDA Techniques
Scale,
Variability,
or Spread
A fundamental task in many statistical analyses is to
characterize the spread, or variability, of a data set. Measures
of scale are simply attempts to estimate this variability.
When assessing the variability of a data set, there are two key
components:
1. How spread out are the data values near the center?
2. How spread out are the tails?
Different numerical summaries will give different weight to
these two elements. The choice of scale estimator is often
driven by which of these components you want to emphasize.
The histogram is an effective graphical technique for showing
both of these components of the spread.
Definitions
of
Variability
For univariate data, there are several common numerical
measures of the spread:
1. variance - the variance is defined as
where is the mean of the data.
The variance is roughly the arithmetic average of the
squared distance from the mean. Squaring the distance
from the mean has the effect of giving greater weight
to values that are further from the mean. For example,
a point 2 units from the mean adds 4 to the above sum
while a point 10 units from the mean adds 100 to the
sum. Although the variance is intended to be an overall
measure of spread, it can be greatly affected by the tail
behavior.
2. standard deviation - the standard deviation is the square
root of the variance. That is,
The standard deviation restores the units of the spread
to the original data units (the variance squares the
units).
3. range - the range is the largest value minus the smallest
value in a data set. Note that this measure is based only
on the lowest and highest extreme values in the sample.
The spread near the center of the data is not captured at
all.
4. average absolute deviation - the average absolute
deviation (AAD) is defined as
where is the mean of the data and |Y| is the absolute
value of Y. This measure does not square the distance
from the mean, so it is less affected by extreme
observations than are the variance and standard
deviation.
5. median absolute deviation - the median absolute
deviation (MAD) is defined as
where is the median of the data and |Y| is the
absolute value of Y. This is a variation of the average
absolute deviation that is even less affected by
extremes in the tail because the data in the tails have
less influence on the calculation of the median than
they do on the mean.
6. interquartile range - this is the value of the 75th
percentile minus the value of the 25th percentile. This
measure of scale attempts to measure the variability of
points near the center.
In summary, the variance, standard deviation, average
absolute deviation, and median absolute deviation measure
both aspects of the variability; that is, the variability near the
center and the variability in the tails. They differ in that the
average absolute deviation and median absolute deviation do
not give undue weight to the tail behavior. On the other hand,
the range only uses the two most extreme points and the
interquartile range only uses the middle portion of the data.
Why
Different
The following example helps to clarify why these alternative
defintions of spread are useful and necessary.
Measures?
This plot shows histograms for 10,000 random numbers
generated from a normal, a double exponential, a Cauchy,
and a Tukey-Lambda distribution.
Normal
Distribution
The standard deviation is 0.997, the median absolute
deviation is 0.681, and the range is 7.87.
behaved tails and a single peak at the center of the
distribution. By symmetric, we mean that the distribution can
be folded about an axis so that the two sides coincide. That
is, it behaves the same to the left and right of some center
point. In this case, the median absolute deviation is a bit less
than the standard deviation due to the downweighting of the
tails. The range of a little less than 8 indicates the extreme
values fall within about 4 standard deviations of the mean. If
a histogram or normal probability plot indicates that your
data are approximated well by a normal distribution, then it is
reasonable to use the standard deviation as the spread
estimator.
Double
Exponential
Distribution
The second histogram is a sample from a double exponential
distribution. The standard deviation is 1.417, the median
absolute deviation is 0.706, and the range is 17.556.
Comparing the double exponential and the normal histograms
shows that the double exponential has a stronger peak at the
center, decays more rapidly near the center, and has much
longer tails. Due to the longer tails, the standard deviation
tends to be inflated compared to the normal. On the other
hand, the median absolute deviation is only slightly larger
than it is for the normal data. The longer tails are clearly
reflected in the value of the range, which shows that the
extremes fall about 6 standard deviations from the mean
compared to about 4 for the normal data.
Cauchy
Distribution
The standard deviation is 998.389, the median absolute
deviation is 1.16, and the range is 118,953.6.
The Cauchy distribution has the interesting property that
collecting more data does not provide a more accurate
estimate for the mean or standard deviation. That is, the
sampling distribution of the means and standard deviation are
equivalent to the sampling distribution of the original data.
That means that for the Cauchy distribution the standard
deviation is useless as a measure of the spread. From the
histogram, it is clear that just about all the data are between
about -5 and 5. However, a few very extreme values cause
both the standard deviation and range to be extremely large.
However, the median absolute deviation is only slightly
larger than it is for the normal distribution. In this case, the
median absolute deviation is clearly the better measure of
spread.
Although the Cauchy distribution is an extreme case, it does
illustrate the importance of heavy tails in measuring the
spread. Extreme values in the tails can distort the standard
deviation. However, these extreme values do not distort the
median absolute deviation since the median absolute
deviation is based on ranks. In general, for data with extreme
values in the tails, the median absolute deviation or
interquartile range can provide a more stable estimate of
spread than the standard deviation.
Tukey-
Lambda
Distribution
The fourth histogram is a sample from a Tukey lambda
distribution with shape parameter = 1.2. The standard
deviation is 0.49, the median absolute deviation is 0.427, and
the range is 1.666.
The Tukey lambda distribution has a range limited to
. That is, it has truncated tails. In this case the
standard deviation and median absolute deviation have closer
values than for the other three examples which have
significant tails.
Robustness Tukey and Mosteller defined two types of robustness where
robustness is a lack of susceptibility to the effects of
nonnormality.
1. Robustness of validity means that the confidence
intervals for a measure of the population spread (e.g.,
the standard deviation) have a 95 % chance of covering
the true value (i.e., the population value) of that
measure of spread regardless of the underlying
distribution.
2. Robustness of efficiency refers to high effectiveness in
the face of non-normal tails. That is, confidence
intervals for the measure of spread tend to be almost as
narrow as the best that could be done if we knew the
true shape of the distribution.
The standard deviation is an example of an estimator that is
the best we can do if the underlying distribution is normal.
However, it lacks robustness of validity. That is, confidence
intervals based on the standard deviation tend to lack
precision if the underlying distribution is in fact not normal.
The median absolute deviation and the interquartile range are
estimates of scale that have robustness of validity. However,
they are not particularly strong for robustness of efficiency.
If histograms and probability plots indicate that your data are
in fact reasonably approximated by a normal distribution,
then it makes sense to use the standard deviation as the
estimate of scale. However, if your data are not normal, and
in particular if there are long tails, then using an alternative
measure such as the median absolute deviation, average
absolute deviation, or interquartile range makes sense. The
range is used in some applications, such as quality control,
for its simplicity. In addition, comparing the range to the
standard deviation gives an indication of the spread of the
data in the tails.
Since the range is determined by the two most extreme points
in the data set, we should be cautious about its use for large
values of N.
Tukey and Mosteller give a scale estimator that has both
robustness of validity and robustness of efficiency. However,
it is more complicated and we do not give the formula here.
generate at least some of the measures of scale discusssed
above.
1.3.5.7. Bartlett's Test

1.3. EDA Techniques
Purpose:
Test for
Homogeneity
of Variances
Bartlett's test (Snedecor and Cochran, 1983) is used to test if k samples have equal
variances. Equal variances across samples is called homogeneity of variances. Some
statistical tests, for example the analysis of variance, assume that variances are equal across
groups or samples. The Bartlett test can be used to verify that assumption.
Bartlett's test is sensitive to departures from normality. That is, if your samples come from
non-normal distributions, then Bartlett's test may simply be testing for non-normality. The
Levene test is an alternative to the Bartlett test that is less sensitive to departures from
normality.
Definition The Bartlett test is defined as:
H
0
:

1
2
=
2
2
= ... =
k
2
H
a
:

i
2

j
2
for at least one pair (i,j).
Test
Statistic:
The Bartlett test statistic is designed to test for equality of variances across
groups against the alternative that variances are unequal for at least two
groups.
In the above, s
i
2
is the variance of the ith group, N is the total sample size, N
i
is the sample size of the ith group, k is the number of groups, and s
p
2
is the
pooled variance. The pooled variance is a weighted average of the group
variances and is defined as:
Significance
Level:

Critical
Region:
The variances are judged to be unequal if,
where is the critical value of the chi-square distribution with k - 1
degrees of freedom and a significance level of .
An alternate definition (Dixon and Massey, 1969) is based on an approximation to the F
distribution. This definition is given in the Product and Process Comparisons chapter
(chapter 7).
Example Bartlett's test was performed for the GEAR.DAT data set. The
data set contains 10 measurements of gear diameter for ten
different batches for a total of 100 measurements.
H
0
:
1
2
=
2
2
= ... =
10
2

H
a
: At least one
i
2
is not equal to the others.
Degrees of freedom: k - 1 = 9
Critical value:
2
1-,k-1
= 16.919
0
if T > 16.919
We are testing the null hypothesis that the batch variances are
all equal. Because the test statistic is larger than the critical
value, we reject the null hypotheses at the 0.05 significance
level and conclude that at least one batch variance is different
from the others.
Question Bartlett's test can be used to answer the following question:
Is the assumption of equal variances valid?
Importance Bartlett's test is useful whenever the assumption of equal
variances is made. In particular, this assumption is made for
the frequently used one-way analysis of variance. In this case,
Bartlett's or Levene's test should be applied to verify the
assumption.
Related
Techniques
Box Plot
Levene Test
Chi-Square Test
Case Study Heat flow meter data
Software The Bartlett test is available in many general purpose
statistical software programs. Both Dataplot code and R code
can be used to generate the analyses in this section.
1.3.5.8. Chi-Square Test for the Variance

1.3. EDA Techniques
Purpose:
Test if the
variance is
equal to a
specified
value
A chi-square test ( Snedecor and Cochran, 1983) can be used to test if the variance
of a population is equal to a specified value. This test can be either a two-sided test
or a one-sided test. The two-sided version tests against the alternative that the true
variance is either less than or greater than the specified value. The one-sided version
only tests in one direction. The choice of a two-sided or one-sided test is determined
by the problem. For example, if we are testing a new process, we may only be
concerned if its variability is greater than the variability of the current process.
Definition The chi-square hypothesis test is defined as:
H
0
:
H
a
:

Test
Statistic:
where N is the sample size and s is the sample standard deviation. The
key element of this formula is the ratio s/
0
which compares the ratio
of the sample standard deviation to the target standard deviation. The
more this ratio deviates from 1, the more likely we are to reject the null
hypothesis.
Significance
Level:
.
Critical
Region:
Reject the null hypothesis that the variance is a specified value,
0
2
, if
where is the critical value of the chi-square distribution with
N - 1 degrees of freedom.
The formula for the hypothesis test can easily be converted to form an interval
estimate for the variance:
A confidence interval for the standard deviation is computed by taking the square
root of the upper and lower limits of the confidence interval for the variance.
Chi-
Square
Test
Example
A chi-square test was performed for the GEAR.DAT data set. The observed variance
for the 100 measurements of gear diameter is 0.00003969 (the standard deviation is
0.0063). We will test the null hypothesis that the true variance is equal to 0.01.
H
0
:
2
= 0.01
H
a
:
2
0.01
Degrees of freedom: N - 1 = 99
Critical values:
2
/2,N-1
= 73.361

2
1-/2,N-1
= 128.422
0
if T < 73.361 or T > 128.422
The test statistic value of 0.3903 is much smaller than the lower critical value, so we
reject the null hypothesis and conclude that the variance is not equal to 0.01.
Questions The chi-square test can be used to answer the following questions:
1. Is the variance equal to some pre-determined threshold value?
2. Is the variance greater than some pre-determined threshold value?
3. Is the variance less than some pre-determined threshold value?
Related
Techniques
F Test
Bartlett Test
Levene Test
Software The chi-square test for the variance is available in many general purpose statistical
software programs. Both Dataplot code and R code can be used to generate the
analyses in this section.
1.3.5.8.1. Data Used for Chi-Square Test for the Variance

1.3. EDA Techniques
1.3.5.8.1. Data Used for Chi-Square Test for the
Variance
Data Used
for Chi-
Square
Test for
the
Variance
Example
The following are the data used for the chi-square test for the
variance example. The first column is gear diameter and the
second column is batch number. Only the first column is used
for this example.
1.006 1.000
0.996 1.000
0.998 1.000
1.000 1.000
0.992 1.000
0.993 1.000
1.002 1.000
0.999 1.000
0.994 1.000
1.000 1.000
0.998 2.000
1.006 2.000
1.000 2.000
1.002 2.000
0.997 2.000
0.998 2.000
0.996 2.000
1.000 2.000
1.006 2.000
0.988 2.000
0.991 3.000
0.987 3.000
0.997 3.000
0.999 3.000
0.995 3.000
0.994 3.000
1.000 3.000
0.999 3.000
0.996 3.000
0.996 3.000
1.005 4.000
1.002 4.000
0.994 4.000
1.000 4.000
0.995 4.000
0.994 4.000
0.998 4.000
0.996 4.000
1.002 4.000
0.996 4.000
0.998 5.000
0.998 5.000
0.982 5.000
0.990 5.000
1.002 5.000
0.984 5.000
0.996 5.000
0.993 5.000
0.980 5.000
0.996 5.000
1.009 6.000
1.013 6.000
1.3.5.8.1. Data Used for Chi-Square Test for the Variance
1.009 6.000
0.997 6.000
0.988 6.000
1.002 6.000
0.995 6.000
0.998 6.000
0.981 6.000
0.996 6.000
0.990 7.000
1.004 7.000
0.996 7.000
1.001 7.000
0.998 7.000
1.000 7.000
1.018 7.000
1.010 7.000
0.996 7.000
1.002 7.000
0.998 8.000
1.000 8.000
1.006 8.000
1.000 8.000
1.002 8.000
0.996 8.000
0.998 8.000
0.996 8.000
1.002 8.000
1.006 8.000
1.002 9.000
0.998 9.000
0.996 9.000
0.995 9.000
0.996 9.000
1.004 9.000
1.004 9.000
0.998 9.000
0.999 9.000
0.991 9.000
0.991 10.000
0.995 10.000
0.984 10.000
0.994 10.000
0.997 10.000
0.997 10.000
0.991 10.000
0.998 10.000
1.004 10.000
0.997 10.000
1.3.5.9. F-Test for Equality of Two Variances

1.3. EDA Techniques
Purpose:
Test if
variances
from two
populations
are equal
An F-test (Snedecor and Cochran, 1983) is used to test if the
variances of two populations are equal. This test can be a
two-tailed test or a one-tailed test. The two-tailed version
tests against the alternative that the variances are not equal.
The one-tailed version only tests in one direction, that is the
variance from the first population is either greater than or less
than (but not both) the second population variance. The
choice is determined by the problem. For example, if we are
testing a new process, we may only be interested in knowing
if the new process is less variable than the old process.
Definition The F hypothesis test is defined as:
H
0
:

1
2
=
2
2
H
a
:

1
2
<
2
2
for a lower one-tailed test
1
2
>
2
2
for an upper one-tailed test
1
2

2
2
for a two-tailed test
Test
Statistic:
F =
where and are the sample variances. The
more this ratio deviates from 1, the stronger the
evidence for unequal population variances.
Significance
Level:

Critical
Region:
The hypothesis that the two variances are equal
is rejected if
F > F
, N
1
-1, N
2
-1
for an upper one-tailed test
F < F
1-, N
1
-1, N
2
-1
for a lower one-tailed
test
F < F
1-/2, N
1
-1, N
2
-1
for a two-tailed test
or
F > F
/2, N
1
-1, N
2
-1
where F
, N
1
-1, N
2
-1
is the critical value of the F
distribution with N
1
-1 and N
2
-1 degrees of
freedom and a significance level of .
In the above formulas for the critical regions,
the Handbook follows the convention that F
is
the upper critical value from the F distribution
and F
1-
is the lower critical value from the F
distribution. Note that this is the opposite of the
designation used by some texts and software
programs.
F Test
Example
The following F-test was generated for the AUTO83B.DAT
data set. The data set contains 480 ceramic strength
measurements for two batches of material. The summary
statistics for each batch are shown below.
BATCH 1:
MEAN = 688.9987

BATCH 2:
MEAN = 611.1559
We are testing the null hypothesis that the variances for the
two batches are equal.
H
0
:
1
2
=
2
2

H
a
:
1
2

2
2

Test statistic: F = 1.123037
Numerator degrees of freedom: N
1
- 1 = 239
Denominator degrees of freedom: N
2
- 1 = 239
Critical values: F(1-/2,N
1
-1,N
2
-1) = 0.7756
F(/2,N
1
-1,N
2
-1) = 1.2894
Rejection region: Reject H
0
if F < 0.7756 or F >
1.2894
The F test indicates that there is not enough evidence to reject
the null hypothesis that the two batch variancess are equal at
the 0.05 significance level.
Questions The F-test can be used to answer the following questions:
1. Do two samples come from populations with equal
variancess?
2. Does a new process, treatment, or test reduce the
variability of the current process?
Related
Techniques
Quantile-Quantile Plot
Bihistogram
Chi-Square Test
Bartlett's Test
Levene Test
Case Study Ceramic strength data.
Software The F-test for equality of two variances is available in many
general purpose statistical software programs. Both Dataplot
code and R code can be used to generate the analyses in this
section.
1.3.5.10. Levene Test for Equality of Variances

1.3. EDA Techniques
Purpose:
Test for
Homogeneity
of Variances
Levene's test ( Levene 1960) is used to test if k samples have
equal variances. Equal variances across samples is called
homogeneity of variance. Some statistical tests, for example
the analysis of variance, assume that variances are equal
across groups or samples. The Levene test can be used to
verify that assumption.
Levene's test is an alternative to the Bartlett test. The Levene
test is less sensitive than the Bartlett test to departures from
normality. If you have strong evidence that your data do in
fact come from a normal, or nearly normal, distribution, then
Bartlett's test has better performance.
Definition The Levene test is defined as:
H
0
:

1
2
=
2
2
= ... =
k
2
H
a
:

i
2

j
2
for at least one pair (i,j).
Test
Statistic:
Given a variable Y with sample of size N
divided into k subgroups, where N
i
is the
sample size of the ith subgroup, the Levene test
statistic is defined as:
where Z
ij
can have one of the following three
definitions:
1.
where is the mean of the ith subgroup.
2.
where is the median of the ith
subgroup.
3.
where is the 10% trimmed mean of
the ith subgroup.
are the group means of the Z
ij
and is the
overall mean of the Z
ij
.
The three choices for defining Z
ij
determine the
robustness and power of Levene's test. By
robustness, we mean the ability of the test to
not falsely detect unequal variances when the
underlying data are not normally distributed and
the variables are in fact equal. By power, we
mean the ability of the test to detect unequal
variances when the variances are in fact
unequal.
Levene's original paper only proposed using the
mean. Brown and Forsythe (1974)) extended
Levene's test to use either the median or the
trimmed mean in addition to the mean. They
performed Monte Carlo studies that indicated
that using the trimmed mean performed best
when the underlying data followed a Cauchy
distribution (i.e., heavy-tailed) and the median
performed best when the underlying data
followed a (i.e., skewed) distribution. Using
the mean provided the best power for
symmetric, moderate-tailed, distributions.
Although the optimal choice depends on the
underlying distribution, the definition based on
the median is recommended as the choice that
provides good robustness against many types of
non-normal data while retaining good power. If
you have knowledge of the underlying
distribution of the data, this may indicate using
one of the other choices.
Significance
Level:

Critical
Region:
The Levene test rejects the hypothesis that the
variances are equal if
W > F
, k-1, N-k
where F
, k-1, N-k
is the upper critical value of
the F distribution with k-1 and N-k degrees of
freedom at a significance level of .
In the above formulas for the critical regions,
the Handbook follows the convention that F
is
the upper critical value from the F distribution
and F
1-
is the lower critical value. Note that
this is the opposite of some texts and software
programs.
Levene's Test
Example
Levene's test, based on the median, was performed for the
GEAR.DAT data set. The data set includes ten measurements
of gear diameter for each of ten batches for a total of 100
measurements.
H
0
:
1
2
= ... =
10
2
H
a
:
1
2
...
10
2
Test statistic: W = 1.705910
Degrees of freedom: k-1 = 10-1 = 9
N-k = 100-10 = 90
Critical value (upper tail): F
,k-1,N-k
= 1.9855
0
if F > 1.9855
We are testing the hypothesis that the group variances are
equal. We fail to reject the null hypothesis at the 0.05
significance level since the value of the Levene test statistic is
less than the critical value. We conclude that there is
insufficient evidence to claim that the variances are not equal.
Question Levene's test can be used to answer the following question:
Is the assumption of equal variances valid?
Related
Techniques
Box Plot
Bartlett Test
Chi-Square Test
Software The Levene test is available in some general purpose
statistical software programs. Both Dataplot code and R code
can be used to generate the analyses in this section.
1.3.5.11. Measures of Skewness and Kurtosis

1.3. EDA Techniques
Skewness
and
Kurtosis
A fundamental task in many statistical analyses is to
characterize the location and variability of a data set. A
further characterization of the data includes skewness and
kurtosis.
Skewness is a measure of symmetry, or more precisely, the
lack of symmetry. A distribution, or data set, is symmetric if
it looks the same to the left and right of the center point.
Kurtosis is a measure of whether the data are peaked or flat
relative to a normal distribution. That is, data sets with high
kurtosis tend to have a distinct peak near the mean, decline
rather rapidly, and have heavy tails. Data sets with low
kurtosis tend to have a flat top near the mean rather than a
sharp peak. A uniform distribution would be the extreme
case.
The histogram is an effective graphical technique for showing
both the skewness and kurtosis of data set.
Definition
of Skewness
For univariate data Y
1
, Y
2
, ..., Y
N
, the formula for skewness
is:
where is the mean, is the standard deviation, and N is the
number of data points. The skewness for a normal
distribution is zero, and any symmetric data should have a
skewness near zero. Negative values for the skewness
indicate data that are skewed left and positive values for the
skewness indicate data that are skewed right. By skewed left,
we mean that the left tail is long relative to the right tail.
Similarly, skewed right means that the right tail is long
relative to the left tail. Some measurements have a lower
bound and are skewed right. For example, in reliability
studies, failure times cannot be negative.
Definition
of Kurtosis
For univariate data Y
1
, Y
2
, ..., Y
N
, the formula for kurtosis is:
where is the mean, is the standard deviation, and N is the
number of data points.
Alternative
Definition
of Kurtosis
The kurtosis for a standard normal distribution is three. For
this reason, some sources use the following definition of
kurtosis (often referred to as "excess kurtosis"):
This definition is used so that the standard normal
distribution has a kurtosis of zero. In addition, with the
second definition positive kurtosis indicates a "peaked"
distribution and negative kurtosis indicates a "flat"
distribution.
Which definition of kurtosis is used is a matter of convention
(this handbook uses the original definition). When using
software to compute the sample kurtosis, you need to be
aware of which convention is being followed. Many sources
use the term kurtosis when they are actually computing
"excess kurtosis", so it may not always be clear.
Examples The following example shows histograms for 10,000 random
numbers generated from a normal, a double exponential, a
Cauchy, and a Weibull distribution.
Normal
Distribution
behaved tails. This is indicated by the skewness of 0.03. The
kurtosis of 2.96 is near the expected value of 3. The
histogram verifies the symmetry.
Double
Exponential
Distribution
The second histogram is a sample from a double exponential
distribution. The double exponential is a symmetric
distribution. Compared to the normal, it has a stronger peak,
more rapid decay, and heavier tails. That is, we would expect
a skewness near zero and a kurtosis higher than 3. The
skewness is 0.06 and the kurtosis is 5.9.
Cauchy
Distribution
For better visual comparison with the other data sets, we
restricted the histogram of the Cauchy distribution to values
between -10 and 10. The full data set for the Cauchy data in
fact has a minimum of approximately -29,000 and a
maximum of approximately 89,000.
Since it is symmetric, we would expect a skewness near zero.
Due to the heavier tails, we might expect the kurtosis to be
larger than for a normal distribution. In fact the skewness is
69.99 and the kurtosis is 6,693. These extremely high values
can be explained by the heavy tails. Just as the mean and
standard deviation can be distorted by extreme values in the
tails, so too can the skewness and kurtosis measures.
Weibull
Distribution
The fourth histogram is a sample from a Weibull distribution
with shape parameter 1.5. The Weibull distribution is a
skewed distribution with the amount of skewness depending
on the value of the shape parameter. The degree of decay as
we move away from the center also depends on the value of
the shape parameter. For this data set, the skewness is 1.08
and the kurtosis is 4.46, which indicates moderate skewness
and kurtosis.
Dealing
with
Skewness
and
Kurtosis
Many classical statistical tests and intervals depend on
normality assumptions. Significant skewness and kurtosis
clearly indicate that data are not normal. If a data set exhibits
significant skewness or kurtosis (as indicated by a histogram
or the numerical measures), what can we do about it?
One approach is to apply some type of transformation to try
to make the data normal, or more nearly normal. The Box-
Cox transformation is a useful technique for trying to
normalize a data set. In particular, taking the log or square
root of a data set is often useful for data that exhibit moderate
right skewness.
Another approach is to use techniques based on distributions
other than the normal. For example, in reliability studies, the
exponential, Weibull, and lognormal distributions are
typically used as a basis for modeling rather than using the
normal distribution. The probability plot correlation
coefficient plot and the probability plot are useful tools for
determining a good distributional model for the data.
Software The skewness and kurtosis coefficients are available in most
general purpose statistical software programs.
1.3.5.12. Autocorrelation

1.3. EDA Techniques
Purpose:
Detect Non-
Randomness,
Time Series
Modeling
The autocorrelation ( Box and Jenkins, 1976) function
can be used for the following two purposes:
1. To detect non-randomness in data.
2. To identify an appropriate time series model if the
data are not random.
Definition Given measurements, Y
1
, Y
2
, ..., Y
N
at time X
1
, X
2
, ..., X
N
,
the lag k autocorrelation function is defined as
Although the time variable, X, is not used in the formula
for autocorrelation, the assumption is that the observations
are equi-spaced.
Autocorrelation is a correlation coefficient. However,
instead of correlation between two different variables, the
correlation is between two values of the same variable at
times X
i
and X
i+k
.
When the autocorrelation is used to detect non-
randomness, it is usually only the first (lag 1)
autocorrelation that is of interest. When the
autocorrelation is used to identify an appropriate time
series model, the autocorrelations are usually plotted for
many lags.
Autocorrelation
Example
Lag-one autocorrelations were computed for the the
LEW.DAT data set.

lag autocorrelation
0. 1.00
1. -0.31
2. -0.74
3. 0.77
4. 0.21
5. -0.90
6. 0.38
7. 0.63
8. -0.77
9. -0.12
10. 0.82
11. -0.40
12. -0.55
13. 0.73
14. 0.07
15. -0.76
16. 0.40
17. 0.48
18. -0.70
19. -0.03
20. 0.70
21. -0.41
22. -0.43
23. 0.67
24. 0.00
25. -0.66
26. 0.42
27. 0.39
28. -0.65
29. 0.03
30. 0.63
31. -0.42
32. -0.36
33. 0.64
34. -0.05
35. -0.60
36. 0.43
37. 0.32
38. -0.64
39. 0.08
40. 0.58
41. -0.45
42. -0.28
43. 0.62
44. -0.10
45. -0.55
46. 0.45
47. 0.25
48. -0.61
49. 0.14

Questions The autocorrelation function can be used to answer the
1. Was this sample data set generated from a random
process?
2. Would a non-linear or time series model be a more
appropriate model for these data than a simple
constant plus error model?
Importance Randomness is one of the key assumptions in
determining if a univariate statistical process is in control.
If the assumptions of constant location and scale,
randomness, and fixed distribution are reasonable, then
the univariate process can be modeled as:
where E
i
is an error term.
If the randomness assumption is not valid, then a different
model needs to be used. This will typically be either a
time series model or a non-linear model (with time as the
independent variable).
Related
Techniques
Run Sequence Plot
Lag Plot
Runs Test
Case Study The heat flow meter data demonstrate the use of
autocorrelation in determining if the data are from a
random process.
Software The autocorrelation capability is available in most general
purpose statistical software programs. Both Dataplot code
and R code can be used to generate the analyses in this
section.
1.3.5.13. Runs Test for Detecting Non-randomness

1.3. EDA Techniques
1.3.5.13. Runs Test for Detecting Non-
randomness
Purpose:
Detect Non-
Randomness
The runs test (Bradley, 1968) can be used to decide if a data
set is from a random process.
A run is defined as a series of increasing values or a series of
decreasing values. The number of increasing, or decreasing,
values is the length of the run. In a random data set, the
probability that the (I+1)th value is larger or smaller than the
Ith value follows a binomial distribution, which forms the basis
of the runs test.
Typical
Analysis
and Test
Statistics
The first step in the runs test is to count the number of runs in
the data sequence. There are several ways to define runs in the
literature, however, in all cases the formulation must produce a
dichotomous sequence of values. For example, a series of 20
coin tosses might produce the following sequence of heads (H)
and tails (T).
H H T T H T H H H H T H H T T T T T H H
The number of runs for this series is nine. There are 11 heads
and 9 tails in the sequence.
Definition We will code values above the median as positive and values
below the median as negative. A run is defined as a series of
consecutive positive (or negative) values. The runs test is
defined as:
H
0
:
the sequence was produced in a random manner
H
a
:
the sequence was not produced in a random
manner
Test
Statistic:
The test statistic is
where R is the observed number of runs, R, is the
expected number of runs, and s
R
is the standard
deviation of the number of runs. The values of R
and s
R
are computed as follows:
where n
1
and n
2
are the number of positive and
negative values in the series.
Significance
Level:

Critical
Region:
The runs test rejects the null hypothesis if
|Z| > Z
1-/2
For a large-sample runs test (where n
1
> 10 and
n
2
> 10), the test statistic is compared to a
standard normal table. That is, at the 5 %
significance level, a test statistic with an absolute
value greater than 1.96 indicates non-
randomness. For a small-sample runs test, there
are tables to determine critical values that depend
on values of n
1
and n
2
(Mendenhall, 1982).
Runs Test
Example
A runs test was performed for 200 measurements of beam
deflection contained in the LEW.DAT data set.

H
0
: the sequence was produced in a random manner
H
a
: the sequence was not produced in a random
manner
Test statistic: Z = 2.6938
Critical value (upper tail): Z
1-/2
= 1.96
0
if |Z| > 1.96
Since the test statistic is greater than the critical value, we
conclude that the data are not random at the 0.05 significance
level.
Question The runs test can be used to answer the following question:
Were these sample data generated from a random
process?
Importance Randomness is one of the key assumptions in determining if a
univariate statistical process is in control. If the assumptions of
constant location and scale, randomness, and fixed distribution
are reasonable, then the univariate process can be modeled as:
where E
i
is an error term.
If the randomness assumption is not valid, then a different
model needs to be used. This will typically be either a times
series model or a non-linear model (with time as the
independent variable).
Related
Techniques
Autocorrelation
Run Sequence Plot
Lag Plot
Case Study Heat flow meter data
Software Most general purpose statistical software programs support a
runs test. Both Dataplot code and R code can be used to
generate the analyses in this section.
1.3.5.14. Anderson-Darling Test

1.3. EDA Techniques
Purpose:
Test for
Distributional
Adequacy
The Anderson-Darling test (Stephens, 1974) is used to test if a
sample of data came from a population with a specific distribution.
It is a modification of the Kolmogorov-Smirnov (K-S) test and
gives more weight to the tails than does the K-S test. The K-S test
is distribution free in the sense that the critical values do not depend
on the specific distribution being tested. The Anderson-Darling test
makes use of the specific distribution in calculating critical values.
This has the advantage of allowing a more sensitive test and the
disadvantage that critical values must be calculated for each
distribution. Currently, tables of critical values are available for the
normal, lognormal, exponential, Weibull, extreme value type I, and
logistic distributions. We do not provide the tables of critical values
in this Handbook (see Stephens 1974, 1976, 1977, and 1979) since
this test is usually applied with a statistical software program that
will print the relevant critical values.
The Anderson-Darling test is an alternative to the chi-square and
Kolmogorov-Smirnov goodness-of-fit tests.
Definition The Anderson-Darling test is defined as:
H
0
: The data follow a specified distribution.
H
a
: The data do not follow the specified distribution
Test
Statistic:
The Anderson-Darling test statistic is defined as
where
F is the cumulative distribution function of the
specified distribution. Note that the Y
i
are the ordered
data.
Significance
Level:
Critical
Region:
The critical values for the Anderson-Darling test are
dependent on the specific distribution that is being
tested. Tabulated values and formulas have been
published (Stephens, 1974, 1976, 1977, 1979) for a
few specific distributions (normal, lognormal,
exponential, Weibull, logistic, extreme value type 1).
The test is a one-sided test and the hypothesis that the
distribution is of a specific form is rejected if the test
statistic, A, is greater than the critical value.
Note that for a given distribution, the Anderson-
Darling statistic may be multiplied by a constant
(which usually depends on the sample size, n). These
constants are given in the various papers by Stephens.
In the sample output below, the test statistic values are
adjusted. Also, be aware that different constants (and
therefore critical values) have been published. You
just need to be aware of what constant was used for a
given set of critical values (the needed constant is
typically given with the critical values).
Sample
Output
We generated 1,000 random numbers for normal, double
exponential, Cauchy, and lognormal distributions. In all four cases,
the Anderson-Darling test was applied to test for a normal
distribution.
The normal random numbers were stored in the variable Y1, the
double exponential random numbers were stored in the variable Y2,
the Cauchy random numbers were stored in the variable Y3, and the
lognormal random numbers were stored in the variable Y4.
Distribution Mean Standard
Deviation
------------ -------- ---------
---------
Normal (Y1) 0.004360
1.001816
Double Exponential (Y2) 0.020349
1.321627
Cauchy (Y3) 1.503854
35.130590
Lognormal (Y4) 1.518372
1.719969

H
0
: the data are normally distributed
H
a
: the data are not normally distributed
Y1 adjusted test statistic: A
2
= 0.2576
2
= 5.8492
2
= 288.7863
2
= 83.3935
Critical value: 0.752
0
if A
2
> 0.752

When the data were generated using a normal distribution, the test
statistic was small and the hypothesis of normality was not rejected.
When the data were generated using the double exponential,
Cauchy, and lognormal distributions, the test statistics were large,
and the hypothesis of an underlying normal distribution was
rejected at the 0.05 significance level.
Questions The Anderson-Darling test can be used to answer the following
questions:
Are the data from a normal distribution?
Are the data from a log-normal distribution?
Are the data from a Weibull distribution?
Are the data from an exponential distribution?
Are the data from a logistic distribution?
Importance Many statistical tests and procedures are based on specific
distributional assumptions. The assumption of normality is
particularly common in classical statistical tests. Much reliability
modeling is based on the assumption that the data follow a Weibull
distribution.
There are many non-parametric and robust techniques that do not
make strong distributional assumptions. However, techniques based
on specific distributional assumptions are in general more powerful
than non-parametric and robust techniques. Therefore, if the
distributional assumptions can be validated, they are generally
preferred.
Related
Techniques
Chi-Square goodness-of-fit Test
Kolmogorov-Smirnov Test
Shapiro-Wilk Normality Test
Probability Plot
Probability Plot Correlation Coefficient Plot
Case Study Josephson junction cryothermometry case study.
Software The Anderson-Darling goodness-of-fit test is available in some
general purpose statistical software programs. Both Dataplot code
and R code can be used to generate the analyses in this section.
1.3.5.15. Chi-Square Goodness-of-Fit Test

1.3. EDA Techniques
Purpose:
Test for
distributional
adequacy
The chi-square test (Snedecor and Cochran, 1989) is used
to test if a sample of data came from a population with a
specific distribution.
An attractive feature of the chi-square goodness-of-fit test
is that it can be applied to any univariate distribution for
which you can calculate the cumulative distribution
function. The chi-square goodness-of-fit test is applied to
binned data (i.e., data put into classes). This is actually not a
restriction since for non-binned data you can simply
calculate a histogram or frequency table before generating
the chi-square test. However, the value of the chi-square
test statistic are dependent on how the data is binned.
Another disadvantage of the chi-square test is that it
requires a sufficient sample size in order for the chi-square
approximation to be valid.
The chi-square test is an alternative to the Anderson-
Darling and Kolmogorov-Smirnov goodness-of-fit tests.
The chi-square goodness-of-fit test can be applied to
discrete distributions such as the binomial and the Poisson.
The Kolmogorov-Smirnov and Anderson-Darling tests are
restricted to continuous distributions.
Additional discussion of the chi-square goodness-of-fit test
is contained in the product and process comparisons chapter
(chapter 7).
Definition The chi-square test is defined for the hypothesis:
H
0
:
The data follow a specified distribution.
H
a
: The data do not follow the specified
distribution.
Test
Statistic:
For the chi-square goodness-of-fit
computation, the data are divided into k bins
and the test statistic is defined as
where is the observed frequency for bin i
and is the expected frequency for bin i.
The expected frequency is calculated by
where F is the cumulative Distribution
function for the distribution being tested, Y
u
is
the upper limit for class i, Y
l
is the lower limit
for class i, and N is the sample size.
This test is sensitive to the choice of bins.
There is no optimal choice for the bin width
(since the optimal bin width depends on the
distribution). Most reasonable choices should
produce similar, but not identical, results. For
the chi-square approximation to be valid, the
expected frequency should be at least 5. This
test is not valid for small samples, and if some
of the counts are less than five, you may need
to combine some bins in the tails.
Significance
Level:
.
Critical
Region:
The test statistic follows, approximately, a
chi-square distribution with (k - c) degrees of
freedom where k is the number of non-empty
cells and c = the number of estimated
parameters (including location and scale
parameters and shape parameters) for the
distribution + 1. For example, for a 3-
parameter Weibull distribution, c = 4.
Therefore, the hypothesis that the data are
from a population with the specified
distribution is rejected if
where is the chi-square critical
value with k - c degrees of freedom and
significance level .
Chi-Square
Test Example
exponential, t with 3 degrees of freedom, and lognormal
distributions. In all cases, a chi-square test with k = 32 bins
was applied to test for normally distributed data. Because
the normal distribution has two parameters, c = 2 + 1 = 3
The normal random numbers were stored in the variable
Y1, the double exponential random numbers were stored in
the variable Y2, the t random numbers were stored in the
variable Y3, and the lognormal random numbers were
stored in the variable Y4.
H
0
H
a
Y1 Test statistic:
2
= 32.256
Y2 Test statistic:
2
= 91.776
Y3 Test statistic:
2
= 101.488
Y4 Test statistic:
2
= 1085.104
Degrees of freedom: k - c = 32 - 3 = 29
Critical value:
2
1-, k-c
= 42.557
0
if
2
> 42.557
As we would hope, the chi-square test fails to reject the null
hypothesis for the normally distributed data set and rejects
the null hypothesis for the three non-normal data sets.
Questions The chi-square test can be used to answer the following
types of questions:
Are the data from a binomial distribution?
particularly common in classical statistical tests. Much
reliability modeling is based on the assumption that the
distribution of the data follows a Weibull distribution.
There are many non-parametric and robust techniques that
are not based on strong distributional assumptions. By non-
parametric, we mean a technique, such as the sign test, that
is not based on a specific distributional assumption. By
robust, we mean a statistical technique that performs well
under a wide range of distributional assumptions. However,
techniques based on specific distributional assumptions are
in general more powerful than these non-parametric and
robust techniques. By power, we mean the ability to detect a
difference when that difference actually exists. Therefore, if
the distributional assumption can be confirmed, the
parametric techniques are generally preferred.
If you are using a technique that makes a normality (or
some other type of distributional) assumption, it is important
to confirm that this assumption is in fact justified. If it is,
the more powerful parametric techniques can be used. If the
distributional assumption is not justified, a non-parametric
or robust technique may be required.
Related
Techniques
Anderson-Darling Goodness-of-Fit Test
Kolmogorov-Smirnov Test
Probability Plots
Software Some general purpose statistical software programs provide
a chi-square goodness-of-fit test for at least some of the
common distributions. Both Dataplot code and R code can
be used to generate the analyses in this section.
1.3.5.16. Kolmogorov-Smirnov Goodness-of-Fit Test

1.3. EDA Techniques
Purpose:
Test for
Distributional
Adequacy
The Kolmogorov-Smirnov test (Chakravart, Laha, and Roy,
1967) is used to decide if a sample comes from a population with
a specific distribution.
The Kolmogorov-Smirnov (K-S) test is based on the empirical
distribution function (ECDF). Given N ordered data points Y
1
,
Y
2
, ..., Y
N
, the ECDF is defined as
where n(i) is the number of points less than Y
i
and the Y
i
are
ordered from smallest to largest value. This is a step function that
increases by 1/N at the value of each ordered data point.
The graph below is a plot of the empirical distribution function
with a normal cumulative distribution function for 100 normal
random numbers. The K-S test is based on the maximum distance
between these two curves.
Characteristics
and
Limitations of
An attractive feature of this test is that the distribution of the K-S
test statistic itself does not depend on the underlying cumulative
distribution function being tested. Another advantage is that it is
the K-S Test an exact test (the chi-square goodness-of-fit test depends on an
adequate sample size for the approximations to be valid). Despite
these advantages, the K-S test has several important limitations:
1. It only applies to continuous distributions.
2. It tends to be more sensitive near the center of the
distribution than at the tails.
3. Perhaps the most serious limitation is that the distribution
must be fully specified. That is, if location, scale, and shape
parameters are estimated from the data, the critical region
of the K-S test is no longer valid. It typically must be
determined by simulation.
Due to limitations 2 and 3 above, many analysts prefer to use the
Anderson-Darling goodness-of-fit test. However, the Anderson-
Darling test is only available for a few specific distributions.
Definition The Kolmogorov-Smirnov test is defined by:
H
0
:
The data follow a specified distribution
H
a
:
The data do not follow the specified distribution
Test
Statistic:
The Kolmogorov-Smirnov test statistic is defined as
where F is the theoretical cumulative distribution of
the distribution being tested which must be a
continuous distribution (i.e., no discrete
distributions such as the binomial or Poisson), and
it must be fully specified (i.e., the location, scale,
and shape parameters cannot be estimated from the
data).
Significance
Level:
.
Critical
Values:
The hypothesis regarding the distributional form is
rejected if the test statistic, D, is greater than the
critical value obtained from a table. There are
several variations of these tables in the literature
that use somewhat different scalings for the K-S
test statistic and critical regions. These alternative
formulations should be equivalent, but it is
necessary to ensure that the test statistic is
calculated in a way that is consistent with how the
critical values were tabulated.
We do not provide the K-S tables in the Handbook
since software programs that perform a K-S test
will provide the relevant critical values.
Technical Note Previous editions of e-Handbook gave the following formula for
the computation of the Kolmogorov-Smirnov goodness of fit
statistic:
This formula is in fact not correct. Note that this formula can be
rewritten as:
This form makes it clear that an upper bound on the difference
between these two formulas is i/N. For actual data, the difference
is likely to be less than the upper bound.
For example, for N = 20, the upper bound on the difference
between these two formulas is 0.05 (for comparison, the 5%
critical value is 0.294). For N = 100, the upper bound is 0.001. In
practice, if you have moderate to large sample sizes (say N 50),
these formulas are essentially equivalent.
Kolmogorov-
Smirnov Test
Example
exponential, t with 3 degrees of freedom, and lognormal
distributions. In all cases, the Kolmogorov-Smirnov test was
applied to test for a normal distribution.
The normal random numbers were stored in the variable Y1, the
double exponential random numbers were stored in the variable
Y2, the t random numbers were stored in the variable Y3, and the
lognormal random numbers were stored in the variable Y4.
H
0
H
a
Y1 test statistic: D = 0.0241492
0
if D > 0.04301

As expected, the null hypothesis is not rejected for the normally
distributed data, but is rejected for the remaining three data sets
that are not normally distributed.
Questions The Kolmogorov-Smirnov test can be used to answer the
following types of questions:
particularly common in classical statistical tests. Much reliability
modeling is based on the assumption that the data follow a
Weibull distribution.
There are many non-parametric and robust techniques that are not
based on strong distributional assumptions. By non-parametric,
we mean a technique, such as the sign test, that is not based on a
specific distributional assumption. By robust, we mean a
statistical technique that performs well under a wide range of
distributional assumptions. However, techniques based on specific
distributional assumptions are in general more powerful than
these non-parametric and robust techniques. By power, we mean
the ability to detect a difference when that difference actually
exists. Therefore, if the distributional assumptions can be
confirmed, the parametric techniques are generally preferred.
If you are using a technique that makes a normality (or some
other type of distributional) assumption, it is important to confirm
that this assumption is in fact justified. If it is, the more powerful
parametric techniques can be used. If the distributional
assumption is not justified, using a non-parametric or robust
technique may be required.
Related
Techniques
Anderson-Darling goodness-of-fit Test
Chi-Square goodness-of-fit Test
Probability Plots
Software Some general purpose statistical software programs support the
Kolmogorov-Smirnov goodness-of-fit test, at least for the more
common distributions. Both Dataplot code and R code can be
used to generate the analyses in this section.
1.3.5.17. Detection of Outliers

1.3. EDA Techniques
Introduction An outlier is an observation that appears to deviate
markedly from other observations in the sample.
Identification of potential outliers is important for the
following reasons.
1. An outlier may indicate bad data. For example, the
data may have been coded incorrectly or an
experiment may not have been run correctly. If it
can be determined that an outlying point is in fact
erroneous, then the outlying value should be deleted
from the analysis (or corrected if possible).
2. In some cases, it may not be possible to determine if
an outlying point is bad data. Outliers may be due to
random variation or may indicate something
scientifically interesting. In any event, we typically
do not want to simply delete the outlying
observation. However, if the data contains
significant outliers, we may need to consider the use
of robust statistical techniques.
Labeling,
Accomodation,
Identification
Iglewicz and Hoaglin distinguish the three following
issues with regards to outliers.
1. outlier labeling - flag potential outliers for further
investigation (i.e., are the potential outliers
erroneous data, indicative of an inappropriate
distributional model, and so on).
2. outlier accomodation - use robust statistical
techniques that will not be unduly affected by
outliers. That is, if we cannot determine that
potential outliers are erroneous observations, do we
need modify our statistical analysis to more
appropriately account for these observations?
3. outlier identification - formally test whether
observations are outliers.
This section focuses on the labeling and identification
issues.
Normality
Assumption
Identifying an observation as an outlier depends on the
underlying distribution of the data. In this section, we limit
the discussion to univariate data sets that are assumed to
follow an approximately normal distribution. If the
normality assumption for the data being tested is not valid,
then a determination that there is an outlier may in fact be
due to the non-normality of the data rather than the
prescence of an outlier.
For this reason, it is recommended that you generate a
normal probability plot of the data before applying an
outlier test. Although you can also perform formal tests for
normality, the prescence of one or more outliers may
cause the tests to reject normality when it is in fact a
reasonable assumption for applying the outlier test.
In addition to checking the normality assumption, the
lower and upper tails of the normal probability plot can be
a useful graphical technique for identifying potential
outliers. In particular, the plot can help determine whether
we need to check for a single outlier or whether we need
to check for multiple outliers.
The box plot and the histogram can also be useful
graphical tools in checking the normality assumption and
in identifying potential outliers.
Single Versus
Multiple
Outliers
Some outlier tests are designed to detect the prescence of a
single outlier while other tests are designed to detect the
prescence of multiple outliers. It is not appropriate to
apply a test for a single outlier sequentially in order to
detect multiple outliers.
In addition, some tests that detect multiple outliers may
require that you specify the number of suspected outliers
exactly.
Masking and
Swamping
Masking can occur when we specify too few outliers in the
test. For example, if we are testing for a single outlier
when there are in fact two (or more) outliers, these
additional outliers may influence the value of the test
statistic enough so that no points are declared as outliers.
On the other hand, swamping can occur when we specify
too many outliers in the test. For example, if we are testing
for two or more outliers when there is in fact only a single
outlier, both points may be declared outliers (many tests
will declare either all or none of the tested points as
outliers).
Due to the possibility of masking and swamping, it is
useful to complement formal outlier tests with graphical
methods. Graphics can often help identify cases where
masking or swamping may be an issue. Swamping and
masking are also the reason that many tests require that the
exact number of outliers being tested must be specified.
Also, masking is one reason that trying to apply a single
outlier test sequentially can fail. For example, if there are
multiple outliers, masking may cause the outlier test for
the first outlier to return a conclusion of no outliers (and
so the testing for any additional outliers is not performed).
Z-Scores and
Modified Z-
Scores
The Z-score of an observation is defined as
with and s denoting the sample mean and sample
standard deviation, respectively. In other words, data is
given in units of how many standard deviations it is from
the mean.
Although it is common practice to use Z-scores to identify
possible outliers, this can be misleading (partiucarly for
small sample sizes) due to the fact that the maximum Z-
score is at most .
Iglewicz and Hoaglin recommend using the modified Z-
score
with MAD denoting the median absolute deviation and
denoting the median.
These authors recommend that modified Z-scores with an
absolute value of greater than 3.5 be labeled as potential
outliers.
Formal
Outlier Tests
A number of formal outlier tests have proposed in the
literature. These can be grouped by the following
characteristics:
What is the distributional model for the data? We
restrict our discussion to tests that assume the data
follow an approximately normal distribution.
Is the test designed for a single outlier or is it
designed for multiple outliers?
If the test is designed for multiple outliers, does the
number of outliers need to be specified exactly or
can we specify an upper bound for the number of
outliers?
The following are a few of the more commonly used
outlier tests for normally distributed data. This list is not
exhaustive (a large number of outlier tests have been
proposed in the literature). The tests given here are
essentially based on the criterion of "distance from the
mean". This is not the only criterion that could be used.
For example, the Dixon test, which is not discussed here,
is based a value being too large (or small) compared to its
nearest neighbor.
1. Grubbs' Test - this is the recommended test when
testing for a single outlier.
2. Tietjen-Moore Test - this is a generalization of the
Grubbs' test to the case of more than one outlier. It
has the limitation that the number of outliers must
be specified exactly.
3. Generalized Extreme Studentized Deviate (ESD)
Test - this test requires only an upper bound on the
suspected number of outliers and is the
recommended test when the exact number of outliers
is not known.
Lognormal
Distribution
The tests discussed here are specifically based on the
assumption that the data follow an approximately normal
disribution. If your data follow an approximately
lognormal distribution, you can transform the data to
normality by taking the logarithms of the data and then
applying the outlier tests discussed here.
Further
Information
Iglewicz and Hoaglin provide an extensive discussion of
the outlier tests given above (as well as some not given
above) and also give a good tutorial on the subject of
outliers. Barnett and Lewis provide a book length
treatment of the subject.
In addition to discussing additional tests for data that
follow an approximately normal distribution, these sources
also discuss the case where the data are not normally
distributed.
1.3.5.18. Yates Algorithm

1.3. EDA Techniques
Purpose:
Estimate
Factor Effects
in a 2-Level
Factorial
Design
Full factorial and fractional factorial designs are common
in designed experiments for engineering and scientific
applications.
In these designs, each factor is assigned two levels. These
are typically called the low and high levels. For
computational purposes, the factors are scaled so that the
low level is assigned a value of -1 and the high level is
assigned a value of +1. These are also commonly referred
to as "-" and "+".
A full factorial design contains all possible combinations
of low/high levels for all the factors. A fractional factorial
design contains a carefully chosen subset of these
combinations. The criterion for choosing the subsets is
discussed in detail in the process improvement chapter.
The Yates algorithm exploits the special structure of these
designs to generate least squares estimates for factor
effects for all factors and all relevant interactions.
The mathematical details of the Yates algorithm are given
in chapter 10 of Box, Hunter, and Hunter (1978). Natrella
(1963) also provides a procedure for testing the
significance of effect estimates.
The effect estimates are typically complemented by a
number of graphical techniques such as the DOE mean
plot and the DOE contour plot ("DOE" represents "design
of experiments"). These are demonstrated in the eddy
current case study.
Yates Order Before performing the Yates algorithm, the data should be
arranged in "Yates order". That is, given k factors, the kth
column consists of 2
k-1
minus signs (i.e., the low level of
the factor) followed by 2
k-1
plus signs (i.e., the high level
of the factor). For example, for a full factorial design with
three factors, the design matrix is
- - -
+ - -
- + -
+ + -
- - +
+ - +
- + +
+ + +

Determining the Yates order for fractional factorial
designs requires knowledge of the confounding structure
of the fractional factorial design.
Yates
Algorithm
The Yates algorithm is demonstrated for the eddy current
data set. The data set contains eight measurements from a
two-level, full factorial design with three factors. The
purpose of the experiment is to identify factors that have
the most effect on eddy current measurements.
In the "Effect" column, we list the main effects and
interactions from our factorial experiment in standard
order. In the "Response" column, we list the measurement
results from our experiment in Yates order.
Effect Response Col 1 Col 2 Col 3
Estimate
------ -------- ----- ----- ----- --
------
Mean 1.70 6.27 10.21 21.27
2.65875
X1 4.57 3.94 11.06 12.41
1.55125
X2 0.55 6.10 5.71 -3.47 -
0.43375
X1*X2 3.39 4.96 6.70 0.51
0.06375
X3 1.51 2.87 -2.33 0.85
0.10625
X1*X3 4.59 2.84 -1.14 0.99
0.12375
X2*X3 0.67 3.08 -0.03 1.19
0.14875
X1*X2*X3 4.29 3.62 0.54 0.57
0.07125
Sum of responses: 21.27
Sum-of-squared responses: 77.7707
Sum-of-squared Col 3: 622.1656
The first four values in Col 1 are obtained by adding
adjacent pairs of responses, for example 4.57 + 1.70 =
6.27, and 3.39 + 0.55 = 3.94. The second four values in
Col 1 are obtained by subtracting the same adjacent pairs
of responses, for example, 4.57 - 1.70 = 2.87, and 3.39 -
0.55 = 2.84. The values in Col 2 are calculated in the same
way, except that we are adding and subtracting adjacent
values from Col 1. Col 3 is computed using adjacent
values from Col 2. Finally, we obtain the "Estimate"
column by dividing the values in Col 3 by the total number
of responses, 8.
We can check our calculations by making sure that the
first value in Col 3 (21.27) is the sum of all the responses.
In addition, the sum-of-squared responses (77.7707)
should equal the sum-of-squared Col 3 values divided by 8
(622.1656/8 = 77.7707).
Practical
Considerations
The Yates algorithm provides a convenient method for
computing effect estimates; however, the same
information is easily obtained from statistical software
using either an analysis of variance or regression
procedure. The methods for analyzing data from a
designed experiment are discussed more fully in the
chapter on Process Improvement.
Graphical
Presentation
The following plots may be useful to complement the
quantitative information from the Yates algorithm.
1. Ordered data plot
2. Ordered absolute effects plot
3. Cumulative residual standard deviation plot
Questions The Yates algorithm can be used to answer the following
question.
1. What is the estimated effect of a factor on the
response?
Related
Techniques
Multi-factor analysis of variance
DOE mean plot
Block plot
DOE contour plot
Case Study The analysis of a full factorial design is demonstrated in
the eddy current case study.
Software All statistical software packages are capable of estimating
effects using an analysis of variance or least squares
regression procedure.
1.3.5.18.1. Defining Models and Prediction Equations
http://www.itl.nist.gov/div898/handbook/eda/section3/eda35i1.htm[6/27/2012 2:02:10 PM]

1.3. EDA Techniques
1.3.5.18.1. Defining Models and Prediction
Equations
For
Orthogonal
Designs,
Parameter
Estimates
Don't
Change as
Additional
Terms Are
Added
In most cases of least-squares fitting, the model coefficients
for previously added terms change depending on what was
successively added. For example, the X1 coefficient might
change depending on whether or not an X2 term was included
in the model. This is not the case when the design is
orthogonal, as is a 2
3
full factorial design. For orthogonal
designs, the estimates for the previously included terms do not
change as additional terms are added. This means the ranked
list of parameter estimates are the least-squares coefficient
estimates for progressively more complicated models.
Example
Prediction
Equation
We use the parameter estimates derived from a least-squares
analysis for the eddy current data set to create an example
prediction equation.
Parameter Estimate
--------- --------
Mean 2.65875
X1 1.55125
X2 -0.43375
X1*X2 0.06375
X3 0.10625
X1*X3 0.12375
X2*X3 0.14875
X1*X2*X3 0.07125
A prediction equation predicts a value of the reponse variable
for given values of the factors. The equation we select can
include all the factors shown above, or it can include a subset
of the factors. For example, one possible prediction equation
using only two factors, X1 and X2, is:
The least-squares parameter estimates in the prediction
equation reflect the change in response for a one-unit change
in the factor value. To obtain "full" effect estimates (as
computed using the Yates algorithm) for the change in factor
levels from -1 to +1, the effect estimates (except for the
intercept) would be multiplied by two.
Remember that the Yates algorithm is just a convenient
1.3.5.18.1. Defining Models and Prediction Equations
method for computing effects, any statistical software package
with least-squares regression capabilities will produce the
same effects as well as many other useful analyses.
Model
Selection
We want to select the most appropriate model for our data
while balancing the following two goals.
1. We want the model to include all important factors.
2. We want the model to be parsimonious. That is, the
model should be as simple as possible.
Note that the residual standard deviation alone is insufficient
for determining the most appropriate model as it will always
be decreased by adding additional factors. The next section
describes a number of approaches for determining which
factors (and interactions) to include in the model.
1.3.5.18.2. Important Factors

1.3. EDA Techniques
Identify
Important
Factors
We want to select the most appropriate model to represent our data. This requires balancing
the following two goals.
1. We want the model to include all important factors.
2. We want the model to be parsimonious. That is, the model should be as simple as
possible.
In short, we want our model to include all the important factors and interactions and to omit
the unimportant factors and interactions.
Seven criteria are utilized to define important factors. These seven criteria are not all equally
important, nor will they yield identical subsets, in which case a consensus subset or a
weighted consensus subset must be extracted. In practice, some of these criteria may not apply
in all situations.
These criteria will be examined in the context of the eddy current data set. The parameter
estimates computed using least-squares analysis are shown below.
Parameter Estimate
--------- --------
Mean 2.65875
X1 1.55125
X2 -0.43375
X1*X2 0.06375
X3 0.10625
X1*X3 0.12375
X2*X3 0.14875
X1*X2*X3 0.07125
In practice, not all of these criteria will be used with every analysis (and some analysts may
have additional criteria). These critierion are given as useful guidelines. Most analysts will
focus on those criteria that they find most useful.
Criteria for
Including
Terms in
the Model
The seven criteria that we can use in determining whether to keep a factor in the model can be
summarized as follows.
1. Parameters: Engineering Significance
2. Parameters: Order of Magnitude
3. Parameters: Statistical Significance
4. Parameters: Probability Plots
5. Effects: Youden Plot
6. Residual Standard Deviation: Engineering Significance
7. Residual Standard Deviation: Statistical Significance
The first four criteria focus on parameter estimates with three numeric criteria and one
graphical criteria. The fifth criteria focuses on effects, which are twice the parameter
estimates. The last two criteria focus on the residual standard deviation of the model. We
discuss each of these seven criteria in detail in the sections that following.
Parameters:
Engineering
Significance
The minimum engineering significant difference is defined as
where is the absolute value of the parameter estimate and is the minimum engineering
significant difference.
That is, declare a factor as "important" if the parameter estimate is greater than some a priori
declared engineering difference. This implies that the engineering staff have in fact stated
what a minimum difference will be. Oftentimes this is not the case. In the absence of an a
priori difference, a good rough rule for the minimum engineering significant is to keep only
those factors whose parameter estimate is greater than, say, 10% of the current production
average. In this case, let's say that the average detector has a sensitivity of 2.5 ohms. This
would suggest that we would declare all factors whose parameter is greater than 10 % of 2.5
ohms = 0.25 ohm to be significant (from an engineering point of view).
Based on this minimum engineering significant difference criterion, we conclude that we
should keep two terms: X1 and X2.
Parameters:
Order of
Magnitude
The order of magnitude criterion is defined as
That is, exclude any factor that is less than 10 % of the maximum parameter size. We may or
may not keep the other factors. This criterion is neither engineering nor statistical, but it does
offer some additional numerical insight. For the current example, the largest parameter is from
X1 (1.55125 ohms), and so 10 % of that is 0.155 ohms, which suggests keeping all factors
whose parameters exceed 0.155 ohms.
Based on the order-of-magnitude criterion, we thus conclude that we should keep two terms:
X1 and X2. A third term, X2*X3 (0.14875), is just slightly under the cutoff level, so we may
consider keeping it based on the other criterion.
Parameters:
Statistical
Significance
Statistical significance is defined as
That is, declare a factor as important if its parameter is more than 2 standard deviations away
from 0 (0, by definition, meaning "no effect").
The "2" comes from normal theory (more specifically, a value of 1.96 yields a 95 %
confidence interval). More precise values would come from t-distribution theory.
The difficulty with this is that in order to invoke this criterion we need the standard deviation,
, of an observation. This is problematic because
1. the engineer may not know ;
2. the experiment might not have replication, and so a model-free estimate of is not
obtainable;
3. obtaining an estimate of by assuming the sometimes- employed assumption of
ignoring 3-term interactions and higher may be incorrect from an engineering point of
view.
For the eddy current example:
1. the engineer did not know ;
2. the design (a 2
3
full factorial) did not have replication;
3. ignoring 3-term interactions and higher interactions leads to an estimate of based on
omitting only a single term: the X1*X2*X3 interaction.
For the eddy current example, if one assumes that the 3-term interaction is nil and hence
represents a single drawing from a population centered at zero, then an estimate of the
standard deviation of a parameter is simply the estimate of the 3-factor interaction (0.07125).
Two standard deviations is thus 0.1425. For this example, the rule is thus to keep all >
0.1425.
This results in keeping three terms: X1 (1.55125), X2 (-0.43375), and X1*X2 (0.14875).
Parameters:
Probability
Plots
Probability plots can be used in the following manner.
1. Normal Probability Plot: Keep a factor as "important" if it is well off the line through
zero on a normal probability plot of the parameter estimates.
2. Half-Normal Probability Plot: Keep a factor as "important" if it is well off the line near
zero on a half-normal probability plot of the absolute value of parameter estimates.
Both of these methods are based on the fact that the least-squares estimates of parameters for
these two-level orthogonal designs are simply half the difference of averages and so the
central limit theorem, loosely applied, suggests that (if no factor were important) the
parameter estimates should have approximately a normal distribution with mean zero and the
absolute value of the estimates should have a half-normal distribution.
Since the half-normal probability plot is only concerned with parmeter magnitudes as opposed
to signed parameters (which are subject to the vagaries of how the initial factor codings +1
and -1 were assigned), the half-normal probability plot is preferred by some over the normal
probability plot.
Normal
Probablity
Plot of
Parameters
The following normal probability plot shows the parameter estimates for the eddy current
data.
For the example at hand, the probability plot clearly shows two factors (X1 and X2) displaced
off the line. All of the remaining five parameters are behaving like random drawings from a
normal distribution centered at zero, and so are deemed to be statistically non-significant. In
conclusion, this rule keeps two factors: X1 (1.55125) and X2 (-0.43375).
Averages:
Youden Plot
A Youden plot can be used in the following way. Keep a factor as "important" if it is
displaced away from the central-tendancy "bunch" in a Youden plot of high and low averages.
By definition, a factor is important when its average response for the low (-1) setting is
significantly different from its average response for the high (+1) setting. (Note that effects are
twice the parameter estimates.) Conversely, if the low and high averages are about the same,
then what difference does it make which setting to use and so why would such a factor be
considered important? This fact in combination with the intrinsic benefits of the Youden plot
for comparing pairs of items leads to the technique of generating a Youden plot of the low
and high averages.
Youden Plot
of Effect
Estimates
The following is the Youden plot of the effect estimatess for the eddy current data.
For the example at hand, the Youden plot clearly shows a cluster of points near the grand
average (2.65875) with two displaced points above (factor 1) and below (factor 2). Based on
the Youden plot, we conclude to keep two factors: X1 (1.55125) and X2 (-0.43375).
Residual
Standard
Deviation:
Engineering
Significance
This criterion is defined as
Residual Standard Deviation > Cutoff
That is, declare a factor as "important" if the cumulative model that includes the factor (and
all larger factors) has a residual standard deviation smaller than an a priori engineering-
specified minimum residual standard deviation.
This criterion is different from the others in that it is model focused. In practice, this criterion
states that starting with the largest parameter, we cumulatively keep adding terms to the model
and monitor how the residual standard deviation for each progressively more complicated
model becomes smaller. At some point, the cumulative model will become complicated
enough and comprehensive enough that the resulting residual standard deviation will drop
below the pre-specified engineering cutoff for the residual standard deviation. At that point,
we stop adding terms and declare all of the model-included terms to be "important" and
everything not in the model to be "unimportant".
This approach implies that the engineer has considered what a minimum residual standard
deviation should be. In effect, this relates to what the engineer can tolerate for the magnitude
of the typical residual (the difference between the raw data and the predicted value from the
model). In other words, how good does the engineer want the prediction equation to be.
Unfortunately, this engineering specification has not always been formulated and so this
criterion can become moot.
In the absence of a prior specified cutoff, a good rough rule for the minimum engineering
residual standard deviation is to keep adding terms until the residual standard deviation just
dips below, say, 5 % of the current production average. For the eddy current data, let's say
that the average detector has a sensitivity of 2.5 ohms. Then this would suggest that we would
keep adding terms to the model until the residual standard deviation falls below 5 % of 2.5
ohms = 0.125 ohms.
Residual
Model Std. Dev.
----------------------------------------------------- ---------
Mean + X1 0.57272
Mean + X1 + X2 0.30429
Mean + X1 + X2 + X2*X3 0.26737
Mean + X1 + X2 + X2*X3 + X1*X3 0.23341
Mean + X1 + X2 + X2*X3 + X1*X3 + X3 0.19121
Mean + X1 + X2 + X2*X3 + X1*X3 + X3 + X1*X2*X3 0.18031
Mean + X1 + X2 + X2*X3 + X1*X3 + X3 + X1*X2*X3 + X1*X2 NA
Based on the minimum residual standard deviation criteria, and we would include all terms in
order to drive the residual standard deviation below 0.125. Again, the 5 % rule is a rough-
and-ready rule that has no basis in engineering or statistics, but is simply a "numerics".
Ideally, the engineer has a better cutoff for the residual standard deviation that is based on
how well he/she wants the equation to peform in practice. If such a number were available,
then for this criterion and data set we would select something less than the entire collection of
terms.
Residual
Standard
Deviation:
Statistical
Significance
This criterion is defined as
Residual Standard Deviation >
where is the standard deviation of an observation under replicated conditions.
That is, declare a term as "important" until the cumulative model that includes the term has a
residual standard deviation smaller than . In essence, we are allowing that we cannot demand
a model fit any better than what we would obtain if we had replicated data; that is, we cannot
demand that the residual standard deviation from any fitted model be any smaller than the
(theoretical or actual) replication standard deviation. We can drive the fitted standard
deviation down (by adding terms) until it achieves a value close to , but to attempt to drive it
down further means that we are, in effect, trying to fit noise.
In practice, this criterion may be difficult to apply because
1. the engineer may not know ;
2. the experiment might not have replication, and so a model-free estimate of is not
obtainable.
For the current case study:
1. the engineer did not know ;
2. the design (a 2
3
full factorial) did not have replication. The most common way of
having replication in such designs is to have replicated center points at the center of the
cube ((X1,X2,X3) = (0,0,0)).
Thus for this current case, this criteria could not be used to yield a subset of "important"
factors.
Conclusions In summary, the seven criteria for specifying "important" factors yielded the following for the
eddy current data:
1. Parameters, Engineering Significance: X1, X2
2. Parameters, Numerically Significant: X1, X2
3. Parameters, Statistically Significant: X1, X2, X2*X3
4. Parameters, Probability Plots: X1, X2
5. Effects, Youden Plot: X1, X2
6. Residual SD, Engineering Significance: all 7 terms
7. Residual SD, Statistical Significance: not applicable
Such conflicting results are common. Arguably, the three most important criteria (listed in
order of most important) are:
4. Parameters, Probability Plots: X1, X2
1. Parameters, Engineering Significance: X1, X2
3. Residual SD, Engineering Significance: all 7 terms
Scanning all of the above, we thus declare the following consensus for the eddy current data:
1. Important Factors: X1 and X2
2. Parsimonious Prediction Equation:
(with a residual standard deviation of 0.30429 ohms)
Note that this is the initial model selection. We still need to perform model validation with a
residual analysis.
1.3.6. Probability Distributions

1.3. EDA Techniques
Probability
Distributions
Probability distributions are a fundamental concept in
statistics. They are used both on a theoretical level and a
practical level.
Some practical uses of probability distributions are:
To calculate confidence intervals for parameters and
to calculate critical regions for hypothesis tests.
For univariate data, it is often useful to determine a
reasonable distributional model for the data.
Statistical intervals and hypothesis tests are often
based on specific distributional assumptions. Before
computing an interval or test based on a distributional
assumption, we need to verify that the assumption is
justified for the given data set. In this case, the
distribution does not need to be the best-fitting
distribution for the data, but an adequate enough
model so that the statistical technique yields valid
conclusions.
Simulation studies with random numbers generated
from using a specific probability distribution are often
needed.
Table of
Contents
1. What is a probability distribution?
2. Related probability functions
3. Families of distributions
4. Location and scale parameters
5. Estimating the parameters of a distribution
6. A gallery of common distributions
7. Tables for probability distributions
1.3.6.1. What is a Probability Distribution

1.3. EDA Techniques
Discrete
Distributions
The mathematical definition of a discrete probability
function, p(x), is a function that satisfies the following
properties.
1. The probability that x can take a specific value is p(x).
That is
2. p(x) is non-negative for all real x.
3. The sum of p(x) over all possible values of x is 1, that
is
where j represents all possible values that x can have
and p
j
is the probability at x
j
.
One consequence of properties 2 and 3 is that 0 <=
p(x) <= 1.
What does this actually mean? A discrete probability
function is a function that can take a discrete number of
values (not necessarily finite). This is most often the non-
negative integers or some subset of the non-negative
integers. There is no mathematical restriction that discrete
probability functions only be defined at integers, but in
practice this is usually what makes sense. For example, if
you toss a coin 6 times, you can get 2 heads or 3 heads but
not 2 1/2 heads. Each of the discrete values has a certain
probability of occurrence that is between zero and one. That
is, a discrete function that allows negative values or values
greater than one is not a probability function. The condition
that the probabilities sum to one means that at least one of
the values has to occur.
Continuous
Distributions
The mathematical definition of a continuous probability
function, f(x), is a function that satisfies the following
properties.
1. The probability that x is between two points a and b is
2. It is non-negative for all real x.
3. The integral of the probability function is one, that is
What does this actually mean? Since continuous probability
functions are defined for an infinite number of points over a
continuous interval, the probability at a single point is
always zero. Probabilities are measured over intervals, not
single points. That is, the area under the curve between two
distinct points defines the probability for that interval. This
means that the height of the probability function can in fact
be greater than one. The property that the integral must
equal one is equivalent to the property for discrete
distributions that the sum of all the probabilities must equal
one.
Probability
Mass
Functions
Versus
Probability
Density
Functions
Discrete probability functions are referred to as probability
mass functions and continuous probability functions are
referred to as probability density functions. The term
probability functions covers both discrete and continuous
distributions. When we are referring to probability functions
in generic terms, we may use the term probability density
functions to mean both discrete and continuous probability
functions.
1.3.6.2. Related Distributions

1.3. EDA Techniques
Probability distributions are typically defined in terms of the
probability density function. However, there are a number of
probability functions used in applications.
Probability
Density
Function
For a continuous function, the probability density function
(pdf) is the probability that the variate has the value x. Since
for continuous distributions the probability at a single point is
zero, this is often expressed in terms of an integral between
two points.
For a discrete distribution, the pdf is the probability that the
variate takes the value x.
The following is the plot of the normal probability density
function.
Cumulative
Distribution
Function
The cumulative distribution function (cdf) is the probability
that the variable takes a value less than or equal to x. That is
For a continuous distribution, this can be expressed
mathematically as
For a discrete distribution, the cdf can be expressed as
The following is the plot of the normal cumulative
The horizontal axis is the allowable domain for the given
probability function. Since the vertical axis is a probability, it
must fall between zero and one. It increases from zero to one
as we go from left to right on the horizontal axis.
Percent
Point
Function
The percent point function (ppf) is the inverse of the
cumulative distribution function. For this reason, the percent
point function is also commonly referred to as the inverse
distribution function. That is, for a distribution function we
calculate the probability that the variable is less than or equal
to x for a given x. For the percent point function, we start
with the probability and compute the corresponding x for the
cumulative distribution. Mathematically, this can be
expressed as
or alternatively
The following is the plot of the normal percent point
function.
Since the horizontal axis is a probability, it goes from zero to
one. The vertical axis goes from the smallest to the largest
value of the cumulative distribution function.
Hazard
Function
The hazard function is the ratio of the probability density
function to the survival function, S(x).
The following is the plot of the normal distribution hazard
function.
Hazard plots are most commonly used in reliability
applications. Note that Johnson, Kotz, and Balakrishnan refer
to this as the conditional failure density function rather than
the hazard function.
Cumulative
Hazard
The cumulative hazard function is the integral of the hazard
function.
Function
This can alternatively be expressed as
The following is the plot of the normal cumulative hazard
function.
Cumulative hazard plots are most commonly used in
reliability applications. Note that Johnson, Kotz, and
Balakrishnan refer to this as the hazard function rather than
the cumulative hazard function.
Survival
Function
Survival functions are most often used in reliability and
related fields. The survival function is the probability that the
variate takes a value greater than x.
The following is the plot of the normal distribution survival
function.
For a survival function, the y value on the graph starts at 1
and monotonically decreases to zero. The survival function
should be compared to the cumulative distribution function.
Inverse
Survival
Function
Just as the percent point function is the inverse of the
cumulative distribution function, the survival function also
has an inverse function. The inverse survival function can be
defined in terms of the percent point function.
The following is the plot of the normal distribution inverse
survival function.
As with the percent point function, the horizontal axis is a
probability. Therefore the horizontal axis goes from 0 to 1
regardless of the particular distribution. The appearance is
similar to the percent point function. However, instead of
going from the smallest to the largest value on the vertical
axis, it goes from the largest to the smallest value.
1.3.6.3. Families of Distributions

1.3. EDA Techniques
Shape
Parameters
Many probability distributions are not a single distribution,
but are in fact a family of distributions. This is due to the
distribution having one or more shape parameters.
Shape parameters allow a distribution to take on a variety of
shapes, depending on the value of the shape parameter. These
distributions are particularly useful in modeling applications
since they are flexible enough to model a variety of data sets.
Example:
Weibull
Distribution
The Weibull distribution is an example of a distribution that
has a shape parameter. The following graph plots the Weibull
pdf with the following values for the shape parameter: 0.5,
1.0, 2.0, and 5.0.
The shapes above include an exponential distribution, a right-
skewed distribution, and a relatively symmetric distribution.
The Weibull distribution has a relatively simple distributional
form. However, the shape parameter allows the Weibull to
assume a wide variety of shapes. This combination of
simplicity and flexibility in the shape of the Weibull
distribution has made it an effective distributional model in
reliability applications. This ability to model a wide variety
of distributional shapes using a relatively simple
distributional form is possible with many other distributional
families as well.
PPCC Plots The PPCC plot is an effective graphical tool for selecting the
member of a distributional family with a single shape
parameter that best fits a given set of data.
1.3.6.4. Location and Scale Parameters

1.3. EDA Techniques
Normal
PDF
A probability distribution is characterized by location and
scale parameters. Location and scale parameters are typically
used in modeling applications.
For example, the following graph is the probability density
function for the standard normal distribution, which has the
location parameter equal to zero and scale parameter equal to
one.
Location
Parameter
The next plot shows the probability density function for a
normal distribution with a location parameter of 10 and a
scale parameter of 1.
The effect of the location parameter is to translate the graph,
relative to the standard normal distribution, 10 units to the
right on the horizontal axis. A location parameter of -10
would have shifted the graph 10 units to the left on the
horizontal axis.
That is, a location parameter simply shifts the graph left or
right on the horizontal axis.
Scale
Parameter
The next plot has a scale parameter of 3 (and a location
parameter of zero). The effect of the scale parameter is to
stretch out the graph. The maximum y value is approximately
0.13 as opposed 0.4 in the previous graphs. The y value, i.e.,
the vertical axis value, approaches zero at about (+/-) 9 as
opposed to (+/-) 3 with the first graph.
In contrast, the next graph has a scale parameter of 1/3
(=0.333). The effect of this scale parameter is to squeeze the
pdf. That is, the maximum y value is approximately 1.2 as
opposed to 0.4 and the y value is near zero at (+/-) 1 as
opposed to (+/-) 3.
The effect of a scale parameter greater than one is to stretch
the pdf. The greater the magnitude, the greater the stretching.
The effect of a scale parameter less than one is to compress
the pdf. The compressing approaches a spike as the scale
parameter goes to zero. A scale parameter of 1 leaves the pdf
unchanged (if the scale parameter is 1 to begin with) and
non-positive scale parameters are not allowed.
Location
and Scale
Together
The following graph shows the effect of both a location and
a scale parameter. The plot has been shifted right 10 units
and stretched by a factor of 3.
Standard
Form
The standard form of any distribution is the form that has
location parameter zero and scale parameter one.
It is common in statistical software packages to only
compute the standard form of the distribution. There are
formulas for converting from the standard form to the form
with other location and scale parameters. These formulas are
independent of the particular probability distribution.
Formulas
for Location
and Scale
Based on
the
Standard
Form
The following are the formulas for computing various
probability functions based on the standard form of the
distribution. The parameter a refers to the location parameter
and the parameter b refers to the scale parameter. Shape
parameters are not included.
Cumulative Distribution
Function
F(x;a,b) = F((x-a)/b;0,1)
Probability Density Function f(x;a,b) = (1/b)f((x-a)/b;0,1)
Percent Point Function G( ;a,b) = a + bG( ;0,1)
Hazard Function h(x;a,b) = (1/b)h((x-a)/b;0,1)
Cumulative Hazard Function H(x;a,b) = H((x-a)/b;0,1)
Survival Function S(x;a,b) = S((x-a)/b;0,1)
Inverse Survival Function Z( ;a,b) = a + bZ( ;0,1)
Random Numbers Y(a,b) = a + bY(0,1)
Relationship
to Mean
and
Standard
Deviation
For the normal distribution, the location and scale parameters
correspond to the mean and standard deviation, respectively.
However, this is not necessarily true for other distributions.
In fact, it is not true for most distributions.
1.3.6.5. Estimating the Parameters of a Distribution

1.3. EDA Techniques
1.3.6.5. Estimating the Parameters of a
Distribution
Model a
univariate
data set
with a
probability
distribution
One common application of probability distributions is
modeling univariate data with a specific probability
distribution. This involves the following two steps:
1. Determination of the "best-fitting" distribution.
2. Estimation of the parameters (shape, location, and scale
parameters) for that distribution.
Various
Methods
There are various methods, both numerical and graphical, for
estimating the parameters of a probability distribution.
1. Method of moments
2. Maximum likelihood
3. Least squares
4. PPCC and probability plots
1.3.6.5.1. Method of Moments

1.3. EDA Techniques
1.3.6.5.1. Method of Moments
Method of
Moments
The method of moments equates sample moments to parameter
estimates. When moment methods are available, they have the
advantage of simplicity. The disadvantage is that they are often
not available and they do not have the desirable optimality
properties of maximum likelihood and least squares estimators.
The primary use of moment estimates is as starting values for
the more precise maximum likelihood and least squares
estimates.
Software Most general purpose statistical software does not include
explicit method of moments parameter estimation commands.
However, when utilized, the method of moment formulas tend
to be straightforward and can be easily implemented in most
statistical software programs.
1.3.6.5.2. Maximum Likelihood

1.3. EDA Techniques
Maximum
Likelihood
Maximum likelihood estimation begins with the
mathematical expression known as a likelihood function of
the sample data. Loosely speaking, the likelihood of a set
of data is the probability of obtaining that particular set of
data given the chosen probability model. This expression
contains the unknown parameters. Those values of the
parameter that maximize the sample likelihood are known
as the maximum likelihood estimates.
The reliability chapter contains some examples of the
likelihood functions for a few of the commonly used
distributions in reliability analysis.
Advantages The advantages of this method are:
Maximum likelihood provides a consistent approach
to parameter estimation problems. This means that
maximum likelihood estimates can be developed for
a large variety of estimation situations. For example,
they can be applied in reliability analysis to
censored data under various censoring models.
Maximum likelihood methods have desirable
mathematical and optimality properties. Specifically,
1. They become minimum variance unbiased
estimators as the sample size increases. By
unbiased, we mean that if we take (a very
large number of) random samples with
replacement from a population, the average
value of the parameter estimates will be
theoretically exactly equal to the population
value. By minimum variance, we mean that
the estimator has the smallest variance, and
thus the narrowest confidence interval, of all
estimators of that type.
2. They have approximate normal distributions
and approximate sample variances that can be
used to generate confidence bounds and
hypothesis tests for the parameters.
Several popular statistical software packages provide
excellent algorithms for maximum likelihood
estimates for many of the commonly used
distributions. This helps mitigate the computational
complexity of maximum likelihood estimation.
Disadvantages The disadvantages of this method are:
The likelihood equations need to be specifically
worked out for a given distribution and estimation
problem. The mathematics is often non-trivial,
particularly if confidence intervals for the
parameters are desired.
The numerical estimation is usually non-trivial.
Except for a few cases where the maximum
likelihood formulas are in fact simple, it is generally
best to rely on high quality statistical software to
obtain maximum likelihood estimates. Fortunately,
high quality maximum likelihood software is
becoming increasingly common.
Maximum likelihood estimates can be heavily biased
for small samples. The optimality properties may not
apply for small samples.
Maximum likelihood can be sensitive to the choice
of starting values.
Software Most general purpose statistical software programs support
maximum likelihood estimation (MLE) in some form.
MLE estimation can be supported in two ways.
1. A software program may provide a generic function
minimization (or equivalently, maximization)
capability. This is also referred to as function
optimization. Maximum likelihood estimation is
essentially a function optimization problem.
This type of capability is particularly common in
mathematical software programs.
2. A software program may provide MLE
computations for a specific problem. For example, it
may generate ML estimates for the parameters of a
Weibull distribution.
Statistical software programs will often provide ML
estimates for many specific problems even when
they do not support general function optimization.
The advantage of function minimization software is that it
can be applied to many different MLE problems. The
drawback is that you have to specify the maximum
likelihood equations to the software. As the functions can
be non-trivial, there is potential for error in entering the
equations.
The advantage of the specific MLE procedures is that
greater efficiency and better numerical stability can often
be obtained by taking advantage of the properties of the
specific estimation problem. The specific methods often
return explicit confidence intervals. In addition, you do not
have to know or specify the likelihood equations to the
software. The disadvantage is that each MLE problem
must be specifically coded.
1.3.6.5.3. Least Squares

1.3. EDA Techniques
1.3.6.5.3. Least Squares
Least Squares Non-linear least squares provides an alternative to
maximum likelihood.
Non-linear least squares software may be available
in many statistical software packages that do not
support maximum likelihood estimates.
It can be applied more generally than maximum
likelihood. That is, if your software provides non-
linear fitting and it has the ability to specify the
probability function you are interested in, then you
can generate least squares estimates for that
distribution. This will allow you to obtain reasonable
estimates for distributions even if the software does
not provide maximum likelihood estimates.
It is not readily applicable to censored data.
It is generally considered to have less desirable
optimality properties than maximum likelihood.
It can be quite sensitive to the choice of starting
values.
Software Non-linear least squares fitting is available in many
general purpose statistical software programs.
1.3.6.5.4. PPCC and Probability Plots

1.3. EDA Techniques
PPCC and
Probability
Plots
The PPCC plot can be used to estimate the shape
parameter of a distribution with a single shape parameter.
After finding the best value of the shape parameter, the
probability plot can be used to estimate the location and
scale parameters of a probability distribution.
It is based on two well-understood concepts.
1. The linearity (i.e., straightness) of the
probability plot is a good measure of the
adequacy of the distributional fit.
2. The correlation coefficient between the points
on the probability plot is a good measure of
the linearity of the probability plot.
It is an easy technique to implement for a wide
variety of distributions with a single shape
parameter. The basic requirement is to be able to
compute the percent point function, which is needed
in the computation of both the probability plot and
the PPCC plot.
The PPCC plot provides insight into the sensitivity
of the shape parameter. That is, if the PPCC plot is
relatively flat in the neighborhood of the optimal
value of the shape parameter, this is a strong
indication that the fitted model will not be sensitive
to small deviations, or even large deviations in some
cases, in the value of the shape parameter.
The maximum correlation value provides a method
for comparing across distributions as well as
identifying the best value of the shape parameter for
a given distribution. For example, we could use the
PPCC and probability fits for the Weibull,
lognormal, and possibly several other distributions.
Comparing the maximum correlation coefficient
achieved for each distribution can help in selecting
which is the best distribution to use.
It is limited to distributions with a single shape
parameter.
PPCC plots are not widely available in statistical
software packages other than Dataplot (Dataplot
provides PPCC plots for 40+ distributions).
Probability plots are generally available. However,
many statistical software packages only provide
them for a limited number of distributions.
Significance levels for the correlation coefficient
(i.e., if the maximum correlation value is above a
given value, then the distribution provides an
adequate fit for the data with a given confidence
level) have only been worked out for a limited
number of distributions.
Other
Graphical
Methods
For reliability applications, the hazard plot and the Weibull
plot are alternative graphical methods that are commonly
used to estimate parameters.
1.3.6.6. Gallery of Distributions

1.3. EDA Techniques
Gallery of
Common
Distributions
Detailed information on a few of the most common
distributions is available below. There are a large number of
distributions used in statistical applications. It is beyond the
scope of this Handbook to discuss more than a few of these.
Two excellent sources for additional detailed information on
a large array of distributions are Johnson, Kotz, and
Balakrishnan and Evans, Hastings, and Peacock. Equations
for the probability functions are given for the standard form
of the distribution. Formulas exist for defining the functions
with location and scale parameters in terms of the standard
form of the distribution.
The sections on parameter estimation are restricted to the
method of moments and maximum likelihood. This is
because the least squares and PPCC and probability plot
estimation procedures are generic. The maximum likelihood
equations are not listed if they involve solving simultaneous
equations. This is because these methods require
sophisticated computer software to solve. Except where the
maximum likelihood estimates are trivial, you should depend
on a statistical software program to compute them.
References are given for those who are interested.
Be aware that different sources may give formulas that are
different from those shown here. In some cases, these are
simply mathematically equivalent formulations. In other
cases, a different parameterization may be used.
Continuous
Distributions
Normal
Distribution
Uniform
Distribution
Cauchy
Distribution
t Distribution F Distribution Chi-Square
Distribution
Exponential
Distribution
Weibull
Distribution
Lognormal
Distribution
Birnbaum-
Saunders
(Fatigue Life)
Distribution
Gamma
Distribution
Double
Exponential
Distribution
Power Normal
Distribution
Power
Lognormal
Distribution
Tukey-Lambda
Distribution
Extreme Value
Type I
Distribution
Beta Distribution
Discrete
Distributions
Binomial
Distribution
Poisson
Distribution
1.3.6.6.1. Normal Distribution

1.3. EDA Techniques
Probability
Density
Function
The general formula for the probability density function of
the normal distribution is
where is the location parameter and is the scale
parameter. The case where = 0 and = 1 is called the
standard normal distribution. The equation for the standard
normal distribution is
Since the general form of probability functions can be
expressed in terms of the standard distribution, all subsequent
formulas in this section are given for the standard form of the
function.
The following is the plot of the standard normal probability
density function.
Cumulative The formula for the cumulative distribution function of the
Distribution
Function
normal distribution does not exist in a simple closed formula.
It is computed numerically.
The following is the plot of the normal cumulative
Percent
Point
Function
The formula for the percent point function of the normal
distribution does not exist in a simple closed formula. It is
computed numerically.
The following is the plot of the normal percent point
function.
Hazard
Function
The formula for the hazard function of the normal
distribution is
where is the cumulative distribution function of the
standard normal distribution and is the probability density
function of the standard normal distribution.
The following is the plot of the normal hazard function.
Cumulative
Hazard
Function
The normal cumulative hazard function can be computed
from the normal cumulative distribution function.
The following is the plot of the normal cumulative hazard
function.
Survival
Function
The normal survival function can be computed from the
normal cumulative distribution function.
The following is the plot of the normal survival function.
Inverse
Survival
Function
The normal inverse survival function can be computed from
the normal percent point function.
The following is the plot of the normal inverse survival
function.
Common
Statistics
Mean The location parameter .
Median The location parameter .
Mode The location parameter .
Range Infinity in both directions.
Standard
Deviation
The scale parameter .
Coefficient of
Variation
Skewness 0
Kurtosis 3
Parameter The location and scale parameters of the normal distribution
Estimation can be estimated with the sample mean and sample standard
deviation, respectively.
Comments For both theoretical and practical reasons, the normal
distribution is probably the most important distribution in
statistics. For example,
Many classical statistical tests are based on the
assumption that the data follow a normal distribution.
This assumption should be tested before applying these
tests.
In modeling applications, such as linear and non-linear
regression, the error term is often assumed to follow a
normal distribution with fixed location and scale.
The normal distribution is used to find significance
levels in many hypothesis tests and confidence
intervals.
Theroretical
Justification
- Central
Limit
Theorem
The normal distribution is widely used. Part of the appeal is
that it is well behaved and mathematically tractable.
However, the central limit theorem provides a theoretical
basis for why it has wide applicability.
The central limit theorem basically states that as the sample
size (N) becomes large, the following occur:
1. The sampling distribution of the mean becomes
approximately normal regardless of the distribution of
the original variable.
2. The sampling distribution of the mean is centered at
the population mean, , of the original variable. In
addition, the standard deviation of the sampling
distribution of the mean approaches .
Software Most general purpose statistical software programs support at
least some of the probability functions for the normal
distribution.
1.3.6.6.2. Uniform Distribution

1.3. EDA Techniques
Probability
Density
Function
The general formula for the probability density function of the uniform
distribution is
where A is the location parameter and (B - A) is the scale parameter. The
case where A = 0 and B = 1 is called the standard uniform distribution.
The equation for the standard uniform distribution is
Since the general form of probability functions can be expressed in terms of
the standard distribution, all subsequent formulas in this section are given
for the standard form of the function.
The following is the plot of the uniform probability density function.
Cumulative
Distribution
Function
The formula for the cumulative distribution function of the uniform
distribution is
The following is the plot of the uniform cumulative distribution function.
Percent
Point
Function
The formula for the percent point function of the uniform distribution is
The following is the plot of the uniform percent point function.
Hazard
Function
The formula for the hazard function of the uniform distribution is
The following is the plot of the uniform hazard function.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the uniform distribution
is
The following is the plot of the uniform cumulative hazard function.
Survival
Function
The uniform survival function can be computed from the uniform
cumulative distribution function.
The following is the plot of the uniform survival function.
Inverse
Survival
Function
The uniform inverse survival function can be computed from the uniform
percent point function.
The following is the plot of the uniform inverse survival function.
Common
Statistics
Mean (A + B)/2
Median (A + B)/2
Range B - A
Standard Deviation
Coefficient of
Variation
Skewness 0
Kurtosis 9/5
Parameter
Estimation
The method of moments estimators for A and B are

The maximum likelihood estimators are usually given in terms of the
parameters a and h where
A = a - h
B = a + h
The maximum likelihood estimators for a and h are

This gives the following maximum likelihood estimators for A and B
Comments The uniform distribution defines equal probability over a given range for a
continuous distribution. For this reason, it is important as a reference
distribution.
One of the most important applications of the uniform distribution is in the
generation of random numbers. That is, almost all random number
generators generate random numbers on the (0,1) interval. For other
distributions, some transformation is applied to the uniform random
numbers.
Software Most general purpose statistical software programs support at least some of
the probability functions for the uniform distribution.
1.3.6.6.3. Cauchy Distribution

1.3. EDA Techniques
Probability
Density
Function
the Cauchy distribution is
where t is the location parameter and s is the scale parameter.
The case where t = 0 and s = 1 is called the standard
Cauchy distribution. The equation for the standard Cauchy
distribution reduces to
function.
The following is the plot of the standard Cauchy probability
density function.
Cumulative
Distribution
The formula for the cumulative distribution function for the
Cauchy distribution is
Function
The following is the plot of the Cauchy cumulative
Percent
Point
Function
The formula for the percent point function of the Cauchy
distribution is
The following is the plot of the Cauchy percent point
function.
Hazard
Function
The Cauchy hazard function can be computed from the
Cauchy probability density and cumulative distribution
functions.
The following is the plot of the Cauchy hazard function.
Cumulative
Hazard
Function
The Cauchy cumulative hazard function can be computed
from the Cauchy cumulative distribution function.
The following is the plot of the Cauchy cumulative hazard
function.
Survival
Function
The Cauchy survival function can be computed from the
Cauchy cumulative distribution function.
The following is the plot of the Cauchy survival function.
Inverse
Survival
Function
The Cauchy inverse survival function can be computed from
the Cauchy percent point function.
The following is the plot of the Cauchy inverse survival
function.
Common
Statistics
Mean The mean is undefined.
Median The location parameter t.
Mode The location parameter t.
Standard
Deviation
The standard deviation is undefined.
Coefficient of
Variation
The coefficient of variation is undefined.
Skewness The skewness is undefined.
Kurtosis The kurtosis is undefined.
Parameter The likelihood functions for the Cauchy maximum likelihood
Estimation estimates are given in chapter 16 of Johnson, Kotz, and
Balakrishnan. These equations typically must be solved
numerically on a computer.
Comments The Cauchy distribution is important as an example of a
pathological case. Cauchy distributions look similar to a
normal distribution. However, they have much heavier tails.
When studying hypothesis tests that assume normality, seeing
how the tests perform on data from a Cauchy distribution is a
good indicator of how sensitive the tests are to heavy-tail
departures from normality. Likewise, it is a good check for
robust techniques that are designed to work well under a wide
variety of distributional assumptions.
The mean and standard deviation of the Cauchy distribution
are undefined. The practical meaning of this is that collecting
1,000 data points gives no more accurate an estimate of the
mean and standard deviation than does a single point.
Software Many general purpose statistical software programs support
at least some of the probability functions for the Cauchy
distribution.
1.3.6.6.4. t Distribution

1.3. EDA Techniques
Probability
Density
Function
The formula for the probability density function of the t
distribution is
where is the beta function and is a positive integer shape
parameter. The formula for the beta function is
In a testing context, the t distribution is treated as a
"standardized distribution" (i.e., no location or scale
parameters). However, in a distributional modeling context
(as with other probability distributions), the t distribution
itself can be transformed with a location parameter, , and a
scale parameter, .
The following is the plot of the t probability density function
for 4 different values of the shape parameter.
These plots all have a similar shape. The difference is in the
heaviness of the tails. In fact, the t distribution with equal
to 1 is a Cauchy distribution. The t distribution approaches a
normal distribution as becomes large. The approximation is
quite good for values of > 30.
Cumulative
Distribution
Function
The formula for the cumulative distribution function of the t
distribution is complicated and is not included here. It is
given in the Evans, Hastings, and Peacock book.
The following are the plots of the t cumulative distribution
function with the same values of as the pdf plots above.
Percent
Point
Function
The formula for the percent point function of the t
distribution does not exist in a simple closed form. It is
The following are the plots of the t percent point function
with the same values of as the pdf plots above.
Other
Probability
Functions
Since the t distribution is typically used to develop hypothesis
tests and confidence intervals and rarely for modeling
applications, we omit the formulas and plots for the hazard,
cumulative hazard, survival, and inverse survival probability
functions.
Common
Statistics
Mean 0 (It is undefined for equal to 1.)
Median 0
Mode 0
Standard
Deviation
It is undefined for equal to 1 or 2.
Coefficient of
Variation
Undefined
Skewness 0. It is undefined for less than or equal
to 3. However, the t distribution is
symmetric in all cases.
Kurtosis
It is undefined for less than or equal to
4.
Parameter
Estimation
Since the t distribution is typically used to develop hypothesis
tests and confidence intervals and rarely for modeling
applications, we omit any discussion of parameter estimation.
Comments The t distribution is used in many cases for the critical
regions for hypothesis tests and in determining confidence
intervals. The most common example is testing if data are
consistent with the assumed process mean.
least some of the probability functions for the t distribution.
1.3.6.6.5. F Distribution

1.3. EDA Techniques
Probability
Density
Function
The F distribution is the ratio of two chi-square distributions
with degrees of freedom and , respectively, where each
chi-square has first been divided by its degrees of freedom.
The formula for the probability density function of the F
distribution is
where and are the shape parameters and is the gamma
function. The formula for the gamma function is
In a testing context, the F distribution is treated as a
(as with other probability distributions), the F distribution
itself can be transformed with a location parameter, , and a
scale parameter, .
The following is the plot of the F probability density function
for 4 different values of the shape parameters.
Cumulative
Distribution
Function
The formula for the Cumulative distribution function of the F
distribution is
where k = / ( + *x) and I
k
is the incomplete beta
function. The formula for the incomplete beta function is
where B is the beta function
The following is the plot of the F cumulative distribution
function with the same values of and as the pdf plots
above.
Percent
Point
Function
The formula for the percent point function of the F
The following is the plot of the F percent point function with
the same values of and as the pdf plots above.
Other
Probability
Functions
Since the F distribution is typically used to develop
hypothesis tests and confidence intervals and rarely for
modeling applications, we omit the formulas and plots for the
hazard, cumulative hazard, survival, and inverse survival
probability functions.
Common
Statistics
The formulas below are for the case where the location
parameter is zero and the scale parameter is one.
Mean
Mode
Range 0 to positive infinity
Standard
Deviation
Coefficient of
Variation
Skewness
Parameter
Estimation
Since the F distribution is typically used to develop
modeling applications, we omit any discussion of parameter
estimation.
Comments The F distribution is used in many cases for the critical
regions for hypothesis tests and in determining confidence
intervals. Two common examples are the analysis of variance
and the F test to determine if the variances of two populations
are equal.
least some of the probability functions for the F distribution.
1.3.6.6.6. Chi-Square Distribution

1.3. EDA Techniques
Probability
Density
Function
The chi-square distribution results when independent
variables with standard normal distributions are squared and
summed. The formula for the probability density function of
the chi-square distribution is
where is the shape parameter and is the gamma function.
The formula for the gamma function is
In a testing context, the chi-square distribution is treated as a
(as with other probability distributions), the chi-square
distribution itself can be transformed with a location
parameter, , and a scale parameter, .
The following is the plot of the chi-square probability density
function for 4 different values of the shape parameter.
Cumulative
Distribution
Function
The formula for the cumulative distribution function of the
chi-square distribution is
where is the gamma function defined above and is the
incomplete gamma function. The formula for the incomplete
gamma function is
The following is the plot of the chi-square cumulative
distribution function with the same values of as the pdf
plots above.
Percent
Point
Function
The formula for the percent point function of the chi-square
The following is the plot of the chi-square percent point
Other
Probability
Functions
Since the chi-square distribution is typically used to develop
modeling applications, we omit the formulas and plots for the
hazard, cumulative hazard, survival, and inverse survival
probability functions.
Common
Statistics
Mean
Median approximately - 2/3 for large
Mode
Standard
Deviation
Coefficient of
Variation
Skewness
Kurtosis
Parameter
Estimation
Since the chi-square distribution is typically used to develop
modeling applications, we omit any discussion of parameter
estimation.
Comments The chi-square distribution is used in many cases for the
critical regions for hypothesis tests and in determining
confidence intervals. Two common examples are the chi-
square test for independence in an RxC contingency table
and the chi-square test to determine if the standard deviation
of a population is equal to a pre-specified value.
least some of the probability functions for the chi-square
distribution.
1.3.6.6.7. Exponential Distribution

1.3. EDA Techniques
Probability
Density
Function
the exponential distribution is
parameter (the scale parameter is often referred to as which
equals ). The case where = 0 and = 1 is called the
standard exponential distribution. The equation for the
standard exponential distribution is
The general form of probability functions can be expressed
in terms of the standard distribution. Subsequent formulas in
this section are given for the 1-parameter (i.e., with scale
parameter) form of the function.
The following is the plot of the exponential probability
density function.
Cumulative
Distribution
exponential distribution is
Function
The following is the plot of the exponential cumulative
Percent
Point
Function
The formula for the percent point function of the exponential
distribution is
The following is the plot of the exponential percent point
function.
Hazard
Function
The formula for the hazard function of the exponential
distribution is
The following is the plot of the exponential hazard function.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the
The following is the plot of the exponential cumulative
hazard function.
Survival
Function
The formula for the survival function of the exponential
distribution is
The following is the plot of the exponential survival function.
Inverse
Survival
Function
The formula for the inverse survival function of the
The following is the plot of the exponential inverse survival
function.
Common
Statistics
Mean
Median
Mode Zero
Range Zero to plus infinity
Standard
Deviation
Coefficient of
Variation
1
Skewness 2
Kurtosis 9
Parameter
Estimation
For the full sample case, the maximum likelihood estimator
of the scale parameter is the sample mean. Maximum
likelihood estimation for the exponential distribution is
discussed in the chapter on reliability (Chapter 8). It is also
discussed in chapter 19 of Johnson, Kotz, and Balakrishnan.
Comments The exponential distribution is primarily used in reliability
applications. The exponential distribution is used to model
data with a constant failure rate (indicated by the hazard plot
which is simply equal to a constant).
least some of the probability functions for the exponential
distribution.
1.3.6.6.8. Weibull Distribution

1.3. EDA Techniques
Probability
Density
Function
The formula for the probability density function of the general Weibull
distribution is
where is the shape parameter, is the location parameter and is the
scale parameter. The case where = 0 and = 1 is called the standard
Weibull distribution. The case where = 0 is called the 2-parameter
Weibull distribution. The equation for the standard Weibull distribution
reduces to
Since the general form of probability functions can be expressed in
terms of the standard distribution, all subsequent formulas in this
section are given for the standard form of the function.
The following is the plot of the Weibull probability density function.
Cumulative
Distribution
Function
The formula for the cumulative distribution function of the Weibull
distribution is
The following is the plot of the Weibull cumulative distribution
Percent
Point
Function
The formula for the percent point function of the Weibull distribution is
The following is the plot of the Weibull percent point function with the
same values of as the pdf plots above.
Hazard
Function
The formula for the hazard function of the Weibull distribution is
The following is the plot of the Weibull hazard function with the same
values of as the pdf plots above.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the Weibull
distribution is
The following is the plot of the Weibull cumulative hazard function
Survival
Function
The formula for the survival function of the Weibull distribution is
The following is the plot of the Weibull survival function with the same
values of as the pdf plots above.
Inverse
Survival
Function
The formula for the inverse survival function of the Weibull
distribution is
The following is the plot of the Weibull inverse survival function with
the same values of as the pdf plots above.
Common
Statistics
The formulas below are with the location parameter equal to zero and
the scale parameter equal to one.
Mean
where is the gamma function
Median
Mode

Range Zero to positive infinity.
Standard Deviation
Coefficient of
Variation
Parameter
Estimation
Maximum likelihood estimation for the Weibull distribution is
discussed in the Reliability chapter (Chapter 8). It is also discussed in
Chapter 21 of Johnson, Kotz, and Balakrishnan.
Comments The Weibull distribution is used extensively in reliability applications
to model failure times.
Software Most general purpose statistical software programs support at least
some of the probability functions for the Weibull distribution.
1.3.6.6.9. Lognormal Distribution

1.3. EDA Techniques
Probability
Density
Function
A variable X is lognormally distributed if Y = LN(X) is
normally distributed with "LN" denoting the natural
logarithm. The general formula for the probability density
function of the lognormal distribution is
where is the shape parameter, is the location parameter
and m is the scale parameter. The case where = 0 and m =
1 is called the standard lognormal distribution. The case
where equals zero is called the 2-parameter lognormal
distribution.
The equation for the standard lognormal distribution is
function.
The following is the plot of the lognormal probability density
function for four values of .
There are several common parameterizations of the
lognormal distribution. The form given here is from Evans,
Hastings, and Peacock.
Cumulative
Distribution
Function
lognormal distribution is
where is the cumulative distribution function of the normal
distribution.
The following is the plot of the lognormal cumulative
plots above.
Percent
Point
Function
The formula for the percent point function of the lognormal
distribution is
where is the percent point function of the normal
distribution.
The following is the plot of the lognormal percent point
Hazard
Function
The formula for the hazard function of the lognormal
distribution is
where is the probability density function of the normal
distribution and is the cumulative distribution function of
the normal distribution.
The following is the plot of the lognormal hazard function
Cumulative
Hazard
Function
distribution.
The following is the plot of the lognormal cumulative hazard
Survival
Function
The formula for the survival function of the lognormal
distribution is
distribution.
The following is the plot of the lognormal survival function
Inverse
Survival
Function
The formula for the inverse survival function of the
where is the percent point function of the normal
distribution.
The following is the plot of the lognormal inverse survival
Common
Statistics
The formulas below are with the location parameter equal to
zero and the scale parameter equal to one.
Mean
Median Scale parameter m (= 1 if scale parameter
not specified).
Mode
Range Zero to positive infinity
Standard
Deviation
Skewness
Kurtosis
Coefficient of
Variation
Parameter
Estimation
The maximum likelihood estimates for the scale parameter,
m, and the shape parameter, , are
and
where
If the location parameter is known, it can be subtracted from
the original data points before computing the maximum
likelihood estimates of the shape and scale parameters.
Comments The lognormal distribution is used extensively in reliability
applications to model failure times. The lognormal and
Weibull distributions are probably the most commonly used
distributions in reliability applications.
least some of the probability functions for the lognormal
distribution.
1.3.6.6.10. Fatigue Life Distribution

1.3. EDA Techniques
1.3.6.6.10. Birnbaum-Saunders (Fatigue Life)
Distribution
Probability
Density
Function
The Birnbaum-Saunders distribution is also commonly known as the
fatigue life distribution. There are several alternative formulations of
the Birnbaum-Saunders distribution in the literature.
The general formula for the probability density function of the
Birnbaum-Saunders distribution is
where is the shape parameter, is the location parameter, is the
scale parameter, is the probability density function of the standard
normal distribution, and is the cumulative distribution function of
the standard normal distribution. The case where = 0 and = 1 is
called the standard Birnbaum-Saunders distribution. The equation
for the standard Birnbaum-Saunders distribution reduces to
Since the general form of probability functions can be expressed in
terms of the standard distribution, all subsequent formulas in this
section are given for the standard form of the function.
The following is the plot of the Birnbaum-Saunders probability density
function.
Cumulative
Distribution
Function
The formula for the cumulative distribution function of the Birnbaum-
Saunders distribution is
where is the cumulative distribution function of the standard normal
distribution. The following is the plot of the Birnbaum-Saunders
cumulative distribution function with the same values of as the pdf
plots above.
Percent
Point
Function
The formula for the percent point function of the Birnbaum-Saunders
distribution is
where is the percent point function of the standard normal
distribution. The following is the plot of the Birnbaum-Saunders
percent point function with the same values of as the pdf plots
above.
Hazard
Function
The Birnbaum-Saunders hazard function can be computed from the
Birnbaum-Saunders probability density and cumulative distribution
functions.
The following is the plot of the Birnbaum-Saunders hazard function
Cumulative
Hazard
Function
The Birnbaum-Saunders cumulative hazard function can be computed
from the Birnbaum-Saunders cumulative distribution function.
The following is the plot of the Birnbaum-Saunders cumulative hazard
Survival
Function
The Birnbaum-Saunders survival function can be computed from the
Birnbaum-Saunders cumulative distribution function.
The following is the plot of the Birnbaum-Saunders survival function
Inverse
Survival
Function
The Birnbaum-Saunders inverse survival function can be computed
from the Birnbaum-Saunders percent point function.
The following is the plot of the gamma inverse survival function with
Common
Statistics
The formulas below are with the location parameter equal to zero and
the scale parameter equal to one.
Mean
Standard Deviation
Coefficient of
Variation
Parameter
Estimation
Maximum likelihood estimation for the Birnbaum-Saunders
distribution is discussed in the Reliability chapter.
Comments The Birnbaum-Saunders distribution is used extensively in reliability
applications to model failure times.
Software Some general purpose statistical software programs, including
Dataplot, support at least some of the probability functions for the
Birnbaum-Saunders distribution. Support for this distribution is likely
to be available for statistical programs that emphasize reliability
applications.
The "bs" package implements support for the Birnbaum-Saunders
distribution for the R package. See
Leiva, V., Hernandez, H., and Riquelme, M. (2006). A New
Package for the Birnbaum-Saunders Distribution. Rnews, 6/4,
35-40. (http://www.r-project.org)
1.3.6.6.11. Gamma Distribution

1.3. EDA Techniques
Probability
Density
Function
the gamma distribution is
where is the shape parameter, is the location parameter,
is the scale parameter, and is the gamma function which
has the formula
The case where = 0 and = 1 is called the standard
gamma distribution. The equation for the standard gamma
distribution reduces to
function.
The following is the plot of the gamma probability density
function.
Cumulative
Distribution
Function
gamma distribution is
where is the gamma function defined above and is
the incomplete gamma function. The incomplete gamma
function has the formula
The following is the plot of the gamma cumulative
plots above.
Percent
Point
Function
The formula for the percent point function of the gamma
The following is the plot of the gamma percent point function
Hazard
Function
The formula for the hazard function of the gamma
distribution is
The following is the plot of the gamma hazard function with
Cumulative
Hazard
Function
gamma distribution is
the incomplete gamma function defined above.
The following is the plot of the gamma cumulative hazard
Survival
Function
The formula for the survival function of the gamma
distribution is
the incomplete gamma function defined above.
The following is the plot of the gamma survival function with
Inverse
Survival
Function
The gamma inverse survival function does not exist in simple
closed form. It is computed numberically.
The following is the plot of the gamma inverse survival
Common
Statistics
The formulas below are with the location parameter equal to
zero and the scale parameter equal to one.
Mean
Mode
Standard
Deviation
Skewness
Kurtosis
Coefficient of
Variation
Parameter
Estimation
The method of moments estimators of the gamma distribution
are
where and s are the sample mean and standard deviation,
respectively.
The equations for the maximum likelihood estimation of the
shape and scale parameters are given in Chapter 18 of Evans,
Hastings, and Peacock and Chapter 17 of Johnson, Kotz, and
Balakrishnan. These equations need to be solved numerically;
this is typically accomplished by using statistical software
packages.
Software Some general purpose statistical software programs support
at least some of the probability functions for the gamma
distribution.
1.3.6.6.12. Double Exponential Distribution

1.3. EDA Techniques
Probability
Density
Function
the double exponential distribution is
standard double exponential distribution. The equation for
the standard double exponential distribution is
function.
The following is the plot of the double exponential
probability density function.
Cumulative
Distribution
double exponential distribution is
Function
cumulative distribution function.
Percent
Point
Function
The formula for the percent point function of the double
The following is the plot of the double exponential percent
point function.
Hazard
Function
The formula for the hazard function of the double exponential
distribution is
The following is the plot of the double exponential hazard
function.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the double
cumulative hazard function.
Survival
Function
The double exponential survival function can be computed
from the cumulative distribution function of the double
exponential distribution.
The following is the plot of the double exponential survival
function.
Inverse
Survival
Function
The formula for the inverse survival function of the double
The following is the plot of the double exponential inverse
survival function.
Common
Statistics
Mean
Median
Mode
Range Negative infinity to positive infinity
Standard
Deviation
Skewness 0
Kurtosis 6
Coefficient of
Variation
Parameter
Estimation
The maximum likelihood estimators of the location and scale
parameters of the double exponential distribution are
where is the sample median.
at least some of the probability functions for the double
exponential distribution.
1.3.6.6.13. Power Normal Distribution

1.3. EDA Techniques
Probability
Density
Function
The formula for the probability density function of the
standard form of the power normal distribution is
where p is the shape parameter (also referred to as the power
parameter), is the cumulative distribution function of the
standard normal distribution, and is the probability density
function of the standard normal distribution.
As with other probability distributions, the power normal
distribution can be transformed with a location parameter, ,
and a scale parameter, . We omit the equation for the
general form of the power normal distribution. Since the
general form of probability functions can be expressed in
terms of the standard distribution, all subsequent formulas in
this section are given for the standard form of the function.
The following is the plot of the power normal probability
density function with four values of p.
Cumulative
Distribution
Function
power normal distribution is
where is the cumulative distribution function of the
standard normal distribution.
The following is the plot of the power normal cumulative
distribution function with the same values of p as the pdf
plots above.
Percent
Point
Function
The formula for the percent point function of the power
where is the percent point function of the standard
normal distribution.
The following is the plot of the power normal percent point
function with the same values of p as the pdf plots above.
Hazard The formula for the hazard function of the power normal
Function distribution is
The following is the plot of the power normal hazard function
with the same values of p as the pdf plots above.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the power
The following is the plot of the power normal cumulative
hazard function with the same values of p as the pdf plots
above.
Survival
Function
The formula for the survival function of the power normal
distribution is
The following is the plot of the power normal survival
Inverse
Survival
Function
The formula for the inverse survival function of the power
The following is the plot of the power normal inverse
survival function with the same values of p as the pdf plots
above.
Common
Statistics
The statistics for the power normal distribution are
complicated and require tables. Nelson discusses the mean,
median, mode, and standard deviation of the power normal
distribution and provides references to the appropriate tables.
support the probability functions for the power normal
distribution.
1.3.6.6.14. Power Lognormal Distribution

1.3. EDA Techniques
Probability
Density
Function
The formula for the probability density function of the standard form of
the power lognormal distribution is
where p (also referred to as the power parameter) and are the shape
parameters, is the cumulative distribution function of the standard
normal distribution, and is the probability density function of the
standard normal distribution.
As with other probability distributions, the power lognormal distribution
can be transformed with a location parameter, , and a scale parameter,
B. We omit the equation for the general form of the power lognormal
distribution. Since the general form of probability functions can be
expressed in terms of the standard distribution, all subsequent formulas
in this section are given for the standard form of the function.
The following is the plot of the power lognormal probability density
function with four values of p and set to 1.
Cumulative
Distribution
Function
The formula for the cumulative distribution function of the power
distribution.
The following is the plot of the power lognormal cumulative
distribution function with the same values of p as the pdf plots above.
Percent
Point
Function
The formula for the percent point function of the power lognormal
distribution is
where is the percent point function of the standard normal
distribution.
The following is the plot of the power lognormal percent point function
with the same values of p as the pdf plots above.
Hazard The formula for the hazard function of the power lognormal distribution
Function is
distribution, and is the probability density function of the standard
Note that this is simply a multiple (p) of the lognormal hazard function.
The following is the plot of the power lognormal hazard function with
the same values of p as the pdf plots above.
Cumulative
Hazard
Function
The formula for the cumulative hazard function of the power lognormal
distribution is
The following is the plot of the power lognormal cumulative hazard
Survival
Function
The formula for the survival function of the power lognormal
distribution is
The following is the plot of the power lognormal survival function with
Inverse
Survival
Function
The formula for the inverse survival function of the power lognormal
distribution is
The following is the plot of the power lognormal inverse survival
Common
Statistics
The statistics for the power lognormal distribution are complicated and
require tables. Nelson discusses the mean, median, mode, and standard
deviation of the power lognormal distribution and provides references to
the appropriate tables.
Parameter
Estimation
Nelson discusses maximum likelihood estimation for the power
lognormal distribution. These estimates need to be performed with
computer software. Software for maximum likelihood estimation of the
parameters of the power lognormal distribution is not as readily
available as for other reliability distributions such as the exponential,
Weibull, and lognormal.
Software Most general purpose statistical software programs do not support the
probability functions for the power lognormal distribution.
1.3.6.6.15. Tukey-Lambda Distribution

1.3. EDA Techniques
Probability
Density
Function
The Tukey-Lambda density function does not have a simple,
closed form. It is computed numerically.
The Tukey-Lambda distribution has the shape parameter .
As with other probability distributions, the Tukey-Lambda
distribution can be transformed with a location parameter, ,
and a scale parameter, . Since the general form of
probability functions can be expressed in terms of the
standard distribution, all subsequent formulas in this section
are given for the standard form of the function.
The following is the plot of the Tukey-Lambda probability
density function for four values of .
Cumulative
Distribution
Function
The Tukey-Lambda distribution does not have a simple,
closed form. It is computed numerically.
The following is the plot of the Tukey-Lambda cumulative
plots above.
Percent
Point
Function
The formula for the percent point function of the standard
form of the Tukey-Lambda distribution is
The following is the plot of the Tukey-Lambda percent point
Other
Probability
Functions
The Tukey-Lambda distribution is typically used to identify
an appropriate distribution (see the comments below) and not
used in statistical models directly. For this reason, we omit
the formulas, and plots for the hazard, cumulative hazard,
survival, and inverse survival functions. We also omit the
common statistics and parameter estimation sections.
Comments The Tukey-Lambda distribution is actually a family of
distributions that can approximate a number of common
distributions. For example,
= -1 approximately Cauchy
= 0 exactly logistic
= 0.14 approximately normal
= 0.5 U-shaped
= 1 exactly uniform (from -1 to +1)
The most common use of this distribution is to generate a
Tukey-Lambda PPCC plot of a data set. Based on the ppcc
plot, an appropriate model for the data is suggested. For
example, if the maximum correlation occurs for a value of
at or near 0.14, then the data can be modeled with a normal
distribution. Values of less than this imply a heavy-tailed
distribution (with -1 approximating a Cauchy). That is, as the
optimal value of goes from 0.14 to -1, increasingly heavy
tails are implied. Similarly, as the optimal value of becomes
greater than 0.14, shorter tails are implied.
As the Tukey-Lambda distribution is a symmetric
distribution, the use of the Tukey-Lambda PPCC plot to
determine a reasonable distribution to model the data only
applies to symmetric distributuins. A histogram of the data
should provide evidence as to whether the data can be
reasonably modeled with a symmetric distribution.
support the probability functions for the Tukey-Lambda
distribution.
1.3.6.6.16. Extreme Value Type I Distribution

1.3. EDA Techniques
Probability
Density
Function
The extreme value type I distribution has two forms. One is
based on the smallest extreme and the other is based on the
largest extreme. We call these the minimum and maximum
cases, respectively. Formulas and plots for both cases are
given. The extreme value type I distribution is also referred to
as the Gumbel distribution.
the Gumbel (minimum) distribution is
standard Gumbel distribution. The equation for the
standard Gumbel distribution (minimum) reduces to
The following is the plot of the Gumbel probability density
function for the minimum case.
the Gumbel (maximum) distribution is
standard Gumbel distribution. The equation for the
standard Gumbel distribution (maximum) reduces to
The following is the plot of the Gumbel probability density
function for the maximum case.
function.
Cumulative
Distribution
Function
Gumbel distribution (minimum) is
The following is the plot of the Gumbel cumulative
distribution function for the minimum case.
Gumbel distribution (maximum) is
The following is the plot of the Gumbel cumulative
distribution function for the maximum case.
Percent
Point
Function
The formula for the percent point function of the Gumbel
distribution (minimum) is
The following is the plot of the Gumbel percent point
The formula for the percent point function of the Gumbel
distribution (maximum) is
The following is the plot of the Gumbel percent point
Hazard
Function
The formula for the hazard function of the Gumbel
The following is the plot of the Gumbel hazard function for
the minimum case.
The formula for the hazard function of the Gumbel
The following is the plot of the Gumbel hazard function for
the maximum case.
Cumulative
Hazard
Function
Gumbel distribution (minimum) is
The following is the plot of the Gumbel cumulative hazard
Gumbel distribution (maximum) is
The following is the plot of the Gumbel cumulative hazard
Survival
Function
The formula for the survival function of the Gumbel
The following is the plot of the Gumbel survival function for
the minimum case.
The formula for the survival function of the Gumbel
The following is the plot of the Gumbel survival function for
the maximum case.
Inverse
Survival
Function
The formula for the inverse survival function of the Gumbel
The following is the plot of the Gumbel inverse survival
The formula for the inverse survival function of the Gumbel
The following is the plot of the Gumbel inverse survival
Common
Statistics
The formulas below are for the maximum order statistic case.
Mean
The constant 0.5772 is Euler's number.
Median
Mode
Range Negative infinity to positive infinity.
Standard
Deviation
Skewness 1.13955
Kurtosis 5.4
Coefficient of
Variation
Parameter
Estimation
The method of moments estimators of the Gumbel
(maximum) distribution are
where and s are the sample mean and standard deviation,
respectively.
The equations for the maximum likelihood estimation of the
shape and scale parameters are discussed in Chapter 15 of
Evans, Hastings, and Peacock and Chapter 22 of Johnson,
Kotz, and Balakrishnan. These equations need to be solved
numerically and this is typically accomplished by using
statistical software packages.
at least some of the probability functions for the extreme
value type I distribution.
1.3.6.6.17. Beta Distribution

1.3. EDA Techniques
Probability
Density
Function
The general formula for the probability density function of the beta
distribution is
where p and q are the shape parameters, a and b are the lower and upper
bounds, respectively, of the distribution, and B(p,q) is the beta function.
The beta function has the formula
The case where a = 0 and b = 1 is called the standard beta distribution.
The equation for the standard beta distribution is
Typically we define the general form of a distribution in terms of location
and scale parameters. The beta is different in that we define the general
distribution in terms of the lower and upper bounds. However, the location
and scale parameters can be defined in terms of the lower and upper limits
as follows:
location = a
scale = b - a
Since the general form of probability functions can be expressed in terms
of the standard distribution, all subsequent formulas in this section are
given for the standard form of the function.
The following is the plot of the beta probability density function for four
different values of the shape parameters.
Cumulative
Distribution
Function
The formula for the cumulative distribution function of the beta distribution
is also called the incomplete beta function ratio (commonly denoted by I
x
)
and is defined as
where B is the beta function defined above.
The following is the plot of the beta cumulative distribution function with
the same values of the shape parameters as the pdf plots above.
Percent
Point
Function
The formula for the percent point function of the beta distribution does not
exist in a simple closed form. It is computed numerically.
The following is the plot of the beta percent point function with the same
values of the shape parameters as the pdf plots above.
Other
Probability
Functions
Since the beta distribution is not typically used for reliability applications,
we omit the formulas and plots for the hazard, cumulative hazard, survival,
and inverse survival probability functions.
Common
Statistics
The formulas below are for the case where the lower limit is zero and the
upper limit is one.
Mean
Mode
Range 0 to 1
Standard Deviation
Coefficient of
Variation
Skewness
Parameter
Estimation
First consider the case where a and b are assumed to be known. For this
case, the method of moments estimates are
where is the sample mean and s
2
is the sample variance. If a and b are
not 0 and 1, respectively, then replace with and s
2
with
in the above equations.
For the case when a and b are known, the maximum likelihood estimates
can be obtained by solving the following set of equations
The maximum likelihood equations for the case when a and b are not
known are given in pages 221-235 of Volume II of Johnson, Kotz, and
Balakrishan.
Software Most general purpose statistical software programs support at least some of
the probability functions for the beta distribution.
1.3.6.6.18. Binomial Distribution

1.3. EDA Techniques
Probability
Mass
Function
The binomial distribution is used when there are exactly two
mutually exclusive outcomes of a trial. These outcomes are
appropriately labeled "success" and "failure". The binomial
distribution is used to obtain the probability of observing x successes
in N trials, with the probability of success on a single trial denoted
by p. The binomial distribution assumes that p is fixed for all trials.
The formula for the binomial probability mass function is
where
The following is the plot of the binomial probability density function
for four values of p and n = 100.
Cumulative
Distribution
Function
The formula for the binomial cumulative probability function is
The following is the plot of the binomial cumulative distribution
Percent
Point
Function
The binomial percent point function does not exist in simple closed
form. It is computed numerically. Note that because this is a discrete
distribution that is only defined for integer values of x, the percent
point function is not smooth in the way the percent point function
typically is for a continuous distribution.
The following is the plot of the binomial percent point function with
Common
Statistics
Mean
Mode
Range 0 to N
Standard Deviation
Coefficient of
Variation
Skewness
Kurtosis
Comments The binomial distribution is probably the most commonly used
discrete distribution.
Parameter
Estimation
The maximum likelihood estimator of p (n is fixed) is
Software Most general purpose statistical software programs support at least
some of the probability functions for the binomial distribution.
1.3.6.6.19. Poisson Distribution

1.3. EDA Techniques
Probability
Mass
Function
The Poisson distribution is used to model the number of
events occurring within a given time interval.
The formula for the Poisson probability mass function is
is the shape parameter which indicates the average number
of events in the given time interval.
The following is the plot of the Poisson probability density
function for four values of .
Cumulative
Distribution
Function
The formula for the Poisson cumulative probability function
is
The following is the plot of the Poisson cumulative
plots above.
Percent
Point
Function
The Poisson percent point function does not exist in simple
closed form. It is computed numerically. Note that because
this is a discrete distribution that is only defined for integer
values of x, the percent point function is not smooth in the
way the percent point function typically is for a continuous
distribution.
The following is the plot of the Poisson percent point
Common
Statistics
Mean
Mode For non-integer , it is the largest integer
less than . For integer , x = and x =
- 1 are both the mode.
Standard
Deviation
Coefficient of
Variation
Skewness
Kurtosis
Parameter
Estimation
The maximum likelihood estimator of is
where is the sample mean.
least some of the probability functions for the Poisson
distribution.
1.3.6.7. Tables for Probability Distributions

1.3. EDA Techniques
Tables Several commonly used tables for probability distributions can
be referenced below.
The values from these tables can also be obtained from most
general purpose statistical software programs. Most
introductory statistics textbooks (e.g., Snedecor and Cochran)
contain more extensive tables than are included here. These
tables are included for convenience.
1. Cumulative distribution function for the standard normal
distribution
2. Upper critical values of Student's t-distribution with
degrees of freedom
3. Upper critical values of the F-distribution with and
degrees of freedom
4. Upper critical values of the chi-square distribution with
degrees of freedom
5. Critical values of t
*
distribution for testing the output of
a linear calibration line at 3 points
6. Upper critical values of the normal PPCC distribution
1.3.6.7.1. Cumulative Distribution Function of the Standard Normal Distribution

1.3. EDA Techniques
1.3.6.7.1. Cumulative Distribution Function of
the Standard Normal Distribution
How to
Use This
Table
The table below contains the area under the standard normal
curve from 0 to z. This can be used to compute the cumulative
distribution function values for the standard normal
distribution.
The table utilizes the symmetry of the normal distribution, so
what in fact is given is
where a is the value of interest. This is demonstrated in the
graph below for a = 0.5. The shaded area of the curve
represents the probability that x is between 0 and a.
This can be clarified by a few simple examples.
1. What is the probability that x is less than or equal to
1.53? Look for 1.5 in the X column, go right to the 0.03
column to find the value 0.43699. Now add 0.5 (for the
probability less than zero) to obtain the final result of
0.93699.
2. What is the probability that x is less than or equal to -
1.53? For negative values, use the relationship
From the first example, this gives 1 - 0.93699 =
0.06301.
3. What is the probability that x is between -1 and 0.5?
Look up the values for 0.5 (0.5 + 0.19146 = 0.69146)
and -1 (1 - (0.5 + 0.34134) = 0.15866). Then subtract
the results (0.69146 - 0.15866) to obtain the result
0.5328.
To use this table with a non-standard normal distribution
(either the location parameter is not 0 or the scale parameter is
not 1), standardize your value by subtracting the mean and
dividing the result by the standard deviation. Then look up the
value for this standardized value.
A few particularly important numbers derived from the table
below, specifically numbers that are commonly used in
significance tests, are summarized in the following table:
p 0.001 0.005 0.010 0.025 0.050 0.100
Z
p
-3.090 -2.576 -2.326 -1.960 -1.645 -1.282
p 0.999 0.995 0.990 0.975 0.950 0.900
Z
p
+3.090 +2.576 +2.326 +1.960 +1.645 +1.282
These are critical values for the normal distribution.
Area under the Normal Curve from 0
to X
X 0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.07 0.08 0.09
0.0 0.00000 0.00399 0.00798 0.01197 0.01595 0.01994 0.02392
0.02790 0.03188 0.03586
0.1 0.03983 0.04380 0.04776 0.05172 0.05567 0.05962 0.06356
0.06749 0.07142 0.07535
0.2 0.07926 0.08317 0.08706 0.09095 0.09483 0.09871 0.10257
0.10642 0.11026 0.11409
0.3 0.11791 0.12172 0.12552 0.12930 0.13307 0.13683 0.14058
0.14431 0.14803 0.15173
0.4 0.15542 0.15910 0.16276 0.16640 0.17003 0.17364 0.17724
0.18082 0.18439 0.18793
0.5 0.19146 0.19497 0.19847 0.20194 0.20540 0.20884 0.21226
0.21566 0.21904 0.22240
0.6 0.22575 0.22907 0.23237 0.23565 0.23891 0.24215 0.24537
0.24857 0.25175 0.25490
0.7 0.25804 0.26115 0.26424 0.26730 0.27035 0.27337 0.27637
0.27935 0.28230 0.28524
0.8 0.28814 0.29103 0.29389 0.29673 0.29955 0.30234 0.30511
0.30785 0.31057 0.31327
0.9 0.31594 0.31859 0.32121 0.32381 0.32639 0.32894 0.33147
0.33398 0.33646 0.33891
1.0 0.34134 0.34375 0.34614 0.34849 0.35083 0.35314 0.35543
0.35769 0.35993 0.36214
1.1 0.36433 0.36650 0.36864 0.37076 0.37286 0.37493 0.37698
0.37900 0.38100 0.38298
1.2 0.38493 0.38686 0.38877 0.39065 0.39251 0.39435 0.39617
0.39796 0.39973 0.40147
1.3 0.40320 0.40490 0.40658 0.40824 0.40988 0.41149 0.41308
0.41466 0.41621 0.41774
1.4 0.41924 0.42073 0.42220 0.42364 0.42507 0.42647 0.42785
0.42922 0.43056 0.43189
1.5 0.43319 0.43448 0.43574 0.43699 0.43822 0.43943 0.44062
0.44179 0.44295 0.44408
1.6 0.44520 0.44630 0.44738 0.44845 0.44950 0.45053 0.45154
0.45254 0.45352 0.45449
1.7 0.45543 0.45637 0.45728 0.45818 0.45907 0.45994 0.46080
0.46164 0.46246 0.46327
1.8 0.46407 0.46485 0.46562 0.46638 0.46712 0.46784 0.46856
0.46926 0.46995 0.47062
1.9 0.47128 0.47193 0.47257 0.47320 0.47381 0.47441 0.47500
0.47558 0.47615 0.47670
2.0 0.47725 0.47778 0.47831 0.47882 0.47932 0.47982 0.48030
0.48077 0.48124 0.48169
2.1 0.48214 0.48257 0.48300 0.48341 0.48382 0.48422 0.48461
0.48500 0.48537 0.48574
2.2 0.48610 0.48645 0.48679 0.48713 0.48745 0.48778 0.48809
0.48840 0.48870 0.48899
2.3 0.48928 0.48956 0.48983 0.49010 0.49036 0.49061 0.49086
0.49111 0.49134 0.49158
2.4 0.49180 0.49202 0.49224 0.49245 0.49266 0.49286 0.49305
0.49324 0.49343 0.49361
2.5 0.49379 0.49396 0.49413 0.49430 0.49446 0.49461 0.49477
0.49492 0.49506 0.49520
2.6 0.49534 0.49547 0.49560 0.49573 0.49585 0.49598 0.49609
0.49621 0.49632 0.49643
2.7 0.49653 0.49664 0.49674 0.49683 0.49693 0.49702 0.49711
0.49720 0.49728 0.49736
2.8 0.49744 0.49752 0.49760 0.49767 0.49774 0.49781 0.49788
0.49795 0.49801 0.49807
2.9 0.49813 0.49819 0.49825 0.49831 0.49836 0.49841 0.49846
0.49851 0.49856 0.49861
3.0 0.49865 0.49869 0.49874 0.49878 0.49882 0.49886 0.49889
0.49893 0.49896 0.49900
3.1 0.49903 0.49906 0.49910 0.49913 0.49916 0.49918 0.49921
0.49924 0.49926 0.49929
3.2 0.49931 0.49934 0.49936 0.49938 0.49940 0.49942 0.49944
0.49946 0.49948 0.49950
3.3 0.49952 0.49953 0.49955 0.49957 0.49958 0.49960 0.49961
0.49962 0.49964 0.49965
3.4 0.49966 0.49968 0.49969 0.49970 0.49971 0.49972 0.49973
0.49974 0.49975 0.49976
3.5 0.49977 0.49978 0.49978 0.49979 0.49980 0.49981 0.49981
0.49982 0.49983 0.49983
3.6 0.49984 0.49985 0.49985 0.49986 0.49986 0.49987 0.49987
0.49988 0.49988 0.49989
3.7 0.49989 0.49990 0.49990 0.49990 0.49991 0.49991 0.49992
0.49992 0.49992 0.49992
3.8 0.49993 0.49993 0.49993 0.49994 0.49994 0.49994 0.49994
0.49995 0.49995 0.49995
3.9 0.49995 0.49995 0.49996 0.49996 0.49996 0.49996 0.49996
0.49996 0.49997 0.49997
4.0 0.49997 0.49997 0.49997 0.49997 0.49997 0.49997 0.49998
0.49998 0.49998 0.49998
1.3.6.7.2. Critical Values of the Student's-t Distribution

1.3. EDA Techniques
1.3.6.7.2. Critical Values of the Student's t
Distribution
How to
Use This
Table
This table contains critical values of the Student's t
distribution computed using the cumulative distribution
function. The t distribution is symmetric so that
t
1-,
= -t
,
.
The t table can be used for both one-sided (lower and upper)
and two-sided tests using the appropriate value of .
The significance level, , is demonstrated in the graph below,
which displays a t distribution with 10 degrees of freedom.
The most commonly used significance level is = 0.05. For a
two-sided test, we compute 1 - /2, or 1 - 0.05/2 = 0.975 when
= 0.05. If the absolute value of the test statistic is greater
than the critical value (0.975), then we reject the null
hypothesis. Due to the symmetry of the t distribution, we only
tabulate the positive critical values in the table below.
Given a specified value for :
1. For a two-sided test, find the column corresponding to
1-/2 and reject the null hypothesis if the absolute value
of the test statistic is greater than the value of t
1-/2,
in
the table below.
2. For an upper, one-sided test, find the column
corresponding to 1- and reject the null hypothesis if the
test statistic is greater than the table value.
3. For a lower, one-sided test, find the column
corresponding to 1- and reject the null hypothesis if the
test statistic is less than the negative of the table value.
Critical values of Student's t distribution with degrees of
freedom
Probability less than the critical value
(t
1-,
)
0.90 0.95 0.975 0.99 0.995
0.999
1. 3.078 6.314 12.706 31.821 63.657
318.313
2. 1.886 2.920 4.303 6.965 9.925
22.327
3. 1.638 2.353 3.182 4.541 5.841
10.215
4. 1.533 2.132 2.776 3.747 4.604
7.173
5. 1.476 2.015 2.571 3.365 4.032
5.893
6. 1.440 1.943 2.447 3.143 3.707
5.208
7. 1.415 1.895 2.365 2.998 3.499
4.782
8. 1.397 1.860 2.306 2.896 3.355
4.499
9. 1.383 1.833 2.262 2.821 3.250
4.296
10. 1.372 1.812 2.228 2.764 3.169
4.143
11. 1.363 1.796 2.201 2.718 3.106
4.024
12. 1.356 1.782 2.179 2.681 3.055
3.929
13. 1.350 1.771 2.160 2.650 3.012
3.852
14. 1.345 1.761 2.145 2.624 2.977
3.787
15. 1.341 1.753 2.131 2.602 2.947
3.733
16. 1.337 1.746 2.120 2.583 2.921
3.686
17. 1.333 1.740 2.110 2.567 2.898
3.646
18. 1.330 1.734 2.101 2.552 2.878
3.610
19. 1.328 1.729 2.093 2.539 2.861
3.579
20. 1.325 1.725 2.086 2.528 2.845
3.552
21. 1.323 1.721 2.080 2.518 2.831
3.527
22. 1.321 1.717 2.074 2.508 2.819
3.505
23. 1.319 1.714 2.069 2.500 2.807
3.485
24. 1.318 1.711 2.064 2.492 2.797
3.467
25. 1.316 1.708 2.060 2.485 2.787
3.450
26. 1.315 1.706 2.056 2.479 2.779
3.435
27. 1.314 1.703 2.052 2.473 2.771
3.421
28. 1.313 1.701 2.048 2.467 2.763
3.408
29. 1.311 1.699 2.045 2.462 2.756
3.396
30. 1.310 1.697 2.042 2.457 2.750
3.385
31. 1.309 1.696 2.040 2.453 2.744
3.375
32. 1.309 1.694 2.037 2.449 2.738
3.365
33. 1.308 1.692 2.035 2.445 2.733
3.356
34. 1.307 1.691 2.032 2.441 2.728
3.348
35. 1.306 1.690 2.030 2.438 2.724
3.340
36. 1.306 1.688 2.028 2.434 2.719
3.333
37. 1.305 1.687 2.026 2.431 2.715
3.326
38. 1.304 1.686 2.024 2.429 2.712
3.319
39. 1.304 1.685 2.023 2.426 2.708
3.313
40. 1.303 1.684 2.021 2.423 2.704
3.307
41. 1.303 1.683 2.020 2.421 2.701
3.301
42. 1.302 1.682 2.018 2.418 2.698
3.296
43. 1.302 1.681 2.017 2.416 2.695
3.291
44. 1.301 1.680 2.015 2.414 2.692
3.286
45. 1.301 1.679 2.014 2.412 2.690
3.281
46. 1.300 1.679 2.013 2.410 2.687
3.277
47. 1.300 1.678 2.012 2.408 2.685
3.273
48. 1.299 1.677 2.011 2.407 2.682
3.269
49. 1.299 1.677 2.010 2.405 2.680
3.265
50. 1.299 1.676 2.009 2.403 2.678
3.261
51. 1.298 1.675 2.008 2.402 2.676
3.258
52. 1.298 1.675 2.007 2.400 2.674
3.255
53. 1.298 1.674 2.006 2.399 2.672
3.251
54. 1.297 1.674 2.005 2.397 2.670
3.248
55. 1.297 1.673 2.004 2.396 2.668
3.245
56. 1.297 1.673 2.003 2.395 2.667
3.242
57. 1.297 1.672 2.002 2.394 2.665
3.239
58. 1.296 1.672 2.002 2.392 2.663
3.237
59. 1.296 1.671 2.001 2.391 2.662
3.234
60. 1.296 1.671 2.000 2.390 2.660
3.232
61. 1.296 1.670 2.000 2.389 2.659
3.229
62. 1.295 1.670 1.999 2.388 2.657
3.227
63. 1.295 1.669 1.998 2.387 2.656
3.225
64. 1.295 1.669 1.998 2.386 2.655
3.223
65. 1.295 1.669 1.997 2.385 2.654
3.220
66. 1.295 1.668 1.997 2.384 2.652
3.218
67. 1.294 1.668 1.996 2.383 2.651
3.216
68. 1.294 1.668 1.995 2.382 2.650
3.214
69. 1.294 1.667 1.995 2.382 2.649
3.213
70. 1.294 1.667 1.994 2.381 2.648
3.211
71. 1.294 1.667 1.994 2.380 2.647
3.209
72. 1.293 1.666 1.993 2.379 2.646
3.207
73. 1.293 1.666 1.993 2.379 2.645
3.206
74. 1.293 1.666 1.993 2.378 2.644
3.204
75. 1.293 1.665 1.992 2.377 2.643
3.202
76. 1.293 1.665 1.992 2.376 2.642
3.201
77. 1.293 1.665 1.991 2.376 2.641
3.199
78. 1.292 1.665 1.991 2.375 2.640
3.198
79. 1.292 1.664 1.990 2.374 2.640
3.197
80. 1.292 1.664 1.990 2.374 2.639
3.195
81. 1.292 1.664 1.990 2.373 2.638
3.194
82. 1.292 1.664 1.989 2.373 2.637
3.193
83. 1.292 1.663 1.989 2.372 2.636
3.191
84. 1.292 1.663 1.989 2.372 2.636
3.190
85. 1.292 1.663 1.988 2.371 2.635
3.189
86. 1.291 1.663 1.988 2.370 2.634
3.188
87. 1.291 1.663 1.988 2.370 2.634
3.187
88. 1.291 1.662 1.987 2.369 2.633
3.185
89. 1.291 1.662 1.987 2.369 2.632
3.184
90. 1.291 1.662 1.987 2.368 2.632
3.183
91. 1.291 1.662 1.986 2.368 2.631
3.182
92. 1.291 1.662 1.986 2.368 2.630
3.181
93. 1.291 1.661 1.986 2.367 2.630
3.180
94. 1.291 1.661 1.986 2.367 2.629
3.179
95. 1.291 1.661 1.985 2.366 2.629
3.178
96. 1.290 1.661 1.985 2.366 2.628
3.177
97. 1.290 1.661 1.985 2.365 2.627
3.176
98. 1.290 1.661 1.984 2.365 2.627
3.175
99. 1.290 1.660 1.984 2.365 2.626
3.175
100. 1.290 1.660 1.984 2.364 2.626
3.174
1.282 1.645 1.960 2.326 2.576
3.090
1.3.6.7.3. Upper Critical Values of the F Distribution

1.3. EDA Techniques
1.3.6.7.3. Upper Critical Values of the F
Distribution
How to
Use This
Table
This table contains the upper critical values of the F
distribution. This table is used for one-sided F tests at the =
0.05, 0.10, and 0.01 levels.
More specifically, a test statistic is computed with and
degrees of freedom, and the result is compared to this table.
For a one-sided test, the null hypothesis is rejected when the
test statistic is greater than the tabled value. This is
demonstrated with the graph of an F distribution with = 10
and = 10. The shaded area of the graph indicates the
rejection region at the significance level. Since this is a one-
sided test, we have probability in the upper tail of exceeding
the critical value and zero in the lower tail. Because the F
distribution is asymmetric, a two-sided test requires a set of of
tables (not included here) that contain the rejection regions for
both the lower and upper tails.
Contents The following tables for from 1 to 100 are included:
1. One sided, 5% significance level, = 1 - 10
Upper critical values of the F distribution
for numerator degrees of freedom and denominator
degrees of freedom
5% significance level
\ 1 2 3 4 5
6 7 8 9 10

1 161.448 199.500 215.707 224.583 230.162
233.986 236.768 238.882 240.543 241.882
2 18.513 19.000 19.164 19.247 19.296
19.330 19.353 19.371 19.385 19.396
3 10.128 9.552 9.277 9.117 9.013
8.941 8.887 8.845 8.812 8.786
4 7.709 6.944 6.591 6.388 6.256
6.163 6.094 6.041 5.999 5.964
5 6.608 5.786 5.409 5.192 5.050
4.950 4.876 4.818 4.772 4.735
6 5.987 5.143 4.757 4.534 4.387
4.284 4.207 4.147 4.099 4.060
7 5.591 4.737 4.347 4.120 3.972
3.866 3.787 3.726 3.677 3.637
8 5.318 4.459 4.066 3.838 3.687
3.581 3.500 3.438 3.388 3.347
9 5.117 4.256 3.863 3.633 3.482
3.374 3.293 3.230 3.179 3.137
10 4.965 4.103 3.708 3.478 3.326
3.217 3.135 3.072 3.020 2.978
11 4.844 3.982 3.587 3.357 3.204
3.095 3.012 2.948 2.896 2.854
12 4.747 3.885 3.490 3.259 3.106
2.996 2.913 2.849 2.796 2.753
13 4.667 3.806 3.411 3.179 3.025
2.915 2.832 2.767 2.714 2.671
14 4.600 3.739 3.344 3.112 2.958
2.848 2.764 2.699 2.646 2.602
15 4.543 3.682 3.287 3.056 2.901
2.790 2.707 2.641 2.588 2.544
16 4.494 3.634 3.239 3.007 2.852
2.741 2.657 2.591 2.538 2.494
17 4.451 3.592 3.197 2.965 2.810
2.699 2.614 2.548 2.494 2.450
18 4.414 3.555 3.160 2.928 2.773
2.661 2.577 2.510 2.456 2.412
19 4.381 3.522 3.127 2.895 2.740
2.628 2.544 2.477 2.423 2.378
20 4.351 3.493 3.098 2.866 2.711
2.599 2.514 2.447 2.393 2.348
21 4.325 3.467 3.072 2.840 2.685
2.573 2.488 2.420 2.366 2.321
22 4.301 3.443 3.049 2.817 2.661
2.549 2.464 2.397 2.342 2.297
23 4.279 3.422 3.028 2.796 2.640
2.528 2.442 2.375 2.320 2.275
24 4.260 3.403 3.009 2.776 2.621
2.508 2.423 2.355 2.300 2.255
25 4.242 3.385 2.991 2.759 2.603
2.490 2.405 2.337 2.282 2.236
26 4.225 3.369 2.975 2.743 2.587
2.474 2.388 2.321 2.265 2.220
27 4.210 3.354 2.960 2.728 2.572
2.459 2.373 2.305 2.250 2.204
28 4.196 3.340 2.947 2.714 2.558
2.445 2.359 2.291 2.236 2.190
29 4.183 3.328 2.934 2.701 2.545
2.432 2.346 2.278 2.223 2.177
30 4.171 3.316 2.922 2.690 2.534
2.421 2.334 2.266 2.211 2.165
31 4.160 3.305 2.911 2.679 2.523
2.409 2.323 2.255 2.199 2.153
32 4.149 3.295 2.901 2.668 2.512
2.399 2.313 2.244 2.189 2.142
33 4.139 3.285 2.892 2.659 2.503
2.389 2.303 2.235 2.179 2.133
34 4.130 3.276 2.883 2.650 2.494
2.380 2.294 2.225 2.170 2.123
35 4.121 3.267 2.874 2.641 2.485
2.372 2.285 2.217 2.161 2.114
36 4.113 3.259 2.866 2.634 2.477
2.364 2.277 2.209 2.153 2.106
37 4.105 3.252 2.859 2.626 2.470
2.356 2.270 2.201 2.145 2.098
38 4.098 3.245 2.852 2.619 2.463
2.349 2.262 2.194 2.138 2.091
39 4.091 3.238 2.845 2.612 2.456
2.342 2.255 2.187 2.131 2.084
40 4.085 3.232 2.839 2.606 2.449
2.336 2.249 2.180 2.124 2.077
41 4.079 3.226 2.833 2.600 2.443
2.330 2.243 2.174 2.118 2.071
42 4.073 3.220 2.827 2.594 2.438
2.324 2.237 2.168 2.112 2.065
43 4.067 3.214 2.822 2.589 2.432
2.318 2.232 2.163 2.106 2.059
44 4.062 3.209 2.816 2.584 2.427
2.313 2.226 2.157 2.101 2.054
45 4.057 3.204 2.812 2.579 2.422
2.308 2.221 2.152 2.096 2.049
46 4.052 3.200 2.807 2.574 2.417
2.304 2.216 2.147 2.091 2.044
47 4.047 3.195 2.802 2.570 2.413
2.299 2.212 2.143 2.086 2.039
48 4.043 3.191 2.798 2.565 2.409
2.295 2.207 2.138 2.082 2.035
49 4.038 3.187 2.794 2.561 2.404
2.290 2.203 2.134 2.077 2.030
50 4.034 3.183 2.790 2.557 2.400
2.286 2.199 2.130 2.073 2.026
51 4.030 3.179 2.786 2.553 2.397
2.283 2.195 2.126 2.069 2.022
52 4.027 3.175 2.783 2.550 2.393
2.279 2.192 2.122 2.066 2.018
53 4.023 3.172 2.779 2.546 2.389
2.275 2.188 2.119 2.062 2.015
54 4.020 3.168 2.776 2.543 2.386
2.272 2.185 2.115 2.059 2.011
55 4.016 3.165 2.773 2.540 2.383
2.269 2.181 2.112 2.055 2.008
56 4.013 3.162 2.769 2.537 2.380
2.266 2.178 2.109 2.052 2.005
57 4.010 3.159 2.766 2.534 2.377
2.263 2.175 2.106 2.049 2.001
58 4.007 3.156 2.764 2.531 2.374
2.260 2.172 2.103 2.046 1.998
59 4.004 3.153 2.761 2.528 2.371
2.257 2.169 2.100 2.043 1.995
60 4.001 3.150 2.758 2.525 2.368
2.254 2.167 2.097 2.040 1.993
61 3.998 3.148 2.755 2.523 2.366
2.251 2.164 2.094 2.037 1.990
62 3.996 3.145 2.753 2.520 2.363
2.249 2.161 2.092 2.035 1.987
63 3.993 3.143 2.751 2.518 2.361
2.246 2.159 2.089 2.032 1.985
64 3.991 3.140 2.748 2.515 2.358
2.244 2.156 2.087 2.030 1.982
65 3.989 3.138 2.746 2.513 2.356
2.242 2.154 2.084 2.027 1.980
66 3.986 3.136 2.744 2.511 2.354
2.239 2.152 2.082 2.025 1.977
67 3.984 3.134 2.742 2.509 2.352
2.237 2.150 2.080 2.023 1.975
68 3.982 3.132 2.740 2.507 2.350
2.235 2.148 2.078 2.021 1.973
69 3.980 3.130 2.737 2.505 2.348
2.233 2.145 2.076 2.019 1.971
70 3.978 3.128 2.736 2.503 2.346
2.231 2.143 2.074 2.017 1.969
71 3.976 3.126 2.734 2.501 2.344
2.229 2.142 2.072 2.015 1.967
72 3.974 3.124 2.732 2.499 2.342
2.227 2.140 2.070 2.013 1.965
73 3.972 3.122 2.730 2.497 2.340
2.226 2.138 2.068 2.011 1.963
74 3.970 3.120 2.728 2.495 2.338
2.224 2.136 2.066 2.009 1.961
75 3.968 3.119 2.727 2.494 2.337
2.222 2.134 2.064 2.007 1.959
76 3.967 3.117 2.725 2.492 2.335
2.220 2.133 2.063 2.006 1.958
77 3.965 3.115 2.723 2.490 2.333
2.219 2.131 2.061 2.004 1.956
78 3.963 3.114 2.722 2.489 2.332
2.217 2.129 2.059 2.002 1.954
79 3.962 3.112 2.720 2.487 2.330
2.216 2.128 2.058 2.001 1.953
80 3.960 3.111 2.719 2.486 2.329
2.214 2.126 2.056 1.999 1.951
81 3.959 3.109 2.717 2.484 2.327
2.213 2.125 2.055 1.998 1.950
82 3.957 3.108 2.716 2.483 2.326
2.211 2.123 2.053 1.996 1.948
83 3.956 3.107 2.715 2.482 2.324
2.210 2.122 2.052 1.995 1.947
84 3.955 3.105 2.713 2.480 2.323
2.209 2.121 2.051 1.993 1.945
85 3.953 3.104 2.712 2.479 2.322
2.207 2.119 2.049 1.992 1.944
86 3.952 3.103 2.711 2.478 2.321
2.206 2.118 2.048 1.991 1.943
87 3.951 3.101 2.709 2.476 2.319
2.205 2.117 2.047 1.989 1.941
88 3.949 3.100 2.708 2.475 2.318
2.203 2.115 2.045 1.988 1.940
89 3.948 3.099 2.707 2.474 2.317
2.202 2.114 2.044 1.987 1.939
90 3.947 3.098 2.706 2.473 2.316
2.201 2.113 2.043 1.986 1.938
91 3.946 3.097 2.705 2.472 2.315
2.200 2.112 2.042 1.984 1.936
92 3.945 3.095 2.704 2.471 2.313
2.199 2.111 2.041 1.983 1.935
93 3.943 3.094 2.703 2.470 2.312
2.198 2.110 2.040 1.982 1.934
94 3.942 3.093 2.701 2.469 2.311
2.197 2.109 2.038 1.981 1.933
95 3.941 3.092 2.700 2.467 2.310
2.196 2.108 2.037 1.980 1.932
96 3.940 3.091 2.699 2.466 2.309
2.195 2.106 2.036 1.979 1.931
97 3.939 3.090 2.698 2.465 2.308
2.194 2.105 2.035 1.978 1.930
98 3.938 3.089 2.697 2.465 2.307
2.193 2.104 2.034 1.977 1.929
99 3.937 3.088 2.696 2.464 2.306
2.192 2.103 2.033 1.976 1.928
100 3.936 3.087 2.696 2.463 2.305
2.191 2.103 2.032 1.975 1.927
\ 11 12 13 14 15
16 17 18 19 20

1 242.983 243.906 244.690 245.364 245.950
246.464 246.918 247.323 247.686 248.013
2 19.405 19.413 19.419 19.424 19.429
19.433 19.437 19.440 19.443 19.446
3 8.763 8.745 8.729 8.715 8.703
8.692 8.683 8.675 8.667 8.660
4 5.936 5.912 5.891 5.873 5.858
5.844 5.832 5.821 5.811 5.803
5 4.704 4.678 4.655 4.636 4.619
4.604 4.590 4.579 4.568 4.558
6 4.027 4.000 3.976 3.956 3.938
3.922 3.908 3.896 3.884 3.874
7 3.603 3.575 3.550 3.529 3.511
3.494 3.480 3.467 3.455 3.445
8 3.313 3.284 3.259 3.237 3.218
3.202 3.187 3.173 3.161 3.150
9 3.102 3.073 3.048 3.025 3.006
2.989 2.974 2.960 2.948 2.936
10 2.943 2.913 2.887 2.865 2.845
2.828 2.812 2.798 2.785 2.774
11 2.818 2.788 2.761 2.739 2.719
2.701 2.685 2.671 2.658 2.646
12 2.717 2.687 2.660 2.637 2.617
2.599 2.583 2.568 2.555 2.544
13 2.635 2.604 2.577 2.554 2.533
2.515 2.499 2.484 2.471 2.459
14 2.565 2.534 2.507 2.484 2.463
2.445 2.428 2.413 2.400 2.388
15 2.507 2.475 2.448 2.424 2.403
2.385 2.368 2.353 2.340 2.328
16 2.456 2.425 2.397 2.373 2.352
2.333 2.317 2.302 2.288 2.276
17 2.413 2.381 2.353 2.329 2.308
2.289 2.272 2.257 2.243 2.230
18 2.374 2.342 2.314 2.290 2.269
2.250 2.233 2.217 2.203 2.191
19 2.340 2.308 2.280 2.256 2.234
2.215 2.198 2.182 2.168 2.155
20 2.310 2.278 2.250 2.225 2.203
2.184 2.167 2.151 2.137 2.124
21 2.283 2.250 2.222 2.197 2.176
2.156 2.139 2.123 2.109 2.096
22 2.259 2.226 2.198 2.173 2.151
2.131 2.114 2.098 2.084 2.071
23 2.236 2.204 2.175 2.150 2.128
2.109 2.091 2.075 2.061 2.048
24 2.216 2.183 2.155 2.130 2.108
2.088 2.070 2.054 2.040 2.027
25 2.198 2.165 2.136 2.111 2.089
2.069 2.051 2.035 2.021 2.007
26 2.181 2.148 2.119 2.094 2.072
2.052 2.034 2.018 2.003 1.990
27 2.166 2.132 2.103 2.078 2.056
2.036 2.018 2.002 1.987 1.974
28 2.151 2.118 2.089 2.064 2.041
2.021 2.003 1.987 1.972 1.959
29 2.138 2.104 2.075 2.050 2.027
2.007 1.989 1.973 1.958 1.945
30 2.126 2.092 2.063 2.037 2.015
1.995 1.976 1.960 1.945 1.932
31 2.114 2.080 2.051 2.026 2.003
1.983 1.965 1.948 1.933 1.920
32 2.103 2.070 2.040 2.015 1.992
1.972 1.953 1.937 1.922 1.908
33 2.093 2.060 2.030 2.004 1.982
1.961 1.943 1.926 1.911 1.898
34 2.084 2.050 2.021 1.995 1.972
1.952 1.933 1.917 1.902 1.888
35 2.075 2.041 2.012 1.986 1.963
1.942 1.924 1.907 1.892 1.878
36 2.067 2.033 2.003 1.977 1.954
1.934 1.915 1.899 1.883 1.870
37 2.059 2.025 1.995 1.969 1.946
1.926 1.907 1.890 1.875 1.861
38 2.051 2.017 1.988 1.962 1.939
1.918 1.899 1.883 1.867 1.853
39 2.044 2.010 1.981 1.954 1.931
1.911 1.892 1.875 1.860 1.846
40 2.038 2.003 1.974 1.948 1.924
1.904 1.885 1.868 1.853 1.839
41 2.031 1.997 1.967 1.941 1.918
1.897 1.879 1.862 1.846 1.832
42 2.025 1.991 1.961 1.935 1.912
1.891 1.872 1.855 1.840 1.826
43 2.020 1.985 1.955 1.929 1.906
1.885 1.866 1.849 1.834 1.820
44 2.014 1.980 1.950 1.924 1.900
1.879 1.861 1.844 1.828 1.814
45 2.009 1.974 1.945 1.918 1.895
1.874 1.855 1.838 1.823 1.808
46 2.004 1.969 1.940 1.913 1.890
1.869 1.850 1.833 1.817 1.803
47 1.999 1.965 1.935 1.908 1.885
1.864 1.845 1.828 1.812 1.798
48 1.995 1.960 1.930 1.904 1.880
1.859 1.840 1.823 1.807 1.793
49 1.990 1.956 1.926 1.899 1.876
1.855 1.836 1.819 1.803 1.789
50 1.986 1.952 1.921 1.895 1.871
1.850 1.831 1.814 1.798 1.784
51 1.982 1.947 1.917 1.891 1.867
1.846 1.827 1.810 1.794 1.780
52 1.978 1.944 1.913 1.887 1.863
1.842 1.823 1.806 1.790 1.776
53 1.975 1.940 1.910 1.883 1.859
1.838 1.819 1.802 1.786 1.772
54 1.971 1.936 1.906 1.879 1.856
1.835 1.816 1.798 1.782 1.768
55 1.968 1.933 1.903 1.876 1.852
1.831 1.812 1.795 1.779 1.764
56 1.964 1.930 1.899 1.873 1.849
1.828 1.809 1.791 1.775 1.761
57 1.961 1.926 1.896 1.869 1.846
1.824 1.805 1.788 1.772 1.757
58 1.958 1.923 1.893 1.866 1.842
1.821 1.802 1.785 1.769 1.754
59 1.955 1.920 1.890 1.863 1.839
1.818 1.799 1.781 1.766 1.751
60 1.952 1.917 1.887 1.860 1.836
1.815 1.796 1.778 1.763 1.748
61 1.949 1.915 1.884 1.857 1.834
1.812 1.793 1.776 1.760 1.745
62 1.947 1.912 1.882 1.855 1.831
1.809 1.790 1.773 1.757 1.742
63 1.944 1.909 1.879 1.852 1.828
1.807 1.787 1.770 1.754 1.739
64 1.942 1.907 1.876 1.849 1.826
1.804 1.785 1.767 1.751 1.737
65 1.939 1.904 1.874 1.847 1.823
1.802 1.782 1.765 1.749 1.734
66 1.937 1.902 1.871 1.845 1.821
1.799 1.780 1.762 1.746 1.732
67 1.935 1.900 1.869 1.842 1.818
1.797 1.777 1.760 1.744 1.729
68 1.932 1.897 1.867 1.840 1.816
1.795 1.775 1.758 1.742 1.727
69 1.930 1.895 1.865 1.838 1.814
1.792 1.773 1.755 1.739 1.725
70 1.928 1.893 1.863 1.836 1.812
1.790 1.771 1.753 1.737 1.722
71 1.926 1.891 1.861 1.834 1.810
1.788 1.769 1.751 1.735 1.720
72 1.924 1.889 1.859 1.832 1.808
1.786 1.767 1.749 1.733 1.718
73 1.922 1.887 1.857 1.830 1.806
1.784 1.765 1.747 1.731 1.716
74 1.921 1.885 1.855 1.828 1.804
1.782 1.763 1.745 1.729 1.714
75 1.919 1.884 1.853 1.826 1.802
1.780 1.761 1.743 1.727 1.712
76 1.917 1.882 1.851 1.824 1.800
1.778 1.759 1.741 1.725 1.710
77 1.915 1.880 1.849 1.822 1.798
1.777 1.757 1.739 1.723 1.708
78 1.914 1.878 1.848 1.821 1.797
1.775 1.755 1.738 1.721 1.707
79 1.912 1.877 1.846 1.819 1.795
1.773 1.754 1.736 1.720 1.705
80 1.910 1.875 1.845 1.817 1.793
1.772 1.752 1.734 1.718 1.703
81 1.909 1.874 1.843 1.816 1.792
1.770 1.750 1.733 1.716 1.702
82 1.907 1.872 1.841 1.814 1.790
1.768 1.749 1.731 1.715 1.700
83 1.906 1.871 1.840 1.813 1.789
1.767 1.747 1.729 1.713 1.698
84 1.905 1.869 1.838 1.811 1.787
1.765 1.746 1.728 1.712 1.697
85 1.903 1.868 1.837 1.810 1.786
1.764 1.744 1.726 1.710 1.695
86 1.902 1.867 1.836 1.808 1.784
1.762 1.743 1.725 1.709 1.694
87 1.900 1.865 1.834 1.807 1.783
1.761 1.741 1.724 1.707 1.692
88 1.899 1.864 1.833 1.806 1.782
1.760 1.740 1.722 1.706 1.691
89 1.898 1.863 1.832 1.804 1.780
1.758 1.739 1.721 1.705 1.690
90 1.897 1.861 1.830 1.803 1.779
1.757 1.737 1.720 1.703 1.688
91 1.895 1.860 1.829 1.802 1.778
1.756 1.736 1.718 1.702 1.687
92 1.894 1.859 1.828 1.801 1.776
1.755 1.735 1.717 1.701 1.686
93 1.893 1.858 1.827 1.800 1.775
1.753 1.734 1.716 1.699 1.684
94 1.892 1.857 1.826 1.798 1.774
1.752 1.733 1.715 1.698 1.683
95 1.891 1.856 1.825 1.797 1.773
1.751 1.731 1.713 1.697 1.682
96 1.890 1.854 1.823 1.796 1.772
1.750 1.730 1.712 1.696 1.681
97 1.889 1.853 1.822 1.795 1.771
1.749 1.729 1.711 1.695 1.680
98 1.888 1.852 1.821 1.794 1.770
1.748 1.728 1.710 1.694 1.679
99 1.887 1.851 1.820 1.793 1.769
1.747 1.727 1.709 1.693 1.678
100 1.886 1.850 1.819 1.792 1.768
1.746 1.726 1.708 1.691 1.676
degrees of freedom
\ 1 2 3 4 5
6 7 8 9 10

1 39.863 49.500 53.593 55.833 57.240
58.204 58.906 59.439 59.858 60.195
2 8.526 9.000 9.162 9.243 9.293
9.326 9.349 9.367 9.381 9.392
3 5.538 5.462 5.391 5.343 5.309
5.285 5.266 5.252 5.240 5.230
4 4.545 4.325 4.191 4.107 4.051
4.010 3.979 3.955 3.936 3.920
5 4.060 3.780 3.619 3.520 3.453
3.405 3.368 3.339 3.316 3.297
6 3.776 3.463 3.289 3.181 3.108
3.055 3.014 2.983 2.958 2.937
7 3.589 3.257 3.074 2.961 2.883
2.827 2.785 2.752 2.725 2.703
8 3.458 3.113 2.924 2.806 2.726
2.668 2.624 2.589 2.561 2.538
9 3.360 3.006 2.813 2.693 2.611
2.551 2.505 2.469 2.440 2.416
10 3.285 2.924 2.728 2.605 2.522
2.461 2.414 2.377 2.347 2.323
11 3.225 2.860 2.660 2.536 2.451
2.389 2.342 2.304 2.274 2.248
12 3.177 2.807 2.606 2.480 2.394
2.331 2.283 2.245 2.214 2.188
13 3.136 2.763 2.560 2.434 2.347
2.283 2.234 2.195 2.164 2.138
14 3.102 2.726 2.522 2.395 2.307
2.243 2.193 2.154 2.122 2.095
15 3.073 2.695 2.490 2.361 2.273
2.208 2.158 2.119 2.086 2.059
16 3.048 2.668 2.462 2.333 2.244
2.178 2.128 2.088 2.055 2.028
17 3.026 2.645 2.437 2.308 2.218
2.152 2.102 2.061 2.028 2.001
18 3.007 2.624 2.416 2.286 2.196
2.130 2.079 2.038 2.005 1.977
19 2.990 2.606 2.397 2.266 2.176
2.109 2.058 2.017 1.984 1.956
20 2.975 2.589 2.380 2.249 2.158
2.091 2.040 1.999 1.965 1.937
21 2.961 2.575 2.365 2.233 2.142
2.075 2.023 1.982 1.948 1.920
22 2.949 2.561 2.351 2.219 2.128
2.060 2.008 1.967 1.933 1.904
23 2.937 2.549 2.339 2.207 2.115
2.047 1.995 1.953 1.919 1.890
24 2.927 2.538 2.327 2.195 2.103
2.035 1.983 1.941 1.906 1.877
25 2.918 2.528 2.317 2.184 2.092
2.024 1.971 1.929 1.895 1.866
26 2.909 2.519 2.307 2.174 2.082
2.014 1.961 1.919 1.884 1.855
27 2.901 2.511 2.299 2.165 2.073
2.005 1.952 1.909 1.874 1.845
28 2.894 2.503 2.291 2.157 2.064
1.996 1.943 1.900 1.865 1.836
29 2.887 2.495 2.283 2.149 2.057
1.988 1.935 1.892 1.857 1.827
30 2.881 2.489 2.276 2.142 2.049
1.980 1.927 1.884 1.849 1.819
31 2.875 2.482 2.270 2.136 2.042
1.973 1.920 1.877 1.842 1.812
32 2.869 2.477 2.263 2.129 2.036
1.967 1.913 1.870 1.835 1.805
33 2.864 2.471 2.258 2.123 2.030
1.961 1.907 1.864 1.828 1.799
34 2.859 2.466 2.252 2.118 2.024
1.955 1.901 1.858 1.822 1.793
35 2.855 2.461 2.247 2.113 2.019
1.950 1.896 1.852 1.817 1.787
36 2.850 2.456 2.243 2.108 2.014
1.945 1.891 1.847 1.811 1.781
37 2.846 2.452 2.238 2.103 2.009
1.940 1.886 1.842 1.806 1.776
38 2.842 2.448 2.234 2.099 2.005
1.935 1.881 1.838 1.802 1.772
39 2.839 2.444 2.230 2.095 2.001
1.931 1.877 1.833 1.797 1.767
40 2.835 2.440 2.226 2.091 1.997
1.927 1.873 1.829 1.793 1.763
41 2.832 2.437 2.222 2.087 1.993
1.923 1.869 1.825 1.789 1.759
42 2.829 2.434 2.219 2.084 1.989
1.919 1.865 1.821 1.785 1.755
43 2.826 2.430 2.216 2.080 1.986
1.916 1.861 1.817 1.781 1.751
44 2.823 2.427 2.213 2.077 1.983
1.913 1.858 1.814 1.778 1.747
45 2.820 2.425 2.210 2.074 1.980
1.909 1.855 1.811 1.774 1.744
46 2.818 2.422 2.207 2.071 1.977
1.906 1.852 1.808 1.771 1.741
47 2.815 2.419 2.204 2.068 1.974
1.903 1.849 1.805 1.768 1.738
48 2.813 2.417 2.202 2.066 1.971
1.901 1.846 1.802 1.765 1.735
49 2.811 2.414 2.199 2.063 1.968
1.898 1.843 1.799 1.763 1.732
50 2.809 2.412 2.197 2.061 1.966
1.895 1.840 1.796 1.760 1.729
51 2.807 2.410 2.194 2.058 1.964
1.893 1.838 1.794 1.757 1.727
52 2.805 2.408 2.192 2.056 1.961
1.891 1.836 1.791 1.755 1.724
53 2.803 2.406 2.190 2.054 1.959
1.888 1.833 1.789 1.752 1.722
54 2.801 2.404 2.188 2.052 1.957
1.886 1.831 1.787 1.750 1.719
55 2.799 2.402 2.186 2.050 1.955
1.884 1.829 1.785 1.748 1.717
56 2.797 2.400 2.184 2.048 1.953
1.882 1.827 1.782 1.746 1.715
57 2.796 2.398 2.182 2.046 1.951
1.880 1.825 1.780 1.744 1.713
58 2.794 2.396 2.181 2.044 1.949
1.878 1.823 1.779 1.742 1.711
59 2.793 2.395 2.179 2.043 1.947
1.876 1.821 1.777 1.740 1.709
60 2.791 2.393 2.177 2.041 1.946
1.875 1.819 1.775 1.738 1.707
61 2.790 2.392 2.176 2.039 1.944
1.873 1.818 1.773 1.736 1.705
62 2.788 2.390 2.174 2.038 1.942
1.871 1.816 1.771 1.735 1.703
63 2.787 2.389 2.173 2.036 1.941
1.870 1.814 1.770 1.733 1.702
64 2.786 2.387 2.171 2.035 1.939
1.868 1.813 1.768 1.731 1.700
65 2.784 2.386 2.170 2.033 1.938
1.867 1.811 1.767 1.730 1.699
66 2.783 2.385 2.169 2.032 1.937
1.865 1.810 1.765 1.728 1.697
67 2.782 2.384 2.167 2.031 1.935
1.864 1.808 1.764 1.727 1.696
68 2.781 2.382 2.166 2.029 1.934
1.863 1.807 1.762 1.725 1.694
69 2.780 2.381 2.165 2.028 1.933
1.861 1.806 1.761 1.724 1.693
70 2.779 2.380 2.164 2.027 1.931
1.860 1.804 1.760 1.723 1.691
71 2.778 2.379 2.163 2.026 1.930
1.859 1.803 1.758 1.721 1.690
72 2.777 2.378 2.161 2.025 1.929
1.858 1.802 1.757 1.720 1.689
73 2.776 2.377 2.160 2.024 1.928
1.856 1.801 1.756 1.719 1.687
74 2.775 2.376 2.159 2.022 1.927
1.855 1.800 1.755 1.718 1.686
75 2.774 2.375 2.158 2.021 1.926
1.854 1.798 1.754 1.716 1.685
76 2.773 2.374 2.157 2.020 1.925
1.853 1.797 1.752 1.715 1.684
77 2.772 2.373 2.156 2.019 1.924
1.852 1.796 1.751 1.714 1.683
78 2.771 2.372 2.155 2.018 1.923
1.851 1.795 1.750 1.713 1.682
79 2.770 2.371 2.154 2.017 1.922
1.850 1.794 1.749 1.712 1.681
80 2.769 2.370 2.154 2.016 1.921
1.849 1.793 1.748 1.711 1.680
81 2.769 2.369 2.153 2.016 1.920
1.848 1.792 1.747 1.710 1.679
82 2.768 2.368 2.152 2.015 1.919
1.847 1.791 1.746 1.709 1.678
83 2.767 2.368 2.151 2.014 1.918
1.846 1.790 1.745 1.708 1.677
84 2.766 2.367 2.150 2.013 1.917
1.845 1.790 1.744 1.707 1.676
85 2.765 2.366 2.149 2.012 1.916
1.845 1.789 1.744 1.706 1.675
86 2.765 2.365 2.149 2.011 1.915
1.844 1.788 1.743 1.705 1.674
87 2.764 2.365 2.148 2.011 1.915
1.843 1.787 1.742 1.705 1.673
88 2.763 2.364 2.147 2.010 1.914
1.842 1.786 1.741 1.704 1.672
89 2.763 2.363 2.146 2.009 1.913
1.841 1.785 1.740 1.703 1.671
90 2.762 2.363 2.146 2.008 1.912
1.841 1.785 1.739 1.702 1.670
91 2.761 2.362 2.145 2.008 1.912
1.840 1.784 1.739 1.701 1.670
92 2.761 2.361 2.144 2.007 1.911
1.839 1.783 1.738 1.701 1.669
93 2.760 2.361 2.144 2.006 1.910
1.838 1.782 1.737 1.700 1.668
94 2.760 2.360 2.143 2.006 1.910
1.838 1.782 1.736 1.699 1.667
95 2.759 2.359 2.142 2.005 1.909
1.837 1.781 1.736 1.698 1.667
96 2.759 2.359 2.142 2.004 1.908
1.836 1.780 1.735 1.698 1.666
97 2.758 2.358 2.141 2.004 1.908
1.836 1.780 1.734 1.697 1.665
98 2.757 2.358 2.141 2.003 1.907
1.835 1.779 1.734 1.696 1.665
99 2.757 2.357 2.140 2.003 1.906
1.835 1.778 1.733 1.696 1.664
100 2.756 2.356 2.139 2.002 1.906
1.834 1.778 1.732 1.695 1.663
\ 11 12 13 14 15
16 17 18 19 20

1 60.473 60.705 60.903 61.073 61.220
61.350 61.464 61.566 61.658 61.740
2 9.401 9.408 9.415 9.420 9.425
9.429 9.433 9.436 9.439 9.441
3 5.222 5.216 5.210 5.205 5.200
5.196 5.193 5.190 5.187 5.184
4 3.907 3.896 3.886 3.878 3.870
3.864 3.858 3.853 3.849 3.844
5 3.282 3.268 3.257 3.247 3.238
3.230 3.223 3.217 3.212 3.207
6 2.920 2.905 2.892 2.881 2.871
2.863 2.855 2.848 2.842 2.836
7 2.684 2.668 2.654 2.643 2.632
2.623 2.615 2.607 2.601 2.595
8 2.519 2.502 2.488 2.475 2.464
2.455 2.446 2.438 2.431 2.425
9 2.396 2.379 2.364 2.351 2.340
2.329 2.320 2.312 2.305 2.298
10 2.302 2.284 2.269 2.255 2.244
2.233 2.224 2.215 2.208 2.201
11 2.227 2.209 2.193 2.179 2.167
2.156 2.147 2.138 2.130 2.123
12 2.166 2.147 2.131 2.117 2.105
2.094 2.084 2.075 2.067 2.060
13 2.116 2.097 2.080 2.066 2.053
2.042 2.032 2.023 2.014 2.007
14 2.073 2.054 2.037 2.022 2.010
1.998 1.988 1.978 1.970 1.962
15 2.037 2.017 2.000 1.985 1.972
1.961 1.950 1.941 1.932 1.924
16 2.005 1.985 1.968 1.953 1.940
1.928 1.917 1.908 1.899 1.891
17 1.978 1.958 1.940 1.925 1.912
1.900 1.889 1.879 1.870 1.862
18 1.954 1.933 1.916 1.900 1.887
1.875 1.864 1.854 1.845 1.837
19 1.932 1.912 1.894 1.878 1.865
1.852 1.841 1.831 1.822 1.814
20 1.913 1.892 1.875 1.859 1.845
1.833 1.821 1.811 1.802 1.794
21 1.896 1.875 1.857 1.841 1.827
1.815 1.803 1.793 1.784 1.776
22 1.880 1.859 1.841 1.825 1.811
1.798 1.787 1.777 1.768 1.759
23 1.866 1.845 1.827 1.811 1.796
1.784 1.772 1.762 1.753 1.744
24 1.853 1.832 1.814 1.797 1.783
1.770 1.759 1.748 1.739 1.730
25 1.841 1.820 1.802 1.785 1.771
1.758 1.746 1.736 1.726 1.718
26 1.830 1.809 1.790 1.774 1.760
1.747 1.735 1.724 1.715 1.706
27 1.820 1.799 1.780 1.764 1.749
1.736 1.724 1.714 1.704 1.695
28 1.811 1.790 1.771 1.754 1.740
1.726 1.715 1.704 1.694 1.685
29 1.802 1.781 1.762 1.745 1.731
1.717 1.705 1.695 1.685 1.676
30 1.794 1.773 1.754 1.737 1.722
1.709 1.697 1.686 1.676 1.667
31 1.787 1.765 1.746 1.729 1.714
1.701 1.689 1.678 1.668 1.659
32 1.780 1.758 1.739 1.722 1.707
1.694 1.682 1.671 1.661 1.652
33 1.773 1.751 1.732 1.715 1.700
1.687 1.675 1.664 1.654 1.645
34 1.767 1.745 1.726 1.709 1.694
1.680 1.668 1.657 1.647 1.638
35 1.761 1.739 1.720 1.703 1.688
1.674 1.662 1.651 1.641 1.632
36 1.756 1.734 1.715 1.697 1.682
1.669 1.656 1.645 1.635 1.626
37 1.751 1.729 1.709 1.692 1.677
1.663 1.651 1.640 1.630 1.620
38 1.746 1.724 1.704 1.687 1.672
1.658 1.646 1.635 1.624 1.615
39 1.741 1.719 1.700 1.682 1.667
1.653 1.641 1.630 1.619 1.610
40 1.737 1.715 1.695 1.678 1.662
1.649 1.636 1.625 1.615 1.605
41 1.733 1.710 1.691 1.673 1.658
1.644 1.632 1.620 1.610 1.601
42 1.729 1.706 1.687 1.669 1.654
1.640 1.628 1.616 1.606 1.596
43 1.725 1.703 1.683 1.665 1.650
1.636 1.624 1.612 1.602 1.592
44 1.721 1.699 1.679 1.662 1.646
1.632 1.620 1.608 1.598 1.588
45 1.718 1.695 1.676 1.658 1.643
1.629 1.616 1.605 1.594 1.585
46 1.715 1.692 1.672 1.655 1.639
1.625 1.613 1.601 1.591 1.581
47 1.712 1.689 1.669 1.652 1.636
1.622 1.609 1.598 1.587 1.578
48 1.709 1.686 1.666 1.648 1.633
1.619 1.606 1.594 1.584 1.574
49 1.706 1.683 1.663 1.645 1.630
1.616 1.603 1.591 1.581 1.571
50 1.703 1.680 1.660 1.643 1.627
1.613 1.600 1.588 1.578 1.568
51 1.700 1.677 1.658 1.640 1.624
1.610 1.597 1.586 1.575 1.565
52 1.698 1.675 1.655 1.637 1.621
1.607 1.594 1.583 1.572 1.562
53 1.695 1.672 1.652 1.635 1.619
1.605 1.592 1.580 1.570 1.560
54 1.693 1.670 1.650 1.632 1.616
1.602 1.589 1.578 1.567 1.557
55 1.691 1.668 1.648 1.630 1.614
1.600 1.587 1.575 1.564 1.555
56 1.688 1.666 1.645 1.628 1.612
1.597 1.585 1.573 1.562 1.552
57 1.686 1.663 1.643 1.625 1.610
1.595 1.582 1.571 1.560 1.550
58 1.684 1.661 1.641 1.623 1.607
1.593 1.580 1.568 1.558 1.548
59 1.682 1.659 1.639 1.621 1.605
1.591 1.578 1.566 1.555 1.546
60 1.680 1.657 1.637 1.619 1.603
1.589 1.576 1.564 1.553 1.543
61 1.679 1.656 1.635 1.617 1.601
1.587 1.574 1.562 1.551 1.541
62 1.677 1.654 1.634 1.616 1.600
1.585 1.572 1.560 1.549 1.540
63 1.675 1.652 1.632 1.614 1.598
1.583 1.570 1.558 1.548 1.538
64 1.673 1.650 1.630 1.612 1.596
1.582 1.569 1.557 1.546 1.536
65 1.672 1.649 1.628 1.610 1.594
1.580 1.567 1.555 1.544 1.534
66 1.670 1.647 1.627 1.609 1.593
1.578 1.565 1.553 1.542 1.532
67 1.669 1.646 1.625 1.607 1.591
1.577 1.564 1.552 1.541 1.531
68 1.667 1.644 1.624 1.606 1.590
1.575 1.562 1.550 1.539 1.529
69 1.666 1.643 1.622 1.604 1.588
1.574 1.560 1.548 1.538 1.527
70 1.665 1.641 1.621 1.603 1.587
1.572 1.559 1.547 1.536 1.526
71 1.663 1.640 1.619 1.601 1.585
1.571 1.557 1.545 1.535 1.524
72 1.662 1.639 1.618 1.600 1.584
1.569 1.556 1.544 1.533 1.523
73 1.661 1.637 1.617 1.599 1.583
1.568 1.555 1.543 1.532 1.522
74 1.659 1.636 1.616 1.597 1.581
1.567 1.553 1.541 1.530 1.520
75 1.658 1.635 1.614 1.596 1.580
1.565 1.552 1.540 1.529 1.519
76 1.657 1.634 1.613 1.595 1.579
1.564 1.551 1.539 1.528 1.518
77 1.656 1.632 1.612 1.594 1.578
1.563 1.550 1.538 1.527 1.516
78 1.655 1.631 1.611 1.593 1.576
1.562 1.548 1.536 1.525 1.515
79 1.654 1.630 1.610 1.592 1.575
1.561 1.547 1.535 1.524 1.514
80 1.653 1.629 1.609 1.590 1.574
1.559 1.546 1.534 1.523 1.513
81 1.652 1.628 1.608 1.589 1.573
1.558 1.545 1.533 1.522 1.512
82 1.651 1.627 1.607 1.588 1.572
1.557 1.544 1.532 1.521 1.511
83 1.650 1.626 1.606 1.587 1.571
1.556 1.543 1.531 1.520 1.509
84 1.649 1.625 1.605 1.586 1.570
1.555 1.542 1.530 1.519 1.508
85 1.648 1.624 1.604 1.585 1.569
1.554 1.541 1.529 1.518 1.507
86 1.647 1.623 1.603 1.584 1.568
1.553 1.540 1.528 1.517 1.506
87 1.646 1.622 1.602 1.583 1.567
1.552 1.539 1.527 1.516 1.505
88 1.645 1.622 1.601 1.583 1.566
1.551 1.538 1.526 1.515 1.504
89 1.644 1.621 1.600 1.582 1.565
1.550 1.537 1.525 1.514 1.503
90 1.643 1.620 1.599 1.581 1.564
1.550 1.536 1.524 1.513 1.503
91 1.643 1.619 1.598 1.580 1.564
1.549 1.535 1.523 1.512 1.502
92 1.642 1.618 1.598 1.579 1.563
1.548 1.534 1.522 1.511 1.501
93 1.641 1.617 1.597 1.578 1.562
1.547 1.534 1.521 1.510 1.500
94 1.640 1.617 1.596 1.578 1.561
1.546 1.533 1.521 1.509 1.499
95 1.640 1.616 1.595 1.577 1.560
1.545 1.532 1.520 1.509 1.498
96 1.639 1.615 1.594 1.576 1.560
1.545 1.531 1.519 1.508 1.497
97 1.638 1.614 1.594 1.575 1.559
1.544 1.530 1.518 1.507 1.497
98 1.637 1.614 1.593 1.575 1.558
1.543 1.530 1.517 1.506 1.496
99 1.637 1.613 1.592 1.574 1.557
1.542 1.529 1.517 1.505 1.495
100 1.636 1.612 1.592 1.573 1.557
1.542 1.528 1.516 1.505 1.494
degrees of freedom
\ 1 2 3 4 5
6 7 8 9 10

1 4052.19 4999.52 5403.34 5624.62 5763.65
5858.97 5928.33 5981.10 6022.50 6055.85
2 98.502 99.000 99.166 99.249 99.300
99.333 99.356 99.374 99.388 99.399
3 34.116 30.816 29.457 28.710 28.237
27.911 27.672 27.489 27.345 27.229
4 21.198 18.000 16.694 15.977 15.522
15.207 14.976 14.799 14.659 14.546
5 16.258 13.274 12.060 11.392 10.967
10.672 10.456 10.289 10.158 10.051
6 13.745 10.925 9.780 9.148 8.746
8.466 8.260 8.102 7.976 7.874
7 12.246 9.547 8.451 7.847 7.460
7.191 6.993 6.840 6.719 6.620
8 11.259 8.649 7.591 7.006 6.632
6.371 6.178 6.029 5.911 5.814
9 10.561 8.022 6.992 6.422 6.057
5.802 5.613 5.467 5.351 5.257
10 10.044 7.559 6.552 5.994 5.636
5.386 5.200 5.057 4.942 4.849
11 9.646 7.206 6.217 5.668 5.316
5.069 4.886 4.744 4.632 4.539
12 9.330 6.927 5.953 5.412 5.064
4.821 4.640 4.499 4.388 4.296
13 9.074 6.701 5.739 5.205 4.862
4.620 4.441 4.302 4.191 4.100
14 8.862 6.515 5.564 5.035 4.695
4.456 4.278 4.140 4.030 3.939
15 8.683 6.359 5.417 4.893 4.556
4.318 4.142 4.004 3.895 3.805
16 8.531 6.226 5.292 4.773 4.437
4.202 4.026 3.890 3.780 3.691
17 8.400 6.112 5.185 4.669 4.336
4.102 3.927 3.791 3.682 3.593
18 8.285 6.013 5.092 4.579 4.248
4.015 3.841 3.705 3.597 3.508
19 8.185 5.926 5.010 4.500 4.171
3.939 3.765 3.631 3.523 3.434
20 8.096 5.849 4.938 4.431 4.103
3.871 3.699 3.564 3.457 3.368
21 8.017 5.780 4.874 4.369 4.042
3.812 3.640 3.506 3.398 3.310
22 7.945 5.719 4.817 4.313 3.988
3.758 3.587 3.453 3.346 3.258
23 7.881 5.664 4.765 4.264 3.939
3.710 3.539 3.406 3.299 3.211
24 7.823 5.614 4.718 4.218 3.895
3.667 3.496 3.363 3.256 3.168
25 7.770 5.568 4.675 4.177 3.855
3.627 3.457 3.324 3.217 3.129
26 7.721 5.526 4.637 4.140 3.818
3.591 3.421 3.288 3.182 3.094
27 7.677 5.488 4.601 4.106 3.785
3.558 3.388 3.256 3.149 3.062
28 7.636 5.453 4.568 4.074 3.754
3.528 3.358 3.226 3.120 3.032
29 7.598 5.420 4.538 4.045 3.725
3.499 3.330 3.198 3.092 3.005
30 7.562 5.390 4.510 4.018 3.699
3.473 3.305 3.173 3.067 2.979
31 7.530 5.362 4.484 3.993 3.675
3.449 3.281 3.149 3.043 2.955
32 7.499 5.336 4.459 3.969 3.652
3.427 3.258 3.127 3.021 2.934
33 7.471 5.312 4.437 3.948 3.630
3.406 3.238 3.106 3.000 2.913
34 7.444 5.289 4.416 3.927 3.611
3.386 3.218 3.087 2.981 2.894
35 7.419 5.268 4.396 3.908 3.592
3.368 3.200 3.069 2.963 2.876
36 7.396 5.248 4.377 3.890 3.574
3.351 3.183 3.052 2.946 2.859
37 7.373 5.229 4.360 3.873 3.558
3.334 3.167 3.036 2.930 2.843
38 7.353 5.211 4.343 3.858 3.542
3.319 3.152 3.021 2.915 2.828
39 7.333 5.194 4.327 3.843 3.528
3.305 3.137 3.006 2.901 2.814
40 7.314 5.179 4.313 3.828 3.514
3.291 3.124 2.993 2.888 2.801
41 7.296 5.163 4.299 3.815 3.501
3.278 3.111 2.980 2.875 2.788
42 7.280 5.149 4.285 3.802 3.488
3.266 3.099 2.968 2.863 2.776
43 7.264 5.136 4.273 3.790 3.476
3.254 3.087 2.957 2.851 2.764
44 7.248 5.123 4.261 3.778 3.465
3.243 3.076 2.946 2.840 2.754
45 7.234 5.110 4.249 3.767 3.454
3.232 3.066 2.935 2.830 2.743
46 7.220 5.099 4.238 3.757 3.444
3.222 3.056 2.925 2.820 2.733
47 7.207 5.087 4.228 3.747 3.434
3.213 3.046 2.916 2.811 2.724
48 7.194 5.077 4.218 3.737 3.425
3.204 3.037 2.907 2.802 2.715
49 7.182 5.066 4.208 3.728 3.416
3.195 3.028 2.898 2.793 2.706
50 7.171 5.057 4.199 3.720 3.408
3.186 3.020 2.890 2.785 2.698
51 7.159 5.047 4.191 3.711 3.400
3.178 3.012 2.882 2.777 2.690
52 7.149 5.038 4.182 3.703 3.392
3.171 3.005 2.874 2.769 2.683
53 7.139 5.030 4.174 3.695 3.384
3.163 2.997 2.867 2.762 2.675
54 7.129 5.021 4.167 3.688 3.377
3.156 2.990 2.860 2.755 2.668
55 7.119 5.013 4.159 3.681 3.370
3.149 2.983 2.853 2.748 2.662
56 7.110 5.006 4.152 3.674 3.363
3.143 2.977 2.847 2.742 2.655
57 7.102 4.998 4.145 3.667 3.357
3.136 2.971 2.841 2.736 2.649
58 7.093 4.991 4.138 3.661 3.351
3.130 2.965 2.835 2.730 2.643
59 7.085 4.984 4.132 3.655 3.345
3.124 2.959 2.829 2.724 2.637
60 7.077 4.977 4.126 3.649 3.339
3.119 2.953 2.823 2.718 2.632
61 7.070 4.971 4.120 3.643 3.333
3.113 2.948 2.818 2.713 2.626
62 7.062 4.965 4.114 3.638 3.328
3.108 2.942 2.813 2.708 2.621
63 7.055 4.959 4.109 3.632 3.323
3.103 2.937 2.808 2.703 2.616
64 7.048 4.953 4.103 3.627 3.318
3.098 2.932 2.803 2.698 2.611
65 7.042 4.947 4.098 3.622 3.313
3.093 2.928 2.798 2.693 2.607
66 7.035 4.942 4.093 3.618 3.308
3.088 2.923 2.793 2.689 2.602
67 7.029 4.937 4.088 3.613 3.304
3.084 2.919 2.789 2.684 2.598
68 7.023 4.932 4.083 3.608 3.299
3.080 2.914 2.785 2.680 2.593
69 7.017 4.927 4.079 3.604 3.295
3.075 2.910 2.781 2.676 2.589
70 7.011 4.922 4.074 3.600 3.291
3.071 2.906 2.777 2.672 2.585
71 7.006 4.917 4.070 3.596 3.287
3.067 2.902 2.773 2.668 2.581
72 7.001 4.913 4.066 3.591 3.283
3.063 2.898 2.769 2.664 2.578
73 6.995 4.908 4.062 3.588 3.279
3.060 2.895 2.765 2.660 2.574
74 6.990 4.904 4.058 3.584 3.275
3.056 2.891 2.762 2.657 2.570
75 6.985 4.900 4.054 3.580 3.272
3.052 2.887 2.758 2.653 2.567
76 6.981 4.896 4.050 3.577 3.268
3.049 2.884 2.755 2.650 2.563
77 6.976 4.892 4.047 3.573 3.265
3.046 2.881 2.751 2.647 2.560
78 6.971 4.888 4.043 3.570 3.261
3.042 2.877 2.748 2.644 2.557
79 6.967 4.884 4.040 3.566 3.258
3.039 2.874 2.745 2.640 2.554
80 6.963 4.881 4.036 3.563 3.255
3.036 2.871 2.742 2.637 2.551
81 6.958 4.877 4.033 3.560 3.252
3.033 2.868 2.739 2.634 2.548
82 6.954 4.874 4.030 3.557 3.249
3.030 2.865 2.736 2.632 2.545
83 6.950 4.870 4.027 3.554 3.246
3.027 2.863 2.733 2.629 2.542
84 6.947 4.867 4.024 3.551 3.243
3.025 2.860 2.731 2.626 2.539
85 6.943 4.864 4.021 3.548 3.240
3.022 2.857 2.728 2.623 2.537
86 6.939 4.861 4.018 3.545 3.238
3.019 2.854 2.725 2.621 2.534
87 6.935 4.858 4.015 3.543 3.235
3.017 2.852 2.723 2.618 2.532
88 6.932 4.855 4.012 3.540 3.233
3.014 2.849 2.720 2.616 2.529
89 6.928 4.852 4.010 3.538 3.230
3.012 2.847 2.718 2.613 2.527
90 6.925 4.849 4.007 3.535 3.228
3.009 2.845 2.715 2.611 2.524
91 6.922 4.846 4.004 3.533 3.225
3.007 2.842 2.713 2.609 2.522
92 6.919 4.844 4.002 3.530 3.223
3.004 2.840 2.711 2.606 2.520
93 6.915 4.841 3.999 3.528 3.221
3.002 2.838 2.709 2.604 2.518
94 6.912 4.838 3.997 3.525 3.218
3.000 2.835 2.706 2.602 2.515
95 6.909 4.836 3.995 3.523 3.216
2.998 2.833 2.704 2.600 2.513
96 6.906 4.833 3.992 3.521 3.214
2.996 2.831 2.702 2.598 2.511
97 6.904 4.831 3.990 3.519 3.212
2.994 2.829 2.700 2.596 2.509
98 6.901 4.829 3.988 3.517 3.210
2.992 2.827 2.698 2.594 2.507
99 6.898 4.826 3.986 3.515 3.208
2.990 2.825 2.696 2.592 2.505
100 6.895 4.824 3.984 3.513 3.206
2.988 2.823 2.694 2.590 2.503
\ 11 12 13 14 15
16 17 18 19 20

1. 6083.35 6106.35 6125.86 6142.70 6157.28
6170.12 6181.42 6191.52 6200.58 6208.74
2. 99.408 99.416 99.422 99.428 99.432
99.437 99.440 99.444 99.447 99.449
3. 27.133 27.052 26.983 26.924 26.872
26.827 26.787 26.751 26.719 26.690
4. 14.452 14.374 14.307 14.249 14.198
14.154 14.115 14.080 14.048 14.020
5. 9.963 9.888 9.825 9.770 9.722
9.680 9.643 9.610 9.580 9.553
6. 7.790 7.718 7.657 7.605 7.559
7.519 7.483 7.451 7.422 7.396
7. 6.538 6.469 6.410 6.359 6.314
6.275 6.240 6.209 6.181 6.155
8. 5.734 5.667 5.609 5.559 5.515
5.477 5.442 5.412 5.384 5.359
9. 5.178 5.111 5.055 5.005 4.962
4.924 4.890 4.860 4.833 4.808
10. 4.772 4.706 4.650 4.601 4.558
4.520 4.487 4.457 4.430 4.405
11. 4.462 4.397 4.342 4.293 4.251
4.213 4.180 4.150 4.123 4.099
12. 4.220 4.155 4.100 4.052 4.010
3.972 3.939 3.909 3.883 3.858
13. 4.025 3.960 3.905 3.857 3.815
3.778 3.745 3.716 3.689 3.665
14. 3.864 3.800 3.745 3.698 3.656
3.619 3.586 3.556 3.529 3.505
15. 3.730 3.666 3.612 3.564 3.522
3.485 3.452 3.423 3.396 3.372
16. 3.616 3.553 3.498 3.451 3.409
3.372 3.339 3.310 3.283 3.259
17. 3.519 3.455 3.401 3.353 3.312
3.275 3.242 3.212 3.186 3.162
18. 3.434 3.371 3.316 3.269 3.227
3.190 3.158 3.128 3.101 3.077
19. 3.360 3.297 3.242 3.195 3.153
3.116 3.084 3.054 3.027 3.003
20. 3.294 3.231 3.177 3.130 3.088
3.051 3.018 2.989 2.962 2.938
21. 3.236 3.173 3.119 3.072 3.030
2.993 2.960 2.931 2.904 2.880
22. 3.184 3.121 3.067 3.019 2.978
2.941 2.908 2.879 2.852 2.827
23. 3.137 3.074 3.020 2.973 2.931
2.894 2.861 2.832 2.805 2.781
24. 3.094 3.032 2.977 2.930 2.889
2.852 2.819 2.789 2.762 2.738
25. 3.056 2.993 2.939 2.892 2.850
2.813 2.780 2.751 2.724 2.699
26. 3.021 2.958 2.904 2.857 2.815
2.778 2.745 2.715 2.688 2.664
27. 2.988 2.926 2.871 2.824 2.783
2.746 2.713 2.683 2.656 2.632
28. 2.959 2.896 2.842 2.795 2.753
2.716 2.683 2.653 2.626 2.602
29. 2.931 2.868 2.814 2.767 2.726
2.689 2.656 2.626 2.599 2.574
30. 2.906 2.843 2.789 2.742 2.700
2.663 2.630 2.600 2.573 2.549
31. 2.882 2.820 2.765 2.718 2.677
2.640 2.606 2.577 2.550 2.525
32. 2.860 2.798 2.744 2.696 2.655
2.618 2.584 2.555 2.527 2.503
33. 2.840 2.777 2.723 2.676 2.634
2.597 2.564 2.534 2.507 2.482
34. 2.821 2.758 2.704 2.657 2.615
2.578 2.545 2.515 2.488 2.463
35. 2.803 2.740 2.686 2.639 2.597
2.560 2.527 2.497 2.470 2.445
36. 2.786 2.723 2.669 2.622 2.580
2.543 2.510 2.480 2.453 2.428
37. 2.770 2.707 2.653 2.606 2.564
2.527 2.494 2.464 2.437 2.412
38. 2.755 2.692 2.638 2.591 2.549
2.512 2.479 2.449 2.421 2.397
39. 2.741 2.678 2.624 2.577 2.535
2.498 2.465 2.434 2.407 2.382
40. 2.727 2.665 2.611 2.563 2.522
2.484 2.451 2.421 2.394 2.369
41. 2.715 2.652 2.598 2.551 2.509
2.472 2.438 2.408 2.381 2.356
42. 2.703 2.640 2.586 2.539 2.497
2.460 2.426 2.396 2.369 2.344
43. 2.691 2.629 2.575 2.527 2.485
2.448 2.415 2.385 2.357 2.332
44. 2.680 2.618 2.564 2.516 2.475
2.437 2.404 2.374 2.346 2.321
45. 2.670 2.608 2.553 2.506 2.464
2.427 2.393 2.363 2.336 2.311
46. 2.660 2.598 2.544 2.496 2.454
2.417 2.384 2.353 2.326 2.301
47. 2.651 2.588 2.534 2.487 2.445
2.408 2.374 2.344 2.316 2.291
48. 2.642 2.579 2.525 2.478 2.436
2.399 2.365 2.335 2.307 2.282
49. 2.633 2.571 2.517 2.469 2.427
2.390 2.356 2.326 2.299 2.274
50. 2.625 2.562 2.508 2.461 2.419
2.382 2.348 2.318 2.290 2.265
51. 2.617 2.555 2.500 2.453 2.411
2.374 2.340 2.310 2.282 2.257
52. 2.610 2.547 2.493 2.445 2.403
2.366 2.333 2.302 2.275 2.250
53. 2.602 2.540 2.486 2.438 2.396
2.359 2.325 2.295 2.267 2.242
54. 2.595 2.533 2.479 2.431 2.389
2.352 2.318 2.288 2.260 2.235
55. 2.589 2.526 2.472 2.424 2.382
2.345 2.311 2.281 2.253 2.228
56. 2.582 2.520 2.465 2.418 2.376
2.339 2.305 2.275 2.247 2.222
57. 2.576 2.513 2.459 2.412 2.370
2.332 2.299 2.268 2.241 2.215
58. 2.570 2.507 2.453 2.406 2.364
2.326 2.293 2.262 2.235 2.209
59. 2.564 2.502 2.447 2.400 2.358
2.320 2.287 2.256 2.229 2.203
60. 2.559 2.496 2.442 2.394 2.352
2.315 2.281 2.251 2.223 2.198
61. 2.553 2.491 2.436 2.389 2.347
2.309 2.276 2.245 2.218 2.192
62. 2.548 2.486 2.431 2.384 2.342
2.304 2.270 2.240 2.212 2.187
63. 2.543 2.481 2.426 2.379 2.337
2.299 2.265 2.235 2.207 2.182
64. 2.538 2.476 2.421 2.374 2.332
2.294 2.260 2.230 2.202 2.177
65. 2.534 2.471 2.417 2.369 2.327
2.289 2.256 2.225 2.198 2.172
66. 2.529 2.466 2.412 2.365 2.322
2.285 2.251 2.221 2.193 2.168
67. 2.525 2.462 2.408 2.360 2.318
2.280 2.247 2.216 2.188 2.163
68. 2.520 2.458 2.403 2.356 2.314
2.276 2.242 2.212 2.184 2.159
69. 2.516 2.454 2.399 2.352 2.310
2.272 2.238 2.208 2.180 2.155
70. 2.512 2.450 2.395 2.348 2.306
2.268 2.234 2.204 2.176 2.150
71. 2.508 2.446 2.391 2.344 2.302
2.264 2.230 2.200 2.172 2.146
72. 2.504 2.442 2.388 2.340 2.298
2.260 2.226 2.196 2.168 2.143
73. 2.501 2.438 2.384 2.336 2.294
2.256 2.223 2.192 2.164 2.139
74. 2.497 2.435 2.380 2.333 2.290
2.253 2.219 2.188 2.161 2.135
75. 2.494 2.431 2.377 2.329 2.287
2.249 2.215 2.185 2.157 2.132
76. 2.490 2.428 2.373 2.326 2.284
2.246 2.212 2.181 2.154 2.128
77. 2.487 2.424 2.370 2.322 2.280
2.243 2.209 2.178 2.150 2.125
78. 2.484 2.421 2.367 2.319 2.277
2.239 2.206 2.175 2.147 2.122
79. 2.481 2.418 2.364 2.316 2.274
2.236 2.202 2.172 2.144 2.118
80. 2.478 2.415 2.361 2.313 2.271
2.233 2.199 2.169 2.141 2.115
81. 2.475 2.412 2.358 2.310 2.268
2.230 2.196 2.166 2.138 2.112
82. 2.472 2.409 2.355 2.307 2.265
2.227 2.193 2.163 2.135 2.109
83. 2.469 2.406 2.352 2.304 2.262
2.224 2.191 2.160 2.132 2.106
84. 2.466 2.404 2.349 2.302 2.259
2.222 2.188 2.157 2.129 2.104
85. 2.464 2.401 2.347 2.299 2.257
2.219 2.185 2.154 2.126 2.101
86. 2.461 2.398 2.344 2.296 2.254
2.216 2.182 2.152 2.124 2.098
87. 2.459 2.396 2.342 2.294 2.252
2.214 2.180 2.149 2.121 2.096
88. 2.456 2.393 2.339 2.291 2.249
2.211 2.177 2.147 2.119 2.093
89. 2.454 2.391 2.337 2.289 2.247
2.209 2.175 2.144 2.116 2.091
90. 2.451 2.389 2.334 2.286 2.244
2.206 2.172 2.142 2.114 2.088
91. 2.449 2.386 2.332 2.284 2.242
2.204 2.170 2.139 2.111 2.086
92. 2.447 2.384 2.330 2.282 2.240
2.202 2.168 2.137 2.109 2.083
93. 2.444 2.382 2.327 2.280 2.237
2.200 2.166 2.135 2.107 2.081
94. 2.442 2.380 2.325 2.277 2.235
2.197 2.163 2.133 2.105 2.079
95. 2.440 2.378 2.323 2.275 2.233
2.195 2.161 2.130 2.102 2.077
96. 2.438 2.375 2.321 2.273 2.231
2.193 2.159 2.128 2.100 2.075
97. 2.436 2.373 2.319 2.271 2.229
2.191 2.157 2.126 2.098 2.073
98. 2.434 2.371 2.317 2.269 2.227
2.189 2.155 2.124 2.096 2.071
99. 2.432 2.369 2.315 2.267 2.225
2.187 2.153 2.122 2.094 2.069
100. 2.430 2.368 2.313 2.265 2.223
2.185 2.151 2.120 2.092 2.067
1.3.6.7.4. Critical Values of the Chi-Square Distribution

1.3. EDA Techniques
1.3.6.7.4. Critical Values of the Chi-Square
Distribution
How to
Use This
Table
This table contains the critical values of the chi-square
distribution. Because of the lack of symmetry of the chi-
square distribution, separate tables are provided for the upper
and lower tails of the distribution.
A test statistic with degrees of freedom is computed from
the data. For upper-tail one-sided tests, the test statistic is
compared with a value from the table of upper-tail critical
values. For two-sided tests, the test statistic is compared with
values from both the table for the upper-tail critical values and
the table for the lower-tail critical values.
The significance level, , is demonstrated with the graph
below which shows a chi-square distribution with 3 degrees of
freedom for a two-sided test at significance level = 0.05. If
the test statistic is greater than the upper-tail critical value or
less than the lower-tail critical value, we reject the null
hypothesis. Specific instructions are given below.
Given a specified value of :
1. For a two-sided test, find the column corresponding to
1-/2 in the table for upper-tail critical values and reject
the null hypothesis if the test statistic is greater than the
tabled value. Similarly, find the column corresponding
to /2 in the table for lower-tail critical values and
reject the null hypothesis if the test statistic is less than
the tabled value.
2. For an upper-tail one-sided test, find the column
corresponding to 1- in the table containing upper-tail
critical and reject the null hypothesis if the test statistic
is greater than the tabled value.
3. For a lower-tail one-sided test, find the column
corresponding to in the lower-tail critical values table
and reject the null hypothesis if the computed test
statistic is less than the tabled value.
Upper-tail critical values of chi-square distribution with
degrees of freedom
Probability less than the critical
value
0.90 0.95 0.975 0.99
0.999
1 2.706 3.841 5.024 6.635
10.828
2 4.605 5.991 7.378 9.210
13.816
3 6.251 7.815 9.348 11.345
16.266
4 7.779 9.488 11.143 13.277
18.467
5 9.236 11.070 12.833 15.086
20.515
6 10.645 12.592 14.449 16.812
22.458
7 12.017 14.067 16.013 18.475
24.322
8 13.362 15.507 17.535 20.090
26.125
9 14.684 16.919 19.023 21.666
27.877
10 15.987 18.307 20.483 23.209
29.588
11 17.275 19.675 21.920 24.725
31.264
12 18.549 21.026 23.337 26.217
32.910
13 19.812 22.362 24.736 27.688
34.528
14 21.064 23.685 26.119 29.141
36.123
15 22.307 24.996 27.488 30.578
37.697
16 23.542 26.296 28.845 32.000
39.252
17 24.769 27.587 30.191 33.409
40.790
18 25.989 28.869 31.526 34.805
42.312
19 27.204 30.144 32.852 36.191
43.820
20 28.412 31.410 34.170 37.566
45.315
21 29.615 32.671 35.479 38.932
46.797
22 30.813 33.924 36.781 40.289
48.268
23 32.007 35.172 38.076 41.638
49.728
24 33.196 36.415 39.364 42.980
51.179
25 34.382 37.652 40.646 44.314
52.620
26 35.563 38.885 41.923 45.642
54.052
27 36.741 40.113 43.195 46.963
55.476
28 37.916 41.337 44.461 48.278
56.892
29 39.087 42.557 45.722 49.588
58.301
30 40.256 43.773 46.979 50.892
59.703
31 41.422 44.985 48.232 52.191
61.098
32 42.585 46.194 49.480 53.486
62.487
33 43.745 47.400 50.725 54.776
63.870
34 44.903 48.602 51.966 56.061
65.247
35 46.059 49.802 53.203 57.342
66.619
36 47.212 50.998 54.437 58.619
67.985
37 48.363 52.192 55.668 59.893
69.347
38 49.513 53.384 56.896 61.162
70.703
39 50.660 54.572 58.120 62.428
72.055
40 51.805 55.758 59.342 63.691
73.402
41 52.949 56.942 60.561 64.950
74.745
42 54.090 58.124 61.777 66.206
76.084
43 55.230 59.304 62.990 67.459
77.419
44 56.369 60.481 64.201 68.710
78.750
45 57.505 61.656 65.410 69.957
80.077
46 58.641 62.830 66.617 71.201
81.400
47 59.774 64.001 67.821 72.443
82.720
48 60.907 65.171 69.023 73.683
84.037
49 62.038 66.339 70.222 74.919
85.351
50 63.167 67.505 71.420 76.154
86.661
51 64.295 68.669 72.616 77.386
87.968
52 65.422 69.832 73.810 78.616
89.272
53 66.548 70.993 75.002 79.843
90.573
54 67.673 72.153 76.192 81.069
91.872
55 68.796 73.311 77.380 82.292
93.168
56 69.919 74.468 78.567 83.513
94.461
57 71.040 75.624 79.752 84.733
95.751
58 72.160 76.778 80.936 85.950
97.039
59 73.279 77.931 82.117 87.166
98.324
60 74.397 79.082 83.298 88.379
99.607
61 75.514 80.232 84.476 89.591
100.888
62 76.630 81.381 85.654 90.802
102.166
63 77.745 82.529 86.830 92.010
103.442
64 78.860 83.675 88.004 93.217
104.716
65 79.973 84.821 89.177 94.422
105.988
66 81.085 85.965 90.349 95.626
107.258
67 82.197 87.108 91.519 96.828
108.526
68 83.308 88.250 92.689 98.028
109.791
69 84.418 89.391 93.856 99.228
111.055
70 85.527 90.531 95.023 100.425
112.317
71 86.635 91.670 96.189 101.621
113.577
72 87.743 92.808 97.353 102.816
114.835
73 88.850 93.945 98.516 104.010
116.092
74 89.956 95.081 99.678 105.202
117.346
75 91.061 96.217 100.839 106.393
118.599
76 92.166 97.351 101.999 107.583
119.850
77 93.270 98.484 103.158 108.771
121.100
78 94.374 99.617 104.316 109.958
122.348
79 95.476 100.749 105.473 111.144
123.594
80 96.578 101.879 106.629 112.329
124.839
81 97.680 103.010 107.783 113.512
126.083
82 98.780 104.139 108.937 114.695
127.324
83 99.880 105.267 110.090 115.876
128.565
84 100.980 106.395 111.242 117.057
129.804
85 102.079 107.522 112.393 118.236
131.041
86 103.177 108.648 113.544 119.414
132.277
87 104.275 109.773 114.693 120.591
133.512
88 105.372 110.898 115.841 121.767
134.746
89 106.469 112.022 116.989 122.942
135.978
90 107.565 113.145 118.136 124.116
137.208
91 108.661 114.268 119.282 125.289
138.438
92 109.756 115.390 120.427 126.462
139.666
93 110.850 116.511 121.571 127.633
140.893
94 111.944 117.632 122.715 128.803
142.119
95 113.038 118.752 123.858 129.973
143.344
96 114.131 119.871 125.000 131.141
144.567
97 115.223 120.990 126.141 132.309
145.789
98 116.315 122.108 127.282 133.476
147.010
99 117.407 123.225 128.422 134.642
148.230
100 118.498 124.342 129.561 135.807
149.449
100 118.498 124.342 129.561 135.807
149.449
Lower-tail critical values of chi-square distribution with
degrees of freedom
Probability less than the critical
value
0.10 0.05 0.025 0.01
0.001
1. .016 .004 .001 .000
.000
2. .211 .103 .051 .020
.002
3. .584 .352 .216 .115
.024
4. 1.064 .711 .484 .297
.091
5. 1.610 1.145 .831 .554
.210
6. 2.204 1.635 1.237 .872
.381
7. 2.833 2.167 1.690 1.239
.598
8. 3.490 2.733 2.180 1.646
.857
9. 4.168 3.325 2.700 2.088
1.152
10. 4.865 3.940 3.247 2.558
1.479
11. 5.578 4.575 3.816 3.053
1.834
12. 6.304 5.226 4.404 3.571
2.214
13. 7.042 5.892 5.009 4.107
2.617
14. 7.790 6.571 5.629 4.660
3.041
15. 8.547 7.261 6.262 5.229
3.483
16. 9.312 7.962 6.908 5.812
3.942
17. 10.085 8.672 7.564 6.408
4.416
18. 10.865 9.390 8.231 7.015
4.905
19. 11.651 10.117 8.907 7.633
5.407
20. 12.443 10.851 9.591 8.260
5.921
21. 13.240 11.591 10.283 8.897
6.447
22. 14.041 12.338 10.982 9.542
6.983
23. 14.848 13.091 11.689 10.196
7.529
24. 15.659 13.848 12.401 10.856
8.085
25. 16.473 14.611 13.120 11.524
8.649
26. 17.292 15.379 13.844 12.198
9.222
27. 18.114 16.151 14.573 12.879
9.803
28. 18.939 16.928 15.308 13.565
10.391
29. 19.768 17.708 16.047 14.256
10.986
30. 20.599 18.493 16.791 14.953
11.588
31. 21.434 19.281 17.539 15.655
12.196
32. 22.271 20.072 18.291 16.362
12.811
33. 23.110 20.867 19.047 17.074
13.431
34. 23.952 21.664 19.806 17.789
14.057
35. 24.797 22.465 20.569 18.509
14.688
36. 25.643 23.269 21.336 19.233
15.324
37. 26.492 24.075 22.106 19.960
15.965
38. 27.343 24.884 22.878 20.691
16.611
39. 28.196 25.695 23.654 21.426
17.262
40. 29.051 26.509 24.433 22.164
17.916
41. 29.907 27.326 25.215 22.906
18.575
42. 30.765 28.144 25.999 23.650
19.239
43. 31.625 28.965 26.785 24.398
19.906
44. 32.487 29.787 27.575 25.148
20.576
45. 33.350 30.612 28.366 25.901
21.251
46. 34.215 31.439 29.160 26.657
21.929
47. 35.081 32.268 29.956 27.416
22.610
48. 35.949 33.098 30.755 28.177
23.295
49. 36.818 33.930 31.555 28.941
23.983
50. 37.689 34.764 32.357 29.707
24.674
51. 38.560 35.600 33.162 30.475
25.368
52. 39.433 36.437 33.968 31.246
26.065
53. 40.308 37.276 34.776 32.018
26.765
54. 41.183 38.116 35.586 32.793
27.468
55. 42.060 38.958 36.398 33.570
28.173
56. 42.937 39.801 37.212 34.350
28.881
57. 43.816 40.646 38.027 35.131
29.592
58. 44.696 41.492 38.844 35.913
30.305
59. 45.577 42.339 39.662 36.698
31.020
60. 46.459 43.188 40.482 37.485
31.738
61. 47.342 44.038 41.303 38.273
32.459
62. 48.226 44.889 42.126 39.063
33.181
63. 49.111 45.741 42.950 39.855
33.906
64. 49.996 46.595 43.776 40.649
34.633
65. 50.883 47.450 44.603 41.444
35.362
66. 51.770 48.305 45.431 42.240
36.093
67. 52.659 49.162 46.261 43.038
36.826
68. 53.548 50.020 47.092 43.838
37.561
69. 54.438 50.879 47.924 44.639
38.298
70. 55.329 51.739 48.758 45.442
39.036
71. 56.221 52.600 49.592 46.246
39.777
72. 57.113 53.462 50.428 47.051
40.519
73. 58.006 54.325 51.265 47.858
41.264
74. 58.900 55.189 52.103 48.666
42.010
75. 59.795 56.054 52.942 49.475
42.757
76. 60.690 56.920 53.782 50.286
43.507
77. 61.586 57.786 54.623 51.097
44.258
78. 62.483 58.654 55.466 51.910
45.010
79. 63.380 59.522 56.309 52.725
45.764
80. 64.278 60.391 57.153 53.540
46.520
81. 65.176 61.261 57.998 54.357
47.277
82. 66.076 62.132 58.845 55.174
48.036
83. 66.976 63.004 59.692 55.993
48.796
84. 67.876 63.876 60.540 56.813
49.557
85. 68.777 64.749 61.389 57.634
50.320
86. 69.679 65.623 62.239 58.456
51.085
87. 70.581 66.498 63.089 59.279
51.850
88. 71.484 67.373 63.941 60.103
52.617
89. 72.387 68.249 64.793 60.928
53.386
90. 73.291 69.126 65.647 61.754
54.155
91. 74.196 70.003 66.501 62.581
54.926
92. 75.100 70.882 67.356 63.409
55.698
93. 76.006 71.760 68.211 64.238
56.472
94. 76.912 72.640 69.068 65.068
57.246
95. 77.818 73.520 69.925 65.898
58.022
96. 78.725 74.401 70.783 66.730
58.799
97. 79.633 75.282 71.642 67.562
59.577
98. 80.541 76.164 72.501 68.396
60.356
99. 81.449 77.046 73.361 69.230
61.137
100. 82.358 77.929 74.222 70.065
61.918
1.3.6.7.5. Critical Values of the t* Distribution

1.3. EDA Techniques
1.3.6.7.5.
Critical Values of the t
*
Distribution
How to
Use This
Table
This table contains upper critical values of the t* distribution
that are appropriate for determining whether or not a
calibration line is in a state of statistical control from
measurements on a check standard at three points in the
calibration interval. A test statistic with degrees of freedom
is compared with the critical value. If the absolute value of the
test statistic exceeds the tabled value, the calibration of the
instrument is judged to be out of control.
Upper critical values of t* distribution at significance level 0.05
for testing the output of a linear calibration line at 3 points

1 37.544 61 2.455
2 7.582 62 2.454
3 4.826 63 2.453
4 3.941 64 2.452
5 3.518 65 2.451
6 3.274 66 2.450
7 3.115 67 2.449
8 3.004 68 2.448
9 2.923 69 2.447
10 2.860 70 2.446
11 2.811 71 2.445
12 2.770 72 2.445
13 2.737 73 2.444
14 2.709 74 2.443
15 2.685 75 2.442
16 2.665 76 2.441
17 2.647 77 2.441
18 2.631 78 2.440
19 2.617 79 2.439
20 2.605 80 2.439
21 2.594 81 2.438
22 2.584 82 2.437
23 2.574 83 2.437
24 2.566 84 2.436
25 2.558 85 2.436
26 2.551 86 2.435
27 2.545 87 2.435
28 2.539 88 2.434
29 2.534 89 2.434
30 2.528 90 2.433
31 2.524 91 2.432
32 2.519 92 2.432
33 2.515 93 2.431
34 2.511 94 2.431
35 2.507 95 2.431
1.3.6.7.5. Critical Values of the t* Distribution
36 2.504 96 2.430
37 2.501 97 2.430
38 2.498 98 2.429
39 2.495 99 2.429
40 2.492 100 2.428
41 2.489 101 2.428
42 2.487 102 2.428
43 2.484 103 2.427
44 2.482 104 2.427
45 2.480 105 2.426
46 2.478 106 2.426
47 2.476 107 2.426
48 2.474 108 2.425
49 2.472 109 2.425
50 2.470 110 2.425
51 2.469 111 2.424
52 2.467 112 2.424
53 2.466 113 2.424
54 2.464 114 2.423
55 2.463 115 2.423
56 2.461 116 2.423
57 2.460 117 2.422
58 2.459 118 2.422
59 2.457 119 2.422
60 2.456 120 2.422
1.3.6.7.6. Critical Values of the Normal PPCC Distribution

1.3. EDA Techniques
1.3.6.7.6. Critical Values of the Normal PPCC
Distribution
How to
Use This
Table
This table contains the critical values of the normal probability
plot correlation coefficient (PPCC) distribution that are
appropriate for determining whether or not a data set came
from a population with approximately a normal distribution. It
is used in conjuction with a normal probability plot. The test
statistic is the correlation coefficient of the points that make up
a normal probability plot. This test statistic is compared with
the critical value below. If the test statistic is less than the
tabulated value, the null hypothesis that the data came from a
population with a normal distribution is rejected.
For example, suppose a set of 50 data points had a correlation
coefficient of 0.985 from the normal probability plot. At the
5% significance level, the critical value is 0.9761. Since 0.985
is greater than 0.9761, we cannot reject the null hypothesis that
the data came from a population with a normal distribution.
Since perferct normality implies perfect correlation (i.e., a
correlation value of 1), we are only interested in rejecting
normality for correlation values that are too low. That is, this
is a lower one-tailed test.
The values in this table were determined from simulation
studies by Filliben and Devaney.
Critical values of the normal PPCC for testing if data come
from a normal distribution
N 0.01 0.05
3 0.8687 0.8790
4 0.8234 0.8666
5 0.8240 0.8786
6 0.8351 0.8880
7 0.8474 0.8970
8 0.8590 0.9043
9 0.8689 0.9115
10 0.8765 0.9173
11 0.8838 0.9223
12 0.8918 0.9267
13 0.8974 0.9310
14 0.9029 0.9343
15 0.9080 0.9376
16 0.9121 0.9405
17 0.9160 0.9433
18 0.9196 0.9452
19 0.9230 0.9479
20 0.9256 0.9498
21 0.9285 0.9515
22 0.9308 0.9535
23 0.9334 0.9548
24 0.9356 0.9564
25 0.9370 0.9575
26 0.9393 0.9590
27 0.9413 0.9600
28 0.9428 0.9615
29 0.9441 0.9622
30 0.9462 0.9634
31 0.9476 0.9644
32 0.9490 0.9652
33 0.9505 0.9661
34 0.9521 0.9671
35 0.9530 0.9678
36 0.9540 0.9686
37 0.9551 0.9693
38 0.9555 0.9700
39 0.9568 0.9704
40 0.9576 0.9712
41 0.9589 0.9719
42 0.9593 0.9723
43 0.9609 0.9730
44 0.9611 0.9734
45 0.9620 0.9739
46 0.9629 0.9744
47 0.9637 0.9748
48 0.9640 0.9753
49 0.9643 0.9758
50 0.9654 0.9761
55 0.9683 0.9781
60 0.9706 0.9797
65 0.9723 0.9809
70 0.9742 0.9822
75 0.9758 0.9831
80 0.9771 0.9841
85 0.9784 0.9850
90 0.9797 0.9857
95 0.9804 0.9864
100 0.9814 0.9869
110 0.9830 0.9881
120 0.9841 0.9889
130 0.9854 0.9897
140 0.9865 0.9904
150 0.9871 0.9909
160 0.9879 0.9915
170 0.9887 0.9919
180 0.9891 0.9923
190 0.9897 0.9927
200 0.9903 0.9930
210 0.9907 0.9933
220 0.9910 0.9936
230 0.9914 0.9939
240 0.9917 0.9941
250 0.9921 0.9943
260 0.9924 0.9945
270 0.9926 0.9947
280 0.9929 0.9949
290 0.9931 0.9951
300 0.9933 0.9952
310 0.9936 0.9954
320 0.9937 0.9955
330 0.9939 0.9956
340 0.9941 0.9957
350 0.9942 0.9958
360 0.9944 0.9959
370 0.9945 0.9960
380 0.9947 0.9961
390 0.9948 0.9962
400 0.9949 0.9963
410 0.9950 0.9964
420 0.9951 0.9965
430 0.9953 0.9966
440 0.9954 0.9966
450 0.9954 0.9967
460 0.9955 0.9968
470 0.9956 0.9968
480 0.9957 0.9969
490 0.9958 0.9969
500 0.9959 0.9970
525 0.9961 0.9972
550 0.9963 0.9973
575 0.9964 0.9974
600 0.9965 0.9975
625 0.9967 0.9976
650 0.9968 0.9977
675 0.9969 0.9977
700 0.9970 0.9978
725 0.9971 0.9979
750 0.9972 0.9980
775 0.9973 0.9980
800 0.9974 0.9981
825 0.9975 0.9981
850 0.9975 0.9982
875 0.9976 0.9982
900 0.9977 0.9983
925 0.9977 0.9983
950 0.9978 0.9984
975 0.9978 0.9984
1000 0.9979 0.9984
1.4. EDA Case Studies

Summary This section presents a series of case studies that demonstrate
the application of EDA methods to specific problems. In some
cases, we have focused on just one EDA technique that
uncovers virtually all there is to know about the data. For other
case studies, we need several EDA techniques, the selection of
which is dictated by the outcome of the previous step in the
analaysis sequence. Note in these case studies how the flow of
the analysis is motivated by the focus on underlying
assumptions and general EDA principles.
Table of
Contents
for Section
4
1. Introduction
2. By Problem Category
1.4.1. Case Studies Introduction

Purpose The purpose of the first eight case studies is to show how
EDA graphics and quantitative measures and tests are
applied to data from scientific processes and to critique
those data with regard to the following assumptions that
typically underlie a measurement process; namely, that the
data behave like:
random drawings
from a fixed distribution
with a fixed location
with a fixed standard deviation
Case studies 9 and 10 show the use of EDA techniques in
distributional modeling and the analysis of a designed
experiment, respectively.
Y
i
= C + E
i
If the above assumptions are satisfied, the process is said
to be statistically "in control" with the core characteristic
of having "predictability". That is, probability statements
can be made about the process, not only in the past, but
also in the future.
An appropriate model for an "in control" process is
Y
i
= C + E
i
where C is a constant (the "deterministic" or "structural"
component), and where E
i
is the error term (or "random"
component).
The constant C is the average value of the process--it is the
primary summary number which shows up on any report.
Although C is (assumed) fixed, it is unknown, and so a
primary analysis objective of the engineer is to arrive at an
estimate of C.
This goal partitions into 4 sub-goals:
1. Is the most common estimator of C, , the best
estimator for C? What does "best" mean?
2. If is best, what is the uncertainty for . In
particular, is the usual formula for the uncertainty of
:
valid? Here, s is the standard deviation of the data
and N is the sample size.
3. If is not the best estimator for C, what is a better
estimator for C (for example, median, midrange,
midmean)?
4. If there is a better estimator, , what is its
uncertainty? That is, what is ?
EDA and the routine checking of underlying assumptions
provides insight into all of the above.
1. Location and variation checks provide information
as to whether C is really constant.
2. Distributional checks indicate whether is the best
estimator. Techniques for distributional checking
include histograms, normal probability plots, and
probability plot correlation coefficient plots.
3. Randomness checks ascertain whether the usual
is valid.
4. Distributional tests assist in determining a better
estimator, if needed.
5. Simulator tools (namely bootstrapping) provide
values for the uncertainty of alternative estimators.
Assumptions
not satisfied
If one or more of the above assumptions is not satisfied,
then we use EDA techniques, or some mix of EDA and
classical techniques, to find a more appropriate model for
the data. That is,
Y
i
= D + E
i
where D is the deterministic part and E is an error
component.
If the data are not random, then we may investigate fitting
some simple time series models to the data. If the constant
location and scale assumptions are violated, we may need
to investigate the measurement process to see if there is an
explanation.
The assumptions on the error term are still quite relevant
in the sense that for an appropriate model the error
component should follow the assumptions. The criterion
for validating the model, or comparing competing models,
is framed in terms of these assumptions.
Multivariable
data
Although the case studies in this chapter utilize univariate
data, the assumptions above are relevant for multivariable
data as well.
If the data are not univariate, then we are trying to find a
model
Y
i
= F(X
1
, ..., X
k
) + E
i
where F is some function based on one or more variables.
The error component, which is a univariate data set, of a
good model should satisfy the assumptions given above.
The criterion for validating and comparing models is based
on how well the error component follows these
assumptions.
The load cell calibration case study in the process
modeling chapter shows an example of this in the
regression context.
First three
case studies
utilize data
with known
characteristics
The first three case studies utilize data that are randomly
generated from the following distributions:
normal distribution with mean 0 and standard
deviation 1
uniform distribution with mean 0 and standard
deviation (uniform over the interval (0,1))
random walk
The other univariate case studies utilize data from
scientific processes. The goal is to determine if
Y
i
= C + E
i
is a reasonable model. This is done by testing the
underlying assumptions. If the assumptions are satisfied,
then an estimate of C and an estimate of the uncertainty of
C are computed. If the assumptions are not satisfied, we
attempt to find a model where the error component does
satisfy the underlying assumptions.
Graphical
methods that
are applied to
the data
To test the underlying assumptions, each data set is
analyzed using four graphical methods that are particularly
suited for this purpose:
1. run sequence plot which is useful for detecting shifts
of location or scale
2. lag plot which is useful for detecting non-
randomness in the data
3. histogram which is useful for trying to determine the
underlying distribution
4. normal probability plot for deciding whether the data
follow the normal distribution
There are a number of other techniques for addressing the
underlying assumptions. However, the four plots listed
above provide an excellent opportunity for addressing all
of the assumptions on a single page of graphics.
Additional graphical techniques are used in certain case
studies to develop models that do have error components
that satisfy the underlying assumptions.
Quantitative
methods that
are applied to
the data
The normal and uniform random number data sets are also
analyzed with the following quantitative techniques, which
are explained in more detail in an earlier section:
1. Summary statistics which include:
mean
standard deviation
autocorrelation coefficient to test for
randomness
normal and uniform probability plot
correlation coefficients (ppcc) to test for a
normal or uniform distribution, respectively
Wilk-Shapiro test for a normal distribution
2. Linear fit of the data as a function of time to assess
drift (test for fixed location)
3. Bartlett test for fixed variance
4. Autocorrelation plot and coefficient to test for
randomness
5. Runs test to test for lack of randomness
6. Anderson-Darling test for a normal distribution
7. Grubbs test for outliers
8. Summary report
Although the graphical methods applied to the normal and
uniform random numbers are sufficient to assess the
validity of the underlying assumptions, the quantitative
techniques are used to show the different flavor of the
graphical and quantitative approaches.
The remaining case studies intermix one or more of these
quantitative techniques into the analysis where appropriate.
1.4.2. Case Studies

1.4.2. Case Studies
Univariate
Y
i
= C +
E
i
Normal Random
Numbers
Uniform Random
Numbers
Random Walk

Josephson Junction
Cryothermometry
Beam Deflections Filter
Transmittance

Standard Resistor Heat Flow Meter
1
Reliability
Fatigue Life of
Aluminum Alloy
Specimens
Multi-
Factor
Ceramic Strength
1.4.2.1. Normal Random Numbers

1.4.2. Case Studies
Normal
Random
Numbers
This example illustrates the univariate analysis of a set of
normal random numbers.
1. Background and Data
2. Graphical Output and Interpretation
3. Quantitative Output and Interpretation
4. Work This Example Yourself
1.4.2.1.1. Background and Data

1.4.2. Case Studies
Generation The normal random numbers used in this case study are from
a Rand Corporation publication.
The motivation for studying a set of normal random numbers
is to illustrate the ideal case where all four underlying
assumptions hold.
Software The analyses used in this case study can be generated using
both Dataplot code and R code.
Data The following is the set of normal random numbers used for
this case study.
-1.2760 -1.2180 -0.4530 -0.3500 0.7230
0.6760 -1.0990 -0.3140 -0.3940 -0.6330
-0.3180 -0.7990 -1.6640 1.3910 0.3820
0.7330 0.6530 0.2190 -0.6810 1.1290
-1.3770 -1.2570 0.4950 -0.1390 -0.8540
0.4280 -1.3220 -0.3150 -0.7320 -1.3480
2.3340 -0.3370 -1.9550 -0.6360 -1.3180
-0.4330 0.5450 0.4280 -0.2970 0.2760
-1.1360 0.6420 3.4360 -1.6670 0.8470
-1.1730 -0.3550 0.0350 0.3590 0.9300
0.4140 -0.0110 0.6660 -1.1320 -0.4100
-1.0770 0.7340 1.4840 -0.3400 0.7890
-0.4940 0.3640 -1.2370 -0.0440 -0.1110
-0.2100 0.9310 0.6160 -0.3770 -0.4330
1.0480 0.0370 0.7590 0.6090 -2.0430
-0.2900 0.4040 -0.5430 0.4860 0.8690
0.3470 2.8160 -0.4640 -0.6320 -1.6140
0.3720 -0.0740 -0.9160 1.3140 -0.0380
0.6370 0.5630 -0.1070 0.1310 -1.8080
-1.1260 0.3790 0.6100 -0.3640 -2.6260
2.1760 0.3930 -0.9240 1.9110 -1.0400
-1.1680 0.4850 0.0760 -0.7690 1.6070
-1.1850 -0.9440 -1.6040 0.1850 -0.2580
-0.3000 -0.5910 -0.5450 0.0180 -0.4850
0.9720 1.7100 2.6820 2.8130 -1.5310
-0.4900 2.0710 1.4440 -1.0920 0.4780
1.2100 0.2940 -0.2480 0.7190 1.1030
1.0900 0.2120 -1.1850 -0.3380 -1.1340
2.6470 0.7770 0.4500 2.2470 1.1510
-1.6760 0.3840 1.1330 1.3930 0.8140
0.3980 0.3180 -0.9280 2.4160 -0.9360
1.0360 0.0240 -0.5600 0.2030 -0.8710
0.8460 -0.6990 -0.3680 0.3440 -0.9260
-0.7970 -1.4040 -1.4720 -0.1180 1.4560
0.6540 -0.9550 2.9070 1.6880 0.7520
-0.4340 0.7460 0.1490 -0.1700 -0.4790
0.5220 0.2310 -0.6190 -0.2650 0.4190
0.5580 -0.5490 0.1920 -0.3340 1.3730
-1.2880 -0.5390 -0.8240 0.2440 -1.0700
0.0100 0.4820 -0.4690 -0.0900 1.1710
1.3720 1.7690 -1.0570 1.6460 0.4810
-0.6000 -0.5920 0.6100 -0.0960 -1.3750
0.8540 -0.5350 1.6070 0.4280 -0.6150
0.3310 -0.3360 -1.1520 0.5330 -0.8330
-0.1480 -1.1440 0.9130 0.6840 1.0430
0.5540 -0.0510 -0.9440 -0.4400 -0.2120
-1.1480 -1.0560 0.6350 -0.3280 -1.2210
0.1180 -2.0450 -1.9770 -1.1330 0.3380
0.3480 0.9700 -0.0170 1.2170 -0.9740
-1.2910 -0.3990 -1.2090 -0.2480 0.4800
0.2840 0.4580 1.3070 -1.6250 -0.6290
-0.5040 -0.0560 -0.1310 0.0480 1.8790
-1.0160 0.3600 -0.1190 2.3310 1.6720
-1.0530 0.8400 -0.2460 0.2370 -1.3120
1.6030 -0.9520 -0.5660 1.6000 0.4650
1.9510 0.1100 0.2510 0.1160 -0.9570
-0.1900 1.4790 -0.9860 1.2490 1.9340
0.0700 -1.3580 -1.2460 -0.9590 -1.2970
-0.7220 0.9250 0.7830 -0.4020 0.6190
1.8260 1.2720 -0.9450 0.4940 0.0500
-1.6960 1.8790 0.0630 0.1320 0.6820
0.5440 -0.4170 -0.6660 -0.1040 -0.2530
-2.5430 -1.3330 1.9870 0.6680 0.3600
1.9270 1.1830 1.2110 1.7650 0.3500
-0.3590 0.1930 -1.0230 -0.2220 -0.6160
-0.0600 -1.3190 0.7850 -0.4300 -0.2980
0.2480 -0.0880 -1.3790 0.2950 -0.1150
-0.6210 -0.6180 0.2090 0.9790 0.9060
-0.0990 -1.3760 1.0470 -0.8720 -2.2000
-1.3840 1.4250 -0.8120 0.7480 -1.0930
-0.4630 -1.2810 -2.5140 0.6750 1.1450
1.0830 -0.6670 -0.2230 -1.5920 -1.2780
0.5030 1.4340 0.2900 0.3970 -0.8370
-0.9730 -0.1200 -1.5940 -0.9960 -1.2440
-0.8570 -0.3710 -0.2160 0.1480 -2.1060
-1.4530 0.6860 -0.0750 -0.2430 -0.1700
-0.1220 1.1070 -1.0390 -0.6360 -0.8600
-0.8950 -1.4580 -0.5390 -0.1590 -0.4200
1.6320 0.5860 -0.4680 -0.3860 -0.3540
0.2030 -1.2340 2.3810 -0.3880 -0.0630
2.0720 -1.4450 -0.6800 0.2240 -0.1200
1.7530 -0.5710 1.2230 -0.1260 0.0340
-0.4350 -0.3750 -0.9850 -0.5850 -0.2030
-0.5560 0.0240 0.1260 1.2500 -0.6150
0.8760 -1.2270 -2.6470 -0.7450 1.7970
-1.2310 0.5470 -0.6340 -0.8360 -0.7190
0.8330 1.2890 -0.0220 -0.4310 0.5820
0.7660 -0.5740 -1.1530 0.5200 -1.0180
-0.8910 0.3320 -0.4530 -1.1270 2.0850
-0.7220 -1.5080 0.4890 -0.4960 -0.0250
0.6440 -0.2330 -0.1530 1.0980 0.7570
-0.0390 -0.4600 0.3930 2.0120 1.3560
0.1050 -0.1710 -0.1100 -1.1450 0.8780
-0.9090 -0.3280 1.0210 -1.6130 1.5600
-1.1920 1.7700 -0.0030 0.3690 0.0520
0.6470 1.0290 1.5260 0.2370 -1.3280
-0.0420 0.5530 0.7700 0.3240 -0.4890
-0.3670 0.3780 0.6010 -1.9960 -0.7380
0.4980 1.0720 1.5670 0.3020 1.1570
-0.7200 1.4030 0.6980 -0.3700 -0.5510
1.4.2.1.2. Graphical Output and Interpretation

1.4.2. Case Studies
Goal The goal of this analysis is threefold:
1. Determine if the univariate model:
is appropriate and valid.
2. Determine if the typical underlying assumptions for
an "in control" measurement process are valid. These
assumptions are:
1. random drawings;
3. with the distribution having a fixed location;
and
4. the distribution having a fixed scale.
3. Determine if the confidence interval
is appropriate and valid where s is the standard
deviation of the original data.
4-Plot of
Data
Interpretation The assumptions are addressed by the graphics shown
above:
1. The run sequence plot (upper left) indicates that the
data do not have any significant shifts in location or
scale over time. The run sequence plot does not show
any obvious outliers.
2. The lag plot (upper right) does not indicate any non-
random pattern in the data.
3. The histogram (lower left) shows that the data are
reasonably symmetric, there do not appear to be
significant outliers in the tails, and that it is
reasonable to assume that the data are from
approximately a normal distribution.
4. The normal probability plot (lower right) verifies that
an assumption of normality is in fact reasonable.
From the above plots, we conclude that the underlying
assumptions are valid and the data follow approximately a
normal distribution. Therefore, the confidence interval form
given previously is appropriate for quantifying the
uncertainty of the population mean. The numerical values
for this model are given in the Quantitative Output and
Interpretation section.
Individual
Plots
Although it is usually not necessary, the plots can be
generated individually to give more detail.
Run
Sequence
Plot
Lag Plot
Histogram
(with
overlaid
Normal PDF)
Normal
Probability
Plot
1.4.2.1.3. Quantitative Output and Interpretation

1.4.2. Case Studies
1.4.2.1.3. Quantitative Output and
Interpretation
Summary
Statistics
As a first step in the analysis, common summary statistics
are computed from the data.
Sample size = 500
Mean = -0.2935997E-02
Median = -0.9300000E-01
Minimum = -0.2647000E+01
Maximum = 0.3436000E+01
Range = 0.6083000E+01
Stan. Dev. = 0.1021041E+01
Location One way to quantify a change in location over time is to fit
a straight line to the data using an index variable as the
independent variable in the regression. For our data, we
assume that data are in sequential run order and that the
data were collected at equally spaced time intervals. In our
regression, we use the index variable X = 1, 2, ..., N, where
N is the number of observations. If there is no significant
drift in the location over time, the slope parameter should
be zero.
Coefficient Estimate Stan. Error
t-Value
B
0
0.699127E-02 0.9155E-01
0.0764
B
1
-0.396298E-04 0.3167E-03
-0.1251

Residual Standard Deviation = 1.02205
Residual Degrees of Freedom = 498
The absolute value of the t-value for the slope parameter is
smaller than the critical value of t
0.975,498
= 1.96. Thus, we
conclude that the slope is not different from zero at the 0.05
significance level.
Variation One simple way to detect a change in variation is with
Bartlett's test, after dividing the data set into several equal-
sized intervals. The choice of the number of intervals is
somewhat arbitrary, although values of four or eight are
reasonable. We will divide our data into four intervals.
H
0
:
1
2
=
2
2
=
3
2
=
4
2

H
a
: At least one
i
2
is not equal to the
others.
Critical value:
2
1-,k-1
= 7.814728
0
if T > 7.814728
In this case, Bartlett's test indicates that the variances are
not significantly different in the four intervals.
Randomness There are many ways in which data can be non-random.
However, most common forms of non-randomness can be
detected with a few simple tests including the lag plot
shown on the previous page.
Another check is an autocorrelation plot that shows the
autocorrelations for various lags. Confidence bands can be
plotted at the 95 % and 99 % confidence levels. Points
outside this band indicate statistically significant values (lag
0 is always 1).
The lag 1 autocorrelation, which is generally the one of
most interest, is 0.045. The critical values at the 5%
significance level are -0.087 and 0.087. Since 0.045 is
within the critical region, the lag 1 autocorrelation is not
statistically significant, so there is no evidence of non-
randomness.
A common test for randomness is the runs test.
H
0
: the sequence was produced in a random
manner
H
a
: the sequence was not produced in a
random manner
Test statistic: Z = -1.0744
Critical value: Z
1-/2
= 1.96
0
if |Z| > 1.96
The runs test fails to reject the null hypothesis that the data
were produced in a random manner.
Distributional
Analysis
Probability plots are a graphical test for assessing if a
particular distribution provides an adequate fit to a data set.
A quantitative enhancement to the probability plot is the
correlation coefficient of the points on the probability plot,
or PPCC. For this data set the PPCC based on a normal
distribution is 0.996. Since the PPCC is greater than the
critical value of 0.987 (this is a tabulated value), the
normality assumption is not rejected.
Chi-square and Kolmogorov-Smirnov goodness-of-fit tests
are alternative methods for assessing distributional
adequacy. The Wilk-Shapiro and Anderson-Darling tests
can be used to test for normality. The results of the
Anderson-Darling test follow.
H
0
H
a
Adjusted test statistic: A
2
= 1.0612
0
if A
2
> 0.787
The Anderson-Darling test rejects the normality assumption
at the 0.05 significance level.
Outlier
Analysis
A test for outliers is the Grubbs test.
H
0
: there are no outliers in the data
H
a
: the maximum value is an outlier
Test statistic: G = 3.368068
Critical value for an upper one-tailed
test: 3.863087
0
if G > 3.863087
For this data set, Grubbs' test does not detect any outliers at
Model Since the underlying assumptions were validated both
graphically and analytically, we conclude that a reasonable
model for the data is:
Y
i
= C + E
i
where C is the estimated value of the mean, -0.00294. We
can express the uncertainty for C as a 95 % confidence
interval (-0.09266, 0.08678).
Univariate
Report
It is sometimes useful and convenient to summarize the
above results in a report.
Analysis of 500 normal random numbers

1: Sample Size = 500

2: Location
Mean = -
0.00294
Standard Deviation of Mean =
0.045663
95% Confidence Interval for Mean = (-
0.09266,0.086779)
Drift with respect to location? = NO

3: Variation
Standard Deviation =
1.021042
95% Confidence Interval for SD =
(0.961437,1.088585)
Drift with respect to variation?
(based on Bartletts test on quarters
of the data) = NO

4: Data are Normal?
(as tested by Anderson-Darling) = YES

5: Randomness
Autocorrelation =
0.045059
Data are Random?
(as measured by autocorrelation) = YES

6: Statistical Control
(i.e., no drift in location or scale,
data are random, distribution is
fixed, here we are testing only for
fixed normal)
Data Set is in Statistical Control? = YES

7: Outliers?
(as determined by Grubbs' test) = NO
1.4.2.1.4. Work This Example Yourself

1.4.2. Case Studies
View
Dataplot
Macro for
this Case
Study
This page allows you to repeat the analysis outlined in the
case study description on the previous page using Dataplot . It
is required that you have already downloaded and installed
Dataplot and configured your browser. to run Dataplot. Output
from each analysis step below will be displayed in one or
more of the Dataplot windows. The four main windows are the
Output window, the Graphics window, the Command History
window, and the data sheet window. Across the top of the
main windows there are menus for executing Dataplot
commands. Across the bottom is a command entry window
where commands can be typed in.
Data Analysis Steps
Results and
Conclusions
Click on the links below to start Dataplot and
run this case study yourself. Each step may use
results from previous steps, so please be patient.
Wait until the software verifies that the current
step is complete before clicking on the next step.
The links in this column
will connect you with
more detailed
information about each
analysis step from the
case study description.
1. Invoke Dataplot and read data.
1. Read in the data.

1. You have read 1
column of numbers
into Dataplot,
variable Y.
2. 4-plot of the data.
1. 4-plot of Y. 1. Based on the 4-
plot, there are no
shifts
in location or
scale, and the data
seem to
follow a normal
distribution.
3. Generate the individual plots.
1. Generate a run sequence plot.
2. Generate a lag plot.
3. Generate a histogram with an
overlaid normal pdf.
4. Generate a normal probability
plot.
1. The run sequence
plot indicates that
there are no
shifts of location or
scale.
2. The lag plot
does not indicate any
significant
patterns (which would
show the data
were not random).
3. The histogram
indicates that a
normal
distribution is a
good
distribution for
these data.
4. The normal
probability plot
verifies
that the normal
distribution is a
reasonable
distribution for
these data.
4. Generate summary statistics,
quantitative
analysis, and print a univariate
report.
1. Generate a table of summary
statistics.
2. Generate the mean, a confidence
interval for the mean, and compute
a linear fit to detect drift in
location.
3. Generate the standard deviation, a
confidence interval for the
standard
deviation, and detect drift in
variation
by dividing the data into quarters
and
computing Barltett's test for
equal
4. Check for randomness by generating
an
autocorrelation plot and a runs
test.
5. Check for normality by computing
the
normal probability plot
correlation
coefficient.
6. Check for outliers using Grubbs'
test.
7. Print a univariate report (this
1. The summary
statistics table
displays
25+ statistics.
2. The mean is -
0.00294 and a 95%
confidence
interval is (-
0.093,0.087).
The linear fit
indicates no drift in
location since
the slope parameter
is
statistically not
significant.
3. The standard
deviation is 1.02
with
a 95% confidence
interval of
(0.96,1.09).
Bartlett's test
indicates no
significant
change in
variation.
4. The lag 1
autocorrelation is
0.04.
From the
autocorrelation plot,
this is
within the 95%
confidence interval
bands.
assumes
steps 2 thru 6 have already been
run).
5. The normal
probability plot
correlation
coefficient is
0.996. At the 5%
level,
we cannot reject
the normality
assumption.
6. Grubbs' test
detects no outliers
at the
5% level.
7. The results are
summarized in a
convenient
report.
1.4.2.2. Uniform Random Numbers

1.4.2. Case Studies
Uniform
Random
Numbers
uniform random numbers.

1.4.2. Case Studies
Generation The uniform random numbers used in this case study are from
a Rand Corporation publication.
The motivation for studying a set of uniform random numbers
is to illustrate the effects of a known underlying non-normal
distribution.
Data The following is the set of uniform random numbers used for
this case study.
.100973 .253376 .520135 .863467 .354876
.809590 .911739 .292749 .375420 .480564
.894742 .962480 .524037 .206361 .040200
.822916 .084226 .895319 .645093 .032320
.902560 .159533 .476435 .080336 .990190
.252909 .376707 .153831 .131165 .886767
.439704 .436276 .128079 .997080 .157361
.476403 .236653 .989511 .687712 .171768
.660657 .471734 .072768 .503669 .736170
.658133 .988511 .199291 .310601 .080545
.571824 .063530 .342614 .867990 .743923
.403097 .852697 .760202 .051656 .926866
.574818 .730538 .524718 .623885 .635733
.213505 .325470 .489055 .357548 .284682
.870983 .491256 .737964 .575303 .529647
.783580 .834282 .609352 .034435 .273884
.985201 .776714 .905686 .072210 .940558
.609709 .343350 .500739 .118050 .543139
.808277 .325072 .568248 .294052 .420152
.775678 .834529 .963406 .288980 .831374
.670078 .184754 .061068 .711778 .886854
.020086 .507584 .013676 .667951 .903647
.649329 .609110 .995946 .734887 .517649
.699182 .608928 .937856 .136823 .478341
.654811 .767417 .468509 .505804 .776974
.730395 .718640 .218165 .801243 .563517
.727080 .154531 .822374 .211157 .825314
.385537 .743509 .981777 .402772 .144323
.600210 .455216 .423796 .286026 .699162
.680366 .252291 .483693 .687203 .766211
.399094 .400564 .098932 .050514 .225685
.144642 .756788 .962977 .882254 .382145
.914991 .452368 .479276 .864616 .283554
.947508 .992337 .089200 .803369 .459826
.940368 .587029 .734135 .531403 .334042
.050823 .441048 .194985 .157479 .543297
.926575 .576004 .088122 .222064 .125507
.374211 .100020 .401286 .074697 .966448
.943928 .707258 .636064 .932916 .505344
.844021 .952563 .436517 .708207 .207317
.611969 .044626 .457477 .745192 .433729
.653945 .959342 .582605 .154744 .526695
.270799 .535936 .783848 .823961 .011833
.211594 .945572 .857367 .897543 .875462
.244431 .911904 .259292 .927459 .424811
.621397 .344087 .211686 .848767 .030711
.205925 .701466 .235237 .831773 .208898
.376893 .591416 .262522 .966305 .522825
.044935 .249475 .246338 .244586 .251025
.619627 .933565 .337124 .005499 .765464
.051881 .599611 .963896 .546928 .239123
.287295 .359631 .530726 .898093 .543335
.135462 .779745 .002490 .103393 .598080
.839145 .427268 .428360 .949700 .130212
.489278 .565201 .460588 .523601 .390922
.867728 .144077 .939108 .364770 .617429
.321790 .059787 .379252 .410556 .707007
.867431 .715785 .394118 .692346 .140620
.117452 .041595 .660000 .187439 .242397
.118963 .195654 .143001 .758753 .794041
.921585 .666743 .680684 .962852 .451551
.493819 .476072 .464366 .794543 .590479
.003320 .826695 .948643 .199436 .168108
.513488 .881553 .015403 .545605 .014511
.980862 .482645 .240284 .044499 .908896
.390947 .340735 .441318 .331851 .623241
.941509 .498943 .548581 .886954 .199437
.548730 .809510 .040696 .382707 .742015
.123387 .250162 .529894 .624611 .797524
.914071 .961282 .966986 .102591 .748522
.053900 .387595 .186333 .253798 .145065
.713101 .024674 .054556 .142777 .938919
.740294 .390277 .557322 .709779 .017119
.525275 .802180 .814517 .541784 .561180
.993371 .430533 .512969 .561271 .925536
.040903 .116644 .988352 .079848 .275938
.171539 .099733 .344088 .461233 .483247
.792831 .249647 .100229 .536870 .323075
.754615 .020099 .690749 .413887 .637919
.763558 .404401 .105182 .161501 .848769
.091882 .009732 .825395 .270422 .086304
.833898 .737464 .278580 .900458 .549751
.981506 .549493 .881997 .918707 .615068
.476646 .731895 .020747 .677262 .696229
.064464 .271246 .701841 .361827 .757687
.649020 .971877 .499042 .912272 .953750
.587193 .823431 .540164 .405666 .281310
.030068 .227398 .207145 .329507 .706178
.083586 .991078 .542427 .851366 .158873
.046189 .755331 .223084 .283060 .326481
.333105 .914051 .007893 .326046 .047594
.119018 .538408 .623381 .594136 .285121
.590290 .284666 .879577 .762207 .917575
.374161 .613622 .695026 .390212 .557817
.651483 .483470 .894159 .269400 .397583
.911260 .717646 .489497 .230694 .541374
.775130 .382086 .864299 .016841 .482774
.519081 .398072 .893555 .195023 .717469
.979202 .885521 .029773 .742877 .525165
.344674 .218185 .931393 .278817 .570568

1.4.2. Case Studies
assumptions are:
1. random drawings;
and
4-Plot of
Data
above:
scale over time.
3. The histogram shows that the frequencies are
relatively flat across the range of the data. This
suggests that the uniform distribution might provide a
better distributional fit than the normal distribution.
4. The normal probability plot verifies that an
assumption of normality is not reasonable. In this
case, the 4-plot should be followed up by a uniform
probability plot to determine if it provides a better fit
to the data. This is shown below.
assumptions are valid. Therefore, the model Y
i
= C + E
i
is
valid. However, since the data are not normally distributed,
using the mean as an estimate of C and the confidence
interval cited above for quantifying its uncertainty are not
valid or appropriate.
Individual
Plots
Although it is usually not necessary, the plots can be
Run
Sequence
Plot
Lag Plot
Histogram
(with
overlaid
Normal PDF)
This plot shows that a normal distribution is a poor fit. The
flatness of the histogram suggests that a uniform
distribution might be a better fit.
Histogram
(with
overlaid
Uniform
PDF)
Since the histogram from the 4-plot suggested that the
uniform distribution might be a good fit, we overlay a
uniform distribution on top of the histogram. This indicates
a much better fit than a normal distribution.
Normal
Probability
Plot
As with the histogram, the normal probability plot shows
that the normal distribution does not fit these data well.
Uniform
Probability
Plot
Since the above plots suggested that a uniform distribution
might be appropriate, we generate a uniform probability
plot. This plot shows that the uniform distribution provides
an excellent fit to the data.
Better Model Since the data follow the underlying assumptions, but with
a uniform distribution rather than a normal distribution, we
would still like to characterize C by a typical value plus or
minus a confidence interval. In this case, we would like to
find a location estimator with the smallest variability.
The bootstrap plot is an ideal tool for this purpose. The
following plots show the bootstrap plot, with the
corresponding histogram, for the mean, median, mid-range,
and median absolute deviation.
Bootstrap
Plots
Mid-Range is
Best
From the above histograms, it is obvious that for these data,
the mid-range is far superior to the mean or median as an
estimate for location.
Using the mean, the location estimate is 0.507 and a 95%
confidence interval for the mean is (0.482,0.534). Using the
mid-range, the location estimate is 0.499 and the 95%
confidence interval for the mid-range is (0.497,0.503).
Although the values for the location are similar, the
difference in the uncertainty intervals is quite large.
Note that in the case of a uniform distribution it is known
theoretically that the mid-range is the best linear unbiased
estimator for location. However, in many applications, the
most appropriate estimator will not be known or it will be
mathematically intractable to determine a valid condfidence
interval. The bootstrap provides a method for determining
(and comparing) confidence intervals in these cases.

1.4.2. Case Studies
Interpretation
Summary
Statistics
are computed for the data.
Sample size = 500
Mean = 0.5078304
Median = 0.5183650
Minimum = 0.0024900
Maximum = 0.9970800
Range = 0.9945900
Stan. Dev. = 0.2943252
Because the graphs of the data indicate the data may not be
normally distributed, we also compute two other statistics
for the data, the normal PPCC and the uniform PPCC.
Normal PPCC = 0.9771602
Uniform PPCC = 0.9995682
The uniform probability plot correlation coefficient (PPCC)
value is larger than the normal PPCC value. This is
evidence that the uniform distribution fits these data better
than does a normal distribution.
be zero.
t-Value
B
0
0.522923 0.2638E-01
19.82
B
1
-0.602478E-04 0.9125E-04
-0.66

The t-value of the slope parameter, -0.66, is smaller than
the critical value of t
0.975,498
= 1.96. Thus, we conclude
that the slope is not different from zero at the 0.05
significance level.
Variation One simple way to detect a change in variation is with a
Bartlett test after dividing the data set into several equal-
sized intervals. However, the Bartlett test is not robust for
non-normality. Since we know this data set is not
approximated well by the normal distribution, we use the
alternative Levene test. In particular, we use the Levene
test based on the median rather the mean. The choice of the
number of intervals is somewhat arbitrary, although values
of four or eight are reasonable. We will divide our data into
four intervals.
H
0
:
1
2
=
2
2
=
3
2
=
4
2

H
a
: At least one
i
2
is not equal to the
others.
Critical value: F
,k-1,N-k
= 2.623
0
if W > 2.623
In this case, the Levene test indicates that the variances are
not significantly different in the four intervals.
detected with a few simple tests including the lag plot
shown on the previous page.
plotted using 95% and 99% confidence levels. Points
0 is always 1).
most interest, is 0.03. The critical values at the 5 %
significance level are -0.087 and 0.087. This indicates that
the lag 1 autocorrelation is not statistically significant, so
there is no evidence of non-randomness.
H
0
manner
H
a
random manner
Critical value: Z
1-/2
= 1.96
0
if |Z| > 1.96
The runs test fails to reject the null hypothesis that the data
were produced in a random manner.
Distributional
Analysis
Probability plots are a graphical test of assessing whether a
distribution is 0.977. Since the PPCC is less than the critical
value of 0.987 (this is a tabulated value), the normality
assumption is rejected.
H
0
H
a
2
= 5.765
0
if A
2
> 0.787
because the value of the test statistic, 5.765, is larger than
the critical value of 0.787 at the 0.05 significance level.
Model Based on the graphical and quantitative analysis, we use the
model
Y
i
= C + E
i
where C is estimated by the mid-range and the uncertainty
interval for C is based on a bootstrap analysis. Specifically,
C = 0.499
95% confidence limit for C = (0.497,0.503)
Univariate
Report

Analysis for 500 uniform random numbers


2: Location
Mean =
0.50783
0.013163
95% Confidence Interval for Mean =
(0.48197,0.533692)

3: Variation
0.294326
(0.277144,0.313796)
(based on Levene's test on quarters
of the data) = NO

4: Distribution
Normal PPCC =
0.9771602
Data are Normal?
(as measured by Normal PPCC) = NO

Uniform PPCC =
0.9995682
Data are Uniform?
(as measured by Uniform PPCC) = YES

5: Randomness
Autocorrelation = -
0.03099
Data are Random?
(as measured by autocorrelation) = YES

data is random, distribution is
fixed uniform)


1.4.2. Case Studies
View
Dataplot
Macro for
this Case
Study
Data Analysis Steps
Results and
Conclusions
more detailed

1. You have read 1
column of numbers
into Dataplot,
variable Y.
plot, there are no
shifts
in location or
scale, and the data
do not
seem to follow a
overlaid uniform pdf.
plot.
6. Generate a uniform probability
plot.
1. The run sequence
plot indicates that
there are no
scale.
2. The lag plot
significant
show the data
were not random).
3. The histogram
indicates that a
normal
distribution is not a
good
distribution for
these data.
4. The histogram
indicates that a
uniform
distribution is a
good
distribution for
these data.
5. The normal
probability plot
verifies
that the normal
distribution is not a
reasonable
distribution for
these data.
6. The uniform
probability plot
verifies
that the uniform
distribution is a
reasonable
distribution for
these data.
4. Generate the bootstrap plot.
1. Generate a bootstrap plot. 1. The bootstrap
plot clearly shows
the superiority
of the mid-range
over the mean
and median as the
location
estimator of choice
for
this problem.
quantitative
report.
statistics.
location.
1. The summary
statistics table
displays
25+ statistics.
2. The mean is
0.5078 and a 95%
confidence
interval is
(0.482,0.534).
The linear fit
location since
the slope parameter
standard
variation
and
computing Barltetts test for equal
an
test.
the
correlation
coefficient.
assumes
run).
is
statistically not
significant.
3. The standard
deviation is 0.29
with
a 95% confidence
interval of
(0.277,0.314).
Levene's test
indicates no
significant
drift in
variation.
4. The lag 1
autocorrelation is -
0.03.
From the
this is
within the 95%
confidence interval
bands.
5. The uniform
probability plot
correlation
coefficient is
0.9995. This
indicates that
the uniform
distribution is a
good fit.
6. The results are
summarized in a
convenient
report.
1.4.2.3. Random Walk

1.4.2. Case Studies
Random
Walk
numbers derived from a random walk.
2. Test Underlying Assumptions
3. Develop Better Model
4. Validate New Model

1.4.2. Case Studies
Generation A random walk can be generated from a set of uniform
random numbers by the formula:
where U is a set of uniform random numbers.
The motivation for studying a set of random walk data is to
illustrate the effects of a known underlying autocorrelation
structure (i.e., non-randomness) in the data.
Data The following is the set of random walk numbers used for this
case study.
-0.399027
-0.645651
-0.625516
-0.262049
-0.407173
-0.097583
0.314156
0.106905
-0.017675
-0.037111
0.357631
0.820111
0.844148
0.550509
0.090709
0.413625
-0.002149
0.393170
0.538263
0.070583
0.473143
0.132676
0.109111
-0.310553
0.179637
-0.067454
-0.190747
-0.536916
-0.905751
-0.518984
-0.579280
-0.643004
-1.014925
-0.517845
-0.860484
-0.884081
-1.147428
-0.657917
-0.470205
-0.798437
-0.637780
-0.666046
-1.093278
-1.089609
-0.853439
-0.695306
-0.206795
-0.507504
-0.696903
-1.116358
-1.044534
-1.481004
-1.638390
-1.270400
-1.026477
-1.123380
-0.770683
-0.510481
-0.958825
-0.531959
-0.457141
-0.226603
-0.201885
-0.078000
0.057733
-0.228762
-0.403292
-0.414237
-0.556689
-0.772007
-0.401024
-0.409768
-0.171804
-0.096501
-0.066854
0.216726
0.551008
0.660360
0.194795
-0.031321
0.453880
0.730594
1.136280
0.708490
1.149048
1.258757
1.102107
1.102846
0.720896
0.764035
1.072312
0.897384
0.965632
0.759684
0.679836
0.955514
1.290043
1.753449
1.542429
1.873803
2.043881
1.728635
1.289703
1.501481
1.888335
1.408421
1.416005
0.929681
1.097632
1.501279
1.650608
1.759718
2.255664
2.490551
2.508200
2.707382
2.816310
3.254166
2.890989
2.869330
3.024141
3.291558
3.260067
3.265871
3.542845
3.773240
3.991880
3.710045
4.011288
4.074805
4.301885
3.956416
4.278790
3.989947
4.315261
4.200798
4.444307
4.926084
4.828856
4.473179
4.573389
4.528605
4.452401
4.238427
4.437589
4.617955
4.370246
4.353939
4.541142
4.807353
4.706447
4.607011
4.205943
3.756457
3.482142
3.126784
3.383572
3.846550
4.228803
4.110948
4.525939
4.478307
4.457582
4.822199
4.605752
5.053262
5.545598
5.134798
5.438168
5.397993
5.838361
5.925389
6.159525
6.190928
6.024970
5.575793
5.516840
5.211826
4.869306
4.912601
5.339177
5.415182
5.003303
4.725367
4.350873
4.225085
3.825104
3.726391
3.301088
3.767535
4.211463
4.418722
4.554786
4.987701
4.993045
5.337067
5.789629
5.726147
5.934353
5.641670
5.753639
5.298265
5.255743
5.500935
5.434664
5.588610
6.047952
6.130557
5.785299
5.811995
5.582793
5.618730
5.902576
6.226537
5.738371
5.449965
5.895537
6.252904
6.650447
7.025909
6.770340
7.182244
6.941536
7.368996
7.293807
7.415205
7.259291
6.970976
7.319743
6.850454
6.556378
6.757845
6.493083
6.824855
6.533753
6.410646
6.502063
6.264585
6.730889
6.753715
6.298649
6.048126
5.794463
5.539049
5.290072
5.409699
5.843266
5.680389
5.185889
5.451353
5.003233
5.102844
5.566741
5.613668
5.352791
5.140087
4.999718
5.030444
5.428537
5.471872
5.107334
5.387078
4.889569
4.492962
4.591042
4.930187
4.857455
4.785815
5.235515
4.865727
4.855005
4.920206
4.880794
4.904395
4.795317
5.163044
4.807122
5.246230
5.111000
5.228429
5.050220
4.610006
4.489258
4.399814
4.606821
4.974252
5.190037
5.084155
5.276501
4.917121
4.534573
4.076168
4.236168
3.923607
3.666004
3.284967
2.980621
2.623622
2.882375
3.176416
3.598001
3.764744
3.945428
4.408280
4.359831
4.353650
4.329722
4.294088
4.588631
4.679111
4.182430
4.509125
4.957768
4.657204
4.325313
4.338800
4.720353
4.235756
4.281361
3.795872
4.276734
4.259379
3.999663
3.544163
3.953058
3.844006
3.684740
3.626058
3.457909
3.581150
4.022659
4.021602
4.070183
4.457137
4.156574
4.205304
4.514814
4.055510
3.938217
4.180232
3.803619
3.553781
3.583675
3.708286
4.005810
4.419880
4.881163
5.348149
4.950740
5.199262
4.753162
4.640757
4.327090
4.080888
3.725953
3.939054
3.463728
3.018284
2.661061
3.099980
3.340274
3.230551
3.287873
3.497652
3.014771
3.040046
3.342226
3.656743
3.698527
3.759707
4.253078
4.183611
4.196580
4.257851
4.683387
4.224290
3.840934
4.329286
3.909134
3.685072
3.356611
2.956344
2.800432
2.761665
2.744913
3.037743
2.787390
2.387619
2.424489
2.247564
2.502179
2.022278
2.213027
2.126914
2.264833
2.528391
2.432792
2.037974
1.699475
2.048244
1.640126
1.149858
1.475253
1.245675
0.831979
1.165877
1.403341
1.181921
1.582379
1.632130
2.113636
2.163129
2.545126
2.963833
3.078901
3.055547
3.287442
2.808189
2.985451
3.181679
2.746144
2.517390
2.719231
2.581058
2.838745
2.987765
3.459642
3.458684
3.870956
4.324706
4.411899
4.735330
4.775494
4.681160
4.462470
3.992538
3.719936
3.427081
3.256588
3.462766
3.046353
3.537430
3.579857
3.931223
3.590096
3.136285
3.391616
3.114700
2.897760
2.724241
2.557346
2.971397
2.479290
2.305336
1.852930
1.471948
1.510356
1.633737
1.727873
1.512994
1.603284
1.387950
1.767527
2.029734
2.447309
2.321470
2.435092
2.630118
2.520330
2.578147
2.729630
2.713100
3.107260
2.876659
2.774242
3.185503
3.403148
3.392646
3.123339
3.164713
3.439843
3.321929
3.686229
3.203069
3.185843
3.204924
3.102996
3.496552
3.191575
3.409044
3.888246
4.273767
3.803540
4.046417
4.071581
3.916256
3.634441
4.065834
3.844651
3.915219
1.4.2.3.2. Test Underlying Assumptions

1.4.2. Case Studies
an "in control" measurement process are valid.
These assumptions are:
1. random drawings;
and
is appropriate and valid, with s denoting the
standard deviation of the original data.
4-Plot of Data
above:
1. The run sequence plot (upper left) indicates
significant shifts in location over time.
2. The lag plot (upper right) indicates significant non-
randomness in the data.
3. When the assumptions of randomness and constant
location and scale are not satisfied, the
distributional assumptions are not meaningful.
Therefore we do not attempt to make any
interpretation of the histogram (lower left) or the
normal probability plot (lower right).
assumptions are seriously violated. Therefore the Y
i
= C +
E
i
model is not valid.
When the randomness assumption is seriously violated, a
time series model may be appropriate. The lag plot often
suggests a reasonable model. For example, in this case the
strongly linear appearance of the lag plot suggests a model
fitting Y
i
versus Y
i-1
might be appropriate. When the data
are non-random, it is helpful to supplement the lag plot
with an autocorrelation plot and a spectral plot. Although
in this case the lag plot is enough to suggest an
appropriate model, we provide the autocorrelation and
spectral plots for comparison.
Autocorrelation
Plot
When the lag plot indicates significant non-randomness, it
can be helpful to follow up with a an autocorrelation plot.
This autocorrelation plot shows significant autocorrelation
at lags 1 through 100 in a linearly decreasing fashion.
Spectral Plot Another useful plot for non-random data is the spectral
plot.
This spectral plot shows a single dominant low frequency
peak.
Quantitative
Output
Although the 4-plot above clearly shows the violation of
the assumptions, we supplement the graphical output with
some quantitative measures.
Summary
Statistics
Sample size = 500
Mean = 3.216681
Median = 3.612030
Minimum = -1.638390
Maximum = 7.415205
Range = 9.053595
Stan. Dev. = 2.078675
We also computed the autocorrelation to be 0.987, which
is evidence of a very strong autocorrelation.
Location One way to quantify a change in location over time is to
fit a straight line to the data using an index variable as the
data were collected at equally spaced time intervals. In
our regression, we use the index variable X = 1, 2, ..., N,
where N is the number of observations. If there is no
significant drift in the location over time, the slope
parameter should be zero.
t-Value
B
0
1.83351 0.1721
10.650
B
1
0.552164E-02 0.5953E-03
9.275

The t-value of the slope parameter, 9.275, is larger than
the critical value of t
0.975,498
= 1.96. Thus, we conclude
that the slope is different from zero at the 0.05
significance level.
non-normality. Since we know this data set is not
approximated well by the normal distribution, we use the
alternative Levene test. In particular, we use the Levene
test based on the median rather the mean. The choice of
the number of intervals is somewhat arbitrary, although
values of four or eight are reasonable. We will divide our
data into four intervals.
H
0
:
1
2
=
2
2
=
3
2
=
4
2

H
a
: At least one
i
2
is not equal to the
others.
Critical value: F
,k-1,N-k
= 2.623
0
if W > 2.623
In this case, the Levene test indicates that the variances
are significantly different in the four intervals since the
test statistic of 10.459 is greater than the 95 % critical
value of 2.623. Therefore we conclude that the scale is not
constant.
Randomness Although the lag 1 autocorrelation coefficient above
clearly shows the non-randomness, we show the output
from a runs test as well.
H
0
manner
H
a
random manner
Critical value: Z
1-/2
= 1.96
0
if |Z| > 1.96
The runs test rejects the null hypothesis that the data were
produced in a random manner at the 0.05 significance
level.
Distributional
Assumptions
Since the quantitative tests show that the assumptions of
randomness and constant location and scale are not met,
the distributional measures will not be meaningful.
Therefore these quantitative tests are omitted.
1.4.2.3.3. Develop A Better Model

1.4.2. Case Studies
1.4.2.3.3. Develop A Better Model
Lag Plot
Suggests
Better
Model
Since the underlying assumptions did not hold, we need to
develop a better model.
The lag plot showed a distinct linear pattern. Given the
definition of the lag plot, Y
i
versus Y
i-1
, a good candidate
model is a model of the form
Fit Output The results of a linear fit of this model generated the
following results.
Coefficient Estimate Stan. Error t-
Value
A
0
0.050165 0.024171
2.075
A
1
0.987087 0.006313
156.350

The slope parameter, A
1
, has a t value of 156.350 which is
statistically significant. Also, the residual standard deviation is
0.2931. This can be compared to the standard deviation shown
in the summary table, which is 2.078675. That is, the fit to the
autoregressive model has reduced the variability by a factor of
7.
Time
Series
Model
This model is an example of a time series model. More
extensive discussion of time series is given in the Process
Monitoring chapter.
1.4.2.3.4. Validate New Model

1.4.2. Case Studies
Plot
Predicted
with Original
Data
The first step in verifying the model is to plot the predicted
values from the fit with the original data.
This plot indicates a reasonably good fit.
Test
Underlying
Assumptions
on the
Residuals
In addition to the plot of the predicted values, the residual
standard deviation from the fit also indicates a significant
improvement for the model. The next step is to validate the
underlying assumptions for the error component, or
residuals, from this model.
4-Plot of
Residuals
above:
1. The run sequence plot (upper left) indicates no
significant shifts in location or scale over time.
2. The lag plot (upper right) exhibits a random
appearance.
3. The histogram shows a relatively flat appearance.
This indicates that a uniform probability distribution
may be an appropriate model for the error component
(or residuals).
4. The normal probability plot clearly shows that the
normal distribution is not an appropriate model for
the error component.
A uniform probability plot can be used to further test the
suggestion that a uniform distribution might be a good
model for the error component.
Uniform
Probability
Plot of
Residuals
Since the uniform probability plot is nearly linear, this
verifies that a uniform distribution is a good model for the
error component.
Conclusions Since the residuals from our model satisfy the underlying
assumptions, we conlude that
where the E
i
follow a uniform distribution is a good model
for this data set. We could simplify this model to
This has the advantage of simplicity (the current point is
simply the previous point plus a uniformly distributed error
term).
Using
Scientific and
Engineering
Knowledge
In this case, the above model makes sense based on our
definition of the random walk. That is, a random walk is
the cumulative sum of uniformly distributed data points. It
makes sense that modeling the current point as the previous
point plus a uniformly distributed error term is about as
good as we can do. Although this case is a bit artificial in
that we knew how the data were constructed, it is common
and desirable to use scientific and engineering knowledge
of the process that generated the data in formulating and
testing models for the data. Quite often, several competing
models will produce nearly equivalent mathematical results.
In this case, selecting the model that best approximates the
scientific understanding of the process is a reasonable
choice.
Time Series
Model
This model is an example of a time series model. More
extensive discussion of time series is given in the Process
Monitoring chapter.

1.4.2. Case Studies
View
Dataplot
Macro for
this Case
Study
Data Analysis Steps
Results and
Conclusions
more detailed

1. You have read 1
column of numbers
into Dataplot,
variable Y.
2. Validate assumptions.
1. 4-plot of Y.
statistics.
3. Generate a linear fit to detect
drift in location.
4. Detect drift in variation by
1. Based on the 4-
plot, there are
shifts
in location and
scale and the data
are not
random.
2. The summary
statistics table
displays
25+ statistics.
dividing the data into quarters
and
computing Levene's test for equal
a runs test.
3. The linear fit
indicates drift in
location since
the slope parameter
is statistically
significant.
4. Levene's test
indicates significant
drift in
variation.
5. The runs test
non-randomness.
3. Generate the randomness plots.
1. Generate an autocorrelation plot.
2. Generate a spectral plot.
1. The
autocorrelation plot
shows
significant
autocorrelation at
lag 1.
2. The spectral plot
shows a single
dominant
low frequency
peak.
4. Fit Y
i
= A0 + A1*Y
i-1
+ E
i
and validate.
1. Generate the fit.
2. Plot fitted line with original
data.
3. Generate a 4-plot of the residuals
from the fit.
4. Generate a uniform probability
plot
of the residuals.
1. The residual
standard deviation
from the
fit is 0.29
(compared to the
standard
deviation of 2.08
from the original
data).
2. The plot of the
predicted values with
the original data
indicates a good fit.
3. The 4-plot
indicates that the
assumptions
of constant
location and scale
are valid.
The lag plot
indicates that the
data are
random. However,
the histogram and
normal
probability plot
indicate that the
uniform
disribution might
be a better model for
the residuals
than the normal
distribution.
4. The uniform
probability plot
verifies
that the
residuals can be fit
by a
uniform
distribution.
1.4.2.4. Josephson Junction Cryothermometry

1.4.2. Case Studies
Josephson
Junction
Cryothermometry
This example illustrates the univariate analysis of
Josephson junction cyrothermometry.

1.4.2. Case Studies
Generation This data set was collected by Bob Soulen of NIST in
October, 1971 as a sequence of observations collected equi-
spaced in time from a volt meter to ascertain the process
temperature in a Josephson junction cryothermometry (low
temperature) experiment. The response variable is voltage
counts.
Motivation The motivation for studying this data set is to illustrate the
case where there is discreteness in the measurements, but the
underlying assumptions hold. In this case, the discreteness is
due to the data being integers.
Data The following are the data used for this case study.
2899 2898 2898 2900 2898
2901 2899 2901 2900 2898
2898 2898 2898 2900 2898
2897 2899 2897 2899 2899
2900 2897 2900 2900 2899
2898 2898 2899 2899 2899
2899 2899 2898 2899 2899
2899 2902 2899 2900 2898
2899 2899 2899 2899 2899
2899 2900 2899 2900 2898
2901 2900 2899 2899 2899
2899 2899 2900 2899 2898
2898 2898 2900 2896 2897
2899 2899 2900 2898 2900
2901 2898 2899 2901 2900
2898 2900 2899 2899 2897
2899 2898 2899 2899 2898
2899 2897 2899 2899 2897
2899 2897 2899 2897 2897
2899 2897 2898 2898 2899
2897 2898 2897 2899 2899
2898 2898 2897 2898 2895
2897 2898 2898 2896 2898
2898 2897 2896 2898 2898
2897 2897 2898 2898 2896
2898 2898 2896 2899 2898
2898 2898 2899 2899 2898
2898 2899 2899 2899 2900
2900 2901 2899 2898 2898
2900 2899 2898 2901 2897
2898 2898 2900 2899 2899
2898 2898 2899 2898 2901
2900 2897 2897 2898 2898
2900 2898 2899 2898 2898
2898 2896 2895 2898 2898
2898 2898 2897 2897 2895
2897 2897 2900 2898 2896
2897 2898 2898 2899 2898
2897 2898 2898 2896 2900
2899 2898 2896 2898 2896
2896 2896 2897 2897 2896
2897 2897 2896 2898 2896
2898 2896 2897 2896 2897
2897 2898 2897 2896 2895
2898 2896 2896 2898 2896
2898 2898 2897 2897 2898
2897 2899 2896 2897 2899
2900 2898 2898 2897 2898
2899 2899 2900 2900 2900
2900 2899 2899 2899 2898
2900 2901 2899 2898 2900
2901 2901 2900 2899 2898
2901 2899 2901 2900 2901
2898 2900 2900 2898 2900
2900 2898 2899 2901 2900
2899 2899 2900 2900 2899
2900 2901 2899 2898 2898
2899 2896 2898 2897 2898
2898 2897 2897 2897 2898
2897 2899 2900 2899 2897
2898 2900 2900 2898 2898
2899 2900 2898 2900 2900
2898 2900 2898 2898 2898
2898 2898 2899 2898 2900
2897 2899 2898 2899 2898
2897 2900 2901 2899 2898
2898 2901 2898 2899 2897
2899 2897 2896 2898 2898
2899 2900 2896 2897 2897
2898 2899 2899 2898 2898
2897 2897 2898 2897 2897
2898 2898 2898 2896 2895
2898 2898 2898 2896 2898
2898 2898 2897 2897 2899
2896 2900 2897 2897 2898
2896 2897 2898 2898 2898
2897 2897 2898 2899 2897
2898 2899 2897 2900 2896
2899 2897 2898 2897 2900
2899 2900 2897 2897 2898
2897 2899 2899 2898 2897
2901 2900 2898 2901 2899
2900 2899 2898 2900 2900
2899 2898 2897 2900 2898
2898 2897 2899 2898 2900
2899 2898 2899 2897 2900
2898 2902 2897 2898 2899
2899 2899 2898 2897 2898
2897 2898 2899 2900 2900
2899 2898 2899 2900 2899
2900 2899 2899 2899 2899
2899 2898 2899 2899 2900
2902 2899 2900 2900 2901
2899 2901 2899 2899 2902
2898 2898 2898 2898 2899
2899 2900 2900 2900 2898
2899 2899 2900 2899 2900
2899 2900 2898 2898 2898
2900 2898 2899 2900 2899
2899 2900 2898 2898 2899
2899 2899 2899 2898 2898
2897 2898 2899 2897 2897
2901 2898 2897 2898 2899
2898 2897 2899 2898 2897
2898 2898 2897 2898 2899
2899 2899 2899 2900 2899
2899 2897 2898 2899 2900
2898 2897 2901 2899 2901
2898 2899 2901 2900 2900
2899 2900 2900 2900 2900
2901 2900 2901 2899 2897
2900 2900 2901 2899 2898
2900 2899 2899 2900 2899
2900 2899 2900 2899 2901
2900 2900 2899 2899 2898
2899 2900 2898 2899 2899
2901 2898 2898 2900 2899
2899 2898 2897 2898 2897
2899 2899 2899 2898 2898
2897 2898 2899 2897 2897
2899 2898 2898 2899 2899
2901 2899 2899 2899 2897
2900 2896 2898 2898 2900
2897 2899 2897 2896 2898
2897 2898 2899 2896 2899
2901 2898 2898 2896 2897
2899 2897 2898 2899 2898
2898 2898 2898 2898 2898
2899 2900 2899 2901 2898
2899 2899 2898 2900 2898
2899 2899 2901 2900 2901
2899 2901 2899 2901 2899
2900 2902 2899 2898 2899
2900 2899 2900 2900 2901
2900 2899 2901 2901 2899
2898 2901 2897 2898 2901
2900 2902 2899 2900 2898
2900 2899 2900 2899 2899
2899 2898 2900 2898 2899
2899 2899 2899 2898 2900

1.4.2. Case Studies
assumptions are:
1. random drawings;
and
4-Plot of
Data
above:
scale over time.
reasonably symmetric, there does not appear to be
significant outliers in the tails, and that it is
reasonable to assume that the data can be fit with a
4. The normal probability plot (lower right) is difficult
to interpret due to the fact that there are only a few
distinct values with many repeats.
The integer data with only a few distinct values and many
repeats accounts for the discrete appearance of several of
the plots (e.g., the lag plot and the normal probability plot).
In this case, the nature of the data makes the normal
probability plot difficult to interpret, especially since each
number is repeated many times. However, the histogram
indicates that a normal distribution should provide an
adequate model for the data.
assumptions are valid and the data can be reasonably
approximated with a normal distribution. Therefore, the
commonly used uncertainty standard is valid and
appropriate. The numerical values for this model are given
in the Quantitative Output and Interpretation section.
Individual
Plots
Although it is normally not necessary, the plots can be
Run
Sequence
Plot
Lag Plot
Histogram
(with
overlaid
Normal PDF)
Normal
Probability
Plot

1.4.2. Case Studies
Interpretation
Summary
Statistics
were computed from the data.
Sample size = 700
Mean = 2898.562
Median = 2899.000
Minimum = 2895.000
Maximum = 2902.000
Range = 7.000
Stan. Dev. = 1.305
Because of the discrete nature of the data, we also compute
the normal PPCC.
Normal PPCC = 0.97484
be zero.
t-Value
B
0
2.898E+03 9.745E-02
29739.288
B
1
1.071E-03 2.409e-04
4.445

The slope parameter, B
1
statistically significant (the critical value is 1.96). However,
the value of the slope is 1.071E-03. Given that the slope is
nearly zero, the assumption of constant location is not
seriously violated even though it is statistically significant.
non-normality. Since the nature of the data (a few distinct
points repeated many times) makes the normality
assumption questionable, we use the alternative Levene
test. In particular, we use the Levene test based on the
median rather the mean. The choice of the number of
intervals is somewhat arbitrary, although values of four or
eight are reasonable. We will divide our data into four
intervals.
H
0
:
1
2
=
2
2
=
3
2
=
4
2

H
a
: At least one
i
2
is not equal to the
others.
Critical value: F
,k-1,N-k
= 2.618
0
if W > 2.618
Since the Levene test statistic value of 1.43 is less than the
95 % critical value of 2.618, we conclude that the variances
are not significantly different in the four intervals.
detected with a few simple tests. The lag plot in the
previous section is a simple graphical technique.
0 is always 1).
most interest, is 0.31. The critical values at the 5 % level of
significance are -0.087 and 0.087. This indicates that the
lag 1 autocorrelation is statistically significant, so there is
some evidence for non-randomness.
H
0
manner
H
a
random manner
Critical value: Z
1-/2
= 1.96
0
if |Z| > 1.96
The runs test indicates non-randomness.
Although the runs test and lag 1 autocorrelation indicate
some mild non-randomness, it is not sufficient to reject the
Y
i
= C + E
i
model. At least part of the non-randomness can
be explained by the discrete nature of the data.
Distributional
Analysis
distribution is 0.975. Since the PPCC is less than the critical
value of 0.987 (this is a tabulated value), the normality
assumption is rejected.
H
0
H
a
2
= 16.858
0
if A
2
> 0.787
because the test statistic, 16.858, is greater than the 95 %
critical value 0.787.
Although the data are not strictly normal, the violation of
the normality assumption is not severe enough to conclude
that the Y
i
= C + E
i
model is unreasonable. At least part of
the non-normality can be explained by the discrete nature
of the data.
Outlier
Analysis
A test for outliers is the Grubbs test.
H
0
H
a
Critical value for a one-tailed test:
3.950619
0
if G > 3.950619
Model Although the randomness and normality assumptions were
mildly violated, we conclude that a reasonable model for
the data is:
In addition, a 95 % confidence interval for the mean value
is (2898.515, 2898.928).
Univariate
Report
Analysis for Josephson Junction Cryothermometry
Data


2: Location
Mean =
2898.562
0.049323
(2898.465,2898.658)
Drift with respect to location? = YES
(Further analysis indicates that
the drift, while statistically
significant, is not practically
significant)

3: Variation
1.30497
(1.240007,1.377169)
of the data) = NO

4: Distribution
Normal PPCC =
0.97484
Data are Normal?
(as measured by Normal PPCC) = NO

5: Randomness
Autocorrelation =
0.314802
Data are Random?
(as measured by autocorrelation) = NO

fixed normal)
Data Set is in Statistical Control? = NO

Note: Although we have violations of
the assumptions, they are mild enough,
and at least partially explained by the
discrete nature of the data, so we may model
the data as if it were in statistical
control

7: Outliers?
(as determined by Grubbs test) = NO

1.4.2. Case Studies
View
Dataplot
Macro for
this Case
Study
Data Analysis Steps
Results and
Conclusions
more detailed

1. You have read 1
column of numbers
into Dataplot,
variable Y.
plot, there are no
shifts
in location or
scale. Due to the
nature
of the data (a
few distinct points
with
many repeats),
the normality
assumption is
questionable.
plot.
1. The run sequence
plot indicates that
there are no
scale.
2. The lag plot
significant
show the data
were not random).
3. The histogram
indicates that a
normal
distribution is a
good
distribution for
these data.
4. The discrete
nature of the data
masks
the normality or
non-normality of the
data somewhat.
The plot indicates
that
a normal
distribution provides
a rough
approximation for
the data.
quantitative
report.
statistics.
location.
standard
variation
and
computing Levene's test for equal
an
test.
the
correlation
coefficient.
1. The summary
statistics table
displays
25+ statistics.
2. The mean is
2898.56 and a 95%
confidence
interval is
(2898.46,2898.66).
The linear fit
indicates no
meaningful drift
in location since
the value of the
slope
parameter is near
zero.
3. The standard
devaition is 1.30
with
a 95% confidence
interval of
(1.24,1.38).
Levene's test
indicates no
significant
drift in
variation.
test.
assumes
run).
4. The lag 1
autocorrelation is
0.31.
This indicates
some mild non-
randomness.
5. The normal
probability plot
correlation
coefficient is
0.975. At the 5%
level,
we reject the
normality assumption.
6. Grubbs' test
detects no outliers
at the
5% level.
7. The results are
summarized in a
convenient
report.
1.4.2.5. Beam Deflections

1.4.2. Case Studies
Beam
Deflection
This example illustrates the univariate analysis of beam
deflection data.
2. Test Underlying Assumptions
3. Develop a Better Model
4. Validate New Model

1.4.2. Case Studies
Generation This data set was collected by H. S. Lew of NIST in 1969 to
measure steel-concrete beam deflections. The response
variable is the deflection of a beam from the center point.
The motivation for studying this data set is to show how the
underlying assumptions are affected by periodic data.
-213
-564
-35
-15
141
115
-420
-360
203
-338
-431
194
-220
-513
154
-125
-559
92
-21
-579
-52
99
-543
-175
162
-457
-346
204
-300
-474
164
-107
-572
-8
83
-541
-224
180
-420
-374
201
-236
-531
83
27
-564
-112
131
-507
-254
199
-311
-495
143
-46
-579
-90
136
-472
-338
202
-287
-477
169
-124
-568
17
48
-568
-135
162
-430
-422
172
-74
-577
-13
92
-534
-243
194
-355
-465
156
-81
-578
-64
139
-449
-384
193
-198
-538
110
-44
-577
-6
66
-552
-164
161
-460
-344
205
-281
-504
134
-28
-576
-118
156
-437
-381
200
-220
-540
83
11
-568
-160
172
-414
-408
188
-125
-572
-32
139
-492
-321
205
-262
-504
142
-83
-574
0
48
-571
-106
137
-501
-266
190
-391
-406
194
-186
-553
83
-13
-577
-49
103
-515
-280
201
300
-506
131
-45
-578
-80
138
-462
-361
201
-211
-554
32
74
-533
-235
187
-372
-442
182
-147
-566
25
68
-535
-244
194
-351
-463
174
-125
-570
15
72
-550
-190
172
-424
-385
198
-218
-536
96

1.4.2. Case Studies
an "in control" measurement process are valid.
1. random drawings;
and
4-Plot of Data
above:
scale over time.
2. The lag plot (upper right) shows that the data are
not random. The lag plot further indicates the
presence of a few outliers.
3. When the randomness assumption is thus seriously
violated, the histogram (lower left) and normal
probability plot (lower right) are ignored since
determining the distribution of the data is only
meaningful when the data are random.
From the above plots we conclude that the underlying
randomness assumption is not valid. Therefore, the model
is not appropriate.
We need to develop a better model. Non-random data can
frequently be modeled using time series mehtodology.
Specifically, the circular pattern in the lag plot indicates
that a sinusoidal model might be appropriate. The
sinusoidal model will be developed in the next section.
Individual Plots The plots can be generated individually for more detail. In
this case, only the run sequence plot and the lag plot are
drawn since the distributional plots are not meaningful.
Run Sequence
Plot
Lag Plot
We have drawn some lines and boxes on the plot to better
isolate the outliers. The following data points appear to be
outliers based on the lag plot.
INDEX Y(i-1) Y(i)
158 -506.00 300.00
157 300.00 201.00
3 -15.00 -35.00
5 115.00 141.00

That is, the third, fifth, 157th, and 158th points appear to
be outliers.
Autocorrelation
Plot
When the lag plot indicates significant non-randomness, it
can be helpful to follow up with a an autocorrelation plot.
This autocorrelation plot shows a distinct cyclic pattern.
As with the lag plot, this suggests a sinusoidal model.
Spectral Plot Another useful plot for non-random data is the spectral
plot.
This spectral plot shows a single dominant peak at a
frequency of 0.3. This frequency of 0.3 will be used in
fitting the sinusoidal model in the next section.
Quantitative
Results
Although the lag plot, autocorrelation plot, and spectral
plot clearly show the violation of the randomness
assumption, we supplement the graphical output with
some quantitative measures.
Summary
Statistics
As a first step in the analysis, summary statistics are
computed from the data.
Sample size = 200
Mean = -177.4350
Median = -162.0000
Minimum = -579.0000
Maximum = 300.0000
Range = 879.0000
Stan. Dev. = 277.3322
Location One way to quantify a change in location over time is to
fit a straight line to the data set using the index variable X
= 1, 2, ..., N, with N denoting the number of observations.
If there is no significant drift in the location, the slope
parameter should be zero.
t-Value
A
0
-178.175 39.47
-4.514
A
1
0.7366E-02 0.34
0.022

The slope parameter, A1, has a t value of 0.022 which is
statistically not significant. This indicates that the slope
can in fact be considered zero.
sized intervals. However, the Bartlett the non-randomness
of this data does not allows us to assume normality, we
use the alternative Levene test. In partiuclar, we use the
Levene test based on the median rather the mean. The
choice of the number of intervals is somewhat arbitrary,
although values of 4 or 8 are reasonable.
H
0
:
1
2
=
2
2
=
3
2
=
4
2

H
a
: At least one
i
2
is not equal to the
others.
Sample size: N = 200
Critical value: F
,k-1,N-k
= 2.651
0
if W > 2.651
In this case, the Levene test indicates that the variances
are not significantly different in the four intervals since
the test statistic value, 0.9378, is less than the critical
value of 2.651.
Randomness A runs test is used to check for randomness
H
0
manner
H
a
random manner
Critical value: Z
1-/2
= 1.96
0
if |Z| > 1.96
The absolute value of the test statistic is larger than the
critical value at the 5 % significance level, so we conclude
that the data are not random.
Distributional
Assumptions
Since the quantitative tests show that the assumptions of
constant scale and non-randomness are not met, the
distributional measures will not be meaningful. Therefore
these quantitative tests are omitted.
1.4.2.5.3. Develop a Better Model

1.4.2. Case Studies
Sinusoidal
Model
The lag plot and autocorrelation plot in the previous section strongly
suggested a sinusoidal model might be appropriate. The basic sinusoidal
model is:
where C is constant defining a mean level, is an amplitude for the sine
function, is the frequency, T
i
is a time variable, and is the phase.
This sinusoidal model can be fit using non-linear least squares.
To obtain a good fit, sinusoidal models require good starting values for
C, the amplitude, and the frequency.
Good Starting
Value for C
A good starting value for C can be obtained by calculating the mean of
the data. If the data show a trend, i.e., the assumption of constant
location is violated, we can replace C with a linear or quadratic least
squares fit. That is, the model becomes
or
Since our data did not have any meaningful change of location, we can
fit the simpler model with C equal to the mean. From the summary
output in the previous page, the mean is -177.44.
Good Starting
Value for
Frequency
The starting value for the frequency can be obtained from the spectral
plot, which shows the dominant frequency is about 0.3.
Complex
Demodulation
Phase Plot
The complex demodulation phase plot can be used to refine this initial
estimate for the frequency.
For the complex demodulation plot, if the lines slope from left to right,
the frequency should be increased. If the lines slope from right to left, it
should be decreased. A relatively flat (i.e., horizontal) slope indicates a
good frequency. We could generate the demodulation phase plot for 0.3
and then use trial and error to obtain a better estimate for the frequency.
To simplify this, we generate 16 of these plots on a single page starting
with a frequency of 0.28, increasing in increments of 0.0025, and
stopping at 0.3175.
Interpretation The plots start with lines sloping from left to right but gradually change
to a right to left slope. The relatively flat slope occurs for frequency
0.3025 (third row, second column). The complex demodulation phase
plot restricts the range from to . This is why the plot appears
to show some breaks.
Good Starting
Values for
Amplitude
The complex demodulation amplitude plot is used to find a good starting
value for the amplitude. In addition, this plot indicates whether or not the
amplitude is constant over the entire range of the data or if it varies. If
the plot is essentially flat, i.e., zero slope, then it is reasonable to assume
a constant amplitude in the non-linear model. However, if the slope
varies over the range of the plot, we may need to adjust the model to be:
That is, we replace with a function of time. A linear fit is specified in
the model above, but this can be replaced with a more elaborate function
if needed.
Complex
Demodulation
Amplitude
Plot
The complex demodulation amplitude plot for this data shows that:
1. The amplitude is fixed at approximately 390.
2. There is a short start-up effect.
3. There is a change in amplitude at around x=160 that should be
investigated for an outlier.
In terms of a non-linear model, the plot indicates that fitting a single
constant for should be adequate for this data set.
Fit Results Using starting estimates of 0.3025 for the frequency, 390 for the
amplitude, and -177.44 for C, the following parameters were estimated.
Coefficient Estimate Stan. Error t-Value
C -178.786 11.02 -16.22
AMP -361.766 26.19 -13.81
FREQ 0.302596 0.1510E-03 2005.00
PHASE 1.46536 0.4909E-01 29.85

Model From the fit results, our proposed model is:
We will evaluate the adequacy of this model in the next section.

1.4.2. Case Studies
4-Plot of
Residuals
The first step in evaluating the fit is to generate a 4-plot of the residuals.
Interpretation The assumptions are addressed by the graphics shown above:
1. The run sequence plot (upper left) indicates that the data do not
have any significant shifts in location. There does seem to be
some shifts in scale. A start-up effect was detected previously by
the complex demodulation amplitude plot. There does appear to be
a few outliers.
2. The lag plot (upper right) shows that the data are random. The
outliers also appear in the lag plot.
3. The histogram (lower left) and the normal probability plot (lower
right) do not show any serious non-normality in the residuals.
However, the bend in the left portion of the normal probability
plot shows some cause for concern.
The 4-plot indicates that this fit is reasonably good. However, we will
attempt to improve the fit by removing the outliers.
Fit Results
with Outliers
Removed
The following parameter estimates were obtained after removing three
outliers.
Coefficient Estimate Stan. Error t-Value
C -178.788 10.57 -16.91
AMP -361.759 25.45 -14.22
FREQ 0.302597 0.1457E-03 2077.00
PHASE 1.46533 0.4715E-01 31.08

New Fit to
Edited Data
The original fit, with a residual standard deviation of 155.84, was:
The new fit, with a residual standard deviation of 148.34, is:
There is minimal change in the parameter estimates and about a 5 %
reduction in the residual standard deviation. In this case, removing the
residuals has a modest benefit in terms of reducing the variability of the
model.
4-Plot for
New Fit
This plot shows that the underlying assumptions are satisfied and
therefore the new fit is a good descriptor of the data.
In this case, it is a judgment call whether to use the fit with or without
the outliers removed.

1.4.2. Case Studies
View
Dataplot
Macro for
this Case
Study
Data Analysis Steps
Results and
Conclusions
more detailed

1. You have read 1
column of numbers
into Dataplot,
variable Y.
2. Validate assumptions.
1. 4-plot of Y.
1. Based on the 4-
plot, there are no
obvious shifts in
location and scale,
but the data are
not random.
2. Based on the run
sequence plot, there
are no obvious
shifts in location
and
4. Generate an autocorrelation plot.
5. Generate a spectral plot.
statistics.
7. Generate a linear fit to detect
drift in location.
8. Detect drift in variation by
dividing the data into quarters
and
computing Levene's test statistic
for
equal standard deviations.
a runs test.
scale.
3. Based on the lag
plot, the data
are not random.
4. The
shows
significant
autocorrelation at
lag 1.
5. The spectral plot
shows a single
dominant
low frequency
peak.
6. The summary
statistics table
displays
25+ statistics.
7. The linear fit
location since
the slope parameter
is not
statistically
significant.
8. Levene's test
indicates no
significant drift
in variation.
9. The runs test
non-randomness.
3. Fit
Y
i
= C + A*SIN(2*PI*omega*t
i
+phi).
1. Generate a complex demodulation
phase plot.
2. Generate a complex demodulation
amplitude plot.
3. Fit the non-linear model.
1. Complex
demodulation phase
plot
indicates a
starting frequency
of 0.3025.
2. Complex
demodulation
amplitude
plot indicates an
amplitude of
390 (but there
is a short start-up
effect).
3. Non-linear fit
generates final
parameter
estimates. The
residual standard
deviation from
the fit is 155.85
(compared to the
standard
deviation of 277.73
from
the original
data).
4. Validate fit.
from the fit.
2. Generate a nonlinear fit with
outliers removed.
from the fit with the outliers
removed.
1. The 4-plot
indicates that the
assumptions
of constant
location and scale
are valid.
The lag plot
indicates that the
data are
random. The
histogram and normal
probability plot
indicate that the
residuals
that the
normality assumption
for the
residuals are not
seriously violated,
although there is
a bend on the
probablity
plot that
warrants attention.
2. The fit after
removing 3 outliers
shows
some marginal
improvement in the
model
(a 5% reduction
in the residual
standard
deviation).
3. The 4-plot of
the model fit after
3 outliers
removed shows
marginal
improvement in
satisfying model
assumptions.
1.4.2.6. Filter Transmittance

1.4.2. Case Studies
Filter
Transmittance
This example illustrates the univariate analysis of filter
transmittance data.

1.4.2. Case Studies
Generation This data set was collected by NIST chemist Radu
Mavrodineaunu in the 1970's from an automatic data
acquisition system for a filter transmittance experiment. The
response variable is transmittance.
The motivation for studying this data set is to show how the
underlying autocorrelation structure in a relatively small data
set helped the scientist detect problems with his automatic
data acquisition system.
2.00180
2.00170
2.00180
2.00190
2.00180
2.00170
2.00150
2.00140
2.00150
2.00150
2.00170
2.00180
2.00180
2.00190
2.00190
2.00210
2.00200
2.00160
2.00140
2.00130
2.00130
2.00150
2.00150
2.00160
2.00150
2.00140
2.00130
2.00140
2.00150
2.00140
2.00150
2.00160
2.00150
2.00160
2.00190
2.00200
2.00200
2.00210
2.00220
2.00230
2.00240
2.00250
2.00270
2.00260
2.00260
2.00260
2.00270
2.00260
2.00250
2.00240

1.4.2. Case Studies
assumptions are:
1. random drawings;
and
4-Plot of
Data
above:
1. The run sequence plot (upper left) indicates a
significant shift in location around x=35.
2. The linear appearance in the lag plot (upper right)
indicates a non-random pattern in the data.
3. Since the lag plot indicates significant non-
randomness, we do not make any interpretation of
either the histogram (lower left) or the normal
probability plot (lower right).
The serious violation of the non-randomness assumption
means that the univariate model
is not valid. Given the linear appearance of the lag plot, the
first step might be to consider a model of the type
However, in this case discussions with the scientist revealed
that non-randomness was entirely unexpected. An
examination of the experimental process revealed that the
sampling rate for the automatic data acquisition system was
too fast. That is, the equipment did not have sufficient time
to reset before the next sample started, resulting in the
current measurement being contaminated by the previous
measurement. The solution was to rerun the experiment
allowing more time between samples.
Simple graphical techniques can be quite effective in
revealing unexpected results in the data. When this occurs,
it is important to investigate whether the unexpected result
is due to problems in the experiment and data collection or
is indicative of unexpected underlying structure in the data.
This determination cannot be made on the basis of statistics
alone. The role of the graphical and statistical analysis is to
detect problems or unexpected results in the data. Resolving
the issues requires the knowledge of the scientist or
engineer.
Individual
Plots
Although it is generally unnecessary, the plots can be
generated individually to give more detail. Since the lag
plot indicates significant non-randomness, we omit the
distributional plots.
Run
Sequence
Plot
Lag Plot

1.4.2. Case Studies
Interpretation
Summary
Statistics
Sample size = 50
Mean = 2.0019
Median = 2.0018
Minimum = 2.0013
Maximum = 2.0027
Range = 0.0014
Stan. Dev. = 0.0004
be zero.
t-Value
B
0
2.00138 0.9695E-04
0.2064E+05
B
1
0.185E-04 0.3309E-05
5.582

Residual Standard Deviation = 0.3376404E-03
1
, has a t value of 5.582, which is
statistically significant. Although the estimated slope,
0.185E-04, is nearly zero, the range of data (2.0013 to
2.0027) is also very small. In this case, we conclude that
there is drift in location, although it is relatively small.
Bartlett test after dividing the data set into several equal
non-normality. Since the normality assumption is
questionable for these data, we use the alternative Levene
intervals.
H
0
:
1
2
=
2
2
=
3
2
=
4
2

H
a
: At least one
i
2
is not equal to the
others.
Critical value: F
,k-1,N-k
= 2.806
0
if W > 2.806
In this case, since the Levene test statistic value of 0.971 is
less than the critical value of 2.806 at the 5 % level, we
conclude that there is no evidence of a change in variation.
detected with a few simple tests. The lag plot in the 4-plot
in the previous seciton is a simple graphical technique.
One check is an autocorrelation plot that shows the
0 is always 1).
most interest, is 0.93. The critical values at the 5 % level
are -0.277 and 0.277. This indicates that the lag 1
autocorrelation is statistically significant, so there is strong
evidence of non-randomness.
H
0
manner
H
a
random manner
Critical value: Z
1-/2
= 1.96
0
if |Z| > 1.96
Because the test statistic is outside of the critical region, we
reject the null hypothesis and conclude that the data are not
random.
Distributional
Analysis
Since we rejected the randomness assumption, the
distributional tests are not meaningful. Therefore, these
quantitative tests are omitted. We also omit Grubbs' outlier
test since it also assumes the data are approximately
normally distributed.
Univariate
Report

Analysis for filter transmittance data

1: Sample Size = 50

2: Location
Mean =
2.001857
0.00006
(2.001735,2.001979)

3: Variation
0.00043
(0.000359,0.000535)
Change in variation?
of the data) = NO

4: Distribution
Distributional tests omitted due to
non-randomness of the data

5: Randomness
Lag One Autocorrelation =
0.937998
Data are Random?

normal)

7: Outliers?
(Grubbs' test omitted) = NO

1.4.2. Case Studies
View
Dataplot
Macro for
this Case
Study
Data Analysis Steps
Results and
Conclusions
more detailed

1. You have read 1
column of numbers
into Dataplot,
variable Y.
plot, there is a
shift
in location and
the data are not
random.
1. Generate a run sequence plot. 1. The run sequence
plot indicates that
there is a shift
in location.
2. The strong linear
pattern of the lag
plot indicates
significant
non-randomness.
quantitative
report.
statistics.
2. Compute a linear fit based on
quarters of the data to detect
drift in location.
3. Compute Levene's test based on
quarters of the data to detect
changes in variation.
an
test.
assumes
run).
1. The summary
statistics table
displays
25+ statistics.
2. The linear fit
indicates a slight
drift in
location since
the slope parameter
is
statistically
significant, but
small.
3. Levene's test
indicates no
significant
drift in
variation.
4. The lag 1
autocorrelation is
0.94.
This is outside
the 95% confidence
interval bands
which indicates
significant
non-randomness.
5. The results are
summarized in a
convenient
report.
1.4.2.7. Standard Resistor

1.4.2. Case Studies
Standard
Resistor
This example illustrates the univariate analysis of standard
resistor data.

1.4.2. Case Studies
Generation This data set was collected by Ron Dziuba of NIST over a 5-
year period from 1980 to 1985. The response variable is
resistor values.
The motivation for studying this data set is to illustrate data
that violate the assumptions of constant location and scale.
27.8680
27.8929
27.8773
27.8530
27.8876
27.8725
27.8743
27.8879
27.8728
27.8746
27.8863
27.8716
27.8818
27.8872
27.8885
27.8945
27.8797
27.8627
27.8870
27.8895
27.9138
27.8931
27.8852
27.8788
27.8827
27.8939
27.8558
27.8814
27.8479
27.8479
27.8848
27.8809
27.8479
27.8611
27.8630
27.8679
27.8637
27.8985
27.8900
27.8577
27.8848
27.8869
27.8976
27.8610
27.8567
27.8417
27.8280
27.8555
27.8639
27.8702
27.8582
27.8605
27.8900
27.8758
27.8774
27.9008
27.8988
27.8897
27.8990
27.8958
27.8830
27.8967
27.9105
27.9028
27.8977
27.8953
27.8970
27.9190
27.9180
27.8997
27.9204
27.9234
27.9072
27.9152
27.9091
27.8882
27.9035
27.9267
27.9138
27.8955
27.9203
27.9239
27.9199
27.9646
27.9411
27.9345
27.8712
27.9145
27.9259
27.9317
27.9239
27.9247
27.9150
27.9444
27.9457
27.9166
27.9066
27.9088
27.9255
27.9312
27.9439
27.9210
27.9102
27.9083
27.9121
27.9113
27.9091
27.9235
27.9291
27.9253
27.9092
27.9117
27.9194
27.9039
27.9515
27.9143
27.9124
27.9128
27.9260
27.9339
27.9500
27.9530
27.9430
27.9400
27.8850
27.9350
27.9120
27.9260
27.9660
27.9280
27.9450
27.9390
27.9429
27.9207
27.9205
27.9204
27.9198
27.9246
27.9366
27.9234
27.9125
27.9032
27.9285
27.9561
27.9616
27.9530
27.9280
27.9060
27.9380
27.9310
27.9347
27.9339
27.9410
27.9397
27.9472
27.9235
27.9315
27.9368
27.9403
27.9529
27.9263
27.9347
27.9371
27.9129
27.9549
27.9422
27.9423
27.9750
27.9339
27.9629
27.9587
27.9503
27.9573
27.9518
27.9527
27.9589
27.9300
27.9629
27.9630
27.9660
27.9730
27.9660
27.9630
27.9570
27.9650
27.9520
27.9820
27.9560
27.9670
27.9520
27.9470
27.9720
27.9610
27.9437
27.9660
27.9580
27.9660
27.9700
27.9600
27.9660
27.9770
27.9110
27.9690
27.9698
27.9616
27.9371
27.9700
27.9265
27.9964
27.9842
27.9667
27.9610
27.9943
27.9616
27.9397
27.9799
28.0086
27.9709
27.9741
27.9675
27.9826
27.9676
27.9703
27.9789
27.9786
27.9722
27.9831
28.0043
27.9548
27.9875
27.9495
27.9549
27.9469
27.9744
27.9744
27.9449
27.9837
27.9585
28.0096
27.9762
27.9641
27.9854
27.9877
27.9839
27.9817
27.9845
27.9877
27.9880
27.9822
27.9836
28.0030
27.9678
28.0146
27.9945
27.9805
27.9785
27.9791
27.9817
27.9805
27.9782
27.9753
27.9792
27.9704
27.9794
27.9814
27.9794
27.9795
27.9881
27.9772
27.9796
27.9736
27.9772
27.9960
27.9795
27.9779
27.9829
27.9829
27.9815
27.9811
27.9773
27.9778
27.9724
27.9756
27.9699
27.9724
27.9666
27.9666
27.9739
27.9684
27.9861
27.9901
27.9879
27.9865
27.9876
27.9814
27.9842
27.9868
27.9834
27.9892
27.9864
27.9843
27.9838
27.9847
27.9860
27.9872
27.9869
27.9602
27.9852
27.9860
27.9836
27.9813
27.9623
27.9843
27.9802
27.9863
27.9813
27.9881
27.9850
27.9850
27.9830
27.9866
27.9888
27.9841
27.9863
27.9903
27.9961
27.9905
27.9945
27.9878
27.9929
27.9914
27.9914
27.9997
28.0006
27.9999
28.0004
28.0020
28.0029
28.0008
28.0040
28.0078
28.0065
27.9959
28.0073
28.0017
28.0042
28.0036
28.0055
28.0007
28.0066
28.0011
27.9960
28.0083
27.9978
28.0108
28.0088
28.0088
28.0139
28.0092
28.0092
28.0049
28.0111
28.0120
28.0093
28.0116
28.0102
28.0139
28.0113
28.0158
28.0156
28.0137
28.0236
28.0171
28.0224
28.0184
28.0199
28.0190
28.0204
28.0170
28.0183
28.0201
28.0182
28.0183
28.0175
28.0127
28.0211
28.0057
28.0180
28.0183
28.0149
28.0185
28.0182
28.0192
28.0213
28.0216
28.0169
28.0162
28.0167
28.0167
28.0169
28.0169
28.0161
28.0152
28.0179
28.0215
28.0194
28.0115
28.0174
28.0178
28.0202
28.0240
28.0198
28.0194
28.0171
28.0134
28.0121
28.0121
28.0141
28.0101
28.0114
28.0122
28.0124
28.0171
28.0165
28.0166
28.0159
28.0181
28.0200
28.0116
28.0144
28.0141
28.0116
28.0107
28.0169
28.0105
28.0136
28.0138
28.0114
28.0122
28.0122
28.0116
28.0025
28.0097
28.0066
28.0072
28.0066
28.0068
28.0067
28.0130
28.0091
28.0088
28.0091
28.0091
28.0115
28.0087
28.0128
28.0139
28.0095
28.0115
28.0101
28.0121
28.0114
28.0121
28.0122
28.0121
28.0168
28.0212
28.0219
28.0221
28.0204
28.0169
28.0141
28.0142
28.0147
28.0159
28.0165
28.0144
28.0182
28.0155
28.0155
28.0192
28.0204
28.0185
28.0248
28.0185
28.0226
28.0271
28.0290
28.0240
28.0302
28.0243
28.0288
28.0287
28.0301
28.0273
28.0313
28.0293
28.0300
28.0344
28.0308
28.0291
28.0287
28.0358
28.0309
28.0286
28.0308
28.0291
28.0380
28.0411
28.0420
28.0359
28.0368
28.0327
28.0361
28.0334
28.0300
28.0347
28.0359
28.0344
28.0370
28.0355
28.0371
28.0318
28.0390
28.0390
28.0390
28.0376
28.0376
28.0377
28.0345
28.0333
28.0429
28.0379
28.0401
28.0401
28.0423
28.0393
28.0382
28.0424
28.0386
28.0386
28.0373
28.0397
28.0412
28.0565
28.0419
28.0456
28.0426
28.0423
28.0391
28.0403
28.0388
28.0408
28.0457
28.0455
28.0460
28.0456
28.0464
28.0442
28.0416
28.0451
28.0432
28.0434
28.0448
28.0448
28.0373
28.0429
28.0392
28.0469
28.0443
28.0356
28.0474
28.0446
28.0348
28.0368
28.0418
28.0445
28.0533
28.0439
28.0474
28.0435
28.0419
28.0538
28.0538
28.0463
28.0491
28.0441
28.0411
28.0507
28.0459
28.0519
28.0554
28.0512
28.0507
28.0582
28.0471
28.0539
28.0530
28.0502
28.0422
28.0431
28.0395
28.0177
28.0425
28.0484
28.0693
28.0490
28.0453
28.0494
28.0522
28.0393
28.0443
28.0465
28.0450
28.0539
28.0566
28.0585
28.0486
28.0427
28.0548
28.0616
28.0298
28.0726
28.0695
28.0629
28.0503
28.0493
28.0537
28.0613
28.0643
28.0678
28.0564
28.0703
28.0647
28.0579
28.0630
28.0716
28.0586
28.0607
28.0601
28.0611
28.0606
28.0611
28.0066
28.0412
28.0558
28.0590
28.0750
28.0483
28.0599
28.0490
28.0499
28.0565
28.0612
28.0634
28.0627
28.0519
28.0551
28.0696
28.0581
28.0568
28.0572
28.0529
28.0421
28.0432
28.0211
28.0363
28.0436
28.0619
28.0573
28.0499
28.0340
28.0474
28.0534
28.0589
28.0466
28.0448
28.0576
28.0558
28.0522
28.0480
28.0444
28.0429
28.0624
28.0610
28.0461
28.0564
28.0734
28.0565
28.0503
28.0581
28.0519
28.0625
28.0583
28.0645
28.0642
28.0535
28.0510
28.0542
28.0677
28.0416
28.0676
28.0596
28.0635
28.0558
28.0623
28.0718
28.0585
28.0552
28.0684
28.0646
28.0590
28.0465
28.0594
28.0303
28.0533
28.0561
28.0585
28.0497
28.0582
28.0507
28.0562
28.0715
28.0468
28.0411
28.0587
28.0456
28.0705
28.0534
28.0558
28.0536
28.0552
28.0461
28.0598
28.0598
28.0650
28.0423
28.0442
28.0449
28.0660
28.0506
28.0655
28.0512
28.0407
28.0475
28.0411
28.0512
28.1036
28.0641
28.0572
28.0700
28.0577
28.0637
28.0534
28.0461
28.0701
28.0631
28.0575
28.0444
28.0592
28.0684
28.0593
28.0677
28.0512
28.0644
28.0660
28.0542
28.0768
28.0515
28.0579
28.0538
28.0526
28.0833
28.0637
28.0529
28.0535
28.0561
28.0736
28.0635
28.0600
28.0520
28.0695
28.0608
28.0608
28.0590
28.0290
28.0939
28.0618
28.0551
28.0757
28.0698
28.0717
28.0529
28.0644
28.0613
28.0759
28.0745
28.0736
28.0611
28.0732
28.0782
28.0682
28.0756
28.0857
28.0739
28.0840
28.0862
28.0724
28.0727
28.0752
28.0732
28.0703
28.0849
28.0795
28.0902
28.0874
28.0971
28.0638
28.0877
28.0751
28.0904
28.0971
28.0661
28.0711
28.0754
28.0516
28.0961
28.0689
28.1110
28.1062
28.0726
28.1141
28.0913
28.0982
28.0703
28.0654
28.0760
28.0727
28.0850
28.0877
28.0967
28.1185
28.0945
28.0834
28.0764
28.1129
28.0797
28.0707
28.1008
28.0971
28.0826
28.0857
28.0984
28.0869
28.0795
28.0875
28.1184
28.0746
28.0816
28.0879
28.0888
28.0924
28.0979
28.0702
28.0847
28.0917
28.0834
28.0823
28.0917
28.0779
28.0852
28.0863
28.0942
28.0801
28.0817
28.0922
28.0914
28.0868
28.0832
28.0881
28.0910
28.0886
28.0961
28.0857
28.0859
28.1086
28.0838
28.0921
28.0945
28.0839
28.0877
28.0803
28.0928
28.0885
28.0940
28.0856
28.0849
28.0955
28.0955
28.0846
28.0871
28.0872
28.0917
28.0931
28.0865
28.0900
28.0915
28.0963
28.0917
28.0950
28.0898
28.0902
28.0867
28.0843
28.0939
28.0902
28.0911
28.0909
28.0949
28.0867
28.0932
28.0891
28.0932
28.0887
28.0925
28.0928
28.0883
28.0946
28.0977
28.0914
28.0959
28.0926
28.0923
28.0950
28.1006
28.0924
28.0963
28.0893
28.0956
28.0980
28.0928
28.0951
28.0958
28.0912
28.0990
28.0915
28.0957
28.0976
28.0888
28.0928
28.0910
28.0902
28.0950
28.0995
28.0965
28.0972
28.0963
28.0946
28.0942
28.0998
28.0911
28.1043
28.1002
28.0991
28.0959
28.0996
28.0926
28.1002
28.0961
28.0983
28.0997
28.0959
28.0988
28.1029
28.0989
28.1000
28.0944
28.0979
28.1005
28.1012
28.1013
28.0999
28.0991
28.1059
28.0961
28.0981
28.1045
28.1047
28.1042
28.1146
28.1113
28.1051
28.1065
28.1065
28.0985
28.1000
28.1066
28.1041
28.0954
28.1090

1.4.2. Case Studies
assumptions are:
1. random drawings;
and
4-Plot of
Data
above:
1. The run sequence plot (upper left) indicates
significant shifts in both location and variation.
Specifically, the location is increasing with time. The
variability seems greater in the first and last third of
the data than it does in the middle third.
2. The lag plot (upper right) shows a significant non-
random pattern in the data. Specifically, the strong
linear appearance of this plot is indicative of a model
that relates Y
t
to Y
t-1
.
3. The distributional plots, the histogram (lower left)
and the normal probability plot (lower right), are not
interpreted since the randomness assumption is so
clearly violated.
The serious violation of the non-randomness assumption
means that the univariate model
is not valid. Given the linear appearance of the lag plot, the
first step might be to consider a model of the type
However, discussions with the scientist revealed the
following:
1. the drift with respect to location was expected.
2. the non-constant variability was not expected.
The scientist examined the data collection device and
determined that the non-constant variation was a seasonal
effect. The high variability data in the first and last thirds
was collected in winter while the more stable middle third
was collected in the summer. The seasonal effect was
determined to be caused by the amount of humidity
affecting the measurement equipment. In this case, the
solution was to modify the test equipment to be less
sensitive to enviromental factors.
Simple graphical techniques can be quite effective in
revealing unexpected results in the data. When this occurs,
it is important to investigate whether the unexpected result
is due to problems in the experiment and data collection, or
is it in fact indicative of an unexpected underlying structure
in the data. This determination cannot be made on the basis
of statistics alone. The role of the graphical and statistical
analysis is to detect problems or unexpected results in the
data. Resolving the issues requires the knowledge of the
scientist or engineer.
Individual
Plots
generated individually to give more detail. Since the lag
plot indicates significant non-randomness, we omit the
distributional plots.
Run
Sequence
Plot
Lag Plot

1.4.2. Case Studies
Interpretation
Summary
Statistics
Sample size = 1000
Mean = 28.01634
Median = 28.02910
Minimum = 27.82800
Maximum = 28.11850
Range = 0.29050
Stan. Dev. = 0.06349
be zero.
t-Value
B
0
27.9114 0.1209E-02
0.2309E+05
B
1
0.20967E-03 0.2092E-05
100.2

1
statistically significant. The value of the slope parameter
estimate is 0.00021. Although this number is nearly zero,
we need to take into account that the original scale of the
data is from about 27.8 to 28.2. In this case, we conclude
that there is a drift in location.
non-normality. Since the normality assumption is
questionable for these data, we use the alternative Levene
intervals.
H
0
:
1
2
=
2
2
=
3
2
=
4
2

H
a
: At least one
i
2
is not equal to the
others.
Critical value: F
,k-1,N-k
= 2.614
0
if W > 2.614
In this case, since the Levene test statistic value of 140.85
is greater than the 5 % significance level critical value of
2.614, we conclude that there is significant evidence of
nonconstant variation.
detected with a few simple tests. The lag plot in the 4-plot
in the previous section is a simple graphical technique.
One check is an autocorrelation plot that shows the
0 is always 1).
greatest interest, is 0.97. The critical values at the 5 %
the lag 1 autocorrelation is statistically significant, so there
is strong evidence of non-randomness.
H
0
manner
H
a
random manner
Critical value: Z
1-/2
= 1.96
0
if |Z| > 1.96
Because the test statistic is outside of the critical region, we
reject the null hypothesis and conclude that the data are not
random.
Distributional
Analysis
Since we rejected the randomness assumption, the
distributional tests are not meaningful. Therefore, these
quantitative tests are omitted. Since the Grubbs' test for
outliers also assumes the approximate normality of the data,
we omit Grubbs' test as well.
Univariate
Report
Analysis for resistor case study


2: Location
Mean =
28.01635
0.002008
(28.0124,28.02029)

3: Variation
0.063495
(0.060829,0.066407)
Change in variation?
of the data) = YES

4: Randomness
Autocorrelation =
0.972158
Data Are Random?

5: Distribution
Distributional test omitted due to
non-randomness of the data

fixed)

7: Outliers?
(Grubbs' test omitted due to
non-randomness of the data)

1.4.2. Case Studies
View
Dataplot
Macro for
this Case
Study
Data Analysis Steps
Results and
Conclusions
Click on the links below to start Dataplot and run
this case study yourself. Each step may use results
from previous steps, so please be patient. Wait
until the software verifies that the current step is
complete before clicking on the next step.
NOTE: This case study has 1,000 points. For
better performance, it is highly recommended that
you check the "No Update" box on the
Spreadsheet window for this case study. This will
suppress subsequent updating of the Spreadsheet
window as the data are created or modified.
more detailed

1. You have read 1
column of numbers
into Dataplot,
variable Y.
plot, there are
shifts
in location and
variation and the
data
are not random.
1. The run
sequence plot
indicates that
there are
shifts of location
and
variation.
2. The lag plot
shows a strong
linear
pattern, which
indicates
significant
non-randomness.
quantitative
report.
statistics.
2. Generate the sample mean, a
confidence
interval for the population mean,
and
compute a linear fit to detect
drift in
location.
3. Generate the sample standard
deviation,
a confidence interval for the
population
standard deviation, and detect
drift in
variation by dividing the data into
quarters and computing Levene's
test for
equal standard deviations.
an
test.
assumes
run).
1. The summary
statistics table
displays
25+ statistics.
2. The mean is
28.0163 and a 95%
confidence
interval is
(28.0124,28.02029).
The linear fit
indicates drift in
location since
the slope parameter
estimate is
statistically
significant.
3. The standard
deviation is 0.0635
with
a 95%
confidence interval
of
(0.060829,0.066407).
Levene's test
indicates
significant
change in
variation.
4. The lag 1
autocorrelation is
0.97.
From the
autocorrelation
plot, this is
outside the 95%
confidence interval
bands,
indicating
significant non-
randomness.
5. The results are
summarized in a
convenient
report.
1.4.2.8. Heat Flow Meter 1

1.4.2. Case Studies
Heat Flow
Meter
Calibration
and
Stability
This example illustrates the univariate analysis of standard
resistor data.

1.4.2. Case Studies
Generation This data set was collected by Bob Zarr of NIST in January,
1990 from a heat flow meter calibration and stability analysis.
The response variable is a calibration factor.
The motivation for studying this data set is to illustrate a well-
behaved process where the underlying assumptions hold and
the process is in statistical control.
9.206343
9.299992
9.277895
9.305795
9.275351
9.288729
9.287239
9.260973
9.303111
9.275674
9.272561
9.288454
9.255672
9.252141
9.297670
9.266534
9.256689
9.277542
9.248205
9.252107
9.276345
9.278694
9.267144
9.246132
9.238479
9.269058
9.248239
9.257439
9.268481
9.288454
9.258452
9.286130
9.251479
9.257405
9.268343
9.291302
9.219460
9.270386
9.218808
9.241185
9.269989
9.226585
9.258556
9.286184
9.320067
9.327973
9.262963
9.248181
9.238644
9.225073
9.220878
9.271318
9.252072
9.281186
9.270624
9.294771
9.301821
9.278849
9.236680
9.233988
9.244687
9.221601
9.207325
9.258776
9.275708
9.268955
9.257269
9.264979
9.295500
9.292883
9.264188
9.280731
9.267336
9.300566
9.253089
9.261376
9.238409
9.225073
9.235526
9.239510
9.264487
9.244242
9.277542
9.310506
9.261594
9.259791
9.253089
9.245735
9.284058
9.251122
9.275385
9.254619
9.279526
9.275065
9.261952
9.275351
9.252433
9.230263
9.255150
9.268780
9.290389
9.274161
9.255707
9.261663
9.250455
9.261952
9.264041
9.264509
9.242114
9.239674
9.221553
9.241935
9.215265
9.285930
9.271559
9.266046
9.285299
9.268989
9.267987
9.246166
9.231304
9.240768
9.260506
9.274355
9.292376
9.271170
9.267018
9.308838
9.264153
9.278822
9.255244
9.229221
9.253158
9.256292
9.262602
9.219793
9.258452
9.267987
9.267987
9.248903
9.235153
9.242933
9.253453
9.262671
9.242536
9.260803
9.259825
9.253123
9.240803
9.238712
9.263676
9.243002
9.246826
9.252107
9.261663
9.247311
9.306055
9.237646
9.248937
9.256689
9.265777
9.299047
9.244814
9.287205
9.300566
9.256621
9.271318
9.275154
9.281834
9.253158
9.269024
9.282077
9.277507
9.284910
9.239840
9.268344
9.247778
9.225039
9.230750
9.270024
9.265095
9.284308
9.280697
9.263032
9.291851
9.252072
9.244031
9.283269
9.196848
9.231372
9.232963
9.234956
9.216746
9.274107
9.273776

1.4.2. Case Studies
assumptions are:
1. random drawings;
and
4-Plot of
Data
above:
scale over time.
reasonably symmetric, there does not appear to be
significant outliers in the tails, and it seems
reasonable to assume that the data are from
approximately a normal distribution.
4. The normal probability plot (lower right) verifies that
an assumption of normality is in fact reasonable.
Individual
Plots
Run
Sequence
Plot
Lag Plot
Histogram
(with
overlaid
Normal PDF)
Normal
Probability
Plot

1.4.2. Case Studies
Interpretation
Summary
Statistics
Sample size = 195
Mean = 9.261460
Median = 9.261952
Minimum = 9.196848
Maximum = 9.327973
Range = 0.131126
Stan. Dev. = 0.022789
be zero.
t-Value
B
0
9.26699 0.3253E-02
2849.
B
1
-0.56412E-04 0.2878E-04
-1.960

1
, has a t value of -1.96 which is
(barely) statistically significant since it is essentially equal
to the 95 % level cutoff of -1.96. However, notice that the
value of the slope parameter estimate is -0.00056. This
slope, even though statistically significant, can essentially
be considered zero.
sized intervals. The choice of the number of intervals is
somewhat arbitrary, although values of four or eight are
reasonable. We will divide our data into four intervals.
2 2 2 2
H
0
:
1
=
2
=
3
=
4

H
a
: At least one
i
2
is not equal to the
others.
Critical value:
2
1-,k-1
= 7.815
0
if T > 7.815
In this case, since the Bartlett test statistic of 3.147 is less
than the critical value at the 5 % significance level of
7.815, we conclude that the variances are not significantly
different in the four intervals. That is, the assumption of
constant scale is valid.
detected with a few simple tests. The lag plot in the
previous section is a simple graphical technique.
0 is always 1).
greatest interest, is 0.281. The critical values at the 5 %
the lag 1 autocorrelation is statistically significant, so there
is evidence of non-randomness.
H
0
manner
H
a
random manner
Critical value: Z
1-/2
= 1.96
0
if |Z| > 1.96
The value of the test statistic is less than -1.96, so we reject
the null hypothesis at the 0.05 significant level and
conclude that the data are not random.
Although the autocorrelation plot and the runs test indicate
some mild non-randomness, the violation of the
randomness assumption is not serious enough to warrant
developing a more sophisticated model. It is common in
practice that some of the assumptions are mildly violated
and it is a judgement call as to whether or not the
violations are serious enough to warrant developing a more
sophisticated model for the data.
Distributional
Analysis
correlation coefficient of the points on the probability plot.
For this data set the correlation coefficient is 0.996. Since
this is greater than the critical value of 0.987 (this is a
tabulated value), the normality assumption is not rejected.
H
0
H
a
2
= 0.129
0
if A
2
> 0.787
The Anderson-Darling test also does not reject the
normality assumption because the test statistic, 0.129, is
less than the critical value at the 5 % significance level of
0.787.
Outlier
Analysis
A test for outliers is the Grubbs' test.
H
0
H
a
Critical value for an upper one-tailed
test: 3.597898
0
if G > 3.597898
Model Since the underlying assumptions were validated both
graphically and analytically, with a mild violation of the
randomness assumption, we conclude that a reasonable
model for the data is:
We can express the uncertainty for C, here estimated by
9.26146, as the 95 % confidence interval
(9.258242,9.26479).
Univariate
Report
above results in a report. The report for the heat flow meter
data follows.

Analysis for heat flow meter data


2: Location
Mean =
9.26146
0.001632
95 % Confidence Interval for Mean =
(9.258242,9.264679)

3: Variation
0.022789
95 % Confidence Interval for SD =
(0.02073,0.025307)
(based on Bartlett's test on quarters
of the data) = NO

4: Randomness
Autocorrelation =
0.280579
Data are Random?

5: Data are Normal?
(as tested by Anderson-Darling) = YES

fixed normal)

7: Outliers?
(as determined by Grubbs' test) = NO


1.4.2. Case Studies
View
Dataplot
Macro for
this Case
Study
Data Analysis Steps
Results and
Conclusions
more detailed

1. You have read 1
column of numbers
into Dataplot,
variable Y.
plot, there are no
shifts
in location or
scale, and the data
seem to
follow a normal
distribution.
plot.
1. The run sequence
plot indicates that
there are no
scale.
2. The lag plot
significant
show the data
were not random).
3. The histogram
indicates that a
normal
distribution is a
good
distribution for
these data.
4. The normal
probability plot
verifies
that the normal
distribution is a
reasonable
distribution for
these data.
quantitative
report.
statistics.
location.
standard
variation
and
computing Bartlett's test for
equal
an
test.
the
correlation
coefficient.
test.
1. The summary
statistics table
displays
25+ statistics.
2. The mean is
9.261 and a 95%
confidence
interval is
(9.258,9.265).
The linear fit
location since
the slope parameter
estimate is
essentially zero.
3. The standard
deviation is 0.023
with
a 95% confidence
interval of
(0.0207,0.0253).
Bartlett's test
indicates no
significant
change in
variation.
4. The lag 1
autocorrelation is
0.28.
From the
this is
statistically
significant at the
95%
level.
assumes
run).
5. The normal
probability plot
correlation
coefficient is
0.999. At the 5%
level,
we cannot reject
the normality
assumption.
6. Grubbs' test
detects no outliers
at the
5% level.
7. The results are
summarized in a
convenient
report.
1.4.2.9. Fatigue Life of Aluminum Alloy Specimens

1.4.2. Case Studies
1.4.2.9. Fatigue Life of Aluminum Alloy
Specimens
Fatigue
Life of
Aluminum
Alloy
Specimens
This example illustrates the univariate analysis of the fatigue
life of aluminum alloy specimens.

1.4.2. Case Studies
Generation This data set comprises measurements of fatigue life
(thousands of cycles until rupture) of rectangular strips of
6061-T6 aluminum sheeting, subjected to periodic loading
with maximum stress of 21,000 psi (pounds per square inch),
as reported by Birnbaum and Saunders (1958).
Purpose of
Analysis
The goal of this case study is to select a probabilistic model,
from among several reasonable alternatives, to describe the
dispersion of the resulting measured values of life-length.
The original study, in the field of statistical reliability analysis,
was concerned with the prediction of failure times of a
material subjected to a load varying in time. It was well-
known that a structure designed to withstand a particular static
load may fail sooner than expected under a dynamic load.
If a realistic model for the probability distribution of lifetime
can be found, then it can be used to estimate the time by
which a part or structure needs to be replaced to guarantee
that the probability of failure does not exceed some maximum
acceptable value, for example 0.1 %, while it is in service.
The chapter of this eHandbook that is concerned with the
assessment of product reliability contains additional material
on statistical methods used in reliability analysis. This case
study is meant to complement that chapter by showing the use
of graphical and other techniques in the model selection stage
of such analysis.
When there is no cogent reason to adopt a particular model, or
when none of the models under consideration seems adequate
for the purpose, one may opt for a non-parametric statistical
method, for example to produce tolerance bounds or
confidence intervals.
A non-parametric method does not rely on the assumption that
the data are like a sample from a particular probability
distribution that is fully specified up to the values of some
adjustable parameters. For example, the Gaussian probability
distribution is a parametric model with two adjustable
parameters.
The price to be paid when using non-parametric methods is
loss of efficiency, meaning that they may require more data
for statistical inference than a parametric counterpart would, if
applicable. For example, non-parametric confidence intervals
for model parameters may be considerably wider than what a
confidence interval would need to be if the underlying
distribution could be identified correctly. Such identification is
what we will attempt in this case study.
It should be noted --- a point that we will stress later in the
development of this case study --- that the very exercise of
selecting a model often contributes substantially to the
uncertainty of the conclusions derived after the selection has
been made.
Software The analyses used in this case study can be generated using R
code.
Data The following data are used for this case study.
370 1016 1235 1419 1567 1820
706 1018 1238 1420 1578 1868
716 1020 1252 1420 1594 1881
746 1055 1258 1450 1602 1890
785 1085 1262 1452 1604 1893
797 1102 1269 1475 1608 1895
844 1102 1270 1478 1630 1910
855 1108 1290 1481 1642 1923
858 1115 1293 1485 1674 1940
886 1120 1300 1502 1730 1945
886 1134 1310 1505 1750 2023
930 1140 1313 1513 1750 2100
960 1199 1315 1522 1763 2130
988 1200 1330 1522 1768 2215
990 1200 1355 1530 1781 2268
1000 1203 1390 1540 1782 2440
1010 1222 1416 1560 1792

1.4.2. Case Studies
Goal The goal of this analysis is to select a probabilistic model to describe the dispersion
of the measured values of fatigue life of specimens of an aluminum alloy described
in [1.4.2.9.1], from among several reasonable alternatives.
Initial Plots
of the Data
Simple diagrams can be very informative about location, spread, and to detect
possibly anomalous data values or particular patterns (clustering, for example).
These include dot-charts, boxplots, and histograms. Since building an effective
histogram requires that a choice be made of bin size, and this choice can be
influential, one may wish to examine a non-parametric estimate of the underlying
probability density.
These several plots variously show that the measurements range from a value
slightly greater than 350,000 to slightly less than 2,500,000 cycles. The boxplot
suggests that the largest measured value may be an outlier.
A recommended first step is to check consistency between the data and what is to be
expected if the data were a sample from a particular probability distribution.
Knowledge about the underlying properties of materials and of relevant industrial
processes typically offer clues as to the models that should be entertained. Graphical
diagnostic techniques can be very useful at this exploratory stage: foremost among
these, for univariate data, is the quantile-quantile plot, or QQ-plot (Wilk and
Gnanadesikan, 1968).
Each data point is represented by one point in the QQ-plot. The ordinate of each of
these points is one data value; if this data value happens to be the kth order statistic
in the sample (that is, the kth largest value), then the corresponding abscissa is the
"typical" value that the kth largest value should have in a sample of the same size as
the data, drawn from a particular distribution. If F denotes the cumulative
probability distribution function of interest, and the sample comprises n values, then
F
-1
[(k - 1/2) / (n + 1/2)] is a reasonable choice for that "typical" value, because it is
an approximation to the median of the kth order statistic in a sample of size n from
this distribution.
The following figure shows a QQ-plot of our data relative to the Gaussian (or,
normal) probability distribution. If the data matched expectations perfectly, then the
points would all fall on a straight line.
In practice, one needs to gauge whether the deviations from such perfect alignment
are commensurate with the natural variability associated with sampling. This can
easily be done by examining how variable QQ-plots of samples from the target
distribution may be.
The following figure shows, superimposed on the QQ-plot of the data, the QQ-plots
of 99 samples of the same size as the data, drawn from a Gaussian distribution with
the same mean and standard deviation as the data.
The fact that the cloud of QQ-plots corresponding to 99 samples from the Gaussian
distribution effectively covers the QQ-plot for the data, suggests that the chances are
better than 1 in 100 that our data are inconsistent with the Gaussian model.
This proves nothing, of course, because even the rarest of events may happen.
However, it is commonly taken to be indicative of an acceptable fit for general
purposes. In any case, one may naturally wonder if an alternative model might not
provide an even better fit.
Knowing the provenance of the data, that they portray strength of a material,
strongly suggests that one may like to examine alternative models, because in many
studies of reliability non-Gaussian models tend to be more appropriate than
Gaussian models.
Candidate
Distributions
There are many probability distributions that could reasonably be entertained as
candidate models for the data. However, we will restrict ourselves to consideration
of the following because these have proven to be useful in reliability studies.
Normal distribution
Gamma distribution
Birnbaum-Saunders distribution
3-parameter Weibull distribution
Approach A very simple approach amounts to comparing QQ-plots of the data for the
candidate models under consideration. This typically involves first fitting the models
to the data, for example employing the method of maximum likelihood [1.3.6.5.2].
The maximum likelihood estimates are the following:
Gaussian: mean 1401, standard deviation 389
Gamma: shape 11.85, rate 0.00846
Birnbaum-Saunders: shape 0.310, scale 1337
3-parameter Weibull: location 181, shape 3.43, scale 1357
The following figure shows how close (or how far) the best fitting probability
densities of the four distributions approximate the non-parametric probability
density estimate. This comparison, however, takes into account neither the fact that
our sample is fairly small (101 measured values), nor that the fitted models
themselves have been estimated from the same data that the non-parametric estimate
was derived from.
These limitations notwithstanding, it is worth examining the corresponding QQ-
plots, shown below, which suggest that the Gaussian and the 3-parameter Weibull
may be the best models.
Model
Selection
A more careful comparison of the merits of the alternative models needs to take into
account the fact that the 3-parameter Weibull model (precisely because it has three
parameters), may be intrinsically more flexible than the others, which all have two
adjustable parameters only.
Two criteria can be employed for a formal comparison: Akaike's Information
Criterion (AIC), and the Bayesian Information Criterion (BIC) (Hastie et. al., 2001).
The smaller the value of either model selection criterion, the better the model:
AIC BIC
GAU 1495 1501
GAM 1499 1504
BS 1507 1512
WEI 1498 1505
On this basis (and according both to AIC and BIC), there seems to be no cogent
reason to replace the Gaussian model by any of the other three. The values of BIC
can also be used to derive an approximate answer to the question of how strongly
the data may support each of these models. Doing this involves the application of
Bayesian statistical methods [8.1.10].
We start from an a priori assignment of equal probabilities to all four models,
indicating that we have no reason to favor one over another at the outset, and then
update these probabilities based on the measured values of lifetime. The updated
probabilities of the four models, called their posterior probabilities, are
approximately proportional to exp(-BIC(GAU)/2), exp(-BIC(GAM)/2), exp(-
BIC(BS)/2), and exp(-BIC(WEI)/2). The values are 76 % for GAU, 16 % for GAM,
0.27 % for BS, and 7.4 % for WEI.
One possible use for the selected model is to answer the question of the age in
service by which a part or structure needs to be replaced to guarantee that the
probability of failure does not exceed some maximum acceptable value, for example
0.1 %.The answer to this question is the 0.1st percentile of the fitted distribution,
that is G
-1
(0.001) = 198 thousand cycles, where, in this case, G
-1
denotes the
inverse of the fitted, Gaussian probability distribution.
To assess the uncertainty of this estimate one may employ the statistical bootstrap
[1.3.3.4]. In this case, this involves drawing a suitably large number of bootstrap
samples from the data, and for each of them applying the model fitting and model
selection exercise described above, ending with the calculation of G
-1
(0.001) for
the best model (which may vary from sample to sample).
The bootstrap samples should be of the same size as the data, with each being drawn
uniformly at random from the data, with replacement. This process, based on 5,000
bootstrap samples, yielded a 95 % confidence interval for the 0.1st percentile
ranging from 40 to 366 thousands of cycles. The large uncertainty is not surprising
given that we are attempting to estimate the largest value that is exceeded with
probability 99.9 %, based on a sample comprising only 101 measured values.
Prediction
Intervals
One more application in this analysis is to evaluate prediction intervals for the
fatigue life of the aluminum alloy specimens. For example, if we were to test three
new specimens using the same process, we would want to know (with 95 %
confidence) the minimum number of cycles for these three specimens. That is, we
need to find a statistical interval [L, ] that contains the fatigue life of all three
future specimens with 95 % confidence. The desired interval is a one-sided, lower
95 % prediction interval. Since tables of factors for constructing L, are widely
available for normal models, we use the results corresponding to the normal model
here for illustration. Specifically, L is computed as
where factor r is given in Table A.14 of Hahn and Meeker (1991) or can be
obtained from an R program.
1.4.2.10. Ceramic Strength

1.4.2. Case Studies
Ceramic
Strength
This case study analyzes the effect of machining factors on the
strength of ceramics.
2. Analysis of the Response Variable
3. Analysis of Batch Effect
4. Analysis of Lab Effect
5. Analysis of Primary Factors

1.4.2. Case Studies
Generation The data for this case study were collected by Said Jahanmir
of the NIST Ceramics Division in 1996 in connection with a
NIST/industry ceramics consortium for strength optimization
of ceramic strength
The motivation for studying this data set is to illustrate the
analysis of multiple factors from a designed experiment
This case study will utilize only a subset of a full study that
was conducted by Lisa Gill and James Filliben of the NIST
Statistical Engineering Division
The response variable is a measure of the strength of the
ceramic material (bonded S
i
nitrate). The complete data set
contains the following variables:
1. Factor 1 = Observation ID, i.e., run number (1 to 960)
2. Factor 2 = Lab (1 to 8)
3. Factor 3 = Bar ID within lab (1 to 30)
4. Factor 4 = Test number (1 to 4)
5. Response Variable = Strength of Ceramic
6. Factor 5 = Table speed (2 levels: 0.025 and 0.125)
7. Factor 6 = Down feed rate (2 levels: 0.050 and 0.125)
8. Factor 7 = Wheel grit size (2 levels: 150 and 80)
9. Factor 8 = Direction (2 levels: longitudinal and
transverse)
10. Factor 9 = Treatment (1 to 16)
11. Factor 10 = Set of 15 within lab (2 levels: 1 and 2)
12. Factor 11 = Replication (2 levels: 1 and 2)
13. Factor 12 = Bar Batch (1 and 2)
The four primary factors of interest are:
1. Table speed (X1)
2. Down feed rate (X2)
3. Wheel grit size (X3)
4. Direction (X4)
For this case study, we are using only half the data.
Specifically, we are using the data with the direction
longitudinal. Therefore, we have only three primary factors
In addition, we are interested in the nuisance factors
1. Lab
2. Batch
Purpose of
Analysis
The goals of this case study are:
1. Determine which of the four primary factors has the
strongest effect on the strength of the ceramic material
2. Estimate the magnitude of the effects
3. Determine the optimal settings for the primary factors
4. Determine if the nuisance factors (lab and batch) have
an effect on the ceramic strength
This case study is an example of a designed experiment. The
Process Improvement chapter contains a detailed discussion of
the construction and analysis of designed experiments. This
case study is meant to complement the material in that chapter
by showing how an EDA approach (emphasizing the use of
graphical techniques) can be used in the analysis of designed
experiments
Data The following are the data used for this case study
Run Lab Batch Y X1 X2 X3
1 1 1 608.781 -1 -1 -1
2 1 2 569.670 -1 -1 -1
3 1 1 689.556 -1 -1 -1
4 1 2 747.541 -1 -1 -1
5 1 1 618.134 -1 -1 -1
6 1 2 612.182 -1 -1 -1
7 1 1 680.203 -1 -1 -1
8 1 2 607.766 -1 -1 -1
9 1 1 726.232 -1 -1 -1
10 1 2 605.380 -1 -1 -1
11 1 1 518.655 -1 -1 -1
12 1 2 589.226 -1 -1 -1
13 1 1 740.447 -1 -1 -1
14 1 2 588.375 -1 -1 -1
15 1 1 666.830 -1 -1 -1
16 1 2 531.384 -1 -1 -1
17 1 1 710.272 -1 -1 -1
18 1 2 633.417 -1 -1 -1
19 1 1 751.669 -1 -1 -1
20 1 2 619.060 -1 -1 -1
21 1 1 697.979 -1 -1 -1
22 1 2 632.447 -1 -1 -1
23 1 1 708.583 -1 -1 -1
24 1 2 624.256 -1 -1 -1
25 1 1 624.972 -1 -1 -1
26 1 2 575.143 -1 -1 -1
27 1 1 695.070 -1 -1 -1
28 1 2 549.278 -1 -1 -1
29 1 1 769.391 -1 -1 -1
30 1 2 624.972 -1 -1 -1
61 1 1 720.186 -1 1 1
62 1 2 587.695 -1 1 1
63 1 1 723.657 -1 1 1
64 1 2 569.207 -1 1 1
65 1 1 703.700 -1 1 1
66 1 2 613.257 -1 1 1
67 1 1 697.626 -1 1 1
68 1 2 565.737 -1 1 1
69 1 1 714.980 -1 1 1
70 1 2 662.131 -1 1 1
71 1 1 657.712 -1 1 1
72 1 2 543.177 -1 1 1
73 1 1 609.989 -1 1 1
74 1 2 512.394 -1 1 1
75 1 1 650.771 -1 1 1
76 1 2 611.190 -1 1 1
77 1 1 707.977 -1 1 1
78 1 2 659.982 -1 1 1
79 1 1 712.199 -1 1 1
80 1 2 569.245 -1 1 1
81 1 1 709.631 -1 1 1
82 1 2 725.792 -1 1 1
83 1 1 703.160 -1 1 1
84 1 2 608.960 -1 1 1
85 1 1 744.822 -1 1 1
86 1 2 586.060 -1 1 1
87 1 1 719.217 -1 1 1
88 1 2 617.441 -1 1 1
89 1 1 619.137 -1 1 1
90 1 2 592.845 -1 1 1
151 2 1 753.333 1 1 1
152 2 2 631.754 1 1 1
153 2 1 677.933 1 1 1
154 2 2 588.113 1 1 1
155 2 1 735.919 1 1 1
156 2 2 555.724 1 1 1
157 2 1 695.274 1 1 1
158 2 2 702.411 1 1 1
159 2 1 504.167 1 1 1
160 2 2 631.754 1 1 1
161 2 1 693.333 1 1 1
162 2 2 698.254 1 1 1
163 2 1 625.000 1 1 1
164 2 2 616.791 1 1 1
165 2 1 596.667 1 1 1
166 2 2 551.953 1 1 1
167 2 1 640.898 1 1 1
168 2 2 636.738 1 1 1
169 2 1 720.506 1 1 1
170 2 2 571.551 1 1 1
171 2 1 700.748 1 1 1
172 2 2 521.667 1 1 1
173 2 1 691.604 1 1 1
174 2 2 587.451 1 1 1
175 2 1 636.738 1 1 1
176 2 2 700.422 1 1 1
177 2 1 731.667 1 1 1
178 2 2 595.819 1 1 1
179 2 1 635.079 1 1 1
180 2 2 534.236 1 1 1
181 2 1 716.926 1 -1 -1
182 2 2 606.188 1 -1 -1
183 2 1 759.581 1 -1 -1
184 2 2 575.303 1 -1 -1
185 2 1 673.903 1 -1 -1
186 2 2 590.628 1 -1 -1
187 2 1 736.648 1 -1 -1
188 2 2 729.314 1 -1 -1
189 2 1 675.957 1 -1 -1
190 2 2 619.313 1 -1 -1
191 2 1 729.230 1 -1 -1
192 2 2 624.234 1 -1 -1
193 2 1 697.239 1 -1 -1
194 2 2 651.304 1 -1 -1
195 2 1 728.499 1 -1 -1
196 2 2 724.175 1 -1 -1
197 2 1 797.662 1 -1 -1
198 2 2 583.034 1 -1 -1
199 2 1 668.530 1 -1 -1
200 2 2 620.227 1 -1 -1
201 2 1 815.754 1 -1 -1
202 2 2 584.861 1 -1 -1
203 2 1 777.392 1 -1 -1
204 2 2 565.391 1 -1 -1
205 2 1 712.140 1 -1 -1
206 2 2 622.506 1 -1 -1
207 2 1 663.622 1 -1 -1
208 2 2 628.336 1 -1 -1
209 2 1 684.181 1 -1 -1
210 2 2 587.145 1 -1 -1
271 3 1 629.012 1 -1 1
272 3 2 584.319 1 -1 1
273 3 1 640.193 1 -1 1
274 3 2 538.239 1 -1 1
275 3 1 644.156 1 -1 1
276 3 2 538.097 1 -1 1
277 3 1 642.469 1 -1 1
278 3 2 595.686 1 -1 1
279 3 1 639.090 1 -1 1
280 3 2 648.935 1 -1 1
281 3 1 439.418 1 -1 1
282 3 2 583.827 1 -1 1
283 3 1 614.664 1 -1 1
284 3 2 534.905 1 -1 1
285 3 1 537.161 1 -1 1
286 3 2 569.858 1 -1 1
287 3 1 656.773 1 -1 1
288 3 2 617.246 1 -1 1
289 3 1 659.534 1 -1 1
290 3 2 610.337 1 -1 1
291 3 1 695.278 1 -1 1
292 3 2 584.192 1 -1 1
293 3 1 734.040 1 -1 1
294 3 2 598.853 1 -1 1
295 3 1 687.665 1 -1 1
296 3 2 554.774 1 -1 1
297 3 1 710.858 1 -1 1
298 3 2 605.694 1 -1 1
299 3 1 701.716 1 -1 1
300 3 2 627.516 1 -1 1
301 3 1 382.133 1 1 -1
302 3 2 574.522 1 1 -1
303 3 1 719.744 1 1 -1
304 3 2 582.682 1 1 -1
305 3 1 756.820 1 1 -1
306 3 2 563.872 1 1 -1
307 3 1 690.978 1 1 -1
308 3 2 715.962 1 1 -1
309 3 1 670.864 1 1 -1
310 3 2 616.430 1 1 -1
311 3 1 670.308 1 1 -1
312 3 2 778.011 1 1 -1
313 3 1 660.062 1 1 -1
314 3 2 604.255 1 1 -1
315 3 1 790.382 1 1 -1
316 3 2 571.906 1 1 -1
317 3 1 714.750 1 1 -1
318 3 2 625.925 1 1 -1
319 3 1 716.959 1 1 -1
320 3 2 682.426 1 1 -1
321 3 1 603.363 1 1 -1
322 3 2 707.604 1 1 -1
323 3 1 713.796 1 1 -1
324 3 2 617.400 1 1 -1
325 3 1 444.963 1 1 -1
326 3 2 689.576 1 1 -1
327 3 1 723.276 1 1 -1
328 3 2 676.678 1 1 -1
329 3 1 745.527 1 1 -1
330 3 2 563.290 1 1 -1
361 4 1 778.333 -1 -1 1
362 4 2 581.879 -1 -1 1
363 4 1 723.349 -1 -1 1
364 4 2 447.701 -1 -1 1
365 4 1 708.229 -1 -1 1
366 4 2 557.772 -1 -1 1
367 4 1 681.667 -1 -1 1
368 4 2 593.537 -1 -1 1
369 4 1 566.085 -1 -1 1
370 4 2 632.585 -1 -1 1
371 4 1 687.448 -1 -1 1
372 4 2 671.350 -1 -1 1
373 4 1 597.500 -1 -1 1
374 4 2 569.530 -1 -1 1
375 4 1 637.410 -1 -1 1
376 4 2 581.667 -1 -1 1
377 4 1 755.864 -1 -1 1
378 4 2 643.449 -1 -1 1
379 4 1 692.945 -1 -1 1
380 4 2 581.593 -1 -1 1
381 4 1 766.532 -1 -1 1
382 4 2 494.122 -1 -1 1
383 4 1 725.663 -1 -1 1
384 4 2 620.948 -1 -1 1
385 4 1 698.818 -1 -1 1
386 4 2 615.903 -1 -1 1
387 4 1 760.000 -1 -1 1
388 4 2 606.667 -1 -1 1
389 4 1 775.272 -1 -1 1
390 4 2 579.167 -1 -1 1
421 4 1 708.885 -1 1 -1
422 4 2 662.510 -1 1 -1
423 4 1 727.201 -1 1 -1
424 4 2 436.237 -1 1 -1
425 4 1 642.560 -1 1 -1
426 4 2 644.223 -1 1 -1
427 4 1 690.773 -1 1 -1
428 4 2 586.035 -1 1 -1
429 4 1 688.333 -1 1 -1
430 4 2 620.833 -1 1 -1
431 4 1 743.973 -1 1 -1
432 4 2 652.535 -1 1 -1
433 4 1 682.461 -1 1 -1
434 4 2 593.516 -1 1 -1
435 4 1 761.430 -1 1 -1
436 4 2 587.451 -1 1 -1
437 4 1 691.542 -1 1 -1
438 4 2 570.964 -1 1 -1
439 4 1 643.392 -1 1 -1
440 4 2 645.192 -1 1 -1
441 4 1 697.075 -1 1 -1
442 4 2 540.079 -1 1 -1
443 4 1 708.229 -1 1 -1
444 4 2 707.117 -1 1 -1
445 4 1 746.467 -1 1 -1
446 4 2 621.779 -1 1 -1
447 4 1 744.819 -1 1 -1
448 4 2 585.777 -1 1 -1
449 4 1 655.029 -1 1 -1
450 4 2 703.980 -1 1 -1
541 5 1 715.224 -1 -1 -1
542 5 2 698.237 -1 -1 -1
543 5 1 614.417 -1 -1 -1
544 5 2 757.120 -1 -1 -1
545 5 1 761.363 -1 -1 -1
546 5 2 621.751 -1 -1 -1
547 5 1 716.106 -1 -1 -1
548 5 2 472.125 -1 -1 -1
549 5 1 659.502 -1 -1 -1
550 5 2 612.700 -1 -1 -1
551 5 1 730.781 -1 -1 -1
552 5 2 583.170 -1 -1 -1
553 5 1 546.928 -1 -1 -1
554 5 2 599.771 -1 -1 -1
555 5 1 734.203 -1 -1 -1
556 5 2 549.227 -1 -1 -1
557 5 1 682.051 -1 -1 -1
558 5 2 605.453 -1 -1 -1
559 5 1 701.341 -1 -1 -1
560 5 2 569.599 -1 -1 -1
561 5 1 759.729 -1 -1 -1
562 5 2 637.233 -1 -1 -1
563 5 1 689.942 -1 -1 -1
564 5 2 621.774 -1 -1 -1
565 5 1 769.424 -1 -1 -1
566 5 2 558.041 -1 -1 -1
567 5 1 715.286 -1 -1 -1
568 5 2 583.170 -1 -1 -1
569 5 1 776.197 -1 -1 -1
570 5 2 345.294 -1 -1 -1
571 5 1 547.099 1 -1 1
572 5 2 570.999 1 -1 1
573 5 1 619.942 1 -1 1
574 5 2 603.232 1 -1 1
575 5 1 696.046 1 -1 1
576 5 2 595.335 1 -1 1
577 5 1 573.109 1 -1 1
578 5 2 581.047 1 -1 1
579 5 1 638.794 1 -1 1
580 5 2 455.878 1 -1 1
581 5 1 708.193 1 -1 1
582 5 2 627.880 1 -1 1
583 5 1 502.825 1 -1 1
584 5 2 464.085 1 -1 1
585 5 1 632.633 1 -1 1
586 5 2 596.129 1 -1 1
587 5 1 683.382 1 -1 1
588 5 2 640.371 1 -1 1
589 5 1 684.812 1 -1 1
590 5 2 621.471 1 -1 1
591 5 1 738.161 1 -1 1
592 5 2 612.727 1 -1 1
593 5 1 671.492 1 -1 1
594 5 2 606.460 1 -1 1
595 5 1 709.771 1 -1 1
596 5 2 571.760 1 -1 1
597 5 1 685.199 1 -1 1
598 5 2 599.304 1 -1 1
599 5 1 624.973 1 -1 1
600 5 2 579.459 1 -1 1
601 6 1 757.363 1 1 1
602 6 2 761.511 1 1 1
603 6 1 633.417 1 1 1
604 6 2 566.969 1 1 1
605 6 1 658.754 1 1 1
606 6 2 654.397 1 1 1
607 6 1 664.666 1 1 1
608 6 2 611.719 1 1 1
609 6 1 663.009 1 1 1
610 6 2 577.409 1 1 1
611 6 1 773.226 1 1 1
612 6 2 576.731 1 1 1
613 6 1 708.261 1 1 1
614 6 2 617.441 1 1 1
615 6 1 739.086 1 1 1
616 6 2 577.409 1 1 1
617 6 1 667.786 1 1 1
618 6 2 548.957 1 1 1
619 6 1 674.481 1 1 1
620 6 2 623.315 1 1 1
621 6 1 695.688 1 1 1
622 6 2 621.761 1 1 1
623 6 1 588.288 1 1 1
624 6 2 553.978 1 1 1
625 6 1 545.610 1 1 1
626 6 2 657.157 1 1 1
627 6 1 752.305 1 1 1
628 6 2 610.882 1 1 1
629 6 1 684.523 1 1 1
630 6 2 552.304 1 1 1
631 6 1 717.159 -1 1 -1
632 6 2 545.303 -1 1 -1
633 6 1 721.343 -1 1 -1
634 6 2 651.934 -1 1 -1
635 6 1 750.623 -1 1 -1
636 6 2 635.240 -1 1 -1
637 6 1 776.488 -1 1 -1
638 6 2 641.083 -1 1 -1
639 6 1 750.623 -1 1 -1
640 6 2 645.321 -1 1 -1
641 6 1 600.840 -1 1 -1
642 6 2 566.127 -1 1 -1
643 6 1 686.196 -1 1 -1
644 6 2 647.844 -1 1 -1
645 6 1 687.870 -1 1 -1
646 6 2 554.815 -1 1 -1
647 6 1 725.527 -1 1 -1
648 6 2 620.087 -1 1 -1
649 6 1 658.796 -1 1 -1
650 6 2 711.301 -1 1 -1
651 6 1 690.380 -1 1 -1
652 6 2 644.355 -1 1 -1
653 6 1 737.144 -1 1 -1
654 6 2 713.812 -1 1 -1
655 6 1 663.851 -1 1 -1
656 6 2 696.707 -1 1 -1
657 6 1 766.630 -1 1 -1
658 6 2 589.453 -1 1 -1
659 6 1 625.922 -1 1 -1
660 6 2 634.468 -1 1 -1
721 7 1 694.430 1 1 -1
722 7 2 599.751 1 1 -1
723 7 1 730.217 1 1 -1
724 7 2 624.542 1 1 -1
725 7 1 700.770 1 1 -1
726 7 2 723.505 1 1 -1
727 7 1 722.242 1 1 -1
728 7 2 674.717 1 1 -1
729 7 1 763.828 1 1 -1
730 7 2 608.539 1 1 -1
731 7 1 695.668 1 1 -1
732 7 2 612.135 1 1 -1
733 7 1 688.887 1 1 -1
734 7 2 591.935 1 1 -1
735 7 1 531.021 1 1 -1
736 7 2 676.656 1 1 -1
737 7 1 698.915 1 1 -1
738 7 2 647.323 1 1 -1
739 7 1 735.905 1 1 -1
740 7 2 811.970 1 1 -1
741 7 1 732.039 1 1 -1
742 7 2 603.883 1 1 -1
743 7 1 751.832 1 1 -1
744 7 2 608.643 1 1 -1
745 7 1 618.663 1 1 -1
746 7 2 630.778 1 1 -1
747 7 1 744.845 1 1 -1
748 7 2 623.063 1 1 -1
749 7 1 690.826 1 1 -1
750 7 2 472.463 1 1 -1
811 7 1 666.893 -1 1 1
812 7 2 645.932 -1 1 1
813 7 1 759.860 -1 1 1
814 7 2 577.176 -1 1 1
815 7 1 683.752 -1 1 1
816 7 2 567.530 -1 1 1
817 7 1 729.591 -1 1 1
818 7 2 821.654 -1 1 1
819 7 1 730.706 -1 1 1
820 7 2 684.490 -1 1 1
821 7 1 763.124 -1 1 1
822 7 2 600.427 -1 1 1
823 7 1 724.193 -1 1 1
824 7 2 686.023 -1 1 1
825 7 1 630.352 -1 1 1
826 7 2 628.109 -1 1 1
827 7 1 750.338 -1 1 1
828 7 2 605.214 -1 1 1
829 7 1 752.417 -1 1 1
830 7 2 640.260 -1 1 1
831 7 1 707.899 -1 1 1
832 7 2 700.767 -1 1 1
833 7 1 715.582 -1 1 1
834 7 2 665.924 -1 1 1
835 7 1 728.746 -1 1 1
836 7 2 555.926 -1 1 1
837 7 1 591.193 -1 1 1
838 7 2 543.299 -1 1 1
839 7 1 592.252 -1 1 1
840 7 2 511.030 -1 1 1
901 8 1 740.833 -1 -1 1
902 8 2 583.994 -1 -1 1
903 8 1 786.367 -1 -1 1
904 8 2 611.048 -1 -1 1
905 8 1 712.386 -1 -1 1
906 8 2 623.338 -1 -1 1
907 8 1 738.333 -1 -1 1
908 8 2 679.585 -1 -1 1
909 8 1 741.480 -1 -1 1
910 8 2 665.004 -1 -1 1
911 8 1 729.167 -1 -1 1
912 8 2 655.860 -1 -1 1
913 8 1 795.833 -1 -1 1
914 8 2 715.711 -1 -1 1
915 8 1 723.502 -1 -1 1
916 8 2 611.999 -1 -1 1
917 8 1 718.333 -1 -1 1
918 8 2 577.722 -1 -1 1
919 8 1 768.080 -1 -1 1
920 8 2 615.129 -1 -1 1
921 8 1 747.500 -1 -1 1
922 8 2 540.316 -1 -1 1
923 8 1 775.000 -1 -1 1
924 8 2 711.667 -1 -1 1
925 8 1 760.599 -1 -1 1
926 8 2 639.167 -1 -1 1
927 8 1 758.333 -1 -1 1
928 8 2 549.491 -1 -1 1
929 8 1 682.500 -1 -1 1
930 8 2 684.167 -1 -1 1
931 8 1 658.116 1 -1 -1
932 8 2 672.153 1 -1 -1
933 8 1 738.213 1 -1 -1
934 8 2 594.534 1 -1 -1
935 8 1 681.236 1 -1 -1
936 8 2 627.650 1 -1 -1
937 8 1 704.904 1 -1 -1
938 8 2 551.870 1 -1 -1
939 8 1 693.623 1 -1 -1
940 8 2 594.534 1 -1 -1
941 8 1 624.993 1 -1 -1
942 8 2 602.660 1 -1 -1
943 8 1 700.228 1 -1 -1
944 8 2 585.450 1 -1 -1
945 8 1 611.874 1 -1 -1
946 8 2 555.724 1 -1 -1
947 8 1 579.167 1 -1 -1
948 8 2 574.934 1 -1 -1
949 8 1 720.872 1 -1 -1
950 8 2 584.625 1 -1 -1
951 8 1 690.320 1 -1 -1
952 8 2 555.724 1 -1 -1
953 8 1 677.933 1 -1 -1
954 8 2 611.874 1 -1 -1
955 8 1 674.600 1 -1 -1
956 8 2 698.254 1 -1 -1
957 8 1 611.999 1 -1 -1
958 8 2 748.130 1 -1 -1
959 8 1 530.680 1 -1 -1
960 8 2 689.942 1 -1 -1
1.4.2.10.2. Analysis of the Response Variable

1.4.2. Case Studies
Numerical
Summary
As a first step in the analysis, common summary statistics are
computed for the response variable.
Sample size = 480
Mean = 650.0773
Median = 646.6275
Minimum = 345.2940
Maximum = 821.6540
Range = 476.3600
Stan. Dev. = 74.6383
4-Plot The next step is generate a 4-plot of the response variable.
This 4-plot shows:
1. The run sequence plot (upper left corner) shows that the
location and scale are relatively constant. It also shows a
few outliers on the low side. Most of the points are in
the range 500 to 750. However, there are about half a
dozen points in the 300 to 450 range that may require
special attention.
A run sequence plot is useful for designed experiments
in that it can reveal time effects. Time is normally a
nuisance factor. That is, the time order on which runs
are made should not have a significant effect on the
response. If a time effect does appear to exist, this
means that there is a potential bias in the experiment that
needs to be investigated and resolved.
2. The lag plot (the upper right corner) does not show any
significant structure. This is another tool for detecting
any potential time effect.
3. The histogram (the lower left corner) shows the
response appears to be reasonably symmetric, but with a
bimodal distribution.
4. The normal probability plot (the lower right corner)
shows some curvature indicating that distributions other
than the normal may provide a better fit.
1.4.2.10.3. Analysis of the Batch Effect

1.4.2. Case Studies
Batch is a
Nuisance
Factor
The two nuisance factors in this experiment are the batch
number and the lab. There are two batches and eight labs.
Ideally, these factors will have minimal effect on the
response variable.
We will investigate the batch factor first.
Bihistogram
This bihistogram shows the following.
1. There does appear to be a batch effect.
2. The batch 1 responses are centered at 700 while the
batch 2 responses are centered at 625. That is, the
batch effect is approximately 75 units.
3. The variability is comparable for the 2 batches.
4. Batch 1 has some skewness in the lower tail. Batch 2
has some skewness in the center of the distribution, but
not as much in the tails compared to batch 1.
5. Both batches have a few low-lying points.
Although we could stop with the bihistogram, we will show a
few other commonly used two-sample graphical techniques
for comparison.
Quantile-
Quantile
Plot
This q-q plot shows the following.
1. Except for a few points in the right tail, the batch 1
values have higher quantiles than the batch 2 values.
This implies that batch 1 has a greater location value
than batch 2.
2. The q-q plot is not linear. This implies that the
difference between the batches is not explained simply
by a shift in location. That is, the variation and/or
skewness varies as well. From the bihistogram, it
appears that the skewness in batch 2 is the most likely
explanation for the non-linearity in the q-q plot.
Box Plot
This box plot shows the following.
1. The median for batch 1 is approximately 700 while the
median for batch 2 is approximately 600.
2. The spread is reasonably similar for both batches,
maybe slightly larger for batch 1.
3. Both batches have a number of outliers on the low side.
Batch 2 also has a few outliers on the high side. Box
plots are a particularly effective method for identifying
the presence of outliers.
Block Plots A block plot is generated for each of the eight labs, with "1"
and "2" denoting the batch numbers. In the first plot, we do
not include any of the primary factors. The next 3 block plots
include one of the primary factors. Note that each of the 3
primary factors (table speed = X1, down feed rate = X2,
wheel grit size = X3) has 2 levels. With 8 labs and 2 levels
for the primary factor, we would expect 16 separate blocks on
these plots. The fact that some of these blocks are missing
indicates that some of the combinations of lab and primary
factor are empty.
These block plots show the following.
1. The mean for batch 1 is greater than the mean for batch
2 in all of the cases above. This is strong evidence that
the batch effect is real and consistent across labs and
primary factors.
Quantitative
Techniques
We can confirm some of the conclusions drawn from the
above graphics by using quantitative techniques. The F-test
can be used to test whether or not the variances from the two
batches are equal and the two sample t-test can be used to
test whether or not the means from the two batches are equal.
Summary statistics for each batch are shown below.
Batch 1:
MEAN = 688.9987
VARIANCE = 4296.6845

Batch 2:
MEAN = 611.1559
VARIANCE = 3825.9544
F-Test The two-sided F-test indicates that the variances for the two
batches are not significantly different at the 5 % level.
H
0
:
1
2
=
2
2

H
a
:
1
2

2
2
Test statistic: F = 1.123
Numerator degrees of freedom:
1
= 239
Denominator degrees of freedom:
2
= 239
Critical values: F
1-/2,
1
,
2
= 0.845
F
/2,
1
,
2
= 1.289
0
if F < 0.845 or F >
1.289
Two Sample
t-Test
Since the F-test indicates that the two batch variances are
equal, we can pool the variances for the two-sided, two-
sample t-test to compare batch means.
H
0
:
1
=
2
H
a
:
1

2
Pooled standard deviation: s
p
= 63.7285
Critical value: t
1-/2,
= 1.965
0
if |T| > 1.965
The t-test indicates that the mean for batch 1 is larger than
the mean for batch 2 at the 5 % significance level.
Conclusions We can draw the following conclusions from the above
analysis.
1. There is in fact a significant batch effect. This batch
effect is consistent across labs and primary factors.
2. The magnitude of the difference is on the order of 75 to
100 (with batch 2 being smaller than batch 1). The
standard deviations do not appear to be significantly
different.
3. There is some skewness in the batches.
This batch effect was completely unexpected by the scientific
investigators in this study.
Note that although the quantitative techniques support the
conclusions of unequal means and equal standard deviations,
they do not show the more subtle features of the data such as
the presence of outliers and the skewness of the batch 2 data.
1.4.2.10.4. Analysis of the Lab Effect

1.4.2. Case Studies
Box Plot The next matter is to determine if there is a lab effect. The
first step is to generate a box plot for the ceramic strength
based on the lab.
1. There is minor variation in the medians for the 8 labs.
2. The scales are relatively constant for the labs.
3. Two of the labs (3 and 5) have outliers on the low side.
Box Plot for
Batch 1
Given that the previous section showed a distinct batch
effect, the next step is to generate the box plots for the two
batches separately.
1. Each of the labs has a median in the 650 to 700 range.
2. The variability is relatively constant across the labs.
3. Each of the labs has at least one outlier on the low side.
Box Plot for
Batch 2
1. The medians are in the range 550 to 600.
2. There is a bit more variability, across the labs, for
batch2 compared to batch 1.
3. Six of the eight labs show outliers on the high side.
Three of the labs show outliers on the low side.
Conclusions We can draw the following conclusions about a possible lab
effect from the above box plots.
1. The batch effect (of approximately 75 to 100 units) on
location dominates any lab effects.
2. It is reasonable to treat the labs as homogeneous.
1.4.2.10.5. Analysis of Primary Factors

1.4.2. Case Studies
Main effects The first step in analyzing the primary factors is to determine which factors
are the most significant. The DOE scatter plot, DOE mean plot, and the DOE
standard deviation plots will be the primary tools, with "DOE" being short
for "design of experiments".
Since the previous pages showed a significant batch effect but a minimal lab
effect, we will generate separate plots for batch 1 and batch 2. However, the
labs will be treated as equivalent.
DOE
Scatter Plot
for Batch 1
This DOE scatter plot shows the following for batch 1.
1. Most of the points are between 500 and 800.
2. There are about a dozen or so points between 300 and 500.
3. Except for the outliers on the low side (i.e., the points between 300
and 500), the distribution of the points is comparable for the 3 primary
factors in terms of location and spread.
DOE Mean
Plot for
Batch 1
This DOE mean plot shows the following for batch 1.
1. The table speed factor (X1) is the most significant factor with an
effect, the difference between the two points, of approximately 35
units.
2. The wheel grit factor (X3) is the next most significant factor with an
effect of approximately 10 units.
3. The feed rate factor (X2) has minimal effect.
DOE SD
Plot for
Batch 1
This DOE standard deviation plot shows the following for batch 1.
1. The table speed factor (X1) has a significant difference in variability
between the levels of the factor. The difference is approximately 20
units.
2. The wheel grit factor (X3) and the feed rate factor (X2) have minimal
differences in variability.
DOE
Scatter Plot
for Batch 2
This DOE scatter plot shows the following for batch 2.
1. Most of the points are between 450 and 750.
2. There are a few outliers on both the low side and the high side.
3. Except for the outliers (i.e., the points less than 450 or greater than
750), the distribution of the points is comparable for the 3 primary
factors in terms of location and spread.
DOE Mean
Plot for
Batch 2
This DOE mean plot shows the following for batch 2.
1. The feed rate (X2) and wheel grit (X3) factors have an approximately
equal effect of about 15 or 20 units.
2. The table speed factor (X1) has a minimal effect.
DOE SD
Plot for
Batch 2
This DOE standard deviation plot shows the following for batch 2.
1. The difference in the standard deviations is roughly comparable for the
three factors (slightly less for the feed rate factor).
Interaction
Effects
The above plots graphically show the main effects. An additonal concern is
whether or not there any significant interaction effects.
Main effects and 2-term interaction effects are discussed in the chapter on
Process Improvement.
In the following DOE interaction plots, the labels on the plot give the
variables and the estimated effect. For example, factor 1 is table speed and it
has an estimated effect of 30.77 (it is actually -30.77 if the direction is taken
into account).
DOE
Interaction
Plot for
Batch 1
The ranked list of factors for batch 1 is:
1. Table speed (X1) with an estimated effect of -30.77.
2. The interaction of table speed (X1) and wheel grit (X3) with an
estimated effect of -20.25.
3. The interaction of table speed (X1) and feed rate (X2) with an
estimated effect of 9.7.
4. Wheel grit (X3) with an estimated effect of -7.18.
5. Down feed (X2) and the down feed interaction with wheel grit (X3)
are essentially zero.
DOE
Interaction
Plot for
Batch 2
The ranked list of factors for batch 2 is:
1. Down feed (X2) with an estimated effect of 18.22.
2. The interaction of table speed (X1) and wheel grit (X3) with an
estimated effect of -16.71.
3. Wheel grit (X3) with an estimated effect of -14.71
4. Remaining main effect and 2-factor interaction effects are essentially
zero.
Conclusions From the above plots, we can draw the following overall conclusions.
1. The batch effect (of approximately 75 units) is the dominant primary
factor.
2. The most important factors differ from batch to batch. See the above
text for the ranked list of factors with the estimated effects.

1.4.2. Case Studies
View
Dataplot
Macro for
this Case
Study
This page allows you to use Dataplot to repeat the analysis
outlined in the case study description on the previous page. It
Data Analysis Steps
Results and
Conclusions
more detailed
1. Read in the data. 1. You have read 1
column of numbers
into Dataplot,
variable Y.
2. Plot of the response variable
1. Numerical summary of Y.
2. 4-plot of Y.
1. The summary shows
the mean strength
is 650.08 and the
standard deviation
of the strength
is 74.64.
2. The 4-plot shows
no drift in
the location and
scale and a
bimodal
distribution.
3. Determine if there is a batch effect.
1. Generate a bihistogram based on
the 2 batches.
2. Generate a q-q plot.
3. Generate a box plot.
4. Generate block plots.
5. Perform a 2-sample t-test for
equal means.
6. Perform an F-test for equal
1. The bihistogram
shows a distinct
batch effect of
approximately
75 units.
2. The q-q plot
shows that batch 1
and batch 2 do
not come from a
common
distribution.
3. The box plot
shows that there is
a batch effect of
approximately
75 to 100 units
and there are
some outliers.
4. The block plot
shows that the batch
effect is
consistent across
labs
and levels of the
primary factor.
5. The t-test
confirms the batch
effect with
respect to the means.
6. The F-test does
not indicate any
significant batch
effect with
respect to the
4. Determine if there is a lab effect.
1. Generate a box plot for the labs
with the 2 batches combined.
for batch 1 only.
for batch 2 only.
1. The box plot
does not show a
significant lab
effect.
2. The box plot
does not show a
significant lab
effect for batch 1.
3. The box plot
does not show a
significant lab
effect for batch 2.
5. Analysis of primary factors.
1. Generate a DOE scatter plot for
batch 1.
1. The DOE scatter
plot shows the
range of the
points and the
2. Generate a DOE mean plot for
batch 1.
3. Generate a DOE sd plot for
batch 1.
4. Generate a DOE scatter plot for
batch 2.
5. Generate a DOE mean plot for
batch 2.
6. Generate a DOE sd plot for
batch 2.
7. Generate a DOE interaction
effects matrix plot for
batch 1.
effects matrix plot for
batch 2.
presence of
outliers.
2. The DOE mean
plot shows that
table speed is
the most
significant
factor for batch 1.
3. The DOE sd plot
shows that
table speed has
the most
variability for
batch 1.
4. The DOE scatter
plot shows
the range of the
points and
the presence of
outliers.
5. The DOE mean
plot shows that
feed rate and
wheel grit are
the most
significant factors
for batch 2.
6. The DOE sd plot
shows that
the variability
is comparable
for all 3 factors
for batch 2.
7. The DOE
interaction effects
matrix plot
provides a ranked
list of factors
with the
estimated
effects.
8. The DOE
interaction effects
matrix plot
provides a ranked
list of factors
with the
estimated
effects.
1.4.3. References For Chapter 1: Exploratory Data Analysis

1.4.3. References For Chapter 1: Exploratory
Data Analysis
Anscombe, F. (1973), Graphs in Statistical Analysis, The American
Statistician, pp. 195-199.
Anscombe, F. and Tukey, J. W. (1963), The Examination and Analysis of
Residuals, Technometrics, pp. 141-160.
Barnett and Lewis (1994), Outliers in Statistical Data, 3rd. Ed., John
Wiley and Sons.
Birnbaum, Z. W. and Saunders, S. C. (1958), A Statistical Model for
Life-Length of Materials, Journal of the American Statistical Association,
53(281), pp. 151-160.
Bloomfield, Peter (1976), Fourier Analysis of Time Series, John Wiley
and Sons.
Box, G. E. P. and Cox, D. R. (1964), An Analysis of Transformations,
Journal of the Royal Statistical Society, pp. 211-243, discussion pp. 244-
252.
Box, G. E. P., Hunter, W. G., and Hunter, J. S. (1978), Statistics for
Experimenters: An Introduction to Design, Data Analysis, and Model
Building, John Wiley and Sons.
Box, G. E. P., and Jenkins, G. (1976), Time Series Analysis: Forecasting
and Control, Holden-Day.
Bradley, (1968). Distribution-Free Statistical Tests, Chapter 12.
Brown, M. B. and Forsythe, A. B. (1974), Journal of the American
Statistical Association, 69, pp. 364-367.
Chakravarti, Laha, and Roy, (1967). Handbook of Methods of Applied
Statistics, Volume I, John Wiley and Sons, pp. 392-394.
Chambers, John, William Cleveland, Beat Kleiner, and Paul Tukey,
(1983), Graphical Methods for Data Analysis, Wadsworth.
Chatfield, C. (1989). The Analysis of Time Series: An Introduction, Fourth
Edition, Chapman & Hall, New York, NY.
Cleveland, William (1985), Elements of Graphing Data, Wadsworth.
Cleveland, William and Marylyn McGill, Editors (1988), Dynamic
Graphics for Statistics, Wadsworth.
Cleveland, William (1993), Visualizing Data, Hobart Press.
Devaney, Judy (1997), Equation Discovery Through Global Self-
Referenced Geometric Intervals and Machine Learning, Ph.d thesis,
George Mason University, Fairfax, VA.
Draper and Smith, (1981). Applied Regression Analysis, 2nd ed., John
Wiley and Sons.
du Toit, Steyn, and Stumpf (1986), Graphical Exploratory Data
Analysis, Springer-Verlag.
Efron and Gong (February 1983), A Leisurely Look at the Bootstrap, the
Jackknife, and Cross Validation, The American Statistician.
Evans, Hastings, and Peacock (2000), Statistical Distributions, 3rd. Ed.,
John Wiley and Sons.
Everitt, Brian (1978), Multivariate Techniques for Multivariate Data,
North-Holland.
Filliben, J. J. (February 1975), The Probability Plot Correlation
Coefficient Test for Normality, Technometrics, pp. 111-117.
Fuller Jr., E. R., Frieman, S. W., Quinn, J. B., Quinn, G. D., and Carter,
W. C. (1994), Fracture Mechanics Approach to the Design of Glass
Aircraft Windows: A Case Study, SPIE Proceedings, Vol. 2286, (Society
of Photo-Optical Instrumentation Engineers (SPIE), Bellingham, WA).
Gill, Lisa (April 1997), Summary Analysis: High Performance Ceramics
Experiment to Characterize the Effect of Grinding Parameters on
Sintered Reaction Bonded Silicon Nitride, Reaction Bonded Silicon
Nitride, and Sintered Silicon Nitride , presented at the NIST - Ceramic
Machining Consortium, 10th Program Review Meeting, April 10, 1997.
Granger and Hatanaka (1964), Spectral Analysis of Economic Time
Series, Princeton University Press.
Grubbs, Frank (1950), Sample Criteria for Testing Outlying
Observations, Annals of Mathematical Statistics, 21(1) pp. 27-58.
Grubbs, Frank (February 1969), Procedures for Detecting Outlying
Observations in Samples, Technometrics, 11(1), pp. 1-21.
Hahn, G. J. and Meeker, W. Q. (1991), Statistical Intervals, John Wiley
and Sons.
Harris, Robert L. (1996), Information Graphics, Management Graphics.
Hastie, T., Tibshirani, R. and Friedman, J. (2001), The Elements of
Statistical Learning: Data Mining, Inference, and Prediction, Springer-
Verlag, New York.
Hawkins, D. M. (1980), Identification of Outliers, Chapman and Hall.
Boris Iglewicz and David Hoaglin (1993), "Volume 16: How to Detect
and Handle Outliers", The ASQC Basic References in Quality Control:
Statistical Techniques, Edward F. Mykytka, Ph.D., Editor.
Jenkins and Watts, (1968), Spectral Analysis and Its Applications,
Holden-Day.
Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate
Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.
Johnson, Kotz, and Kemp, (1992), Univariate Discrete Distributions,
2nd. Ed., John Wiley and Sons.
Kuo, Way and Pierson, Marcia Martens, Eds. (1993), Quality Through
Engineering Design", specifically, the article Filliben, Cetinkunt, Yu, and
Dommenz (1993), Exploratory Data Analysis Techniques as Applied to a
High-Precision Turning Machine, Elsevier, New York, pp. 199-223.
Levene, H. (1960). In Contributions to Probability and Statistics: Essays
in Honor of Harold Hotelling, I. Olkin et al. eds., Stanford University
Press, pp. 278-292.
McNeil, Donald (1977), Interactive Data Analysis, John Wiley and Sons.
Mendenhall, William and Reinmuth, James (1982), Statistics for
Management and Ecomonics, Fourth Edition, Duxbury Press.
Mosteller, Frederick and Tukey, John (1977), Data Analysis and
Regression, Addison-Wesley.
Natrella, Mary (1963), Experimental Statistics, National Bureau of
Standards Handbook 91.
Nelson, Wayne (1982), Applied Life Data Analysis, Addison-Wesley.
Nelson, Wayne and Doganaksoy, Necip (1992), A Computer Program
POWNOR for Fitting the Power-Normal and -Lognormal Models to Life
or Strength Data from Specimens of Various Sizes, NISTIR 4760, U.S.
Department of Commerce, National Institute of Standards and
Technology.
Neter, Wasserman, and Kunter (1990). Applied Linear Statistical Models,
3rd ed., Irwin.
Pepi, John W., (1994), Failsafe Design of an All BK-7 Glass Aircraft
Window, SPIE Proceedings, Vol. 2286, (Society of Photo-Optical
Instrumentation Engineers (SPIE), Bellingham, WA).
The RAND Corporation (1955), A Million Random Digits with 100,000
Normal Deviates, Free Press.
Rosner, Bernard (May 1983), Percentage Points for a Generalized ESD
Many-Outlier Procedure,Technometrics, 25(2), pp. 165-172.
Ryan, Thomas (1997), Modern Regression Methods, John Wiley.
Scott, David (1992), Multivariate Density Estimation: Theory, Practice,
and Visualization , John Wiley and Sons.
Snedecor, George W. and Cochran, William G. (1989), Statistical
Methods, Eighth Edition, Iowa State University Press.
Stefansky, W. (1972), Rejecting Outliers in Factorial Designs,
Technometrics, 14, pp. 469-479.
Stephens, M. A. (1974). EDF Statistics for Goodness of Fit and Some
Comparisons, Journal of the American Statistical Association, 69, pp.
730-737.
Stephens, M. A. (1976). Asymptotic Results for Goodness-of-Fit
Statistics with Unknown Parameters, Annals of Statistics, 4, pp. 357-369.
Stephens, M. A. (1977). Goodness of Fit for the Extreme Value
Distribution, Biometrika, 64, pp. 583-588.
Stephens, M. A. (1977). Goodness of Fit with Special Reference to Tests
for Exponentiality , Technical Report No. 262, Department of Statistics,
Stanford University, Stanford, CA.
Stephens, M. A. (1979). Tests of Fit for the Logistic Distribution Based
on the Empirical Distribution Function, Biometrika, 66, pp. 591-595.
Tietjen and Moore (August 1972), Some Grubbs-Type Statistics for the
Detection of Outliers, Technometrics, 14(3), pp. 583-597.
Tufte, Edward (1983), The Visual Display of Quantitative Information,
Graphics Press.
Tukey, John (1977), Exploratory Data Analysis, Addison-Wesley.
Velleman, Paul and Hoaglin, David (1981), The ABC's of EDA:
Applications, Basics, and Computing of Exploratory Data Analysis,
Duxbury.
Wainer, Howard (1981), Visual Revelations, Copernicus.
Wilk, M. B. and Gnanadesikan, R. (1968), Probability Plotting Methods
for the Analysis of Data, Biometrika, 5(5), pp. 1-19.
2. Measurement Process Characterization
http://www.itl.nist.gov/div898/handbook/mpc/mpc.htm[6/27/2012 1:52:32 PM]

1. Characterization
1. Issues
2. Check standards
2. Control
1. Issues
2. Bias and long-term variability
3. Short-term variability
3. Calibration
1. Issues
2. Artifacts
3. Designs
4. Catalog of designs
5. Artifact control
6. Instruments
7. Instrument control
4. Gauge R & R studies
1. Issues
2. Design
3. Data collection
4. Variability
5. Bias
6. Uncertainty
5. Uncertainty analysis
1. Issues
2. Approach
3. Type A evaluations
4. Type B evaluations
5. Propagation of error
6. Error budget
7. Expanded uncertainties
8. Uncorrected bias
6. Case Studies
1. Gauge study
2. Check standard
3. Type A uncertainty
4. Type B uncertainty
Detailed table of contents
References for Chapter 2
http://www.itl.nist.gov/div898/handbook/mpc/mpc_d.htm[6/27/2012 1:50:07 PM]

2. Measurement Process Characterization - Detailed Table of
Contents
1. Characterization [2.1.]
1. What are the issues for characterization? [2.1.1.]
1. Purpose [2.1.1.1.]
2. Reference base [2.1.1.2.]
3. Bias and Accuracy [2.1.1.3.]
4. Variability [2.1.1.4.]
2. What is a check standard? [2.1.2.]
1. Assumptions [2.1.2.1.]
2. Data collection [2.1.2.2.]
3. Analysis [2.1.2.3.]
2. Statistical control of a measurement process [2.2.]
1. What are the issues in controlling the measurement process? [2.2.1.]
2. How are bias and variability controlled? [2.2.2.]
1. Shewhart control chart [2.2.2.1.]
1. EWMA control chart [2.2.2.1.1.]
3. Monitoring bias and long-term variability [2.2.2.3.]
4. Remedial actions [2.2.2.4.]
3. How is short-term variability controlled? [2.2.3.]
1. Control chart for standard deviations [2.2.3.1.]
3. Monitoring short-term precision [2.2.3.3.]
4. Remedial actions [2.2.3.4.]
3. Calibration [2.3.]
1. Issues in calibration [2.3.1.]
1. Reference base [2.3.1.1.]
2. Reference standards [2.3.1.2.]
2. What is artifact (single-point) calibration? [2.3.2.]
3. What are calibration designs? [2.3.3.]
1. Elimination of special types of bias [2.3.3.1.]
1. Left-right (constant instrument) bias [2.3.3.1.1.]
2. Bias caused by instrument drift [2.3.3.1.2.]
2. Solutions to calibration designs [2.3.3.2.]
1. General matrix solutions to calibration designs [2.3.3.2.1.]
3. Uncertainties of calibrated values [2.3.3.3.]
1. Type A evaluations for calibration designs [2.3.3.3.1.]
2. Repeatability and level-2 standard deviations [2.3.3.3.2.]
3. Combination of repeatability and level-2 standard deviations [2.3.3.3.3.]
4. Calculation of standard deviations for 1,1,1,1 design [2.3.3.3.4.]
5. Type B uncertainty [2.3.3.3.5.]
6. Expanded uncertainties [2.3.3.3.6.]
4. Catalog of calibration designs [2.3.4.]
1. Mass weights [2.3.4.1.]
1. Design for 1,1,1 [2.3.4.1.1.]
2. Design for 1,1,1,1 [2.3.4.1.2.]
3. Design for 1,1,1,1,1 [2.3.4.1.3.]
4. Design for 1,1,1,1,1,1 [2.3.4.1.4.]
5. Design for 2,1,1,1 [2.3.4.1.5.]
6. Design for 2,2,1,1,1 [2.3.4.1.6.]
7. Design for 2,2,2,1,1 [2.3.4.1.7.]
8. Design for 5,2,2,1,1,1 [2.3.4.1.8.]
9. Design for 5,2,2,1,1,1,1 [2.3.4.1.9.]
10. Design for 5,3,2,1,1,1 [2.3.4.1.10.]
11. Design for 5,3,2,1,1,1,1 [2.3.4.1.11.]
12. Design for 5,3,2,2,1,1,1 [2.3.4.1.12.]
13. Design for 5,4,4,3,2,2,1,1 [2.3.4.1.13.]
14. Design for 5,5,2,2,1,1,1,1 [2.3.4.1.14.]
15. Design for 5,5,3,2,1,1,1 [2.3.4.1.15.]
16. Design for 1,1,1,1,1,1,1,1 weights [2.3.4.1.16.]
17. Design for 3,2,1,1,1 weights [2.3.4.1.17.]
18. Design for 10 and 20 pound weights [2.3.4.1.18.]
2. Drift-elimination designs for gage blocks [2.3.4.2.]
1. Doiron 3-6 Design [2.3.4.2.1.]
2. Doiron 3-9 Design [2.3.4.2.2.]
3. Doiron 4-8 Design [2.3.4.2.3.]
4. Doiron 4-12 Design [2.3.4.2.4.]
5. Doiron 5-10 Design [2.3.4.2.5.]
6. Doiron 6-12 Design [2.3.4.2.6.]
7. Doiron 7-14 Design [2.3.4.2.7.]
8. Doiron 8-16 Design [2.3.4.2.8.]
9. Doiron 9-18 Design [2.3.4.2.9.]
10. Doiron 10-20 Design [2.3.4.2.10.]
11. Doiron 11-22 Design [2.3.4.2.11.]
3. Designs for electrical quantities [2.3.4.3.]
1. Left-right balanced design for 3 standard cells [2.3.4.3.1.]
5. Left-right balanced design for 4 references and 4 test items [2.3.4.3.5.]
6. Design for 8 references and 8 test items [2.3.4.3.6.]
7. Design for 4 reference zeners and 2 test zeners [2.3.4.3.7.]
8. Design for 4 reference zeners and 3 test zeners [2.3.4.3.8.]
9. Design for 3 references and 1 test resistor [2.3.4.3.9.]
10. Design for 4 references and 1 test resistor [2.3.4.3.10.]
4. Roundness measurements [2.3.4.4.]
1. Single trace roundness design [2.3.4.4.1.]
2. Multiple trace roundness designs [2.3.4.4.2.]
5. Designs for angle blocks [2.3.4.5.]
1. Design for 4 angle blocks [2.3.4.5.1.]
6. Thermometers in a bath [2.3.4.6.]
7. Humidity standards [2.3.4.7.]
1. Drift-elimination design for 2 reference weights and 3 cylinders [2.3.4.7.1.]
5. Control of artifact calibration [2.3.5.]
1. Control of precision [2.3.5.1.]
1. Example of control chart for precision [2.3.5.1.1.]
2. Control of bias and long-term variability [2.3.5.2.]
1. Example of Shewhart control chart for mass calibrations [2.3.5.2.1.]
2. Example of EWMA control chart for mass calibrations [2.3.5.2.2.]
6. Instrument calibration over a regime [2.3.6.]
1. Models for instrument calibration [2.3.6.1.]
3. Assumptions for instrument calibration [2.3.6.3.]
4. What can go wrong with the calibration procedure [2.3.6.4.]
1. Example of day-to-day changes in calibration [2.3.6.4.1.]
5. Data analysis and model validation [2.3.6.5.]
1. Data on load cell #32066 [2.3.6.5.1.]
6. Calibration of future measurements [2.3.6.6.]
7. Uncertainties of calibrated values [2.3.6.7.]
1. Uncertainty for quadratic calibration using propagation of error [2.3.6.7.1.]
2. Uncertainty for linear calibration using check standards [2.3.6.7.2.]
3. Comparison of check standard analysis and propagation of error [2.3.6.7.3.]
7. Instrument control for linear calibration [2.3.7.]
1. Control chart for a linear calibration line [2.3.7.1.]
4. Gauge R & R studies [2.4.]
1. What are the important issues? [2.4.1.]
2. Design considerations [2.4.2.]
3. Data collection for time-related sources of variability [2.4.3.]
1. Simple design [2.4.3.1.]
2. 2-level nested design [2.4.3.2.]
3. 3-level nested design [2.4.3.3.]
4. Analysis of variability [2.4.4.]
1. Analysis of repeatability [2.4.4.1.]
2. Analysis of reproducibility [2.4.4.2.]
3. Analysis of stability [2.4.4.3.]
1. Example of calculations [2.4.4.4.4.]
5. Analysis of bias [2.4.5.]
1. Resolution [2.4.5.1.]
2. Linearity of the gauge [2.4.5.2.]
3. Drift [2.4.5.3.]
4. Differences among gauges [2.4.5.4.]
5. Geometry/configuration differences [2.4.5.5.]
6. Remedial actions and strategies [2.4.5.6.]
6. Quantifying uncertainties from a gauge study [2.4.6.]
5. Uncertainty analysis [2.5.]
1. Issues [2.5.1.]
2. Approach [2.5.2.]
1. Steps [2.5.2.1.]
3. Type A evaluations [2.5.3.]
1. Type A evaluations of random components [2.5.3.1.]
1. Type A evaluations of time-dependent effects [2.5.3.1.1.]
2. Measurement configuration within the laboratory [2.5.3.1.2.]
2. Material inhomogeneity [2.5.3.2.]
1. Data collection and analysis [2.5.3.2.1.]
3. Type A evaluations of bias [2.5.3.3.]
1. Inconsistent bias [2.5.3.3.1.]
2. Consistent bias [2.5.3.3.2.]
3. Bias with sparse data [2.5.3.3.3.]
4. Type B evaluations [2.5.4.]
1. Standard deviations from assumed distributions [2.5.4.1.]
5. Propagation of error considerations [2.5.5.]
1. Formulas for functions of one variable [2.5.5.1.]
2. Formulas for functions of two variables [2.5.5.2.]
3. Propagation of error for many variables [2.5.5.3.]
6. Uncertainty budgets and sensitivity coefficients [2.5.6.]
1. Sensitivity coefficients for measurements on the test item [2.5.6.1.]
2. Sensitivity coefficients for measurements on a check standard [2.5.6.2.]
3. Sensitivity coefficients for measurements from a 2-level design [2.5.6.3.]
4. Sensitivity coefficients for measurements from a 3-level design [2.5.6.4.]
5. Example of uncertainty budget [2.5.6.5.]
7. Standard and expanded uncertainties [2.5.7.]
1. Degrees of freedom [2.5.7.1.]
8. Treatment of uncorrected bias [2.5.8.]
1. Computation of revised uncertainty [2.5.8.1.]
6. Case studies [2.6.]
1. Gauge study of resistivity probes [2.6.1.]
1. Background and data [2.6.1.1.]
1. Database of resistivity measurements [2.6.1.1.1.]
2. Analysis and interpretation [2.6.1.2.]
3. Repeatability standard deviations [2.6.1.3.]
4. Effects of days and long-term stability [2.6.1.4.]
5. Differences among 5 probes [2.6.1.5.]
6. Run gauge study example using Dataplot [2.6.1.6.]
7. Dataplot macros [2.6.1.7.]
2. Check standard for resistivity measurements [2.6.2.]
1. Database for resistivity check standard [2.6.2.1.1.]
1. Repeatability and level-2 standard deviations [2.6.2.2.1.]
3. Control chart for probe precision [2.6.2.3.]
4. Control chart for bias and long-term variability [2.6.2.4.]
5. Run check standard example yourself [2.6.2.5.]
3. Evaluation of type A uncertainty [2.6.3.]
1. Database of resistivity measurements [2.6.3.1.1.]
2. Measurements on wiring configurations [2.6.3.1.2.]
1. Difference between 2 wiring configurations [2.6.3.2.1.]
3. Run the type A uncertainty analysis using Dataplot [2.6.3.3.]
4. Evaluation of type B uncertainty and propagation of error [2.6.4.]
7. References [2.7.]
2.1. Characterization
http://www.itl.nist.gov/div898/handbook/mpc/section1/mpc1.htm[6/27/2012 1:50:12 PM]

The primary goal of this section is to lay the groundwork for
understanding the measurement process in terms of the errors
that affect the process.
What are the issues for characterization?
1. Purpose
2. Reference base
3. Bias and Accuracy
4. Variability
What is a check standard?
1. Assumptions
2. Data collection
3. Analysis
2.1.1. What are the issues for characterization?

'Goodness' of
measurements
A measurement process can be thought of as a well-run
production process in which measurements are the output.
The 'goodness' of measurements is the issue, and goodness
is characterized in terms of the errors that affect the
measurements.
Bias,
variability
and
uncertainty
The goodness of measurements is quantified in terms of
Bias
Short-term variability or instrument precision
Day-to-day or long-term variability
Uncertainty
Requires
ongoing
statistical
control
program
The continuation of goodness is guaranteed by a statistical
control program that controls both
Short-term variability or instrument precision
Long-term variability which controls bias and day-
to-day variability of the process
Scope is
limited to
ongoing
processes
The techniques in this chapter are intended primarily for
ongoing processes. One-time tests and special tests or
destructive tests are difficult to characterize. Examples of
ongoing processes are:
Calibration where similar test items are measured on
a regular basis
Certification where materials are characterized on a
regular basis
Production where the metrology (tool) errors may be
significant
Special studies where data can be collected over the
life of the study
Application to
production
processes
The material in this chapter is pertinent to the study of
production processes for which the size of the metrology
(tool) error may be an important consideration. More
specific guidance on assessing metrology errors can be
found in the section on gauge studies.
2.1.1.1. Purpose

2.1.1.1. Purpose
Purpose is
to
understand
and
quantify
the effect
of error on
reported
values
The purpose of characterization is to develop an understanding
of the sources of error in the measurement process and how
they affect specific measurement results. This section provides
the background for:
identifying sources of error in the measurement process
understanding and quantifying errors in the
measurement process
codifying the effects of these errors on a specific
reported value in a statement of uncertainty
Important
concepts
Characterization relies upon the understanding of certain
underlying concepts of measurement systems; namely,
reference base (authority) for the measurement
bias
variability
check standard
Reported
value is a
generic
term that
identifies
the result
that is
transmitted
to the
customer
The reported value is the measurement result for a particular
test item. It can be:
a single measurement
an average of several measurements
a least-squares prediction from a model
a combination of several measurement results that are
related by a physical model
2.1.1.2. Reference base

Ultimate
authority
The most critical element of any measurement process is the
relationship between a single measurement and the reference
base for the unit of measurement. The reference base is the
ultimate source of authority for the measurement unit.
For
fundamental
units
Reference bases for fundamental units of measurement
(length, mass, temperature, voltage, and time) and some
derived units (such as pressure, force, flow rate, etc.) are
maintained by national and regional standards laboratories.
Consensus values from interlaboratory tests or
instrumentation/standards as maintained in specific
environments may serve as reference bases for other units of
measurement.
For
comparison
purposes
A reference base, for comparison purposes, may be based on
an agreement among participating laboratories or
organizations and derived from
measurements made with a standard test method
measurements derived from an interlaboratory test
2.1.1.3. Bias and Accuracy

Definition of
Accuracy and
Bias
Accuracy is a qualitative term referring to whether there is
agreement between a measurement made on an object and
its true (target or reference) value. Bias is a quantitative
term describing the difference between the average of
measurements made on the same object and its true value.
In particular, for a measurement laboratory, bias is the
difference (generally unknown) between a laboratory's
average value (over time) for a test item and the average
that would be achieved by the reference laboratory if it
undertook the same measurements on the same test item.
Depiction of
bias and
unbiased
measurements
Unbiased measurements relative to the target
Biased measurements relative to the target
Identification
of bias
Bias in a measurement process can be identified by:
1. Calibration of standards and/or instruments by a
reference laboratory, where a value is assigned to the
client's standard based on comparisons with the
reference laboratory's standards.
2. Check standards , where violations of the control
limits on a control chart for the check standard
suggest that re-calibration of standards or instruments
is needed.
3. Measurement assurance programs, where artifacts
from a reference laboratory or other qualified agency
are sent to a client and measured in the client's
environment as a 'blind' sample.
4. Interlaboratory comparisons, where reference
standards or materials are circulated among several
laboratories.
Reduction of Bias can be eliminated or reduced by calibration of
bias standards and/or instruments. Because of costs and time
constraints, the majority of calibrations are performed by
secondary or tertiary laboratories and are related to the
reference base via a chain of intercomparisons that start at
the reference laboratory.
Bias can also be reduced by corrections to in-house
measurements based on comparisons with artifacts or
instruments circulated for that purpose (reference
materials).
Caution Errors that contribute to bias can be present even where all
equipment and standards are properly calibrated and under
control. Temperature probably has the most potential for
introducing this type of bias into the measurements. For
example, a constant heat source will introduce serious
errors in dimensional measurements of metal objects.
Temperature affects chemical and electrical measurements
as well.
Generally speaking, errors of this type can be identified
only by those who are thoroughly familiar with the
measurement technology. The reader is advised to consult
the technical literature and experts in the field for guidance.
2.1.1.4. Variability

Sources of
time-dependent
variability
Variability is the tendency of the measurement process to
produce slightly different measurements on the same test
item, where conditions of measurement are either stable
or vary over time, temperature, operators, etc. In this
chapter we consider two sources of time-dependent
variability:
Short-term variability ascribed to the precision of
the instrument
Long-term variability related to changes in
environment and handling techniques
Depiction of
two
measurement
processes with
the same short-
term variability
over six days
where process
1 has large
between-day
variability and
process 2 has
negligible
between-day
variability
Process 1 Process
2
Large between-day variability Small between-
day variability

Distributions of short-term measurements over
6 days where distances from the centerlines
illustrate between-day variability
Short-term
variability
Short-term errors affect the precision of the instrument.
Even very precise instruments exhibit small changes
caused by random errors. It is useful to think in terms of
measurements performed with a single instrument over
minutes or hours; this is to be understood, normally, as
the time that it takes to complete a measurement
sequence.
Terminology Four terms are in common usage to describe short-term
phenomena. They are interchangeable.
1. precision
2. repeatability
3. within-time variability
4. short-term variability
Precision is
quantified by a
standard
deviation
The measure of precision is a standard deviation. Good
precision implies a small standard deviation. This
standard deviation is called the short-term standard
deviation of the process or the repeatability standard
deviation.
Caution --
long-term
variability may
be dominant
With very precise instrumentation, it is not unusual to
find that the variability exhibited by the measurement
process from day-to-day often exceeds the precision of
the instrument because of small changes in environmental
conditions and handling techniques which cannot be
controlled or corrected in the measurement process. The
measurement process is not completely characterized
until this source of variability is quantified.
Terminology Three terms are in common usage to describe long-term
phenomena. They are interchangeable.
1. day-to-day variability
2. long-term variability
3. reproducibility
Caution --
regarding term
'reproducibility'
The term 'reproducibility' is given very specific
definitions in some national and international standards.
However, the definitions are not always in agreement.
Therefore, it is used here only in a generic sense to
indicate variability across days.
Definitions in
this Handbook
We adopt precise definitions and provide data collection
and analysis techniques in the sections on check standards
and measurement control for estimating:
Level-1 standard deviation for short-term
variability
Level-2 standard deviation for day-to-day
variability
In the section on gauge studies, the concept of variability
is extended to include very long-term measurement
variability:
Level-1 standard deviation for short-term
variability
Level-2 standard deviation for day-to-day
variability
Level-3 standard deviation for very long-term
variability
We refer to the standard deviations associated with these
three kinds of uncertainty as "Level 1, 2, and 3 standard
deviations", respectively.
Long-term
variability is
quantified by a
standard
deviation
The measure of long-term variability is the standard
deviation of measurements taken over several days,
weeks or months.
The simplest method for doing this assessment is by
analysis of a check standard database. The measurements
on the check standards are structured to cover a long time
interval and to capture all sources of variation in the
measurement process.
2.1.2. What is a check standard?

A check
standard is
useful for
gathering
data on the
process
Check standard methodology is a tool for collecting data on
the measurement process to expose errors that afflict the
process over time. Time-dependent sources of error are
evaluated and quantified from the database of check
standard measurements. It is a device for controlling the
bias and long-term variability of the process once a
baseline for these quantities has been established from
historical data on the check standard.
Think in
terms of data
A check
standard can
be an artifact
or defined
quantity
The check standard should be thought of in terms of a
database of measurements. It can be defined as an artifact
or as a characteristic of the measurement process whose
value can be replicated from measurements taken over the
life of the process. Examples are:
measurements on a stable artifact
differences between values of two reference
standards as estimated from a calibration experiment
values of a process characteristic, such as a bias
term, which is estimated from measurements on
reference standards and/or test items.
An artifact check standard must be close in material
content and geometry to the test items that are measured in
the workload. If possible, it should be one of the test items
from the workload. Obviously, it should be a stable artifact
and should be available to the measurement process at all
times.
Solves the
difficulty of
sampling the
process
Measurement processes are similar to production processes
in that they are continual and are expected to produce
identical results (within acceptable limits) over time,
instruments, operators, and environmental conditions.
However, it is difficult to sample the output of the
measurement process because, normally, test items change
with each measurement sequence.
Surrogate for
unseen
measurements
Measurements on the check standard, spaced over time at
regular intervals, act as surrogates for measurements that
could be made on test items if sufficient time and resources
were available.

Case study:
Resistivity
check
standard
Before applying the quality control procedures
recommended in this chapter to check standard data, basic
assumptions should be examined. The basic assumptions
underlying the quality control procedures are:
1. The data come from a single statistical distribution.
2. The distribution is a normal distribution.
3. The errors are uncorrelated over time.
An easy method for checking the assumption of a single
normal distribution is to construct a histogram of the check
standard data. The histogram should follow a bell-shaped
pattern with a single hump. Types of anomalies that
indicate a problem with the measurement system are:
1. a double hump indicating that errors are being drawn
from two or more distributions;
2. long tails indicating outliers in the process;
3. flat pattern or one with humps at either end
indicating that the measurement process in not in
control or not properly specified.
Another graphical method for testing the normality
assumption is a probability plot. The points are expected to
fall approximately on a straight line if the data come from
a normal distribution. Outliers, or data from other
distributions, will produce an S-shaped curve.
A graphical method for testing for correlation among
measurements is a time-lag plot. Correlation will
frequently not be a problem if measurements are properly
structured over time. Correlation problems generally occur
when measurements are taken so close together in time that
the instrument cannot properly recover from one
measurement to the next. Correlations over time are
usually present but are often negligible.
2.1.2.2. Data collection

Schedule for
making
measurements
A schedule for making check standard measurements over time (once a
day, twice a week, or whatever is appropriate for sampling all conditions
of measurement) should be set up and adhered to. The check standard
measurements should be structured in the same way as values reported on
the test items. For example, if the reported values are averages of two
repetitions made within 5 minutes of each other, the check standard
values should be averages of the two measurements made in the same
manner.
Exception One exception to this rule is that there should be at least J = 2 repetitions
per day. Without this redundancy, there is no way to check on the short-
term precision of the measurement system.
Depiction of
schedule for
making check
standard
measurements
with four
repetitions
per day over
K days on the
surface of a
silicon wafer
with the
repetitions
randomized
at various
positions on
the wafer
K days - 4 repetitions
2-level design for measurement process
Case study:
Resistivity
check
standard for
measurements
on silicon
wafers
The values for the check standard should be recorded along with pertinent
environmental readings and identifications for all other significant
factors. The best way to record this information is in one file with one
line or row (on a spreadsheet) of information in fixed fields for each
check standard measurement. A list of typical entries follows.
1. Identification for check standard
2. Date
3. Identification for the measurement design (if applicable)
4. Identification for the instrument
5. Check standard value
6. Short-term standard deviation from J repetitions
7. Degrees of freedom
8. Operator identification
9. Environmental readings (if pertinent)
2.1.2.3. Analysis

2.1.2.3. Analysis
Short-term
or level-1
standard
deviations
from J
repetitions
An analysis of the check standard data is the basis for
quantifying random errors in the measurement process --
particularly time-dependent errors.
Given that we have a database of check standard
measurements as described in data collection where
represents the jth repetition on the kth day, the mean for the
kth day is

and the short-term (level-1) standard deviation with v = J - 1
degrees of freedom is
.
Drawback
of short-
term
standard
deviations
An individual short-term standard deviation will not be a
reliable estimate of precision if the degrees of freedom is less
than ten, but the individual estimates can be pooled over the K
days to obtain a more reliable estimate. The pooled level-1
standard deviation estimate with v = K(J - 1) degrees of
freedom is
.
This standard deviation can be interpreted as quantifying the
basic precision of the instrumentation used in the measurement
process.
Process
(level-2)
The level-2 standard deviation of the check standard is
appropriate for representing the process variability. It is
2.1.2.3. Analysis
standard
deviation
computed with v = K - 1 degrees of freedom as:
where

is the grand mean of the KJ check standard measurements.
Use in
quality
control
The check standard data and standard deviations that are
described in this section are used for controlling two aspects
of a measurement process:
1. Control of short-term variability
2. Control of bias and long-term variability
Case
study:
Resistivity
check
standard
For an example, see the case study for resistivity where
several check standards were measured J = 6 times per day
over several days.
2.2. Statistical control of a measurement process

The purpose of this section is to outline the steps that can be
taken to exercise statistical control over the measurement
process and demonstrate the validity of the uncertainty
statement. Measurement processes can change both with
respect to bias and variability. A change in instrument
precision may be readily noted as measurements are being
recorded, but changes in bias or long-term variability are
difficult to catch when the process is looking at a multitude of
artifacts over time.
What are the issues for control of a measurement process?
1. Purpose
2. Assumptions
3. Role of the check standard
How are bias and long-term variability controlled?
1. Shewhart control chart
2. Exponentially weighted moving average control chart
3. Data collection and analysis
4. Control procedure
5. Remedial actions & strategies
How is short-term variability controlled?
1. Control chart for standard deviations
3. Control procedure
4. Remedial actions and strategies
2.2.1. What are the issues in controlling the measurement process?

2.2.1. What are the issues in controlling the
measurement process?
Purpose is to
guarantee
the
'goodness' of
measurement
results
The purpose of statistical control is to guarantee the
'goodness' of measurement results within predictable limits
and to validate the statement of uncertainty of the
measurement result.
Statistical control methods can be used to test the
measurement process for change with respect to bias and
variability from its historical levels. However, if the
measurement process is improperly specified or calibrated,
then the control procedures can only guarantee
comparability among measurements.
Assumption
of normality
is not
stringent
The assumptions that relate to measurement processes apply
to statistical control; namely that the errors of measurement
are uncorrelated over time and come from a population with
a single distribution. The tests for control depend on the
assumption that the underlying distribution is normal
(Gaussian), but the test procedures are robust to slight
departures from normality. Practically speaking, all that is
required is that the distribution of measurements be bell-
shaped and symmetric.
Check
standard is
mechanism
for
controlling
the process
Measurements on a check standard provide the mechanism
for controlling the measurement process.
Measurements on the check standard should produce
identical results except for the effect of random errors, and
tests for control are basically tests of whether or not the
random errors from the process continue to be drawn from
the same statistical distribution as the historical data on the
check standard.
Changes that can be monitored and tested with the check
standard database are:
1. Changes in bias and long-term variability
2. Changes in instrument precision or short-term
variability
2.2.1. What are the issues in controlling the measurement process?
2.2.2. How are bias and variability controlled?

Bias and
variability
are controlled
by monitoring
measurements
on a check
standard over
time
Bias and long-term variability are controlled by monitoring
measurements on a check standard over time. A change in the
measurement on the check standard that persists at a constant
level over several measurement sequences indicates possible:
1. Change or damage to the reference standards
2. Change or damage to the check standard artifact
3. Procedural change that vitiates the assumptions of the
measurement process
A change in the variability of the measurements on the check
standard can be due to one of many causes such as:
1. Loss of environmental controls
2. Change in handling techniques
3. Severe degradation in instrumentation.
The control procedure monitors the progress of measurements on
the check standard over time and signals when a significant
change occurs. There are two control chart procedures that are
suitable for this purpose.
Shewhart
Chart is easy
to implement
The Shewhart control chart has the advantage of being intuitive
and easy to implement. It is characterized by a center line and
symmetric upper and lower control limits. The chart is good for
detecting large changes but not for quickly detecting small
changes (of the order of one-half to one standard deviation) in the
process.
Depiction of
Shewhart
control chart
In the simplistic illustration of a Shewhart control chart shown
below, the measurements are within the control limits with the
exception of one measurement which exceeds the upper control
limit.
EWMA Chart
is better for
detecting
small changes
The EWMA control chart (exponentially weighted moving
average) is more difficult to implement but should be considered
if the goal is quick detection of small changes. The decision
process for the EWMA chart is based on an exponentially
decreasing (over time) function of prior measurements on the
check standard while the decision process for the Shewhart chart
is based on the current measurement only.
Example of
EWMA Chart
In the EWMA control chart below, the red dots represent the
measurements. Control is exercised via the exponentially weighted
moving average (shown as the curved line) which, in this case, is
approaching its upper control limit.
Artifacts for
process
control must
be stable and
available
Case study:
Resistivity
The check standard artifacts for controlling the bias or long-term
variability of the process must be of the same type and geometry
as items that are measured in the workload. The artifacts must be
stable and available to the measurement process on a continuing
basis. Usually, one artifact is sufficient. It can be:
1. An individual item drawn at random from the workload
2. A specific item reserved by the laboratory for the purpose.
Topic covered
in this
section>
The topics covered in this section include:
1. Shewhart control chart methodology
2. EWMA control chart methodology
3. Data collection & analysis
4. Monitoring
5. Remedies and strategies for dealing with out-of-control
signals.
2.2.2.1. Shewhart control chart

Example of
Shewhart
control chart
for mass
calibrations
The Shewhart control chart has a baseline and upper and
lower limits, shown as dashed lines, that are symmetric
about the baseline. Measurements are plotted on the chart
versus a time line. Measurements that are outside the limits
are considered to be out of control.
Baseline is
the average
from
historical
data
The baseline for the control chart is the accepted value, an
average of the historical check standard values. A
minimum of 100 check standard values is required to
establish an accepted value.
Caution -
control limits
are computed
from the
process
standard
deviation --
not from
rational
subsets
The upper (UCL) and lower (LCL) control limits are:
UCL = Accepted value + k*process standard
deviation
LCL = Accepted value - k*process standard
deviation
where the process standard deviation is the standard
deviation computed from the check standard database.
Individual
measurements
cannot be
assessed
using the
standard
deviation
from short-
term
repetitions
This procedure is an individual observations control chart.
The previously described control charts depended on
rational subsets, which use the standard deviations
computed from the rational subsets to calculate the control
limits. For a measurement process, the subgroups would
consist of short-term repetitions which can characterize the
precision of the instrument but not the long-term variability
of the process. In measurement science, the interest is in
assessing individual measurements (or averages of short-
term repetitions). Thus, the standard deviation over time is
the appropriate measure of variability.
Choice of k
depends on
number of
measurements
we are
willing to
To achieve tight control of the measurement process, set
k = 2
in which case approximately 5% of the measurements from
a process that is in control will produce out-of-control
reject
signals. This assumes that there is a sufficiently large
number of degrees of freedom (>100) for estimating the
process standard deviation.
To flag only those measurements that are egregiously out of
control, set
k = 3
in which case approximately 1% of the measurements from
an in-control process will produce out-of-control signals.
2.2.2.1.1. EWMA control chart

Small
changes only
become
obvious over
time
Because it takes time for the patterns in the data to emerge,
a permanent shift in the process may not immediately cause
individual violations of the control limits on a Shewhart
control chart. The Shewhart control chart is not powerful for
detecting small changes, say of the order of 1 - 1/2 standard
deviations. The EWMA (exponentially weighted moving
average) control chart is better suited to this purpose.
Example of
EWMA
control chart
for mass
calibrations
The exponentially weighted moving average (EWMA) is a
statistic for monitoring the process that averages the data in
a way that gives less and less weight to data as they are
further removed in time from the current measurement. The
data
Y
1
, Y
2
, ... , Y
t
are the check standard measurements ordered in time. The
EWMA statistic at time t is computed recursively from
individual data points, with the first EWMA statistic,
EWMA
1
, being the arithmetic average of historical data.
Control
mechanism
for EWMA
The EWMA control chart can be made sensitive to small
changes or a gradual drift in the process by the choice of the
weighting factor, . A weighting factor of 0.2 - 0.3 is
usually suggested for this purpose (Hunter), and 0.15 is also
a popular choice.
Limits for the
control chart
The target or center line for the control chart is the average
of historical data. The upper (UCL) and lower (LCL) limits
are
where s times the radical expression is a good
approximation to the standard deviation of the EWMA
statistic and the factor k is chosen in the same way as for
the Shewhart control chart -- generally to be 2 or 3.
Procedure
for
implementing
the EWMA
control chart
The implementation of the EWMA control chart is the same
as for any other type of control procedure. The procedure is
built on the assumption that the "good" historical data are
representative of the in-control process, with future data
from the same process tested for agreement with the
historical data. To start the procedure, a target (average) and
process standard deviation are computed from historical
check standard data. Then the procedure enters the
monitoring stage with the EWMA statistics computed and
tested against the control limits. The EWMA statistics are
weighted averages, and thus their standard deviations are
smaller than the standard deviations of the raw data and the
corresponding control limits are narrower than the control
limits for the Shewhart individual observations chart.

Measurements
should cover
a sufficiently
long time
period to
cover all
environmental
conditions
A schedule should be set up for making measurements on the artifact
(check standard) chosen for control purposes. The measurements are
structured to sample all environmental conditions in the laboratory and all
other sources of influence on the measurement result, such as operators
and instruments.
For high-precision processes where the uncertainty of the result must be
guaranteed, a measurement on the check standard should be included
with every measurement sequence, if possible, and at least once a day.
For each occasion, J measurements are made on the check standard. If
there is no interest in controlling the short-term variability or precision of
the instrument, then one measurement is sufficient. However, a dual
purpose is served by making two or three measurements that track both
the bias and the short-term variability of the process with the same
database.
Depiction of
check
standard
measurements
with J = 4
repetitions
per day on the
surface of a
silicon wafer
over K days
where the
repetitions
are
randomized
over position
on the wafer
2-level design for measurements on a check standard
Notation For J measurements on each of K days, the measurements are denoted by
The check
standard
value is
The check standard value for the kth day is
defined as an
average of
short-term
repetitions
Accepted
value of check
standard
The accepted value, or baseline for the control chart, is
Process
standard
deviation
The process standard deviation is
Caution Check standard measurements should be structured in the same way as
values reported on the test items. For example, if the reported values are
averages of two measurements made within 5 minutes of each other, the
check standard values should be averages of the two measurements made
in the same manner.
Database
Case study:
Resistivity
Averages and short-term standard deviations computed from J repetitions
should be recorded in a file along with identifications for all significant
factors. The best way to record this information is to use one file with
one line (row in a spreadsheet) of information in fixed fields for each
group. A list of typical entries follows:
1. Month
2. Day
3. Year
4. Check standard identification
5. Identification for the measurement design (if applicable)
6. Instrument identification
8. Repeatability (short-term) standard deviation from J repetitions
2.2.2.3. Monitoring bias and long-term variability

Monitoring
stage
Once the baseline and control limits for the control chart have been determined from
historical data, and any bad observations removed and the control limits recomputed, the
measurement process enters the monitoring stage. A Shewhart control chart and EWMA
control chart for monitoring a mass calibration process are shown below. For the
purpose of comparing the two techniques, the two control charts are based on the same
data where the baseline and control limits are computed from the data taken prior to
1985. The monitoring stage begins at the start of 1985. Similarly, the control limits for
both charts are 3-standard deviation limits. The check standard data and analysis are
explained more fully in another section.
Shewhart
control chart
of
measurements
of kilogram
check
standard
showing
outliers and a
shift in the
process that
occurred after
1985
EWMA chart
for
measurements
on kilogram
check
standard
In the EWMA control chart below, the control data after 1985 are shown in green, and
the EWMA statistics are shown as black dots superimposed on the raw data. The
EWMA statistics, and not the raw data, are of interest in looking for out-of-control
signals. Because the EWMA statistic is a weighted average, it has a smaller standard
deviation than a single control measurement, and, therefore, the EWMA control limits
are narrower than the limits for the Shewhart control chart shown above.
showing
multiple
violations of
the control
limits for the
EWMA
statistics
Measurements
that exceed
the control
limits require
action
The control strategy is based on the predictability of future measurements from
historical data. Each new check standard measurement is plotted on the control chart in
real time. These values are expected to fall within the control limits if the process has
not changed. Measurements that exceed the control limits are probably out-of-control
and require remedial action. Possible causes of out-of-control signals need to be
understood when developing strategies for dealing with outliers.
Signs of
significant
trends or
shifts
The control chart should be viewed in its entirety on a regular basis] to identify drift or
shift in the process. In the Shewhart control chart shown above, only a few points
exceed the control limits. The small, but significant, shift in the process that occurred
after 1985 can only be identified by examining the plot of control measurements over
time. A re-analysis of the kilogram check standard data shows that the control limits for
the Shewhart control chart should be updated based on the the data after 1985. In the
EWMA control chart, multiple violations of the control limits occur after 1986. In the
calibration environment, the incidence of several violations should alert the control
engineer that a shift in the process has occurred, possibly because of damage or change
in the value of a reference standard, and the process requires review.
2.2.2.4. Remedial actions

Consider
possible
causes for
out-of-
control
signals and
take
corrective
long-term
actions
There are many possible causes of out-of-control signals.
A. Causes that do not warrant corrective action for the
process (but which do require that the current measurement
be discarded) are:
1. Chance failure where the process is actually in-
control
2. Glitch in setting up or operating the measurement
process
3. Error in recording of data
B. Changes in bias can be due to:
1. Damage to artifacts
2. Degradation in artifacts (wear or build-up of dirt and
mineral deposits)
C. Changes in long-term variability can be due to:
1. Degradation in the instrumentation
2. Changes in environmental conditions
3. Effect of a new or inexperienced operator
4-step
strategy for
short-term
An immediate strategy for dealing with out-of-control
signals associated with high precision measurement
processes should be pursued as follows:
Repeat
measurements
1. Repeat the measurement sequence to establish
whether or not the out-of-control signal was simply a
chance occurrence, glitch, or whether it flagged a
permanent change or trend in the process.
Discard
measurements
on test items
2. With high precision processes, for which a check
standard is measured along with the test items, new
values should be assigned to the test items based on
new measurement data.
Check for 3. Examine the patterns of recent data. If the process is
drift gradually drifting out of control because of
degradation in instrumentation or artifacts, then:
Instruments may need to be repaired
Reference artifacts may need to be
recalibrated.
Reevaluate 4. Reestablish the process value and control limits from
more recent data if the measurement process cannot
be brought back into control.
2.2.3. How is short-term variability controlled?

Emphasis
on
instruments
Short-term variability or instrument precision is controlled by
monitoring standard deviations from repeated measurements
on the instrument(s) of interest. The database can come from
measurements on a single artifact or a representative set of
artifacts.
Artifacts -
Case
study:
Resistivity
The artifacts must be of the same type and geometry as items
that are measured in the workload, such as:
1. Items from the workload
2. A single check standard chosen for this purpose
3. A collection of artifacts set aside for this specific
purpose
Concepts
covered in
this section
The concepts that are covered in this section include:
1. Control chart methodology for standard deviations
3. Monitoring
4. Remedies and strategies for dealing with out-of-control
signals
2.2.3.1. Control chart for standard deviations

Degradation
of
instrument
or
anomalous
behavior on
one
occasion
Changes in the precision of the instrument, particularly
anomalies and degradation, must be addressed. Changes in
precision can be detected by a statistical control procedure
based on the F-distribution where the short-term standard
deviations are plotted on the control chart.
The base line for this type of control chart is the pooled
standard deviation, s
1
, as defined in Data collection and
analysis.
Example of
control
chart for a
mass
balance
Only the upper control limit, UCL, is of interest for detecting
degradation in the instrument. As long as the short-term
standard deviations fall within the upper control limit
established from historical data, there is reason for
confidence that the precision of the instrument has not
degraded (i.e., common cause variations).
The control
limit is
based on the
F-
distribution
The control limit is
where the quantity under the radical is the upper critical
value from the F table with degrees of freedom (J - 1) and
K(J - 1). The numerator degrees of freedom, v1 = (J -1), are
associated with the standard deviation computed from the
current measurements, and the denominator degrees of
freedom, v2 = K(J -1), correspond to the pooled standard
deviation of the historical data. The probability is chosen
to be small, say 0.05.
The justification for this control limit, as opposed to the
more conventional standard deviation control limit, is that we
are essentially performing the following hypothesis test:
H
0
:
1
=
2

H
a
:
2
>
1
where
1
is the population value for the s
1
defined above
and
2
is the population value for the standard deviation of
the current values being tested. Generally, s
1
is based on
sufficient historical data that it is reasonable to make the
assumption that
1
is a "known" value.
The upper control limit above is then derived based on the
standard F test for equal standard deviations. Justification
and details of this derivation are given in Cameron and
Hailes (1974).
Sample
Code
Sample code for computing the F value for the case where
= 0.05, J = 6, and K = 6, is available for both Dataplot and
R.

Case
study:
Resistivity
A schedule should be set up for making measurements with a
single instrument (once a day, twice a week, or whatever is
appropriate for sampling all conditions of measurement).
Short-term
standard
deviations
The measurements are denoted

where there are J measurements on each of K occasions. The
average for the kth occasion is:

The short-term (repeatability) standard deviation for the kth
occasion is:

with (J-1) degrees of freedom.
Pooled
standard
deviation
The repeatability standard deviations are pooled over the K
occasions to obtain an estimate with K(J - 1) degrees of
freedom of the level-1 standard deviation
Note: The same notation is used for the repeatability standard
deviation whether it is based on one set of measurements or
pooled over several sets.
Database The individual short-term standard deviations along with
identifications for all significant factors are recorded in a file.
The best way to record this information is by using one file
with one line (row in a spreadsheet) of information in fixed
fields for each group. A list of typical entries follows.
1. Identification of test item or check standard
2. Date
3. Short-term standard deviation
5. Instrument
6. Operator
2.2.3.3. Monitoring short-term precision

Monitoring
future
precision
Once the base line and control limit for the control chart have been determined
from historical data, the measurement process enters the monitoring stage. In
the control chart shown below, the control limit is based on the data taken prior
to 1985.
Each new
standard
deviation is
monitored on
the control
chart
Each new short-term standard deviation based on J measurements is plotted on
the control chart; points that exceed the control limits probably indicate lack of
statistical control. Drift over time indicates degradation of the instrument.
Points out of control require remedial action, and possible causes of out of
control signals need to be understood when developing strategies for dealing
with outliers.
Control chart
for precision
for a mass
balance from
historical
standard
deviations for
the balance
with 3
degrees of
freedom each.
The control
chart
identifies two
outliers and
slight
degradation
over time in
the precision
of the balance
TIME IN YEARS
Monitoring
where the
number of
There is no requirement that future standard deviations be based on J, the
number of measurements in the historical database. However, a change in the
number of measurements leads to a change in the test for control, and it may
measurements
are different
from J
not be convenient to draw a control chart where the control limits are changing
with each new measurement sequence.
For a new standard deviation based on J' measurements, the precision of the
instrument is in control if
.
Notice that the numerator degrees of freedom, v1 = J'- 1, changes but the
denominator degrees of freedom, v2 = K(J - 1), remains the same.

Examine
possible
causes
A. Causes that do not warrant corrective action (but which
do require that the current measurement be discarded) are:
1. Chance failure where the precision is actually in
control
2. Glitch in setting up or operating the measurement
process
3. Error in recording of data
B. Changes in instrument performance can be due to:
1. Degradation in electronics or mechanical components
2. Changes in environmental conditions
3. Effect of a new or inexperienced operator
Repeat
measurements
Repeat the measurement sequence to establish whether or
not the out-of-control signal was simply a chance
occurrence, glitch, or whether it flagged a permanent
change or trend in the process.
Assign new
value to test
item
With high precision processes, for which the uncertainty
must be guaranteed, new values should be assigned to the
test items based on new measurement data.
Check for
degradation
Examine the patterns of recent standard deviations. If the
process is gradually drifting out of control because of
degradation in instrumentation or artifacts, instruments may
need to be repaired or replaced.
2.3. Calibration

2.3. Calibration
The purpose of this section is to outline the procedures for
calibrating artifacts and instruments while guaranteeing the
'goodness' of the calibration results. Calibration is a
measurement process that assigns values to the property of an
artifact or to the response of an instrument relative to
reference standards or to a designated measurement process.
The purpose of calibration is to eliminate or reduce bias in the
user's measurement system relative to the reference base. The
calibration procedure compares an "unknown" or test item(s)
or instrument with reference standards according to a specific
algorithm.
What are the issues for calibration?
1. Artifact or instrument calibration
2. Reference base
3. Reference standard(s)
What is artifact (single-point) calibration?
1. Purpose
2. Assumptions
3. Bias
4. Calibration model
What are calibration designs?
1. Purpose
2. Assumptions
3. Properties of designs
4. Restraint
5. Check standard in a design
6. Special types of bias (left-right effect & linear drift)
7. Solutions to calibration designs
8. Uncertainty of calibrated values
Catalog of calibration designs
1. Mass weights
2. Gage blocks
3. Electrical standards - saturated standard cells, zeners,
resistors
4. Roundness standards
5. Angle blocks
2.3. Calibration
6. Indexing tables
7. Humidity cylinders
Control of artifact calibration
1. Control of the precision of the calibrating instrument
What is instrument calibration over a regime?
1. Models for instrument calibration
2. Data collection
3. Assumptions
4. What can go wrong with the calibration procedure?
5. Data analysis and model validation
6. Calibration of future measurements
7. Uncertainties of calibrated values
1. From propagation of error for a quadratic
calibration
2. From check standard measurements for a linear
calibration
3. Comparison of check standard technique and
propagation of error
Control of instrument calibration
1. Control chart for linear calibration
2. Critical values of t* statistic
2.3.1. Issues in calibration

2.3. Calibration
Calibration
reduces
bias
Calibration is a measurement process that assigns values to
the property of an artifact or to the response of an instrument
relative to reference standards or to a designated measurement
process. The purpose of calibration is to eliminate or reduce
bias in the user's measurement system relative to the reference
base.
Artifact &
instrument
calibration
The calibration procedure compares an "unknown" or test
item(s) or instrument with reference standards according to a
specific algorithm. Two general types of calibration are
considered in this Handbook:
artifact calibration at a single point
instrument calibration over a regime
Types of
calibration
not
discussed
The procedures in this Handbook are appropriate for
calibrations at secondary or lower levels of the traceability
chain where reference standards for the unit already exist.
Calibration from first principles of physics and reciprocity
calibration are not discussed.

2.3. Calibration
Ultimate
authority
The most critical element of any measurement process is the
relationship between a single measurement and the
reference base for the unit of measurement. The reference
base is the ultimate source of authority for the measurement
unit.
Base and
derived units
of
measurement
The base units of measurement in the Le Systeme
International d'Unites (SI) are (Taylor):
kilogram - mass
meter - length
second - time
ampere - electric current
kelvin - thermodynamic temperature
mole - amount of substance
candela - luminous intensity
These units are maintained by the Bureau International des
Poids et Mesures in Paris. Local reference bases for these
units and SI derived units such as:
pascal - pressure
newton - force
hertz - frequency
ohm - resistance
degrees Celsius - Celsius temperature, etc.
are maintained by national and regional standards
laboratories.
Other
sources
Consensus values from interlaboratory tests or
instrumentation/standards as maintained in specific
environments may serve as reference bases for other units of
measurement.
2.3.1.2. Reference standards

2.3. Calibration
2.3.1.2. Reference standards
Primary
reference
standards
A reference standard for a unit of measurement is an artifact
that embodies the quantity of interest in a way that ties its
value to the reference base.
At the highest level, a primary reference standard is assigned a
value by direct comparison with the reference base. Mass is
the only unit of measurement that is defined by an artifact. The
kilogram is defined as the mass of a platinum-iridium
kilogram that is maintained by the Bureau International des
Poids et Mesures in Sevres, France.
Primary reference standards for other units come from
realizations of the units embodied in artifact standards. For
example, the reference base for length is the meter which is
defined as the length of the path by light in vacuum during a
time interval of 1/299,792,458 of a second.
Secondary
reference
standards
Secondary reference standards are calibrated by comparing
with primary standards using a high precision comparator and
making appropriate corrections for non-ideal conditions of
measurement.
Secondary reference standards for mass are stainless steel
kilograms, which are calibrated by comparing with a primary
standard on a high precision balance and correcting for the
buoyancy of air. In turn these weights become the reference
standards for assigning values to test weights.
Secondary reference standards for length are gage blocks,
which are calibrated by comparing with primary gage block
standards on a mechanical comparator and correcting for
temperature. In turn, these gage blocks become the reference
standards for assigning values to test sets of gage blocks.
2.3.2. What is artifact (single-point) calibration?

2.3. Calibration
Purpose Artifact calibration is a measurement process that assigns
values to the property of an artifact relative to a reference
standard(s). The purpose of calibration is to eliminate or
reduce bias in the user's measurement system relative to the
reference base.
The calibration procedure compares an "unknown" or test
item(s) with a reference standard(s) of the same nominal
value (hence, the term single-point calibration) according to
a specific algorithm called a calibration design.
Assumptions The calibration procedure is based on the assumption that
individual readings on test items and reference standards are
subject to:
Bias that is a function of the measuring system or
instrument
Random error that may be uncontrollable
What is
bias?
The operational definition of bias is that it is the difference
between values that would be assigned to an artifact by the
client laboratory and the laboratory maintaining the reference
standards. Values, in this sense, are understood to be the
long-term averages that would be achieved in both
laboratories.
Calibration
model for
eliminating
bias
requires a
reference
standard
that is very
close in
value to the
test item
One approach to eliminating bias is to select a reference
standard that is almost identical to the test item; measure the
two artifacts with a comparator type of instrument; and take
the difference of the two measurements to cancel the bias.
The only requirement on the instrument is that it be linear
over the small range needed for the two artifacts.
The test item has value X*, as yet to be assigned, and the
reference standard has an assigned value R*. Given a
measurement, X, on the test item and a measurement, R, on
the reference standard,
,
the difference between the test item and the reference is
estimated by
,
and the value of the test item is reported as
.
Need for
redundancy
leads to
calibration
designs
A deficiency in relying on a single difference to estimate D
is that there is no way of assessing the effect of random
errors. The obvious solution is to:
Repeat the calibration measurements J times
Average the results
Compute a standard deviation from the J results
Schedules of redundant intercomparisons involving
measurements on several reference standards and test items
in a connected sequence are called calibration designs and
are discussed in later sections.
2.3.3. What are calibration designs?

2.3. Calibration
Calibration
designs are
redundant
schemes for
intercomparing
reference
standards and
test items
Calibration designs are redundant schemes for
intercomparing reference standards and test items in such
a way that the values can be assigned to the test items
based on known values of reference standards. Artifacts
that traditionally have been calibrated using calibration
designs are:
mass weights
resistors
voltage standards
length standards
angle blocks
indexing tables
liquid-in-glass thermometers, etc.
Outline of
section
The topics covered in this section are:
Designs for elimination of left-right bias and linear
drift
Solutions to calibration designs
Uncertainties of calibrated values
A catalog of calibration designs is provided in the next
section.
Assumptions
for calibration
designs include
demands on
the quality of
the artifacts
The assumptions that are necessary for working with
calibration designs are that:
Random errors associated with the measurements
are independent.
All measurements come from a distribution with the
same standard deviation.
Reference standards and test items respond to the
measuring environment in the same manner.
Handling procedures are consistent from item to
item.
Reference standards and test items are stable during
the time of measurement.
Bias is canceled by taking the difference between
measurements on the test item and the reference
standard.
Important
concept -
Restraint
The restraint is the known value of the reference standard
or, for designs with two or more reference standards, the
restraint is the summation of the values of the reference
standards.
Requirements
& properties of
designs
Basic requirements are:
The differences must be nominally zero.
The design must be solvable for individual items
given the restraint.
It is possible to construct designs which do not have these
properties. This will happen, for example, if reference
standards are only compared among themselves and test
items are only compared among themselves without any
intercomparisons.
Practical
considerations
determine a
'good' design
We do not apply 'optimality' criteria in constructing
calibration designs because the construction of a 'good'
design depends on many factors, such as convenience in
manipulating the test items, time, expense, and the
maximum load of the instrument.
The number of measurements should be small.
The degrees of freedom should be greater than
three.
The standard deviations of the estimates for the test
items should be small enough for their intended
purpose.
Check
standard in a
design
Designs listed in this Handbook have provision for a
check standard in each series of measurements. The check
standard is usually an artifact, of the same nominal size,
type, and quality as the items to be calibrated. Check
standards are used for:
Controlling the calibration process
Quantifying the uncertainty of calibrated results
Estimates that
can be
computed from
a design
Calibration designs are solved by a restrained least-
squares technique (Zelen) which gives the following
estimates:
Values for individual reference standards
Values for individual test items
Value for the check standard
Repeatability standard deviation and degrees of
freedom
Standard deviations associated with values for
reference standards and test items
2.3.3.1. Elimination of special types of bias

2.3. Calibration
Assumptions
which may
be violated
Two of the usual assumptions relating to calibration
measurements are not always valid and result in biases.
Bias is canceled by taking the difference between the
measurement on the test item and the measurement on
the reference standard
Reference standards and test items remain stable
throughout the measurement sequence
Ideal
situation
In the ideal situation, bias is eliminated by taking the
difference between a measurement X on the test item and a
measurement R on the reference standard. However, there are
situations where the ideal is not satisfied:
Left-right (or constant instrument) bias
Bias caused by instrument drift
2.3.3.1.1. Left-right (constant instrument) bias

2.3. Calibration
Left-right
bias which is
not
eliminated by
differencing
A situation can exist in which a bias, P, which is constant
and independent of the direction of measurement, is
introduced by the measurement instrument itself. This type
of bias, which has been observed in measurements of
standard voltage cells (Eicke & Cameron) and is not
eliminated by reversing the direction of the current, is
shown in the following equations.
Elimination
of left-right
bias requires
two
measurements
in reverse
direction
The difference between the test and the reference can be
estimated without bias only by taking the difference
between the two measurements shown above where P
cancels in the differencing so that
.
The value of
the test item
depends on
the known
value of the
reference
standard, R*
The test item, X, can then be estimated without bias by
and P can be estimated by
.
Calibration
designs that
are left-right
balanced
This type of scheme is called left-right balanced and the
principle is extended to create a catalog of left-right
balanced designs for intercomparing reference standards
among themselves. These designs are appropriate ONLY for
comparing reference standards in the same environment, or
enclosure, and are not appropriate for comparing, say,
across standard voltage cells in two boxes.
1. Left-right balanced design for a group of 3 artifacts
2.3.3.1.2. Bias caused by instrument drift

2.3. Calibration
Bias caused
by linear drift
over the time
of
measurement
The requirement that reference standards and test items be
stable during the time of measurement cannot always be
met because of changes in temperature caused by body
heat, handling, etc.
Representation
of linear drift
Linear drift for an even number of measurements is
represented by
..., -5d, -3d, -1d, +1d, +3d, +5d, ...
and for an odd number of measurements by
..., -3d, -2d, -1d, 0d, +1d, +2d, +3d, ... .
Assumptions
for drift
elimination
The effect can be mitigated by a drift-elimination scheme
(Cameron/Hailes) which assumes:
Linear drift over time
Equally spaced measurements in time
Example of
drift-
elimination
scheme
An example is given by substitution weighing where scale
deflections on a balance are observed for X, a test weight,
and R, a reference weight.
Estimates of
drift-free
difference and
size of drift
The drift-free difference between the test and the reference
is estimated by
and the size of the drift is estimated by
Calibration
designs for
eliminating
linear drift
This principle is extended to create a catalog of drift-
elimination designs for multiple reference standards and
test items. These designs are listed under calibration
designs for gauge blocks because they have traditionally
been used to counteract the effect of temperature build-up
in the comparator during calibration.
2.3.3.2. Solutions to calibration designs

2.3. Calibration
Solutions for
designs listed
in the catalog
Solutions for all designs that are cataloged in this Handbook are included
with the designs. Solutions for other designs can be computed from the
instructions on the following page given some familiarity with matrices.
Measurements
for the 1,1,1
design
The use of the tables shown in the catalog are illustrated for three
artifacts; namely, a reference standard with known value R* and a check
standard and a test item with unknown values. All artifacts are of the
same nominal size. The design is referred to as a 1,1,1 design for
n = 3 difference measurements
m = 3 artifacts
Convention
for showing
the
measurement
sequence and
identifying the
reference and
check
standards
The convention for showing the measurement sequence is shown below.
Nominal values are underlined in the first line showing that this design is
appropriate for comparing three items of the same nominal size such as
three one-kilogram weights. The reference standard is the first artifact,
the check standard is the second, and the test item is the third.
1 1 1
Y(1) = + -
Y(2) = + -
Y(3) = + -
Restraint +
Check standard +
Limitation of
this design
This design has degrees of freedom
v = n - m + 1 = 1
Convention
for showing
least-squares
estimates for
individual
items
The table shown below lists the coefficients for finding the estimates for
the individual items. The estimates are computed by taking the cross-
product of the appropriate column for the item of interest with the
column of measurement data and dividing by the divisor shown at the
top of the table.
SOLUTION MATRIX
DIVISOR = 3
OBSERVATIONS 1 1 1
Y(1) 0 -2 -1
Y(2) 0 -1 -2
Y(3) 0 1 -1
R* 3 3 3
Solutions for
individual
items from the
table above
For example, the solution for the reference standard is shown under the
first column; for the check standard under the second column; and for
the test item under the third column. Notice that the estimate for the
reference standard is guaranteed to be R*, regardless of the measurement
results, because of the restraint that is imposed on the design. The
estimates are as follows:
Convention
for showing
standard
deviations for
individual
items and
combinations
of items
The standard deviations are computed from two tables of factors as
shown below. The standard deviations for combinations of items include
appropriate covariance terms.
FACTORS FOR REPEATABILITY STANDARD DEVIATIONS
WT FACTOR
K1 1 1 1
1 0.0000 +
1 0.8165 +
1 0.8165 +
2 1.4142 + +
1 0.8165 +
FACTORS FOR BETWEEN-DAY STANDARD DEVIATIONS
WT FACTOR
K2 1 1 1
1 0.0000 +
1 1.4142 +
1 1.4142 +
2 2.4495 + +
1 1.4142 +

Unifying
equation
The standard deviation for each item is computed using the unifying
equation:
Standard
deviations for
1,1,1 design
from the
For the 1,1,1 design, the standard deviations are:
tables of
factors
Process
standard
deviations
must be
known from
historical
data
In order to apply these equations, we need an estimate of the standard
deviation, s
days
, that describes day-to-day changes in the measurement
process. This standard deviation is in turn derived from the level-2
2
, for the check standard. This standard deviation is
estimated from historical data on the check standard; it can be negligible,
in which case the calculations are simplified.
The repeatability standard deviation s
1
, is estimated from historical data,
usually from data of several designs.
Steps in
computing
standard
deviations
The steps in computing the standard deviation for a test item are:
Compute the repeatability standard deviation from the design or
historical data.
Compute the standard deviation of the check standard from
historical data.
Locate the factors, K
1
and K
2
for the check standard; for the
1,1,1 design the factors are 0.8165 and 1.4142, respectively, where
the check standard entries are last in the tables.
Apply the unifying equation to the check standard to estimate the
standard deviation for days. Notice that the standard deviation of
the check standard is the same as the level-2 standard deviation,
s
2
, that is referred to on some pages. The equation for the between-
days standard deviation from the unifying equation is
.
Thus, for the example above
.
This is the number that is entered into the NIST mass calibration
software as the between-time standard deviation. If you are using
this software, this is the only computation that you need to make
because the standard deviations for the test items are computed
automatically by the software.
If the computation under the radical sign gives a negative number,
set s
days
=0. (This is possible and indicates that there is no
contribution to uncertainty from day-to-day effects.)
For completeness, the computations of the standard deviations for
the test item and for the sum of the test and the check standard
using the appropriate factors are shown below.
2.3.3.2.1. General matrix solutions to calibration designs

2.3. Calibration
2.3.3. Calibration designs
2.3.3.2. General solutions to calibration designs
Requirements Solutions for all designs that are cataloged in this Handbook are included with the
designs. Solutions for other designs can be computed from the instructions below
given some familiarity with matrices. The matrix manipulations that are required for
the calculations are:
transposition (indicated by ')
multiplication
inversion
Notation n = number of difference measurements
m = number of artifacts
(n - m + 1) = degrees of freedom
X= (nxm) design matrix
r'= (mx1) vector identifying the restraint
= (mx1) vector identifying ith item of interest consisting of a 1 in the ith
position and zeros elsewhere
R*= value of the reference standard
Y= (mx1) vector of observed difference measurements
Convention
for showing
the
measurement
sequence
The convention for showing the measurement sequence is illustrated with the three
measurements that make up a 1,1,1 design for 1 reference standard, 1 check
standard, and 1 test item. Nominal values are underlined in the first line .
1 1 1
Y(1) = + -
Y(2) = + -
Y(3) = + -
Matrix
algebra for
solving a
design
The (mxn) design matrix X is constructed by replacing the pluses (+), minues (-)
and blanks with the entries 1, -1, and 0 respectively.
The (mxm) matrix of normal equations, X'X, is formed and augmented by the
restraint vector to form an (m+1)x(m+1) matrix, A:
Inverse of
design matrix
The A matrix is inverted and shown in the form:
where Q is an mxm matrix that, when multiplied by s
2
, yields the usual variance-
covariance matrix.
Estimates of
values of
individual
artifacts
The least-squares estimates for the values of the individual artifacts are contained in
the (mx1) matrix, B, where
where Q is the upper left element of the A
-1
matrix shown above. The structure of
the individual estimates is contained in the QX' matrix; i.e. the estimate for the ith
item can be computed from XQ and Y by
Cross multiplying the ith column of XQ with Y
And adding R*(nominal test)/(nominal restraint)
Clarify with
an example
We will clarify the above discussion with an example from the mass calibration
process at NIST. In this example, two NIST kilograms are compared with a
customer's unknown kilogram.
The design matrix, X, is
The first two columns represent the two NIST kilograms while the third column
represents the customers kilogram (i.e., the kilogram being calibrated).
The measurements obtained, i.e., the Y matrix, are
The measurements are the differences between two measurements, as specified by
the design matrix, measured in grams. That is, Y(1) is the difference in measurement
between NIST kilogram one and NIST kilogram two, Y(2) is the difference in
measurement between NIST kilogram one and the customer kilogram, and Y(3) is
the difference in measurement between NIST kilogram two and the customer
kilogram.
The value of the reference standard, R
*
, is 0.82329.
Then
If there are three weights with known values for weights one and two, then
r = [ 1 1 0 ]
Thus
and so
From A
-1
, we have
We then compute QX'
We then compute B = QX'Y + h'R
*
This yields the following least-squares coefficient estimates:
Standard
deviations of
estimates
The standard deviation for the ith item is:
where
The process standard deviation, which is a measure of the overall precision of the
(NIST) mass calibrarion process,
is the residual standard deviation from the design, and s
days
is the standard
deviation for days, which can only be estimated from check standard measurements.
Example We continue the example started above. Since n = 3 and m = 3, the formula reduces
to:
Substituting the values shown above for X, Y, and Q results in
and
Y'(I - XQX')Y = 0.0000083333
Finally, taking the square root gives
s
1
= 0.002887
The next step is to compute the standard deviation of item 3 (the customers
kilogram), that is s
item
3
. We start by substitituting the values for X and Q and
computing D
Next, we substitute = [0 0 1] and = 0.02111
2
(this value is taken from a
check standard and not computed from the values given in this example).
We obtain the following computations
and
and
2.3.3.3. Uncertainties of calibrated values

2.3. Calibration
Uncertainty
analysis
follows the
ISO
principles
This section discusses the calculation of uncertainties of
calibrated values from calibration designs. The discussion
follows the guidelines in the section on classifying and
combining components of uncertainty. Two types of
evaluations are covered.
1. type A evaluations of time-dependent sources of
random error
2. type B evaluations of other sources of error
The latter includes, but is not limited to, uncertainties from
sources that are not replicated in the calibration design such
as uncertainties of values assigned to reference standards.
Uncertainties
for test items
Uncertainties associated with calibrated values for test items
from designs require calculations that are specific to the
individual designs. The steps involved are outlined below.
Outline for
the section
on
uncertainty
analysis
Historical perspective
Assumptions
Example of more realistic model
Computation of repeatability standard deviations
Computation of level-2 standard deviations
Combination of repeatability and level-2 standard
deviations
Example of computations for 1,1,1,1 design
Type B uncertainty associated with the restraint
Expanded uncertainty of calibrated values
2.3.3.3.1. Type A evaluations for calibration designs

2.3. Calibration
2.3.3.3.1. Type A evaluations for calibration
designs
Change over
time
Type A evaluations for calibration processes must take into
account changes in the measurement process that occur
over time.
Historically,
uncertainties
considered
only
instrument
imprecision
Historically, computations of uncertainties for calibrated
values have treated the precision of the comparator
instrument as the primary source of random uncertainty in
the result. However, as the precision of instrumentation has
improved, effects of other sources of variability have begun
to show themselves in measurement processes. This is not
universally true, but for many processes, instrument
imprecision (short-term variability) cannot explain all the
variation in the process.
Effects of
environmental
changes
Effects of humidity, temperature, and other environmental
conditions which cannot be closely controlled or corrected
must be considered. These tend to exhibit themselves over
time, say, as between-day effects. The discussion of
between-day (level-2) effects relating to gauge studies
carries over to the calibration setting, but the computations
are not as straightforward.
Assumptions
which are
specific to
this section
The computations in this section depend on specific
assumptions:
1. Short-term effects associated with instrument
response
come from a single distribution
vary randomly from measurement to
measurement within a design.
2. Day-to-day effects
come from a single distribution
vary from artifact to artifact but remain
constant for a single calibration
vary from calibration to calibration
These These assumptions have proved useful for characterizing
assumptions
have proved
useful but
may need to
be expanded
in the future
high precision measurement processes, but more
complicated models may eventually be needed which take
the relative magnitudes of the test items into account. For
example, in mass calibration, a 100 g weight can be
compared with a summation of 50g, 30g and 20 g weights
in a single measurement. A sophisticated model might
consider the size of the effect as relative to the nominal
masses or volumes.
Example of
the two
models for a
design for
calibrating
test item
using 1
reference
standard
To contrast the simple model with the more complicated
model, a measurement of the difference between X, the test
item, with unknown and yet to be determined value, X*,
and a reference standard, R, with known value, R*, and the
reverse measurement are shown below.
Model (1) takes into account only instrument imprecision
so that:
(1)
with the error terms random errors that come from the
imprecision of the measuring instrument.
Model (2) allows for both instrument imprecision and
level-2 effects such that:
(2)
where the delta terms explain small changes in the values
of the artifacts that occur over time. For both models, the
value of the test item is estimated as
Standard
deviations
from both
models
For model (l), the standard deviation of the test item is
For model (2), the standard deviation of the test item is
.
Note on
relative
contributions
of both
components
to uncertainty
In both cases, is the repeatability standard deviation that
describes the precision of the instrument and is the
level-2 standard deviation that describes day-to-day
changes. One thing to notice in the standard deviation for
the test item is the contribution of relative to the total
uncertainty. If is large relative to , or dominates, the
uncertainty will not be appreciably reduced by adding
measurements to the calibration design.
2.3.3.3.2. Repeatability and level-2 standard deviations

2.3. Calibration
2.3.3.3.2. Repeatability and level-2 standard
deviations
Repeatability
standard
deviation
comes from
the data of a
single design
The repeatability standard deviation of the instrument can
be computed in two ways.
1. It can be computed as the residual standard deviation
from the design and should be available as output
from any software package that reduces data from
calibration designs. The matrix equations for this
computation are shown in the section on solutions to
calibration designs. The standard deviation has
degrees of freedom
v = n - m + 1
for n difference measurements and m items.
Typically the degrees of freedom are very small. For
two differences measurements on a reference
standard and test item, the degrees of freedom is
v=1.
A more
reliable
estimate
comes from
pooling over
historical
data
2. A more reliable estimate of the standard deviation
can be computed by pooling variances from K
calibrations (and then taking its square root) using
the same instrument (assuming the instrument is in
statistical control). The formula for the pooled
estimate is
Level-2
standard
deviation is
estimated
from check
standard
measurements
The level-2 standard deviation cannot be estimated from
the data of the calibration design. It cannot generally be
estimated from repeated designs involving the test items.
The best mechanism for capturing the day-to-day effects is
a check standard, which is treated as a test item and
included in each calibration design. Values of the check
standard, estimated over time from the calibration design,
are used to estimate the standard deviation.
Assumptions The check standard value must be stable over time, and the
measurements must be in statistical control for this
procedure to be valid. For this purpose, it is necessary to
keep a historical record of values for a given check
standard, and these values should be kept by instrument
and by design.
Computation
of level-2
standard
deviation
Given K historical check standard values,
the standard deviation of the check standard values is
computed as
where
with degrees of freedom v = K - 1.
2.3.3.3.3. Combination of repeatability and level-2 standard deviations

2.3. Calibration
2.3.3.3.3. Combination of repeatability and
level-2 standard deviations
Standard
deviation
of test item
depends on
several
factors
The final question is how to combine the repeatability
standard deviation and the standard deviation of the check
standard to estimate the standard deviation of the test item.
This computation depends on:
structure of the design
position of the check standard in the design
position of the reference standards in the design
position of the test item in the design
Derivations
require
matrix
algebra
Tables for estimating standard deviations for all test items are
reported along with the solutions for all designs in the catalog.
The use of the tables for estimating the standard deviations
for test items is illustrated for the 1,1,1,1 design. Matrix
equations can be used for deriving estimates for designs that
are not in the catalog.
The check standard for each design is either an additional test
item in the design, other than the test items that are submitted
for calibration, or it is a construction, such as the difference
between two reference standards as estimated by the design.
2.3.3.3.4. Calculation of standard deviations for 1,1,1,1 design

2.3. Calibration
2.3.3.3.4. Calculation of standard deviations for 1,1,1,1
design
Design with
2 reference
standards
and 2 test
items
An example is shown below for a 1,1,1,1 design for two reference standards,
R
1
and R
2
, and two test items, X
1
and X
2
, and six difference measurements.
The restraint, R*, is the sum of values of the two reference standards, and the
check standard, which is independent of the restraint, is the difference
between the values of the reference standards. The design and its solution are
reproduced below.
Check
standard is
the
difference
between the
2 reference
standards
OBSERVATIONS 1 1 1 1
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) + -
Y(5) + -
Y(6) + -
RESTRAINT + +

CHECK STANDARD + -

DEGREES OF FREEDOM = 3
SOLUTION MATRIX
DIVISOR = 8
Y(1) 2 -2 0 0
Y(2) 1 -1 -3 -1
Y(3) 1 -1 -1 -3
Y(4) -1 1 -3 -1
Y(5) -1 1 -1 -3
Y(6) 0 0 2 -2
R* 4 4 4 4
Explanation
of solution
matrix
The solution matrix gives values for the test items of
Factors for
computing
contributions
of
repeatability
and level-2
standard
deviations to
uncertainty
WT FACTOR
K
1
1 1 1 1
1 0.3536 +
1 0.3536 +
1 0.6124 +
1 0.6124 +
0 0.7071 + -

FACTORS FOR LEVEL-2 STANDARD DEVIATIONS
WT FACTOR
K
2
1 1 1 1
1 0.7071 +
1 0.7071 +
1 1.2247 +
1 1.2247 +
0 1.4141 + -
The first table shows factors for computing the contribution of the
repeatability standard deviation to the total uncertainty. The second table
shows factors for computing the contribution of the between-day standard
deviation to the uncertainty. Notice that the check standard is the last entry in
each table.
Unifying
equation
The unifying equation is:
Standard
deviations
are
computed
using the
factors from
the tables
with the
unifying
equation
The steps in computing the standard deviation for a test item are:
Compute the repeatability standard deviation from historical data.
Compute the standard deviation of the check standard from historical
data.
1
and K
2
, for the check standard.
Compute the between-day variance (using the unifying equation for
the check standard). For this example,
.
If this variance estimate is negative, set = 0. (This is possible
and indicates that there is no contribution to uncertainty from day-to-
day effects.)
1
and K
2
, for the test items, and compute the
standard deviations using the unifying equation. For this example,
and
2.3.3.3.5. Type B uncertainty

2.3. Calibration
2.3.3.3.5. Type B uncertainty
Type B
uncertainty
associated
with the
restraint
The reference standard is assumed to have known value, R*,
for the purpose of solving the calibration design. For the
purpose of computing a standard uncertainty, it has a type B
uncertainty that contributes to the uncertainty of the test item.
The value of R* comes from a higher-level calibration
laboratory or process, and its value is usually reported along
with its uncertainty, U. If the laboratory also reports the k
factor for computing U, then the standard deviation of the
restraint is
If k is not reported, then a conservative way of proceeding is
to assume k = 2.
Situation
where the
test is
different in
size from
the
reference
Usually, a reference standard and test item are of the same
nominal size and the calibration relies on measuring the small
difference between the two; for example, the intercomparison
of a reference kilogram compared with a test kilogram. The
calibration may also consist of an intercomparison of the
reference with a summation of artifacts where the summation
is of the same nominal size as the reference; for example, a
reference kilogram compared with 500 g + 300 g + 200 g test
weights.
Type B
uncertainty
for the test
artifact
The type B uncertainty that accrues to the test artifact from the
uncertainty of the reference standard is proportional to their
nominal sizes; i.e.,
2.3.3.3.6. Expanded uncertainties

2.3. Calibration
Standard
uncertainty
The standard uncertainty for the test item is
Expanded
uncertainty
The expanded uncertainty is computed as
where k is either the critical value from the t table for degrees of freedom v
or k is set equal to 2.
Problem of
the degrees of
freedom
The calculation of degrees of freedom, v, can be a problem. Sometimes it
can be computed using the Welch-Satterthwaite approximation and the
structure of the uncertainty of the test item. Degrees of freedom for the
standard deviation of the restraint is assumed to be infinite. The coefficients
in the Welch-Satterthwaite formula must all be positive for the
approximation to be reliable.
Standard
deviation for
test item from
the 1,1,1,1
design
For the 1,1,1,1 design, the standard deviation of the test items can be
rewritten by substituting in the equation
so that the degrees of freedom depends only on the degrees of freedom in
the standard deviation of the check standard. This device may not work
satisfactorily for all designs.
Standard
uncertainty
from the
1,1,1,1 design
To complete the calculation shown in the equation at the top of the page,
the nominal value of the test item (which is equal to 1) is divided by the
nominal value of the restraint (which is also equal to 1), and the result is
squared. Thus, the standard uncertainty is
Degrees of Therefore, the degrees of freedom is approximated as
freedom using
the Welch-
Satterthwaite
approximation
where n - 1 is the degrees of freedom associated with the check standard
uncertainty. Notice that the standard deviation of the restraint drops out of
the calculation because of an infinite degrees of freedom.
2.3.4. Catalog of calibration designs

2.3. Calibration
Important
concept -
Restraint
The designs are constructed for measuring differences
among reference standards and test items, singly or in
combinations. Values for individual standards and test
items can be computed from the design only if the value
(called the restraint = R*) of one or more reference
standards is known. The methodology for constructing and
solving calibration designs is described briefly in matrix
solutions and in more detail in a NIST publication.
(Cameron et al.).
Designs
listed in this
catalog
Designs are listed by traditional subject area although many
of the designs are appropriate generally for
intercomparisons of artifact standards.
Designs for mass weights
Drift-eliminating designs for gage blocks
Left-right balanced designs for electrical standards
Designs for roundness standards
Designs for angle blocks
Drift-eliminating design for thermometers in a bath
Drift-eliminating designs for humidity cylinders
Properties of
designs in
this catalog
Basic requirements are:
1. The differences must be nominally zero.
2. The design must be solvable for individual items
given the restraint.
Other desirable properties are:
1. The number of measurements should be small.
2. The degrees of freedom should be greater than zero.
3. The standard deviations of the estimates for the test
items should be small enough for their intended
purpose.
Information:
Design
Solution
Given
n = number of difference measurements
m = number of artifacts (reference standards + test
items) to be calibrated
Factors for
computing
standard
deviations
the following information is shown for each design:
Design matrix -- (n x m)
Vector that identifies standards in the restraint -- (1 x
m)
Degrees of freedom = (n - m + 1)
Solution matrix for given restraint -- (n x m)
Table of factors for computing standard deviations
Convention
for showing
the
measurement
sequence
Nominal sizes of standards and test items are shown at the
top of the design. Pluses (+) indicate items that are
measured together; and minuses (-) indicate items are not
measured together. The difference measurements are
constructed from the design of pluses and minuses. For
example, a 1,1,1 design for one reference standard and two
test items of the same nominal size with three
measurements is shown below:
1 1 1
Y(1) = + -
Y(2) = + -
Y(3) = + -
Solution
matrix
Example and
interpretation
The cross-product of the column of difference
measurements and R* with a column from the solution
matrix, divided by the named divisor, gives the value for an
individual item. For example,
Solution matrix
Divisor = 3

1 1 1
Y(1) 0 -2 -1
Y(2) 0 -1 -2
Y(3) 0 +1 -1
R* +3 +3 +3
implies that estimates for the restraint and the two test items
are:
Interpretation
of table of
factors
The factors in this table provide information on precision.
The repeatability standard deviation, , is multiplied by the
appropriate factor to obtain the standard deviation for an
individual item or combination of items. For example,

Sum Factor 1 1
1
1 0.0000 +
1 0.8166 +
1 0.8166
+
2 1.4142 +
+
implies that the standard deviations for the estimates are:
2.3.4.1. Mass weights

2.3. Calibration
Tie to
kilogram
reference
standards
Near-accurate mass measurements require a sequence of
designs that relate the masses of individual weights to a
reference kilogram(s) standard ( Jaeger & Davis). Weights
generally come in sets, and an entire set may require several
series to calibrate all the weights in the set.
Example
of weight
set
A 5,3,2,1 weight set would have the following weights:
1000 g
500g, 300g, 200g, 100g
50g, 30g 20g, 10g
5g, 3g, 2g, 1g
0.5g, 0.3g, 0.2g, 0.1g
Depiction
of a design
with three
series for
calibrating
a 5,3,2,1
weight set
with
weights
between 1
kg and 10
g
First
series
using
The calibrations start with a comparison of the one kilogram
test weight with the reference kilograms (see the graphic
above). The 1,1,1,1 design requires two kilogram reference
1,1,1,1
design
standards with known values, R1* and R2*. The fourth
kilogram in this design is actually a summation of the 500,
300, 200 g weights which becomes the restraint in the next
series.
The restraint for the first series is the known average mass of
the reference kilograms,
The design assigns values to all weights including the
individual reference standards. For this design, the check
standard is not an artifact standard but is defined as the
difference between the values assigned to the reference
kilograms by the design; namely,
2nd series
using
5,3,2,1,1,1
design
The second series is a 5,3,2,1,1,1 design where the restraint
over the 500g, 300g and 200g weights comes from the value
assigned to the summation in the first series; i.e.,
The weights assigned values by this series are:
500g, 300g, 200 g and 100g test weights
100 g check standard (2nd 100g weight in the design)
Summation of the 50g, 30g, 20g weights.
Other
starting
points
The calibration sequence can also start with a 1,1,1 design.
This design has the disadvantage that it does not have
provision for a check standard.
Better
choice of
design
A better choice is a 1,1,1,1,1 design which allows for two
reference kilograms and a kilogram check standard which
occupies the 4th position among the weights. This is preferable
to the 1,1,1,1 design but has the disadvantage of requiring the
laboratory to maintain three kilogram standards.
Important
detail
The solutions are only applicable for the restraints as shown.
Designs
for
decreasing
weight sets
1. 1,1,1 design
2. 1,1,1,1 design
3. 1,1,1,1,1 design
4. 1,1,1,1,1,1 design
5. 2,1,1,1 design
6. 2,2,1,1,1 design
7. 2,2,2,1,1 design
8. 5,2,2,1,1,1 design
9. 5,2,2,1,1,1,1 design
10. 5,3,2,1,1,1 design
11. 5,3,2,1,1,1,1 design
12. 5,3,2,2,1,1,1 design
13. 5,4,4,3,2,2,1,1 design
14. 5,5,2,2,1,1,1,1 design
15. 5,5,3,2,1,1,1 design
16. 1,1,1,1,1,1,1,1 design
17. 3,2,1,1,1 design
Design for
pound
weights
1. 1,2,2,1,1 design
Designs
for
increasing
weight sets
1. 1,1,1 design
2. 1,1,1,1 design
3. 5,3,2,1,1 design
4. 5,3,2,1,1,1 design
5. 5,2,2,1,1,1 design
6. 3,2,1,1,1 design
2.3.4.1.1. Design for 1,1,1

2.3. Calibration
2.3.4.1.1. Design for 1,1,1
Design 1,1,1
OBSERVATIONS 1 1 1
Y(1) + -
Y(2) + -
Y(3) + -
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 3
OBSERVATIONS 1 1 1
Y(1) 0 -2 -1
Y(2) 0 -1 -2
Y(3) 0 1 -1
R* 3 3 3
R* = value of reference weight
WT FACTOR
1 1 1
1 0.0000 +
1 0.8165 +
1 0.8165 +
2 1.4142 + +
1 0.8165 +
WT FACTOR
1 1 1
1 0.0000 +
1 1.4142 +
1 1.4142 +
2 2.4495 + +
1 1.4142 +
Explanation of notation and interpretation of tables
2.3.4.1.2. Design for 1,1,1,1

2.3. Calibration
2.3.4.1.2. Design for 1,1,1,1
Design 1,1,1,1
OBSERVATIONS 1
1 1 1
Y(1) +
-
Y(2) +
-
Y(3) +
-
Y(4)
+ -
Y(5)
+ -
Y(6)
+ -
RESTRAINT +
+
CHECK STANDARD +
-
DEGREES OF
FREEDOM = 3

SOLUTION MATRIX

DIVISOR = 8
OBSERVATIONS
1 1 1
1
Y(1)
2 -2 0
0
Y(2)
1 -1 -3
-1
Y(3)
1 -1 -1
-3
Y(4)
-1 1 -3
-1
Y(5)
-1 1 -1
-3
Y(6)
0 0 2
-2
R*
4 4 4
2.3.4.1.2. Design for 1,1,1,1
4
R* = sum of
two reference
standards
FACTORS FOR
REPEATABILITY
STANDARD
DEVIATIONS
WT FACTOR
K1 1
1 1 1
1 0.3536 +
1 0.3536
+
1 0.6124
+
1 0.6124
+
0 0.7071 +
-

FACTORS FOR
BETWEEN-DAY
STANDARD
DEVIATIONS
WT FACTOR
K2 1
1 1 1
1 0.7071 +
1 0.7071
+
1 1.2247
+
1 1.2247
+
0 1.4141 +
-
Explanation of
notation and
interpretation of
tables
2.3.4.1.3. Design for 1,1,1,1,1

2.3. Calibration
2.3.4.1.3. Design for 1,1,1,1,1
CASE 1: CHECK STANDARD =
DIFFERENCE BETWEEN
FIRST TWO WEIGHTS
1
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) +
-
Y(5) + -
Y(6) + -
Y(7) +
-
Y(8) + -
Y(9) +
-
Y(10) +
-
RESTRAINT + +
CHECK STANDARD + -

SOLUTION MATRIX

DIVISOR = 10
OBSERVATIONS 1 1
1 1 1
Y(1) 2 -2
0 0 0
Y(2) 1 -1
-3 -1 -1
Y(3) 1 -1
-1 -3 -1
Y(4) 1 -1
-1 -1 -3
Y(5) -1 1
-3 -1 -1
Y(6) -1 1
-1 -3 -1
Y(7) -1 1
-1 -1 -3
Y(8) 0 0
2 -2 0
Y(9) 0 0
CASE 2: CHECK STANDARD =
FOURTH WEIGHT
1
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) +
-
Y(5) + -
Y(6) + -
Y(7) +
-
Y(8) + -
Y(9) +
-
Y(10) +
-
RESTRAINT + +
CHECK STANDARD +

SOLUTION MATRIX

DIVISOR = 10
OBSERVATIONS 1 1
1 1 1
Y(1) 2 -2
0 0 0
Y(2) 1 -1
-3 -1 -1
Y(3) 1 -1
-1 -3 -1
Y(4) 1 -1
-1 -1 -3
Y(5) -1 1
-3 -1 -1
Y(6) -1 1
-1 -3 -1
Y(7) -1 1
-1 -1 -3
Y(8) 0 0
2 -2 0
Y(9) 0 0
2 0 -2
2.3.4.1.3. Design for 1,1,1,1,1
2 0 -2
Y(10) 0 0
0 2 -2
R* 5 5
5 5 5
R* = sum of two reference
standards
FACTORS FOR REPEATABILITY
STANDARD DEVIATIONS
WT FACTOR
K1 1 1 1 1
1
1 0.3162 +
1 0.3162 +
1 0.5477 +
1 0.5477 +
1 0.5477
+
2 0.8944 +
+
3 1.2247 + +
+
0 0.6325 + -

FACTORS FOR BETWEEN-DAY
STANDARD DEVIATIONS
WT FACTOR
K2 1 1 1 1
1
1 0.7071 +
1 0.7071 +
1 1.2247 +
1 1.2247 +
1 1.2247
+
2 2.0000 +
+
3 2.7386 + +
+
0 1.4142 + -

Y(10) 0 0
0 2 -2
R* 5 5
5 5 5
R* = sum of two reference
standards
STANDARD DEVIATIONS
WT FACTOR
K1 1 1 1 1
1
1 0.3162 +
1 0.3162 +
1 0.5477 +
1 0.5477 +
1 0.5477
+
2 0.8944 +
+
3 1.2247 + +
+
1 0.5477 +

STANDARD DEVIATIONS
WT FACTOR
K2 1 1 1 1
1
1 0.7071 +
1 0.7071 +
1 1.2247 +
1 1.2247 +
1 1.2247
+
2 2.0000 +
+
3 2.7386 + +
+
1 1.2247 +
Explanation of notation and
interpretation of tables
2.3.4.1.4. Design for 1,1,1,1,1,1

2.3. Calibration
2.3.4.1.4. Design for 1,1,1,1,1,1
Design 1,1,1,1,1,1
OBSERVATIONS 1 1 1 1 1 1
X(1) + -
X(2) + -
X(3) + -
X(4) + -
X(5) + -
X(6) + -
X(7) + -
X(8) + -
X(9) + -
X(10) + -
X(11) + -
X(12) + -
X(13) + -
X(14) + -
X(15) + -
RESTRAINT + +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 8
Y(1) 1 -1 0 0 0 0
Y(2) 1 0 -1 0 0 0
Y(3) 1 0 0 -1 0 0
Y(4) 1 0 0 0 -1 0
Y(5) 2 1 1 1 1 0
Y(6) 0 1 -1 0 0 0
Y(7) 0 1 0 -1 0 0
Y(8) 0 1 0 0 -1 0
Y(9) 1 2 1 1 1 0
Y(10) 0 0 1 -1 0 0
Y(11) 0 0 1 0 -1 0
Y(12) 1 1 2 1 1 0
Y(13) 0 0 0 1 -1 0
Y(14) 1 1 1 2 1 0
Y(15) 1 1 1 1 2 0
R* 6 6 6 6 6 6
R* = sum of two reference standards
FACTORS FOR COMPUTING REPEATABILITY STANDARD DEVIATIONS
WT FACTOR
1 1 1 1 1 1
1 0.2887 +
2.3.4.1.4. Design for 1,1,1,1,1,1
1 0.2887 +
1 0.5000 +
1 0.5000 +
1 0.5000 +
1 0.5000 +
2 0.8165 + +
3 1.1180 + + +
4 1.4142 + + + +
1 0.5000 +
FACTORS FOR COMPUTING BETWEEN-DAY STANDARD DEVIATIONS
WT FACTOR
1 1 1 1 1 1
1 0.7071 +
1 0.7071 +
1 1.2247 +
1 1.2247 +
1 1.2247 +
1 1.2247 +
2 2.0000 + +
3 2.7386 + + +
4 3.4641 + + + +
1 1.2247 +
2.3.4.1.5. Design for 2,1,1,1

2.3. Calibration
2.3.4.1.5. Design for 2,1,1,1
Design 2,1,1,1
Y(1) + - -
Y(2) + - -
Y(3) + - -
Y(4) + -
Y(5) + -
Y(6) + -
RESTRAINT +
CHECK STANDARD +

SOLUTION MATRIX
DIVISOR = 4
Y(1) 0 -1 0 -1
Y(2) 0 0 -1 -1
Y(3) 0 -1 -1 0
Y(4) 0 1 0 -1
Y(5) 0 1 -1 0
Y(6) 0 0 1 -1
R* 4 2 2 2
R* = value of the reference standard
WT FACTOR
2 1 1 1
2 0.0000 +
1 0.5000 +
1 0.5000 +
1 0.5000 +
2 0.7071 + +
3 0.8660 + + +
1 0.5000 +
WT FACTOR
2 1 1 1
2 0.0000 +
1 1.1180 +
1 1.1180 +
1 1.1180 +
2 1.7321 + +
2.3.4.1.5. Design for 2,1,1,1
3 2.2913 + + +
1 1.1180 +
2.3.4.1.6. Design for 2,2,1,1,1

2.3. Calibration
2.3.4.1.6. Design for 2,2,1,1,1
Design 2,2,1,1,1
OBSERVATIONS 2 2 1 1 1
Y(1) + - - +
Y(2) + - - +
Y(3) + - + -
Y(4) + -
Y(5) + - -
Y(6) + - -
Y(7) + - -
Y(8) + - -
Y(9) + - -
Y(10) + - -
RESTRAINT + + +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 275
Y(1) 47 -3 -44 66 11
Y(2) 25 -25 0 -55 55
Y(3) 3 -47 44 -11 -66
Y(4) 25 -25 0 0 0
Y(5) 29 4 -33 -33 22
Y(6) 29 4 -33 22 -33
Y(7) 7 -18 11 -44 -44
Y(8) 4 29 -33 -33 22
Y(9) 4 29 -33 22 -33
Y(10) -18 7 11 -44 -44
R* 110 110 55 55 55
R* = sum of three reference standards
WT FACTOR
2 2 1 1 1
2 0.2710 +
2 0.2710 +
1 0.3347 +
1 0.4382 +
1 0.4382 +
2 0.6066 + +
3 0.5367 + + +
1 0.4382 +
2.3.4.1.6. Design for 2,2,1,1,1
WT FACTOR
2 2 1 1 1
2 0.8246 +
2 0.8246 +
1 0.8485 +
1 1.0583 +
1 1.0583 +
2 1.5748 + +
3 1.6971 + + +
1 1.0583 +
2.3.4.1.7. Design for 2,2,2,1,1

2.3. Calibration
2.3.4.1.7. Design for 2,2,2,1,1
Design 2,2,2,1,1
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) + - -
Y(5) + - -
Y(6) + - -
Y(7) + -
RESTRAINT + +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 16
Y(1) 4 -4 0 0 0
Y(2) 2 -2 -6 -1 -1
Y(3) -2 2 -6 -1 -1
Y(4) 2 -2 -2 -3 -3
Y(5) -2 2 -2 -3 -3
Y(6) 0 0 4 -2 -2
Y(7) 0 0 0 8 -8
R* 8 8 8 4 4
R* = sum of the two reference standards
WT FACTOR
2 2 2 1 1
2 0.3536 +
2 0.3536 +
2 0.6124 +
1 0.5863 +
1 0.5863 +
2 0.6124 + +
4 1.0000 + + +
1 0.5863 +
WT FACTOR
2 2 2 1 1
2 0.7071 +
2 0.7071 +
2.3.4.1.7. Design for 2,2,2,1,1
2 1.2247 +
1 1.0607 +
1 1.0607 +
2 1.5811 + +
4 2.2361 + + +
1 1.0607 +
2.3.4.1.8. Design for 5,2,2,1,1,1

2.3. Calibration
2.3.4.1.8. Design for 5,2,2,1,1,1
Design 5,2,2,1,1,1
Y(1) + - - - - +
Y(2) + - - - + -
Y(3) + - - + - -
Y(4) + - - - -
Y(5) + - - - -
Y(6) + - + -
Y(7) + - - +
Y(8) + - + -
RESTRAINT + + + +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 70
Y(1) 15 -8 -8 1 1 21
Y(2) 15 -8 -8 1 21 1
Y(3) 5 -12 -12 19 -1 -1
Y(4) 0 2 12 -14 -14 -14
Y(5) 0 12 2 -14 -14 -14
Y(6) -5 8 -12 9 -11 -1
Y(7) 5 12 -8 -9 1 11
Y(8) 0 10 -10 0 10 -10
R* 35 14 14 7 7 7
R* = sum of the four reference standards
WT FACTOR
5 2 2 1 1 1
5 0.3273 +
2 0.3854 +
2 0.3854 +
1 0.4326 +
1 0.4645 +
1 0.4645 +
1 0.4645 +
WT FACTOR
5 2 2 1 1 1
5 1.0000 +
2.3.4.1.8. Design for 5,2,2,1,1,1
2 0.8718 +
2 0.8718 +
1 0.9165 +
1 1.0198 +
1 1.0198 +
1 1.0198 +
2.3.4.1.9. Design for 5,2,2,1,1,1,1

2.3. Calibration
2.3.4.1.9. Design for 5,2,2,1,1,1,1
Design 5,2,2,1,1,1,1
OBSERVATIONS 5 2 2 1 1 1 1
Y(1) + - - -
Y(2) + - - -
Y(3) + - - -
Y(4) + - - -
Y(5) + + - - -
Y(6) + + - - -
Y(7) + + - - - -
Y(8) + -
Y(9) + -
Y(10) + -
RESTRAINT + + + +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 60
1
Y(1) 12 0 0 -12 0 0
0
Y(2) 6 -4 -4 2 -12 3
3
Y(3) 6 -4 -4 2 3 -12
3
Y(4) 6 -4 -4 2 3 3 -
12
Y(5) -6 28 -32 10 -6 -6
-6
Y(6) -6 -32 28 10 -6 -6
-6
Y(7) 6 8 8 -22 -6 -6
-6
Y(8) 0 0 0 0 15 -15
0
Y(9) 0 0 0 0 15 0 -
15
Y(10) 0 0 0 0 0 15 -
15
R* 30 12 12 6 6 6
6
R* = sum of the four reference standards
2.3.4.1.9. Design for 5,2,2,1,1,1,1
WT FACTOR
5 2 2 1 1 1 1
5 0.3162 +
2 0.7303 +
2 0.7303 +
1 0.4830 +
1 0.4472 +
1 0.4472 +
1 0.4472 +
2 0.5477 + +
3 0.5477 + + +
1 0.4472 +
WT FACTOR
5 2 2 1 1 1 1
5 1.0000 +
2 0.8718 +
2 0.8718 +
1 0.9165 +
1 1.0198 +
1 1.0198 +
1 1.0198 +
2 1.4697 + +
3 1.8330 + + +
1 1.0198 +
2.3.4.1.10. Design for 5,3,2,1,1,1
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341a.htm[6/27/2012 1:50:47 PM]

2.3. Calibration
2.3.4.1.10. Design for 5,3,2,1,1,1
Y(1) + - - + -
Y(2) + - - + -
Y(3) + - - - +
Y(4) + - -
Y(5) + - - - -
Y(6) + - + - -
Y(7) + - - + -
Y(8) + - - - +
Y(9) + - -
Y(10) + - -
Y(11) + - -
RESTRAINT + + +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 920
Y(1) 100 -68 -32 119 -111 4
Y(2) 100 -68 -32 4 119 -111
Y(3) 100 -68 -32 -111 4 119
Y(4) 100 -68 -32 4 4 4
Y(5) 60 -4 -56 -108 -108 -108
Y(6) -20 124 -104 128 -102 -102
Y(7) -20 124 -104 -102 128 -102
Y(8) -20 124 -104 -102 -102 128
Y(9) -20 -60 80 -125 -125 -10
Y(10) -20 -60 80 -125 -10 -125
Y(11) -20 -60 80 -10 -125 -125
R* 460 276 184 92 92 92
R* = sum of the three reference standards
WT FACTOR
5 3 2 1 1 1
5 0.2331 +
3 0.2985 +
2 0.2638 +
1 0.3551 +
1 0.3551 +
1 0.3551 +
2 0.5043 + +
3 0.6203 + + +
1 0.3551 +
2.3.4.1.10. Design for 5,3,2,1,1,1
WT FACTOR
5 3 2 1 1 1
5 0.8660 +
3 0.8185 +
2 0.8485 +
1 1.0149 +
1 1.0149 +
1 1.0149 +
2 1.4560 + +
3 1.8083 + + +
1 1.0149 +
2.3.4.1.11. Design for 5,3,2,1,1,1,1
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341b.htm[6/27/2012 1:50:48 PM]

2.3. Calibration
2.3.4.1.11. Design for 5,3,2,1,1,1,1
Design 5,3,2,1,1,1,1
Y(1) + - -
Y(2) + - - -
Y(3) + - - -
Y(4) + - - - -
Y(5) + - - - -
Y(6) + - - - -
Y(7) + - - - -
Y(8) + - -
Y(9) + - -
Y(10) + - -
Y(11) + - -
RESTRAINT + + +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 40
1
Y(1) 20 -4 -16 12 12 12
12
Y(2) 0 -4 4 -8 -8 2
2
Y(3) 0 -4 4 2 2 -8
-8
Y(4) 0 0 0 -5 -5 -10
10
Y(5) 0 0 0 -5 -5 10 -
10
Y(6) 0 0 0 -10 10 -5
-5
Y(7) 0 0 0 10 -10 -5
-5
Y(8) 0 4 -4 -12 8 3
3
Y(9) 0 4 -4 8 -12 3
3
Y(10) 0 4 -4 3 3 -12
8
Y(11) 0 4 -4 3 3 8 -
12
R* 20 12 8 4 4 4
4
2.3.4.1.11. Design for 5,3,2,1,1,1,1
WT FACTOR
5 3 2 1 1 1 1
5 0.5000 +
3 0.2646 +
2 0.4690 +
1 0.6557 +
1 0.6557 +
1 0.6557 +
1 0.6557 +
2 0.8485 + +
3 1.1705 + + +
4 1.3711 + + + +
1 0.6557 +
FACTORS FOR LEVEL-2 STANDARD DEVIATIONS
WT FACTOR
5 3 2 1 1 1 1
5 0.8660 +
3 0.8185 +
2 0.8485 +
1 1.0149 +
1 1.0149 +
1 1.0149 +
1 1.0149 +
2 1.4560 + +
3 1.8083 + + +
4 2.1166 + + + +
1 1.0149 +
2.3.4.1.12. Design for 5,3,2,2,1,1,1
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341c.htm[6/27/2012 1:50:48 PM]

2.3. Calibration
2.3.4.1.12. Design for 5,3,2,2,1,1,1
Y(1) + - -
Y(2) + - -
Y(3) + - - -
Y(4) + - - -
Y(5) + - - -
Y(6) + - -
Y(7) + - -
Y(8) + - -
Y(9) + - - -
Y(10) + -
Y(11) + -
Y(12) - +
RESTRAINT + + +
CHECK STANDARDS +
SOLUTION MATRIX
DIVISOR = 10
1
Y(1) 2 0 -2 2 0 0
0
Y(2) 0 -6 6 -4 -2 -2
-2
Y(3) 1 1 -2 0 -1 1
1
Y(4) 1 1 -2 0 1 -1
1
Y(5) 1 1 -2 0 1 1
-1
Y(6) -1 1 0 -2 -1 1
1
Y(7) -1 1 0 -2 1 -1
1
Y(8) -1 1 0 -2 1 1
-1
Y(9) 0 -2 2 2 -4 -4
-4
Y(10) 0 0 0 0 2 -2
0
Y(11) 0 0 0 0 0 2
-2
Y(12) 0 0 0 0 -2 0
2
R* 5 3 2 2 1 1
1
2.3.4.1.12. Design for 5,3,2,2,1,1,1
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341c.htm[6/27/2012 1:50:48 PM]
WT FACTOR
5 3 2 2 1 1 1
5 0.3162 +
3 0.6782 +
2 0.7483 +
2 0.6000 +
1 0.5831 +
1 0.5831 +
1 0.5831 +
3 0.8124 + +
4 1.1136 + + +
1 0.5831 +
WT FACTOR
5 3 2 2 1 1 1
5 0.8660 +
3 0.8185 +
2 0.8485 +
2 1.0583 +
1 1.0149 +
1 1.0149 +
1 1.0149 +
3 1.5067 + +
4 1.8655 + + +
1 1.0149 +
2.3.4.1.13. Design for 5,4,4,3,2,2,1,1
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341d.htm[6/27/2012 1:50:49 PM]

2.3. Calibration
2.3.4.1.13. Design for 5,4,4,3,2,2,1,1
OBSERVATIONS 5 4 4 3 2 2 1 1
Y(1) + + - - - - -
Y(2) + + - - - - -
Y(3) + - -
Y(4) + - -
Y(5) + - -
Y(6) + - -
Y(7) + - - -
Y(8) + - - -
Y(9) + - -
Y(10) + - -
Y(11) + - -
Y(12) + - -
RESTRAINT + +
CHECK STANDARD + -
SOLUTION MATRIX
DIVISOR = 916
1 1
Y(1) 232 325 123 8 -37 135
-1 1
Y(2) 384 151 401 108 73 105
101 -101
Y(3) 432 84 308 236 168 204 -
144 144
Y(4) 608 220 196 400 440 -120
408 -408
Y(5) 280 258 30 136 58 234 -
246 246
Y(6) 24 -148 68 64 -296 164
-8 8
Y(7) -104 -122 -142 28 214 -558 -
118 118
Y(8) -512 -354 -382 -144 -250 -598
18 -18
Y(9) 76 -87 139 -408 55 443
51 -51
Y(10) -128 26 -210 -36 -406 194 -
110 110
Y(11) -76 87 -139 -508 -55 473 -
51 51
Y(12) -300 -440 -392 116 36 -676
100 -100
R* 1224 696 720 516 476 120
508 408
2.3.4.1.13. Design for 5,4,4,3,2,2,1,1
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341d.htm[6/27/2012 1:50:49 PM]
R* = sum of the two reference standards (for going-up
calibrations)
WT FACTOR
5 4 4 3 2 2 1 1
5 1.2095 +
4 0.8610 +
4 0.9246 +
3 0.9204 +
2 0.8456 +
2 1.4444 +
1 0.5975 +
1 0.5975 +
4 1.5818 + +
7 1.7620 + + +
11 2.5981 + + + +
15 3.3153 + + + + +
20 4.4809 + + + + + +
0 1.1950 + -
WT FACTOR
5 4 4 3 2 2 1 1
5 2.1380 +
4 1.4679 +
4 1.4952 +
3 1.2785 +
2 1.2410 +
2 1.0170 +
1 0.7113 +
1 0.7113 +
4 1.6872 + +
7 2.4387 + + +
11 3.4641 + + + +
15 4.4981 + + + + +
20 6.2893 + + + + + +
0 1.4226 + -
2.3.4.1.14. Design for 5,5,2,2,1,1,1,1
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341e.htm[6/27/2012 1:50:50 PM]

2.3. Calibration
2.3.4.1.14. Design for 5,5,2,2,1,1,1,1
Design 5,5,2,2,1,1,1,1
Y(1) + - - -
Y(2) + - - -
Y(3) + - - -
Y(4) + - - -
Y(5) + + - - - -
Y(6) + - -
Y(7) + - -
Y(8) + - -
Y(9) + - -
Y(10) + -
Y(11) + -
RESTRAINT + +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 120
1 1
Y(1) 30 -30 -12 -12 -22 -10
10 -2
Y(2) -30 30 -12 -12 -10 -22
-2 10
Y(3) 30 -30 -12 -12 10 -2 -
22 -10
Y(4) -30 30 -12 -12 -2 10 -
10 -22
Y(5) 0 0 6 6 -12 -12 -
12 -12
Y(6) -30 30 33 -27 -36 24 -
36 24
Y(7) 30 -30 33 -27 24 -36
24 -36
Y(8) 0 0 -27 33 -18 6
6 -18
Y(9) 0 0 -27 33 6 -18 -
18 6
Y(10) 0 0 0 0 32 8 -
32 -8
Y(11) 0 0 0 0 8 32
-8 -32
R* 60 60 24 24 12 12
12 12
2.3.4.1.14. Design for 5,5,2,2,1,1,1,1
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341e.htm[6/27/2012 1:50:50 PM]
WT FACTOR
5 5 2 2 1 1 1 1
5 0.6124 +
5 0.6124 +
2 0.5431 +
2 0.5431 +
1 0.5370 +
1 0.5370 +
1 0.5370 +
1 0.5370 +
2 0.6733 + +
4 0.8879 + + +
6 0.8446 + + + +
11 1.0432 + + + + +
16 0.8446 + + + + + +
1 0.5370 +
FACTORS FOR COMPUTING LEVEL-2 STANDARD DEVIATIONS
WT FACTOR
5 5 2 2 1 1 1 1
5 0.7071 +
5 0.7071 +
2 1.0392 +
2 1.0392 +
1 1.0100 +
1 1.0100 +
1 1.0100 +
1 1.0100 +
2 1.4422 + +
4 1.8221 + + +
6 2.1726 + + + +
11 2.2847 + + + + +
16 2.1726 + + + + + +
1 1.0100 +
2.3.4.1.15. Design for 5,5,3,2,1,1,1
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341f.htm[6/27/2012 1:50:50 PM]

2.3. Calibration
2.3.4.1.15. Design for 5,5,3,2,1,1,1
OBSERVATIONS 5 5 3
2 1 1 1
Y(1) + -
-
Y(2) + -
-
Y(3) +
- - - -
Y(4) +
- - - -
Y(5) + -
- -
Y(6) + -
- -
Y(7) + -
- -
Y(8) + -
- -
Y(9) + -
- -
Y(10) + -
- -
RESTRAINT + +
CHECK STANDARD
+
DEGREES OF FREEDOM =
4

SOLUTION MATRIX

DIVISOR = 10
OBSERVATIONS 5
5 3 2 1
1 1
Y(1) 1 -
1 -2 -3 1
1 1
Y(2) -1
1 -2 -3 1
1 1
Y(3) 1 -
1 2 -2 -1
-1 -1
Y(4) -1
1 2 -2 -1
-1 -1
Y(5) 1 -
1 -1 1 -2
2.3.4.1.15. Design for 5,5,3,2,1,1,1
-2 3
Y(6) 1 -
1 -1 1 -2
3 -2
Y(7) 1 -
1 -1 1 3
-2 -2
Y(8) -1
1 -1 1 -2
-2 3
Y(9) -1
1 -1 1 -2
3 -2
Y(10) -1
1 -1 1 3
-2 -2
R* 5
5 3 2 1
1 1
R* = sum of the two
reference standards
STANDARD DEVIATIONS
WT FACTOR
5 5 3
2 1 1 1
5 0.3162 +
5 0.3162 +
3 0.4690 +
2 0.5657
+
1 0.6164
+
1 0.6164
+
1 0.6164
+
3 0.7874
+ +
6 0.8246 +
+ +
11 0.8832 + +
+ +
16 0.8246 + + +
+ +
1 0.6164
+
STANDARD DEVIATIONS
WT FACTOR
5 5 3
2 1 1 1
5 0.7071 +
5 0.7071 +
3 1.0863 +
2 1.0392
+
1 1.0100
+
1 1.0100
+
1 1.0100
+
3 1.4765
+ +
6 1.9287 +
+ +
11 2.0543 + +
+ +
16 1.9287 + + +
+ +
1 1.0100
+
2.3.4.1.15. Design for 5,5,3,2,1,1,1
Explanation of notation and
interpretation of tables
2.3.4.1.16. Design for 1,1,1,1,1,1,1,1 weights
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341g.htm[6/27/2012 1:50:51 PM]

2.3. Calibration
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) + -
Y(5) + -
Y(6) + -
Y(7) + -
Y(8) + -
Y(9) + -
Y(10) + -
Y(11) + -
Y(12) + -
RESTRAINT + +
CHECK STANDARD +

SOLUTION MATRIX
DIVISOR = 12
1 1
Y(1) 1 -1 -6 0 0 0
0 0
Y(2) 1 -1 0 -6 0 0
0 0
Y(3) 1 -1 0 0 -6 0
0 0
Y(4) 1 -1 0 0 0 -6
0 0
Y(5) 1 -1 0 0 0 0
-6 0
Y(6) 1 -1 0 0 0 0
0 -6
Y(7) -1 1 -6 0 0 0
0 0
Y(8) -1 1 0 -6 0 0
0 0
Y(9) -1 1 0 0 -6 0
0 0
Y(10) -1 1 0 0 0 -6
0 0
Y(11) -1 1 0 0 0 0
-6 0
Y(12) -1 1 0 0 0 0
0 -6
R* 6 6 6 6 6 6
6 6
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341g.htm[6/27/2012 1:50:51 PM]
WT K1 1 1 1 1 1 1 1 1
1 0.2887 +
1 0.2887 +
1 0.7071 +
1 0.7071 +
1 0.7071 +
1 0.7071 +
1 0.7071 +
1 0.7071 +
2 1.0000 + +
3 1.2247 + + +
4 1.4142 + + + +
5 1.5811 + + + + +
6 1.7321 + + + + + +
1 0.7071 +
WT K2 1 1 1 1 1 1 1 1
1 0.7071 +
1 0.7071 +
1 1.2247 +
1 1.2247 +
1 1.2247 +
1 1.2247 +
1 1.2247 +
1 1.2247 +
2 2.0000 + +
3 2.7386 + + +
4 3.4641 + + + +
5 4.1833 + + + + +
6 4.8990 + + + + + +
1 1.2247 +

2.3.4.1.17. Design for 3,2,1,1,1 weights
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341h.htm[6/27/2012 1:50:51 PM]

2.3. Calibration
Y(1) + - -
Y(2) + - -
Y(3) + - -
Y(4) + - - -
Y(5) + - -
Y(6) + - -
Y(7) + - -
Y(8) + -
Y(9) + -
Y(10) + -
RESTRAINT + +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 25
Y(1) 3 -3 -4 1 1
Y(2) 3 -3 1 -4 1
Y(3) 3 -3 1 1 -4
Y(4) 1 -1 -3 -3 -3
Y(5) -2 2 -4 -4 1
Y(6) -2 2 -4 1 -4
Y(7) -2 2 1 -4 -4
Y(8) 0 0 5 -5 0
Y(9) 0 0 5 0 -5
Y(10) 0 0 0 5 -5
R* 15 10 5 5 5
WT K1 3 2 1 1 1
3 0.2530 +
2 0.2530 +
1 0.4195 +
1 0.4195 +
1 0.4195 +
2 0.5514 + +
3 0.6197 + + +
1 0.4195 +
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341h.htm[6/27/2012 1:50:51 PM]
WT K2 3 2 1 1 1
3 0.7211 +
2 0.7211 +
1 1.0392 +
1 1.0392 +
1 1.0392 +
2 1.5232 + +
3 1.9287 + + +
1 1.0392 +

2.3.4.1.18. Design for 10-and 20-pound weights
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341i.htm[6/27/2012 1:50:52 PM]

2.3. Calibration
Y(1) + -
Y(2) + -
Y(3) + - +
Y(4) + - +
Y(5) + - +
Y(6) + - +
Y(7) + -

RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 24
Y(1) 0 -12 -12 -16 -8
Y(2) 0 -12 -12 -8 -16
Y(3) 0 -9 -3 -4 4
Y(4) 0 -3 -9 4 -4
Y(5) 0 -9 -3 4 -4
Y(6) 0 -3 -9 -4 4
Y(7) 0 6 -6 0 0
R* 24 48 48 24 24
R* = Value of the reference standard
WT K1 1 2 2 1 1
2 0.9354 +
2 0.9354 +
1 0.8165 +
1 0.8165 +
4 1.7321 + +
5 2.3805 + + +
6 3.0000 + + + +
1 0.8165 +
WT K2 1 2 2 1 1
2 2.2361 +
2 2.2361 +
1 1.4142 +
1 1.4142 +
4 4.2426 + +
5 5.2915 + + +
http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc341i.htm[6/27/2012 1:50:52 PM]
6 6.3246 + + + +
1 1.4142 +
2.3.4.2. Drift-elimination designs for gauge blocks

2.3. Calibration
2.3.4.2. Drift-elimination designs for gauge
blocks
Tie to the
defined
unit of
length
The unit of length in many industries is maintained and
disseminated by gauge blocks. The highest accuracy
calibrations of gauge blocks are done by laser intererometry
which allows the transfer of the unit of length to a gauge
piece. Primary standards laboratories maintain master sets of
English gauge blocks and metric gauge blocks which are
calibrated in this manner. Gauge blocks ranging in sizes from
0.1 to 20 inches are required to support industrial processes in
the United States.
Mechanical
comparison
of gauge
blocks
However, the majority of gauge blocks are calibrated by
comparison with master gauges using a mechanical
comparator specifically designed for measuring the small
difference between two blocks of the same nominal length.
The measurements are temperature corrected from readings
taken directly on the surfaces of the blocks. Measurements on
2 to 20 inch blocks require special handling techniques to
minimize thermal effects. A typical calibration involves a set
of 81 gauge blocks which are compared one-by-one with
master gauges of the same nominal size.
Calibration
designs for
gauge
blocks
Calibration designs allow comparison of several gauge blocks
of the same nominal size to one master gauge in a manner
that promotes economy of operation and minimizes wear on
the master gauge. The calibration design is repeated for each
size until measurements on all the blocks in the test sets are
completed.
Problem of
thermal
drift
Measurements on gauge blocks are subject to drift from heat
build-up in the comparator. This drift must be accounted for
in the calibration experiment or the lengths assigned to the
blocks will be contaminated by the drift term.
Elimination
of linear
drift
The designs in this catalog are constructed so that the
solutions are immune to linear drift if the measurements are
equally spaced over time. The size of the drift is the average
of the n difference measurements. Keeping track of drift from
design to design is useful because a marked change from its
usual range of values may indicate a problem with the
measurement system.
Assumption
for Doiron
designs
Mechanical measurements on gauge blocks take place
successively with one block being inserted into the
comparator followed by a second block and so on. This
scenario leads to the assumption that the individual
measurements are subject to drift (Doiron). Doiron lists
designs meeting this criterion which also allow for:
two master blocks, R1 and R2
one check standard = difference between R1 and R2
one - nine test blocks
Properties
of drift-
elimination
designs
that use 1
master
block
The designs are constructed to:
Be immune to linear drift
Minimize the standard deviations for test blocks (as
much as possible)
Spread the measurements on each block throughout the
design
Be completed in 5-10 minutes to keep the drift at the 5
nm level
Caution Because of the large number of gauge blocks that are being
intercompared and the need to eliminate drift, the Doiron
designs are not completely balanced with respect to the test
blocks. Therefore, the standard deviations are not equal for all
blocks. If all the blocks are being calibrated for use in one
facility, it is easiest to quote the largest of the standard
deviations for all blocks rather than try to maintain a separate
record on each block.
Definition
of master
block and
check
standard
At the National Institute of Standards and Technology
(NIST), the first two blocks in the design are NIST masters
which are designated R1 and R2, respectively. The R1 block
is a steel block, and the R2 block is a chrome-carbide block.
If the test blocks are steel, the reference is R1; if the test
blocks are chrome-carbide, the reference is R2. The check
standard is always the difference between R1 and R2 as
estimated from the design and is independent of R1 and R2.
The designs are listed in this section of the catalog as:
1. Doiron design for 3 gauge blocks - 6 measurements
Properties
of designs
that use 2
master
blocks
Historical designs for gauge blocks (Cameron and Hailes)
work on the assumption that the difference measurements are
contaminated by linear drift. This assumption is more
restrictive and covers the case of drift in successive
measurements but produces fewer designs. The
Cameron/Hailes designs meeting this criterion allow for:
two reference (master) blocks, R1 and R2
check standard = difference between the two master
blocks
and assign equal uncertainties to values of all test blocks.
The designs are listed in this section of the catalog as:
1. Cameron-Hailes design for 2 masters + 2 test blocks
Important
concept -
check
standard
The check standards for the designs in this section are not
artifact standards but constructions from the design. The value
of one master block or the average of two master blocks is the
restraint for the design, and values for the masters, R1 and R2,
are estimated from a set of measurements taken according to
the design. The check standard value is the difference
between the estimates, R1 and R2. Measurement control is
exercised by comparing the current value of the check
standard with its historical average.
2.3.4.2.1. Doiron 3-6 Design

2.3. Calibration
2.3.4.2. Drift-elimination designs for gage blocks
Doiron 3-6 design
OBSERVATIONS 1 1 1
Y(1) + -
Y(2) - +
Y(3) + -
Y(4) - +
Y(5) - +
Y(6) + -
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 6
OBSERVATIONS 1 1 1
Y(1) 0 -2 -1
Y(2) 0 1 2
Y(3) 0 1 -1
Y(4) 0 2 1
Y(5) 0 -1 1
Y(6) 0 -1 -2
R* 6 6 6
NOM FACTOR
1 1 1
1 0.0000 +
1 0.5774 +
1 0.5774 +
1 0.5774 +


2.3. Calibration
Doiron 3-9 Design
OBSERVATIONS 1 1 1
Y(1) + -
Y(2) - +
Y(3) + -
Y(4) - +
Y(5) - +
Y(6) + -
Y(7) - +
Y(8) - +
Y(9) + -
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 9
OBSERVATIONS 1 1 1
Y(1) 0 -2 -1
Y(2) 0 -1 1
Y(3) 0 -1 -2
Y(4) 0 2 1
Y(5) 0 1 2
Y(6) 0 1 -1
Y(7) 0 2 1
Y(8) 0 -1 1
Y(9) 0 -1 -2
R(1) 9 9 9
FACTORS FOR COMPUTING REPEATABILITY STANDARD
DEVIATIONS
NOM FACTOR
1 1 1
1 0.0000 +
1 0.4714 +
1 0.4714 +
1 0.4714 +


2.3. Calibration
Doiron 4-8 Design
Y(1) + -
Y(2) + -
Y(3) - +
Y(4) + -
Y(5) - +
Y(6) - +
Y(7) + -
Y(8) - +
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 8
Y(1) 0 -3 -2 -1
Y(2) 0 1 2 -1
Y(3) 0 1 2 3
Y(4) 0 1 -2 -1
Y(5) 0 3 2 1
Y(6) 0 -1 -2 1
Y(7) 0 -1 -2 -3
Y(8) 0 -1 2 1
R* 8 8 8 8
NOM FACTOR
1 1 1 1
1 0.0000 +
1 0.6124 +
1 0.7071 +
1 0.6124 +
1 0.6124 +


2.3. Calibration
Doiron 4-12 Design
Y(1) + -
Y(2) + +
Y(3) + -
Y(4) - +
Y(5) + -
Y(6) - +
Y(7) + -
Y(8) + -
Y(9) + -
Y(10) - +
Y(11) - +
Y(12) - +
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 8
Y(1) 0 -2 -1 -1
Y(2) 0 1 1 2
Y(3) 0 0 1 -1
Y(4) 0 2 1 1
Y(5) 0 1 -1 0
Y(6) 0 -1 0 1
Y(7) 0 -1 -2 -1
Y(8) 0 1 0 -1
Y(9) 0 -1 -1 -2
Y(10) 0 -1 1 0
Y(11) 0 1 2 1
Y(12) 0 0 -1 1
R* 6 6 6 4

NOM FACTOR
1 1 1 1
1 0.0000 +
1 0.5000 +
1 0.5000 +
1 0.5000 +
1 0.5000 +


2.3. Calibration
Doiron 5-10 Design
Y(1) + -
Y(2) - +
Y(3) + -
Y(4) - +
Y(5) - +
Y(6) + -
Y(7) - +
Y(8) + -
Y(9) - +
Y(10) + -
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 90
Y(1) 0 -50 -10 -10 -30
Y(2) 0 20 4 -14 30
Y(3) 0 -10 -29 -11 -15
Y(4) 0 -20 5 5 15
Y(5) 0 0 -18 18 0
Y(6) 0 -10 -11 -29 -15
Y(7) 0 10 29 11 15
Y(8) 0 -20 14 -4 -30
Y(9) 0 10 11 29 15
Y(10) 0 20 -5 -5 -15
R* 90 90 90 90 90
NOM FACTOR
1 1 1 1 1
1 0.0000 +
1 0.7454 +
1 0.5676 +
1 0.5676 +
1 0.7071 +
1 0.7454 +


2.3. Calibration
Doiron 6-12 Design
Y(1) + -
Y(2) - +
Y(3) - +
Y(4) - +
Y(5) - +
Y(6) + -
Y(7) + -
Y(8) + -
Y(9) + -
Y(10) - +
Y(11) + -
Y(12) - +
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 360
1
Y(1) 0 -136 -96 -76 -72
-76
Y(2) 0 -4 -24 -79 72
11
Y(3) 0 -20 -120 -35 0
55
Y(4) 0 4 24 -11 -72
79
Y(5) 0 -60 0 75 0
-15
Y(6) 0 20 120 -55 0
35
Y(7) 0 -76 -96 -61 -72
-151
Y(8) 0 64 24 4 -72
4
Y(9) 0 40 -120 -20 0
-20
Y(10) 0 72 72 72 144
72
Y(11) 0 60 0 15 0
-75
Y(12) 0 76 96 151 72
61
R* 360 360 360 360 360
360
NOM FACTOR
1 1 1 1 1 1
1 0.0000 +
1 0.6146 +
1 0.7746 +
1 0.6476 +
1 0.6325 +
1 0.6476 +
1 0.6146 +


2.3. Calibration
Doiron 7-14 Design
Y(1) + -
Y(2) - +
Y(3) + -
Y(4) + -
Y(5) + -
Y(6) - +
Y(7) + -
Y(8) + -
Y(9) + -
Y(10) - +
Y(11) - +
Y(12) - +
Y(13) - +
Y(14) - +
RESTRAINT +
CHECK STANDARD +
PARAMETER VALUES
DIVISOR = 1015
1 1
Y(1) 0 -406 -203 -203 -203
-203 -203
Y(2) 0 0 -35 -210 35
210 0
Y(3) 0 0 175 35 -175
-35 0
Y(4) 0 203 -116 29 -116
29 -261
Y(5) 0 -203 -229 -214 -264
-424 -174
Y(6) 0 0 -175 -35 175
35 0
Y(7) 0 203 -61 -221 -26
-11 29
Y(8) 0 0 305 90 130
55 -145
Y(9) 0 0 220 15 360
-160 145
Y(10) 0 203 319 174 319
174 464
Y(11) 0 -203 26 11 61
221 -29
Y(12) 0 0 -360 160 -220
-15 -145
Y(13) 0 203 264 424 229
214 174
Y(14) 0 0 -130 -55 -305
-90 145
R* 1015 1015 1015 1015 1015
1015 1015
NOM FACTOR
1 1 1 1 1 1 1
1 0.0000 +
1 0.6325 +
1 0.7841 +
1 0.6463 +
1 0.7841 +
1 0.6463 +
1 0.6761 +
1 0.6325 +


2.3. Calibration
Doiron 8-16 Design
Y(1) + -
Y(2) + -
Y(3) - +
Y(4) - +
Y(5) + -
Y(6) - +
Y(7) - +
Y(8) - +
Y(9) - +
Y(10) - +
Y(11) + -
Y(12) - +
Y(13) - +
Y(14) - +
Y(15) + -
Y(16) + -
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 2852
1 1 1
Y(1) 0 -1392 -620 -472 -516
-976 -824 -916
Y(2) 0 60 248 -78 96
878 -112 -526
Y(3) 0 352 124 -315 278
255 864 289
Y(4) 0 516 992 470 1396
706 748 610
Y(5) 0 -356 620 35 286
-979 -96 -349
Y(6) 0 92 0 23 -138
253 -552 667
Y(7) 0 -148 -992 335 -522
-407 -104 -81
Y(8) 0 -416 372 113 190
995 16 177
Y(9) 0 308 -248 170 -648
134 756 342
Y(10) 0 472 620 955 470
585 640 663
Y(11) 0 476 -124 -191 -94
-117 -128 -703
Y(12) 0 -104 -620 -150 404
-286 4 -134
Y(13) 0 472 620 955 470
585 640 663
Y(14) 0 444 124 -292 140
508 312 956
Y(15) 0 104 620 150 -404
286 -4 134
Y(16) 0 568 -124 -168 -232
136 -680 -36
R* 2852 2852 2852 2852 2852
2852 2852 2852
R* = value of reference block
WT FACTOR
1 1 1 1 1 1 1 1
1 0.0000 +
1 0.6986 +
1 0.7518 +
1 0.5787 +
1 0.6996 +
1 0.8313 +
1 0.7262 +
1 0.7534 +
1 0.6986 +


2.3. Calibration
Doiron 9-18 Design
OBSERVATIONS 1 1 1 1 1 1 1 1 1
Y(1) + -
Y(2) - +
Y(3) + -
Y(4) - +
Y(5) + -
Y(6) - +
Y(7) + -
Y(8) + -
Y(9) - +
Y(10) + -
Y(11) - +
Y(12) - +
Y(13) - +
Y(14) + -
Y(15) - +
Y(16) + -
Y(17) - +
Y(18) + -
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 8247
1 1 1 1
Y(1) 0 -3680 -2305 -2084 -1175 -
1885 -1350 -1266 -654
Y(2) 0 -696 -1422 -681 -1029
-984 -2586 -849 1203
Y(3) 0 1375 -3139 196 -491 -
1279 -1266 -894 -540
Y(4) 0 -909 -222 -1707 1962
-432 675 633 327
Y(5) 0 619 1004 736 -329
2771 -378 -1674 -513
Y(6) 0 -1596 -417 1140 342
303 42 186 57
Y(7) 0 955 2828 496 -401
971 -1689 -411 -525
Y(8) 0 612 966 741 1047
1434 852 2595 -1200
Y(9) 0 1175 1666 1517 3479
1756 2067 2085 1038
Y(10) 0 199 -1276 1036 -239 -
3226 -801 -1191 -498
Y(11) 0 654 1194 711 1038
1209 1719 1722 2922
Y(12) 0 91 494 -65 -1394
887 504 2232 684
Y(13) 0 2084 1888 3224 1517
2188 1392 1452 711
Y(14) 0 1596 417 -1140 -342
-303 -42 -186 -57
Y(15) 0 175 950 -125 -1412
437 2238 486 681
Y(16) 0 -654 -1194 -711 -1038 -
1209 -1719 -1722 -2922
Y(17) 0 -420 -2280 300 90
2250 -423 483 15
Y(18) 0 84 456 -60 -18
-450 1734 -1746 -3
R* 8247 8247 8247 8247 8247
8247 8247 8247 8247
NOM FACTOR
1 1 1 1 1 1 1 1 1
1 0.0000 +
1 0.6680 +
1 0.8125 +
1 0.6252 +
1 0.6495 +
1 0.8102 +
1 0.7225 +
1 0.7235 +
1 0.5952 +
1 0.6680 +

2.3.4.2.10. Doiron 10-20 Design

2.3. Calibration
2.3.4.2.10. Doiron 10-20 Design
Doiron 10-20 Design
1
Y(1) + -
Y(2) + -
Y(3) -
+
Y(4) + -
Y(5) + -
Y(6) + -
Y(7) + -
Y(8) -
+
Y(9) + -
Y(10) +
-
Y(11) + -
Y(12) + -
Y(13) +
-
Y(14) - +
Y(15) + -
Y(16) + -
Y(17) - +
Y(18) + -
Y(19) - +
Y(20) - +
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 33360
1 1 1 1
Y(1) 0 -15300 -9030 -6540 -5970 -
9570 -7770 -6510 -9240
Y(2) 0 1260 1594 1716 3566
3470 9078 -5678 -24
Y(3) 0 -960 -2856 -7344 -2664 -
1320 -1992 -1128 336
Y(4) 0 -3600 -1536 816 5856 -
9120 -1632 -1728 -3744
Y(5) 0 6060 306 -1596 -906 -
1050 -978 -2262 -8376
Y(6) 0 2490 8207 -8682 -1187
1165 2769 2891 588
Y(7) 0 -2730 809 -1494 -869 -
2885 903 6557 -8844
Y(8) 0 5580 7218 11412 6102
2.3.4.2.10. Doiron 10-20 Design
6630 6366 5514 8472
Y(9) 0 1800 -2012 -408 -148
7340 -7524 -1916 1872
Y(10) 0 3660 1506 -3276 774
3990 2382 3258 9144
Y(11) 0 -1800 -3548 408 5708 -
1780 -9156 -3644 -1872
Y(12) 0 6270 -9251 -3534 -1609
455 -3357 -3023 516
Y(13) 0 960 2856 7344 2664
1320 1992 1128 -336
Y(14) 0 -330 -391 186 -2549 -
7925 -2457 1037 6996
Y(15) 0 2520 8748 3432 1572
1380 1476 -5796 -48
Y(16) 0 -5970 -7579 -8766 -15281 -
9425 -9573 -6007 -6876
Y(17) 0 -1260 -7154 -1716 1994
2090 7602 118 24
Y(18) 0 570 2495 9990 -6515 -
1475 -1215 635 1260
Y(19) 0 6510 9533 6642 6007
7735 9651 15329 8772
Y(20) 0 -5730 85 1410 3455
8975 3435 1225 1380
R* 33360 33360 33360 33360 33360
33360 33360 33360 33360
NOM FACTOR
1 1 1 1 1 1 1 1 1
1
1 0.0000 +
1 0.6772 +
1 0.7403 +
1 0.7498 +
1 0.6768 +
1 0.7456 +
1 0.7493 +
1 0.6779 +
1 0.7267 +
1 0.6961
+
1 0.6772 +

2.3.4.2.11. Doiron 11-22 Design

2.3. Calibration
2.3.4.2.11. Doiron 11-22 Design
Doiron 11-22 Design
1 1
Y(1) + -
Y(2) + -
Y(3) +
-
Y(4) + -
Y(5) + -
Y(6)
+ -
Y(7) - +
Y(8) - +
Y(9) + -
Y(10) + -
Y(11) + -
Y(12) - +
Y(13) + -
Y(14) -
+
Y(15) + -
Y(16) +
-
Y(17) + -
Y(18) -
+
Y(19) + -
Y(20) - +
Y(21) -
+
Y(22) +
-
RESTRAINT +
CHECK STANDARD +
SOLUTION MATRIX
DIVISOR = 55858
1 1 1 1 1
Y(1) 0 -26752 -18392 -15532 -9944 -
8778 -14784 -15466 -16500 -10384 -17292
Y(2) 0 1166 1119 3976 12644 -
11757 -1761 2499 1095 -2053 1046
Y(3) 0 5082 4446 3293 4712
160 5882 15395 3527 -9954 487
Y(4) 0 -968 -1935 10496 2246 -
635 -4143 -877 -13125 -643 -1060
Y(5) 0 8360 -18373 -8476 -3240 -
3287 -8075 -1197 -9443 -1833 -2848
Y(6) 0 -6908 -7923 -9807 -2668
2.3.4.2.11. Doiron 11-22 Design
431 -4753 -1296 -10224 9145 -18413
Y(7) 0 1716 3084 6091 404 -
2452 -10544 -2023 15073 332 5803
Y(8) 0 9944 13184 15896 24476
11832 13246 14318 13650 9606 12274
Y(9) 0 2860 12757 -11853 -2712
145 3585 860 578 -293 -2177
Y(10) 0 -8778 -12065 -11920 -11832 -
23589 -15007 -11819 -12555 -11659 -11228
Y(11) 0 11286 1729 -271 -4374 -
3041 -3919 -14184 -180 -3871 1741
Y(12) 0 -3608 -13906 -4734 62
2942 11102 2040 -2526 604 -2566
Y(13) 0 -6006 -10794 -7354 -1414
8582 -18954 -6884 -10862 -1162 -6346
Y(14) 0 -9460 1748 6785 2330
2450 2790 85 6877 4680 16185
Y(15) 0 5588 10824 19965 -8580
88 6028 1485 11715 2904 10043
Y(16) 0 -792 5803 3048 1376
1327 5843 1129 15113 -1911 -10100
Y(17) 0 -682 6196 3471 -1072
3188 15258 -10947 6737 -1434 2023
Y(18) 0 10384 12217 12510 9606
11659 12821 14255 13153 24209 15064
Y(19) 0 1892 10822 -1357 -466 -
490 -558 -17 -12547 -936 -3237
Y(20) 0 5522 3479 -93 -10158 -
13 5457 15332 3030 4649 3277
Y(21) 0 1760 -3868 -13544 -3622 -
692 -1700 -252 -1988 2554 11160
Y(22) 0 -1606 -152 -590 2226
11930 2186 -2436 -598 -12550 -3836
R* 55858 55858 55858 55858 55858
55858 55858 55858 55858 55858 55858

NOM FACTOR
1 1 1 1 1 1 1 1 1
1 1
1 0.0000 +
1 0.6920 +
1 0.8113 +
1 0.8013 +
1 0.6620 +
1 0.6498 +
1 0.7797 +
1 0.7286 +
1 0.8301 +
1 0.6583
+
1 0.6920 +

2.3.4.3. Designs for electrical quantities

2.3. Calibration
Standard
cells
Banks of saturated standard cells that are nominally one volt
are the basis for maintaining the unit of voltage in many
laboratories.
Bias
problem
It has been observed that potentiometer measurements of the
difference between two saturated standard cells, connected in
series opposition, are effected by a thermal emf which remains
constant even when the direction of the circuit is reversed.
Designs
for
eliminating
bias
A calibration design for comparing standard cells can be
constructed to be left-right balanced so that:
A constant bias, P, does not contaminate the estimates
for the individual cells.
P is estimated as the average of difference
measurements.
Designs
for
electrical
quantities
Designs are given for the following classes of electrical
artifacts. These designs are left-right balanced and may be
appropriate for artifacts other than electrical standards.
Saturated standard reference cells
Saturated standard test cells
Zeners
Resistors
Standard
cells in a
single box
Left-right balanced designs for comparing standard cells
among themselves where the restraint is over all reference
cells are listed below. These designs are not appropriate for
assigning values to test cells.
Estimates for individual standard cells and the bias term, P,
are shown under the heading, 'SOLUTION MATRIX'. These
designs also have the advantage of requiring a change of
connections to only one cell at a time.
1. Design for 3 standard cells
Test cells Calibration designs for assigning values to test cells in a
common environment on the basis of comparisons with
reference cells with known values are shown below. The
designs in this catalog are left-right balanced.
1. Design for 4 test cells and 4 reference cells
2. Design for 8 test cells and 8 reference cells
Zeners Increasingly, zeners are replacing saturated standard cells as
artifacts for maintaining and disseminating the volt. Values are
assigned to test zeners, based on a group of reference zeners,
using calibration designs.
1. Design for 4 reference zeners and 2 test zeners
2. Design for 4 reference zeners and 3 test zeners
Standard
resistors
Designs for comparing standard resistors that are used for
maintaining and disseminating the ohm are listed in this
section.
1. Design for 3 reference resistors and 1 test resistor
2. Design for 4 reference resistors and 1 test resistor
2.3.4.3.1. Left-right balanced design for 3 standard cells

2.3. Calibration
2.3.4.3.1. Left-right balanced design for 3
standard cells
Design 1,1,1
CELLS
OBSERVATIONS 1 1 1
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) - +
Y(5) - +
Y(6) - +
RESTRAINT + + +
SOLUTION MATRIX
DIVISOR = 6
OBSERVATIONS 1 1 1 P
Y(1) 1 -1 0 1
Y(2) 1 0 -1 1
Y(3) 0 1 -1 1
Y(4) -1 1 0 1
Y(5) -1 0 1 1
Y(6) 0 -1 1 1
R* 2 2 2 0
R* = AVERAGE VALUE OF 3 REFERENCE CELLS
P = LEFT-RIGHT BIAS
FACTORS FOR COMPUTING STANDARD DEVIATIONS
V FACTOR CELLS
1 1 1
1 0.3333 +
1 0.3333 +
1 0.3333 +

2.3. Calibration
standard cells
Design 1,1,1,1
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) + -
Y(5) + -
Y(6) - +
Y(7) - +
Y(8) - +
Y(9) - +
Y(10) - +
Y(11) - +
Y(12) + -
RESTRAINT + + + +
SOLUTION MATRIX
DIVISOR = 8
OBSERVATIONS 1 1 1 1 P
Y(1) 1 -1 0 0 1
Y(2) 1 0 -1 0 1
Y(3) 0 1 -1 0 1
Y(4) 0 1 0 -1 1
Y(5) 0 0 1 -1 1
Y(6) -1 0 1 0 1
Y(7) 0 -1 1 0 1
Y(8) 0 -1 0 1 1
Y(9) -1 0 0 1 1
Y(10) 0 0 -1 1 1
Y(11) -1 1 0 0 1
Y(12) 1 0 0 -1 1
R* 2 2 2 2 0
P = LEFT-RIGHT BIAS
V FACTOR CELLS
1 1 1 1
1 0.3062 +
1 0.3062 +
1 0.3062 +
1 0.3062 +

2.3. Calibration
standard cells
Design 1,1,1,1,1
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) + -
Y(5) + -
Y(6) + -
Y(7) + -
Y(8) - +
Y(9) - +
Y(10) - +
RESTRAINT + + + + +
SOLUTION MATRIX
DIVISOR = 5
OBSERVATIONS 1 1 1 1 1 P
Y(1) 1 -1 0 0 0 1
Y(2) 1 0 -1 0 0 1
Y(3) 0 1 -1 0 0 1
Y(4) 0 1 0 -1 0 1
Y(5) 0 0 1 -1 0 1
Y(6) 0 0 1 0 -1 1
Y(7) 0 0 0 1 -1 1
Y(8) -1 0 0 1 0 1
Y(9) -1 0 0 0 1 1
Y(10) 0 -1 0 0 1 1
R* 1 1 1 1 1 0
P = LEFT-RIGHT BIAS
V FACTOR CELLS
1 1 1 1 1
1 0.4000 +
1 0.4000 +
1 0.4000 +
1 0.4000 +
1 0.4000 +

2.3. Calibration
standard cells
Design 1,1,1,1,1,1
CELLS
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) + -
Y(5) + -
Y(6) + -
Y(7) + -
Y(8) + -
Y(9) + -
Y(10) - +
Y(11) - +
Y(12) - +
Y(13) + -
Y(14) + -
Y(15) + -
RESTRAINT + + + + + +
SOLUTION MATRIX
DIVISOR = 6
P
Y(1) 1 -1 0 0 0 0
1
Y(2) 1 0 -1 0 0 0
1
Y(3) 0 1 -1 0 0 0
1
Y(4) 0 1 0 -1 0 0
1
Y(5) 0 0 1 -1 0 0
1
Y(6) 0 0 1 0 -1 0
1
Y(7) 0 0 0 1 -1 0
1
Y(8) 0 0 0 1 0 -1
1
Y(9) 0 0 0 0 1 -1
1
Y(10) -1 0 0 0 1 0
1
Y(11) -1 0 0 0 0 1
1
Y(12) 0 -1 0 0 0 1
1
Y(13) 1 0 0 -1 0 0
1
Y(14) 0 1 0 0 -1 0
1
Y(15) 0 0 1 0 0 -1
1
R* 1 1 1 1 1 1
0
P = LEFT-RIGHT BIAS
V FACTOR CELLS
1 1 1 1 1 1
1 0.3727 +
1 0.3727 +
1 0.3727 +
1 0.3727 +
1 0.3727 +
1 0.3727 +
2.3.4.3.5. Left-right balanced design for 4 references and 4 test items

2.3. Calibration
references and 4 test items
Design for 4 references and 4 test items.
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) + -
Y(5) + -
Y(6) + -
Y(7) + -
Y(8) + -
Y(9) - +
Y(10) - +
Y(11) - +
Y(12) - +
Y(13) - +
Y(14) - +
Y(15) - +
Y(16) - +
RESTRAINT + + + +
SOLUTION MATRIX
DIVISOR = 16
1 1 P
Y(1) 3 -1 -1 -1 -4 0
0 0 1
Y(2) 3 -1 -1 -1 0 0
-4 0 1
Y(3) -1 -1 3 -1 0 0
-4 0 1
Y(4) -1 -1 3 -1 -4 0
0 0 1
Y(5) -1 3 -1 -1 0 -4
0 0 1
Y(6) -1 3 -1 -1 0 0
0 -4 1
Y(7) -1 -1 -1 3 0 0
0 -4 1
Y(8) -1 -1 -1 3 0 -4
0 0 1
Y(9) -3 1 1 1 0 4
0 0 1
Y(10) -3 1 1 1 0 0
0 4 1
Y(11) 1 1 -3 1 0 0
0 4 1
Y(12) 1 1 -3 1 0 4
2.3.4.3.5. Left-right balanced design for 4 references and 4 test items
0 0 1
Y(13) 1 -3 1 1 4 0
0 0 1
Y(14) 1 -3 1 1 0 0
4 0 1
Y(15) 1 1 1 -3 0 0
4 0 1
Y(16) 1 1 1 -3 4 0
0 0 1
R* 4 4 4 4 4 4
4 4 0
R* = AVERAGE VALUE OF REFERENCE CELLS
P = ESTIMATE OF LEFT-RIGHT BIAS
V FACTORS CELLS
1 1 1 1 1 1 1 1
1 0.4330 +
1 0.4330 +
1 0.4330 +
1 0.4330 +
1 0.5000 +
1 0.5000 +
1 0.5000 +
1 0.5000 +
2.3.4.3.6. Design for 8 references and 8 test items

2.3. Calibration
2.3.4.3.6. Design for 8 references and 8 test
items
Design for 8 references and 8 test items.
TEST CELLS REFERENCE
CELLS
OBSERVATIONS 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1
Y(1) + -
Y(2) - +
Y(3) - +
Y(4) +
-
Y(5) +
-
Y(6) -
+
Y(7) -
+
Y(8) +
-
Y(9) + -
Y(10) + -
Y(11) - +
Y(12) -
+
Y(13) +
-
Y(14) +
-
Y(15) -
+
Y(16) -
+
RESTRAINT + + +
+ + + + +
SOLUTION MATRIX FOR TEST CELLS
DIVISOR = 16
1 1
Y(1) 8 4 0 -4 -6 6
2 -2
Y(2) -8 4 0 -4 -6 6
2 -2
Y(3) 4 -8 -4 0 2 6
-6 -2
Y(4) 4 8 -4 0 2 6
-6 -2
Y(5) 0 -4 8 4 2 -2
-6 6
Y(6) 0 -4 -8 4 2 -2
-6 6
Y(7) -4 0 4 -8 -6 -2
2 6
Y(8) -4 0 4 8 -6 -2
2 6
Y(9) -6 -2 2 6 8 -4
0 4
Y(10) -6 6 2 -2 -4 8
4 0
Y(11) -6 6 2 -2 -4 -8
4 0
Y(12) 2 6 -6 -2 0 4
-8 -4
Y(13) 2 6 -6 -2 0 4
8 -4
Y(14) 2 -2 -6 6 4 0
-4 8
Y(15) 2 -2 -6 6 4 0
-4 -8
Y(16) -6 -2 2 6 -8 -4
0 4
R 2 2 2 2 2 2
2 2
SOLUTION MATRIX FOR REFERENCE
CELLS
DIVISOR = 16
1 1 P
Y(1) -7 7 5 3 1 -1
-3 -5 1
Y(2) -7 7 5 3 1 -1
-3 -5 1
Y(3) 3 5 7 -7 -5 -3
-1 1 1
Y(4) 3 5 7 -7 -5 -3
-1 1 1
Y(5) 1 -1 -3 -5 -7 7
5 3 1
Y(6) 1 -1 -3 -5 -7 7
5 3 1
Y(7) -5 -3 -1 1 3 5
7 -7 1
Y(8) -5 -3 -1 1 3 5
7 -7 1
Y(9) -7 -5 -3 -1 1 3
5 7 1
Y(10) -5 -7 7 5 3 1
-1 -3 1
Y(11) -5 -7 7 5 3 1
-1 -3 1
Y(12) 1 3 5 7 -7 -5
-3 -1 1
Y(13) 1 3 5 7 -7 -5
-3 -1 1
Y(14) 3 1 -1 -3 -5 -7
7 5 1
Y(15) 3 1 -1 -3 -5 -7
7 5 1
Y(16) -7 -5 -3 -1 1 3
5 7 1
R* 2 2 2 2 2 2
2 2 0
P = ESTIMATE OF LEFT-RIGHT BIAS
FACTORS FOR COMPUTING STANDARD DEVIATIONS FOR TEST CELLS
V FACTORS TEST CELLS
1 1 1 1 1 1 1 1
1 1.1726 +
1 1.1726 +
1 1.1726 +
1 1.1726 +
1 1.1726 +
1 1.1726 +
1 1.1726 +
1 1.1726 +
2.3.4.3.7. Design for 4 reference zeners and 2 test zeners

2.3. Calibration
2.3.4.3.7. Design for 4 reference zeners and 2
test zeners
Design for 4 references zeners and 2 test zeners.
ZENERS
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) + -
Y(5) + -
Y(6) + -
Y(7) + -
Y(8) + -
Y(9) - +
Y(10) - +
Y(11) - +
Y(12) - +
Y(13) - +
Y(14) - +
Y(15) - +
Y(16) - +
RESTRAINT + + + +
CHECK STANDARD + -
SOLUTION MATRIX
DIVISOR = 16
P
Y(1) 3 -1 -1 -1 -2 0
1
Y(2) 3 -1 -1 -1 0 -2
1
Y(3) -1 3 -1 -1 -2 0
1
Y(4) -1 3 -1 -1 0 -2
1
Y(5) -1 -1 3 -1 -2 0
1
Y(6) -1 -1 3 -1 0 -2
1
Y(7) -1 -1 -1 3 -2 0
1
Y(8) -1 -1 -1 3 0 -2
1
Y(9) 1 1 1 -3 2 0
1
Y(10) 1 1 1 -3 0 2
1
Y(11) 1 1 -3 1 2 0
1
Y(12) 1 1 -3 1 0 2
1
Y(13) 1 -3 1 1 2 0
1
Y(14) 1 -3 1 1 0 2
1
Y(15) -3 1 1 1 2 0
1
Y(16) -3 1 1 1 0 2
1
R* 4 4 4 4 4 4
0
R* = AVERAGE VALUE OF 4 REFERENCE STANDARDS
P = LEFT-RIGHT EFFECT

V FACTORS ZENERS
1 1 1 1 1 1 P
1 0.4330 +
1 0.4330 +
1 0.4330 +
1 0.4330 +
1 0.3536 +
1 0.3536 +
1 0.2500 +

2.3. Calibration
2.3.4.3.8. Design for 4 reference zeners and 3
test zeners
Design for 4 references and 3 test zeners.
ZENERS
Y(1) - +
Y(2) - +
Y(3) + -
Y(4) + -
Y(5) + -
Y(6) + -
Y(7) - +
Y(8) - +
Y(9) - +
Y(10) - +
Y(11) - +
Y(12) - +
Y(13) + -
Y(14) + -
Y(15) + -
Y(16) + -
Y(17) + -
Y(18) - +
RESTRAINT + + + +
CHECK STANDARD + -
SOLUTION MATRIX
DIVISOR = 1260
1 P
Y(1) -196 196 -56 56 0 0
0 70
Y(2) -160 -20 160 20 0 0
0 70
Y(3) 20 160 -20 -160 0 0
0 70
Y(4) 143 -53 -17 -73 0 0 -
315 70
Y(5) 143 -53 -17 -73 0 -315
0 70
Y(6) 143 -53 -17 -73 -315 0
0 70
Y(7) 53 -143 73 17 315 0
0 70
Y(8) 53 -143 73 17 0 315
0 70
Y(9) 53 -143 73 17 0 0
315 70
Y(10) 17 73 -143 53 0 0
315 70
Y(11) 17 73 -143 53 0 315
0 70
Y(12) 17 73 -143 53 315 0
0 70
Y(13) -73 -17 -53 143 -315 0
0 70
Y(14) -73 -17 -53 143 0 -315
0 70
Y(15) -73 -17 -53 143 0 0 -
315 70
Y(16) 56 -56 196 -196 0 0
0 70
Y(17) 20 160 -20 -160 0 0
0 70
Y(18) -160 -20 160 20 0 0
0 70
R* 315 315 315 315 315 315
315 0
R* = Average value of the 4 reference zeners
P = left-right effect
V K1 1 1 1 1 1 1 1
1 0.5000 +
1 0.5000 +
1 0.5000 +
2 0.7071 + +
3 0.8660 + + +
0 0.5578 + -

2.3.4.3.9. Design for 3 references and 1 test resistor

2.3. Calibration
resistor
Design 1,1,1,1
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) - +
Y(5) - +
Y(6) - +
RESTRAINT + + +
SOLUTION MATRIX
DIVISOR = 6
Y(1) 1 -2 1 1
Y(2) 1 1 -2 1
Y(3) 0 0 0 -3
Y(4) 0 0 0 3
Y(5) -1 -1 2 -1
Y(6) -1 2 -1 -1
R 2 2 2 2
R = AVERAGE VALUE OF 3 REFERENCE RESISTORS
OHM FACTORS RESISTORS
1 1 1 1
1 0.3333 +
1 0.5270 +
1 0.5270 +
1 0.7817 +
2.3.4.3.10. Design for 4 references and 1 test resistor

2.3. Calibration
resistor
Design 1,1,1,1,1
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) + -
Y(5) - +
Y(6) - +
Y(7) - +
Y(8) - +
RESTRAINT + + + +
SOLUTION MATRIX
DIVISOR = 8
Y(1) 3 -1 -1 -1 -1
Y(2) -1 3 -1 -1 -1
Y(3) -1 -1 3 -1 -1
Y(4) -1 -1 -1 3 -1
Y(5) 1 1 1 -3 1
Y(6) 1 1 -3 1 1
Y(7) 1 -3 1 1 1
Y(8) -3 1 1 1 1
R 2 2 2 2 2
R = AVERAGE VALUE OF REFERENCE RESISTORS
OHM FACTORS
1 1 1 1 1
1 0.6124 +
1 0.6124 +
1 0.6124 +
1 0.6124 +
1 0.3536 +
2.3.4.4. Roundness measurements

2.3. Calibration
Roundness
measurements
Measurements of roundness require 360 traces of the workpiece made
with a turntable-type instrument or a stylus-type instrument. A least
squares fit of points on the trace to a circle define the parameters of
noncircularity of the workpiece. A diagram of the measurement method
is shown below.
The diagram
shows the
trace and Y,
the distance
from the
spindle center
to the trace at
the angle.
A least
squares circle
fit to data at
equally
spaced angles
gives
estimates of P
- R, the
noncircularity,
where R =
radius of the
circle and P =
distance from
the center of
the circle to
the trace.
Low precision
measurements
Some measurements of roundness do not require a high level of
precision, such as measurements on cylinders, spheres, and ring gages
where roundness is not of primary importance. For this purpose, a
single trace is made of the workpiece.
Weakness of
single trace
method
The weakness of this method is that the deviations contain both the
spindle error and the workpiece error, and these two errors cannot be
separated with the single trace. Because the spindle error is usually
small and within known limits, its effect can be ignored except when
the most precise measurements are needed.
High precision
measurements
High precision measurements of roundness are appropriate where an
object, such as a hemisphere, is intended to be used primarily as a
roundness standard.
Measurement
method
The measurement sequence involves making multiple traces of the
roundness standard where the standard is rotated between traces. Least-
squares analysis of the resulting measurements enables the
noncircularity of the spindle to be separated from the profile of the
standard.
Choice of
measurement
method
A synopsis of the measurement method and the estimation technique
are given in this chapter for:
Single-trace method
Multiple-trace method
The reader is encouraged to obtain a copy of the publication on
roundness (Reeve) for a more complete description of the measurement
method and analysis.
2.3.4.4.1. Single-trace roundness design

2.3. Calibration
Low precision
measurements
Some measurements of roundness do not require a high
level of precision, such as measurements on cylinders,
spheres, and ring gages where roundness is not of primary
importance. The diagram of the measurement method
shows the trace and Y, the distance from the spindle center
to the trace at the angle. A least-squares circle fit to data at
equally spaced angles gives estimates of P - R, the
noncircularity, where R = radius of the circle and P =
distance from the center of the circle to the trace.
Single trace
method
For this purpose, a single trace covering exactly 360 is
made of the workpiece and measurements at angles
of the distance between the center of the spindle and the
trace, are made at
equally spaced angles. A least-squares circle fit to the data
gives the following estimators of the parameters of the
circle.
.
Noncircularity
of workpiece
The deviation of the trace from the circle at angle ,
which defines the noncircularity of the workpiece, is
estimated by:
Weakness of
single trace
method
The weakness of this method is that the deviations contain
both the spindle error and the workpiece error, and these
two errors cannot be separated with the single trace.
Because the spindle error is usually small and within
known limits, its effect can be ignored except when the
most precise measurements are needed.
2.3.4.4.2. Multiple-trace roundness designs

2.3. Calibration
High
precision
measurements
High precision roundness measurements are required when
an object, such as a hemisphere, is intended to be used
primarily as a roundness standard. The method outlined on
this page is appropriate for either a turntable-type
instrument or a spindle-type instrument.
Measurement
method
The measurement sequence involves making multiple
traces of the roundness standard where the standard is
rotated between traces. Least-squares analysis of the
resulting measurements enables the noncircularity of the
spindle to be separated from the profile of the standard.
The reader is referred to the publication on the subject
(Reeve) for details covering measurement techniques and
analysis.
Method of n
traces
The number of traces that are made on the workpiece is
arbitrary but should not be less than four. The workpiece is
centered as well as possible under the spindle. The mark on
the workpiece which denotes the zero angular position is
aligned with the zero position of the spindle as shown in
the graph. A trace is made with the workpiece in this
position. The workpiece is then rotated clockwise by 360/n
degrees and another trace is made. This process is
continued until n traces have been recorded.
Mathematical
model for
estimation
For i = 1,...,n, the ith angular position is denoted by
Definition of
terms relating
to distances
to the least
squares circle
The deviation from the least squares circle (LSC) of the
workpiece at the position is .
The deviation of the spindle from its LSC at the
position is .
Terms
relating to
For the jth graph, let the three parameters that define the
LSC be given by
parameters of
least squares
circle
defining the radius R, a, and b as shown in the graph. In an
idealized measurement system these parameters would be
constant for all j. In reality, each rotation of the workpiece
causes it to shift a small amount vertically and horizontally.
To account for this shift, separate parameters are needed
for each trace.
Correction
for
obstruction to
stylus
Let be the observed distance (in polar graph units) from
the center of the jth graph to the point on the curve that
corresponds to the position of the spindle. If K is the
magnification factor of the instrument in microinches/polar
graph unit and is the angle between the lever arm of the
stylus and the tangent to the workpiece at the point of
contact (which normally can be set to zero if there is no
obstruction), the transformed observations to be used in the
estimation equations are:
.
Estimates for
parameters
The estimation of the individual parameters is obtained as a
least-squares solution that requires six restraints which
essentially guarantee that the sum of the vertical and
horizontal deviations of the spindle from the center of the
LSC are zero. The expressions for the estimators are as
follows:
where
Finally, the standard deviations of the profile estimators are
given by:
Computation
of standard
deviation
The computation of the residual standard deviation of the
fit requires, first, the computation of the predicted values,
The residual standard deviation with v = n*n - 5n + 6
2.3.4.5. Designs for angle blocks

2.3. Calibration
Purpose The purpose of this section is to explain why calibration of angle
blocks of the same size in groups is more efficient than calibration
of angle blocks individually.
Calibration
schematic for
five angle
blocks
showing the
reference as
block 1 in the
center of the
diagram, the
check
standard as
block 2 at the
top; and the
test blocks as
blocks 3, 4,
and 5.
A schematic of a calibration scheme for 1 reference block, 1 check
standard, and three test blocks is shown below. The reference
block, R, is shown in the center of the diagram and the check
standard, C, is shown at the top of the diagram.
Block sizes Angle blocks normally come in sets of
1, 3, 5, 20, and 30 seconds
1, 3, 5, 20, 30 minutes
1, 3, 5, 15, 30, 45 degrees
and blocks of the same nominal size from 4, 5 or 6 different sets
can be calibrated simultaneously using one of the designs shown in
this catalog.
Design for 4 angle blocks
Restraint The solution to the calibration design depends on the known value
of a reference block, which is compared with the test blocks. The
reference block is designated as block 1 for the purpose of this
discussion.
Check
standard
It is suggested that block 2 be reserved for a check standard that is
maintained in the laboratory for quality control purposes.
Calibration
scheme
A calibration scheme developed by Charles Reeve (Reeve) at the
National Institute of Standards and Technology for calibrating
customer angle blocks is explained on this page. The reader is
encouraged to obtain a copy of the publication for details on the
calibration setup and quality control checks for angle block
calibrations.
Series of
measurements
for calibrating
4, 5, and 6
angle blocks
simultaneously
For all of the designs, the measurements are made in groups of
seven starting with the measurements of blocks in the following
order: 2-3-2-1-2-4-2. Schematically, the calibration design is
completed by counter-clockwise rotation of the test blocks about
the reference block, one-at-a-time, with 7 readings for each series
reduced to 3 difference measurements. For n angle blocks
(including the reference block), this amounts to n - 1 series of 7
readings. The series for 4, 5, and 6 angle blocks are shown below.
Measurements
for 4 angle
blocks
Series 1: 2-3-2-1-2-4-2
Series 2: 4-2-4-1-4-3-4
Series 3: 3-4-3-1-3-2-3
Measurements
for 5 angle
blocks (see
diagram)
Series 1: 2-3-2-1-2-4-2
Series 2: 5-2-5-1-5-3-5
Series 3: 4-5-4-1-4-2-4
Series 4: 3-4-3-1-3-5-3
Measurements
for 6 angle
blocks
Series 1: 2-3-2-1-2-4-2
Series 2: 6-2-6-1-6-3-6
Series 3: 5-6-5-1-5-2-5
Series 4: 4-5-4-1-4-6-4
Series 5: 3-4-3-1-3-5-3
Equations for
the
measurements
in the first
series showing
error sources
The equations explaining the seven measurements for the first
series in terms of the errors in the measurement system are:
Z
11
= B + X
1
+ error
11

Z
12
= B + X
2
+ d + error
12

Z
13
= B + X
3
+ 2d + error
13

Z
14
= B + X
4
+ 3d + error
14

Z = B + X + 4d + error
15 5 15
Z
16
= B + X
6
+ 5d + error
16

Z
17
= B + X
7
+ 6d + error
17

with B a bias associated with the instrument, d is a linear drift
factor, X is the value of the angle block to be determined; and the
error terms relate to random errors of measurement.
Calibration
procedure
depends on
difference
measurements
The check block, C, is measured before and after each test block,
and the difference measurements (which are not the same as the
difference measurements for calibrations of mass weights, gage
blocks, etc.) are constructed to take advantage of this situation.
Thus, the 7 readings are reduced to 3 difference measurements for
the first series as follows:
For all series, there are 3(n - 1) difference measurements, with the
first subscript in the equations above referring to the series number.
The difference measurements are free of drift and instrument bias.
Design matrix As an example, the design matrix for n = 4 angle blocks is shown
below.
1 1 1 1
0 1 -1 0
-1 1 0 0
0 1 0 -1
0 -1 0 1
-1 0 0 1
0 0 -1 1
0 0 1 -1
-1 0 1 0
0 -1 1 0
The design matrix is shown with the solution matrix for
identification purposes only because the least-squares solution is
weighted (Reeve) to account for the fact that test blocks are
measured twice as many times as the reference block. The weight
matrix is not shown.
Solutions to
the calibration
designs
measurements
Solutions to the angle block designs are shown on the following
pages. The solution matrix and factors for the repeatability standard
deviation are to be interpreted as explained in solutions to
calibration designs . As an example, the solution for the design for
n=4 angle blocks is as follows:
The solution for the reference standard is shown under the first
column of the solution matrix; for the check standard under the
second column; for the first test block under the third column; and
for the second test block under the fourth column. Notice that the
estimate for the reference block is guaranteed to be R*, regardless
of the measurement results, because of the restraint that is imposed
on the design. Specifically,
Solutions are correct only for the restraint as shown.
Calibrations
can be run for
top and
bottom faces
of blocks
The calibration series is run with the blocks all face "up" and is
then repeated with the blocks all face "down", and the results
averaged. The difference between the two series can be large
compared to the repeatability standard deviation, in which case a
between-series component of variability must be included in the
calculation of the standard deviation of the reported average.
Calculation of
standard
deviations
when the
blocks are
measured in
two
orientations
For n blocks, the differences between the values for the blocks
measured in the top ( denoted by "t") and bottom (denoted by "b")
positions are denoted by:
The standard deviation of the average (for each block) is calculated
from these differences to be:
Standard
deviations
when the
blocks are
measured in
only one
If the blocks are measured in only one orientation, there is no way
to estimate the between-series component of variability and the
standard deviation for the value of each block is computed as
s
test
= K
1
s
1
orientation where K
1
is shown under "Factors for computing repeatability
standard deviations" for each design and is the repeatability
standard deviation as estimated from the design. Because this
standard deviation may seriously underestimate the uncertainty, a
better approach is to estimate the standard deviation from the data
on the check standard over time. An expanded uncertainty is
computed according to the ISO guidelines.
2.3.4.5.1. Design for 4 angle blocks

2.3. Calibration
DESIGN MATRIX
1 1 1 1
Y(1) 0 1 -1 0
Y(2) -1 1 0 0
Y(3) 0 1 0 -1
Y(4) 0 -1 0 1
Y(5) -1 0 0 1
Y(6) 0 0 -1 1
Y(7) 0 0 1 -1
Y(8) -1 0 1 0
Y(9) 0 -1 1 0
REFERENCE +
CHECK STANDARD +

SOLUTION MATRIX
DIVISOR = 24
Y(11) 0 2.2723000 -5.0516438 -
1.2206578
Y(12) 0 9.3521166 7.3239479
7.3239479
Y(13) 0 2.2723000 -1.2206578 -
5.0516438
Y(21) 0 -5.0516438 -1.2206578
2.2723000
Y(22) 0 7.3239479 7.3239479
9.3521166
Y(23) 0 -1.2206578 -5.0516438
2.2723000
Y(31) 0 -1.2206578 2.2723000 -
5.0516438
Y(32) 0 7.3239479 9.3521166
7.3239479
Y(33) 0 -5.0516438 2.2723000 -
1.2206578
R* 1 1. 1. 1.
R* = VALUE OF REFERENCE ANGLE BLOCK
SIZE K1
1 1 1 1
1 0.0000 +
1 0.9749 +
1 0.9749 +
1 0.9749 +
1 0.9749 +

2.3. Calibration
DESIGN MATRIX
1 1 1 1 1

0 1 -1 0 0
-1 1 0 0 0
0 1 0 -1 0
0 -1 0 0 1
-1 0 0 0 1
0 0 -1 0 1
0 0 0 1 -1
-1 0 0 1 0
0 -1 0 1 0
0 0 1 -1 0
-1 0 1 0 0
0 0 1 0 -1

REFERENCE +
CHECK STANDARD +

SOLUTION MATRIX
DIVISOR = 24
Y(11) 0.00000 3.26463 -5.48893 -0.21200 -1.56370
Y(12) 0.00000 7.95672 5.38908 5.93802 4.71618
Y(13) 0.00000 2.48697 -0.89818 -4.80276 -0.78603
Y(21) 0.00000 -5.48893 -0.21200 -1.56370 3.26463
Y(22) 0.00000 5.38908 5.93802 4.71618 7.95672
Y(23) 0.00000 -0.89818 -4.80276 -0.78603 2.48697
Y(31) 0.00000 -0.21200 -1.56370 3.26463 -5.48893
Y(32) 0.00000 5.93802 4.71618 7.95672 5.38908
Y(33) 0.00000 -4.80276 -0.78603 2.48697 -0.89818
Y(41) 0.00000 -1.56370 3.26463 -5.48893 -0.21200
Y(42) 0.00000 4.71618 7.95672 5.38908 5.93802
Y(43) 0.00000 -0.78603 2.48697 -0.89818 -4.80276
R* 1. 1. 1. 1. 1.
SIZE K1
1 1 1 1 1
1 0.0000 +
1 0.7465 +
1 0.7465 +
1 0.7456 +
1 0.7456 +
1 0.7465 +

2.3. Calibration
DESIGN MATRIX
1 1 1 1 1 1

0 1 -1 0 0 0
-1 1 0 0 0 0
0 1 0 -1 0 0
0 -1 0 0 0 1
-1 0 0 0 0 1
0 0 -1 0 0 1
0 0 0 0 1 -1
-1 0 0 0 1 0
0 -1 0 0 1 0
0 0 0 1 -1 0
-1 0 0 1 0 0
0 0 0 1 0 -1
0 0 1 -1 0 0
-1 0 1 0 0 0
0 0 1 0 -1 0

REFERENCE +
CHECK STANDARD +

SOLUTION MATRIX
DIVISOR = 24
1
Y(11) 0.0000 3.2929 -5.2312 -0.7507 -0.6445
-0.6666
Y(12) 0.0000 6.9974 4.6324 4.6495 3.8668
3.8540
Y(13) 0.0000 3.2687 -0.7721 -5.2098 -0.6202
-0.6666
Y(21) 0.0000 -5.2312 -0.7507 -0.6445 -0.6666
3.2929
Y(22) 0.0000 4.6324 4.6495 3.8668 3.8540
6.9974
Y(23) 0.0000 -0.7721 -5.2098 -0.6202 -0.6666
3.2687
Y(31) 0.0000 -0.7507 -0.6445 -0.6666 3.2929
-5.2312
Y(32) 0.0000 4.6495 3.8668 3.8540 6.9974
4.6324
Y(33) 0.0000 -5.2098 -0.6202 -0.6666 3.2687
-0.7721
Y(41) 0.0000 -0.6445 -0.6666 3.2929 -5.2312
-0.7507
Y(42) 0.0000 3.8668 3.8540 6.9974 4.6324
4.6495
Y(43) 0.0000 -0.6202 -0.6666 3.2687 -0.7721
-5.2098
Y(51) 0.0000 -0.6666 3.2929 -5.2312 -0.7507
-0.6445
Y(52) 0.0000 3.8540 6.9974 4.6324 4.6495
3.8668
Y(53) 0.0000 -0.6666 3.2687 -0.7721 -5.2098
-0.6202
R* 1. 1. 1. 1. 1.
1.
SIZE K1
1 1 1 1 1 1
1 0.0000 +
1 0.7111 +
1 0.7111 +
1 0.7111 +
1 0.7111 +
1 0.7111 +
1 0.7111 +
2.3.4.6. Thermometers in a bath

2.3. Calibration
Measurement
sequence
Calibration of liquid in glass thermometers is usually carried
out in a controlled bath where the temperature in the bath is
increased steadily over time to calibrate the thermometers
over their entire range. One way of accounting for the
temperature drift is to measure the temperature of the bath
with a standard resistance thermometer at the beginning,
middle and end of each run of K test thermometers. The test
thermometers themselves are measured twice during the run
in the following time sequence:
where R
1
, R
2
, R
3
represent the measurements on the
standard resistance thermometer and T
1
, T
2
, ... , T
K
and T'
1
,
T'
2
, ... , T'
K
represent the pair of measurements on the K test
thermometers.
Assumptions
regarding
temperature
The assumptions for the analysis are that:
Equal time intervals are maintained between
measurements on the test items.
Temperature increases by with each interval.
A temperature change of is allowed for the reading
of the resistance thermometer in the middle of the
run.
Indications
for test
thermometers
It can be shown (Cameron and Hailes) that the average
reading for a test thermometer is its indication at the
temperature implied by the average of the three resistance
readings. The standard deviation associated with this
indication is calculated from difference readings where
is the difference for the ith thermometer. This difference is
an estimate of .
Estimates of
drift
The estimates of the shift due to the resistance thermometer
and temperature drift are given by:
Standard
deviations
The residual variance is given by
.
The standard deviation of the indication assigned to the ith
test thermometer is
and the standard deviation for the estimates of shift and
drift are
respectively.
2.3.4.7. Humidity standards

2.3. Calibration
Humidity
standards
The calibration of humidity standards usually involves the
comparison of reference weights with cylinders containing
moisture. The designs shown in this catalog are drift-
eliminating and may be suitable for artifacts other than
humidity cylinders.
List of
designs
2 reference weights and 3 cylinders
2.3.4.7.1. Drift-elimination design for 2 reference weights and 3 cylinders

2.3. Calibration
2.3.4.7.1. Drift-elimination design for 2
reference weights and 3 cylinders
Y(1) + -
Y(2) + -
Y(3) + -
Y(4) + -
Y(5) - +
Y(6) - +
Y(7) + -
Y(8) + -
Y(9) - +
Y(10) + -
RESTRAINT + +
CHECK STANDARD + -
SOLUTION MATRIX
DIVISOR = 10
Y(1) 2 -2 0 0 0
Y(2) 0 0 0 2 -2
Y(3) 0 0 2 -2 0
Y(4) -1 1 -3 -1 -1
Y(5) -1 1 1 1 3
Y(6) -1 1 1 3 1
Y(7) 0 0 2 0 -2
Y(8) -1 1 -1 -3 -1
Y(9) 1 -1 1 1 3
Y(10) 1 -1 -3 -1 -1
R* 5 5 5 5 5
R* = average value of the two reference weights

WT K1 1 1 1 1 1
1 0.5477 +
1 0.5477 +
1 0.5477 +
2 0.8944 + +
3 1.2247 + + +
0 0.6325 + -
2.3.4.7.1. Drift-elimination design for 2 reference weights and 3 cylinders
2.3.5. Control of artifact calibration

2.3. Calibration
Purpose The purpose of statistical control in the calibration process is
to guarantee the 'goodness' of calibration results within
predictable limits and to validate the statement of uncertainty
of the result. Two types of control can be imposed on a
calibration process that makes use of statistical designs:
1. Control of instrument precision or short-term variability
Example of a Shewhart control chart
Example of an EWMA control chart
Short-term
standard
deviation
The short-term standard deviation from each design is the
basis for controlling instrument precision. Because the
measurements for a single design are completed in a short
time span, this standard deviation estimates the basic precision
of the instrument. Designs should be chosen to have enough
measurements so that the standard deviation from the design
has at least 3 degrees of freedom where the degrees of
freedom are (n - m + 1) with
n = number of difference measurements
m = number of artifacts.
Check
standard
Measurements on a check standard provide the mechanism for
controlling the bias and long-term variability of the calibration
process. The check standard is treated as one of the test items
in the calibration design, and its value as computed from each
calibration run is the basis for accepting or rejecting the
calibration. All designs cataloged in this Handbook have
provision for a check standard.
The check standard should be of the same type and geometry
as items that are measured in the designs. These artifacts must
be stable and available to the calibration process on a
continuing basis. There should be a check standard at each
critical level of measurement. For example, for mass
calibrations there should be check standards at the 1 kg; 100 g,
10 g, 1 g, 0.1 g levels, etc. For gage blocks, there should be
check standards at all nominal lengths.
A check standard can also be a mathematical construction,
such as the computed difference between the calibrated values
of two reference standards in a design.
Database
of check
standard
values
The creation and maintenance of the database of check
standard values is an important aspect of the control process.
The results from each calibration run are recorded in the
database. The best way to record this information is in one file
with one line (row in a spreadsheet) of information in fixed
fields for each calibration run. A list of typical entries follows:
1. Date
2. Identification for check standard
3. Identification for the calibration design
4. Identification for the instrument
6. Repeatability standard deviation from design
9. Flag for out-of-control signal
2.3.5.1. Control of precision

2.3. Calibration
Control
parameters
from
historical
data
A modified control chart procedure is used for controlling
instrument precision. The procedure is designed to be
implemented in real time after a baseline and control limit for
the instrument of interest have been established from the
database of short-term standard deviations. A separate control
chart is required for each instrument -- except where
instruments are of the same type with the same basic
precision, in which case they can be treated as one.
The baseline is the process standard deviation that is pooled
from k = 1, ..., K individual repeatability standard deviations,
, in the database, each having degrees of freedom. The
pooled repeatability standard deviation is
with degrees of freedom
.
Control
procedure
is invoked
in real-
time for
each
calibration
run
The control procedure compares each new repeatability
standard deviation that is recorded for the instrument with an
upper control limit, UCL. Usually, only the upper control limit
is of interest because we are primarily interested in detecting
degradation in the instrument's precision. A possible
complication is that the control limit is dependent on the
degrees of freedom in the new standard deviation and is
computed as follows:
.
The quantity under the radical is the upper percentage point
from the F table where is chosen small to be, say, 0.05. The
other two terms refer to the degrees of freedom in the new
standard deviation and the degrees of freedom in the process
standard deviation.
Limitation
of
graphical
method
The graphical method of plotting every new estimate of
repeatability on a control chart does not work well when the
UCL can change with each calibration design, depending on
the degrees of freedom. The algebraic equivalent is to test if
the new standard deviation exceeds its control limit, in which
case the short-term precision is judged to be out of control
and the current calibration run is rejected. For more guidance,
see Remedies and strategies for dealing with out-of-control
signals.
As long as the repeatability standard deviations are in control,
there is reason for confidence that the precision of the
instrument has not degraded.
Case
study:
Mass
balance
precision
It is recommended that the repeatability standard deviations be
plotted against time on a regular basis to check for gradual
degradation in the instrument. Individual failures may not
trigger a suspicion that the instrument is in need of adjustment
or tuning.
2.3.5.1.1. Example of control chart for precision

2.3. Calibration
Example of a
control chart
for precision
of a mass
balance
Mass calibrations usually start with the comparison of kilograms standards using a high
precision balance as a comparator. Many of the measurements at the kilogram level that were
made at NIST between 1975 and 1989 were made on balance #12 using a 1,1,1,1 calibration
design. The redundancy in the calibration design produces estimates for the individual
kilograms and a repeatability standard deviation with three degrees of freedom for each
calibration run. These standard deviations estimate the precision of the balance.
Need for
monitoring
precision
The precision of the balance is monitored to check for:
1. Slow degradation in the balance
2. Anomalous behavior at specific times
Monitoring
technique for
standard
deviations
The standard deviations over time and many calibrations are tracked and monitored using a
control chart for standard deviations. The database and control limits are updated on a yearly
or bi-yearly basis and standard deviations for each calibration run in the next cycle are
compared with the control limits. In this case, the standard deviations from 117 calibrations
between 1975 and 1985 were pooled to obtain a repeatability standard deviation with v =
3*117 = 351 degrees of freedom, and the control limits were computed at the 1 %
significance level.
Control chart
for precision
The following control chart for precision for balance #12 can be generated using both
Dataplot code and R code.
Interpretation
of the control
chart
The control chart shows that the precision of the balance remained in control through the first
five months of 1988 with only two violations of the control limits. For those occasions, the
calibrations were discarded and repeated. Clearly, for the second violation, something
significant occurred that invalidated the calibration results.
Further
interpretation
of the control
chart
However, it is also clear from the pattern of standard deviations over time that the precision of
the balance was gradually degrading and more and more points were approaching the control
limits. This finding led to a decision to replace this balance for high accuracy calibrations.
2.3.5.2. Control of bias and long-term variability

2.3. Calibration
2.3.5.2. Control of bias and long-term
variability
Control
parameters
are estimated
using
historical
data
A control chart procedure is used for controlling bias and
long-term variability. The procedure is designed to be
implemented in real time after a baseline and control limits
for the check standard of interest have been established
from the database of check standard values. A separate
control chart is required for each check standard. The
control procedure outlined here is based on a Shewhart
control chart with upper and lower control limits that are
symmetric about the average. The EWMA control
procedure that is sensitive to small changes in the process is
discussed on another page.
For a
Shewhart
control
procedure,
the average
and standard
deviation of
historical
check
standard
values are
the
parameters of
interest
The check standard values are denoted by
The baseline is the process average which is computed from
the check standard values as
The process standard deviation is
with K - 1 degrees of freedom.
The control
limits depend
on the t
distribution
and the
degrees of
freedom in
the process
standard
deviation
If has been computed from historical data, the upper and
lower control limits are:
where t
1-/2, K-1
denotes the 1-/2 critical value from the t
table with v = K - 1 degrees of freedom.
Sample code Sample code for computing the t value for a conservative
case where = 0.05, J = 6, and K = 6, is available for both
Dataplot and R.
Simplification
for large
degrees of
freedom
It is standard practice to use a value of 3 instead of a
critical value from the t table, given the process standard
deviation has large degrees of freedom, say, v > 15.
The control
procedure is
invoked in
real-time and
a failure
implies that
the current
calibration
should be
rejected
The control procedure compares the check standard value,
C, from each calibration run with the upper and lower
control limits. This procedure should be implemented in
real time and does not necessarily require a graphical
presentation. The check standard value can be compared
algebraically with the control limits. The calibration run is
judged to be out-of-control if either:
C > UCL
or
C < LCL
Actions to be
taken
If the check standard value exceeds one of the control
limits, the process is judged to be out of control and the
current calibration run is rejected. The best strategy in this
situation is to repeat the calibration to see if the failure was
a chance occurrence. Check standard values that remain in
control, especially over a period of time, provide
confidence that no new biases have been introduced into the
measurement process and that the long-term variability of
the process has not changed.
Out-of-
control
signals that
recur require
investigation
Out-of-control signals, particularly if they recur, can be
symptomatic of one of the following conditions:
Change or damage to the reference standard(s)
Change or damage to the check standard
Change in the long-term variability of the calibration
process
For more guidance, see Remedies and strategies for dealing
with out-of-control signals.
Caution - be
sure to plot
the data
If the tests for control are carried out algebraically, it is
recommended that, at regular intervals, the check standard
values be plotted against time to check for drift or
anomalies in the measurement process.
2.3.5.2.1. Example of Shewhart control chart for mass calibrations

2.3. Calibration
Example of a
control chart
for mass
calibrations
at the
kilogram
level
Mass calibrations usually start with the comparison of four kilogram standards using a high
precision balance as a comparator. Many of the measurements at the kilogram level that were
made at NIST between 1975 and 1989 were made on balance #12 using a 1,1,1,1 calibration
design. The restraint for this design is the known average of two kilogram reference standards.
The redundancy in the calibration design produces individual estimates for the two test
kilograms and the two reference standards.
Check
standard
There is no slot in the 1,1,1,1 design for an artifact check standard when the first two
kilograms are reference standards; the third kilogram is a test weight; and the fourth is a
summation of smaller weights that act as the restraint in the next series. Therefore, the check
standard is a computed difference between the values of the two reference standards as
estimated from the design. The convention with mass calibrations is to report the correction to
nominal, in this case the correction to 1000 g, as shown in the control charts below.
Need for
monitoring
The kilogram check standard is monitored to check for:
1. Long-term degradation in the calibration process
2. Anomalous behavior at specific times
Monitoring
technique for
check
standard
values
Check standard values over time and many calibrations are tracked and monitored using a
Shewhart control chart. The database and control limits are updated when needed and check
standard values for each calibration run in the next cycle are compared with the control limits.
In this case, the values from 117 calibrations between 1975 and 1985 were averaged to obtain
a baseline and process standard deviation with v = 116 degrees of freedom. Control limits are
computed with a factor of
k = 3 to identify truly anomalous data points.
Control chart
of kilogram
check
standard
measurements
showing a
change in the
process after
1985
Interpretation
of the control
chart
The control chart shows only two violations of the control limits. For those occasions, the
calibrations were discarded and repeated. The configuration of points is unacceptable if many
points are close to a control limit and there is an unequal distribution of data points on the two
sides of the control chart -- indicating a change in either:
process average which may be related to a change in the reference standards
or
variability which may be caused by a change in the instrument precision or may be the
result of other factors on the measurement process.
Small
changes only
become
obvious over
time
Unfortunately, it takes time for the patterns in the data to emerge because individual violations
of the control limits do not necessarily point to a permanent shift in the process. The Shewhart
control chart is not powerful for detecting small changes, say of the order of at most one
standard deviation, which appears to be approximately the case in this application. This level
of change might seem insignificant, but the calculation of uncertainties for the calibration
process depends on the control limits.
Re-
establishing
the limits
based on
recent data
and EWMA
option
If the limits for the control chart are re-calculated based on the data after 1985, the extent of
the change is obvious. Because the exponentially weighted moving average (EWMA) control
chart is capable of detecting small changes, it may be a better choice for a high precision
process that is producing many control values.
Revised
control chart
based on
check
standard
measurements
after 1985
Sample code The original and revised Shewhart control charts can be generated using both Dataplot code
and R code.
2.3.5.2.2. Example of EWMA control chart for mass calibrations

2.3. Calibration
Small
changes only
become
obvious over
time
Unfortunately, it takes time for the patterns in the data to emerge because individual violations
of the control limits do not necessarily point to a permanent shift in the process. The Shewhart
control chart is not powerful for detecting small changes, say of the order of at most one
standard deviation, which appears to be the case for the calibration data shown on the
previous page. The EWMA (exponentially weighted moving average) control chart is better
suited for this purpose.
Explanation
of EWMA
statistic at
the kilogram
level
The exponentially weighted moving average (EWMA) is a statistic for monitoring the process
that averages the data in a way that gives less and less weight to data as they are further
removed in time from the current measurement. The EWMA statistic at time t is computed
recursively from individual data points which are ordered in time to be
where the first EWMA statistic is the average of historical data.
Control
mechanism
for EWMA
The EWMA control chart can be made sensitive to small changes or a gradual drift in the
process by the choice of the weighting factor, . A weighting factor between 0.2 - 0.3 has
been suggested for this purpose (Hunter), and 0.15 is another popular choice.
Limits for the
control chart
The target or center line for the control chart is the average of historical data. The upper
(UCL) and lower (LCL) limits are
where s is the standard deviation of the historical data; the function under the radical is a good
approximation to the component of the standard deviation of the EWMA statistic that is a
function of time; and k is the multiplicative factor, defined in the same manner as for the
Shewhart control chart, which is usually taken to be 3.
Example of
EWMA chart
for check
The target (average) and process standard deviation are computed from the check standard
data taken prior to 1985. The computation of the EWMA statistic begins with the data taken at
the start of 1985. In the control chart below, the control data after 1985 are shown in green,
standard data
for kilogram
calibrations
showing
multiple
violations of
the control
limits for the
EWMA
statistics
and the EWMA statistics are shown as black dots superimposed on the raw data. The control
limits are calculated according to the equation above where the process standard deviation, s
= 0.03065 mg and k = 3. The EWMA statistics, and not the raw data, are of interest in looking
for out-of-control signals. Because the EWMA statistic is a weighted average, it has a smaller
standard deviation than a single control measurement, and, therefore, the EWMA control
limits are narrower than the limits for a Shewhart control chart.
The EWMA control chart for mass calibrations can be generated using both Dataplot code and
R code.
Interpretation
of the control
chart
The EWMA control chart shows many violations of the control limits starting at
approximately the mid-point of 1986. This pattern emerges because the process average has
actually shifted about one standard deviation, and the EWMA control chart is sensitive to
small changes.
2.3.6. Instrument calibration over a regime

2.3. Calibration
Topics This section discusses the creation of a calibration curve for calibrating instruments (gauges)
whose responses cover a large range. Topics are:
Models for instrument calibration
Data collection
Assumptions
Conditions that can invalidate the calibration procedure
Data analysis and model validation
Calibration of future measurements
Uncertainties of calibrated values
Purpose of
instrument
calibration
Instrument calibration is intended to eliminate or reduce bias in an instrument's readings over
a range for all continuous values. For this purpose, reference standards with known values for
selected points covering the range of interest are measured with the instrument in question.
Then a functional relationship is established between the values of the standards and the
corresponding measurements. There are two basic situations.
Instruments
which require
correction for
bias
The instrument reads in the same units as the reference standards. The purpose of the
calibration is to identify and eliminate any bias in the instrument relative to the defined
unit of measurement. For example, optical imaging systems that measure the width of
lines on semiconductors read in micrometers, the unit of interest. Nonetheless, these
instruments must be calibrated to values of reference standards if line width
measurements across the industry are to agree with each other.
Instruments
whose
measurements
act as
surrogates for
other
measurements
The instrument reads in different units than the reference standards. The purpose of the
calibration is to convert the instrument readings to the units of interest. An example is
densitometer measurements that act as surrogates for measurements of radiation dosage.
For this purpose, reference standards are irradiated at several dosage levels and then
measured by radiometry. The same reference standards are measured by densitometer.
The calibrated results of future densitometer readings on medical devices are the basis
for deciding if the devices have been sterilized at the proper radiation level.
Basic steps
for correcting
the
instrument for
bias
The calibration method is the same for both situations and requires the following basic steps:
Selection of reference standards with known values to cover the range of interest.
Measurements on the reference standards with the instrument to be calibrated.
Functional relationship between the measured and known values of the reference
standards (usually a least-squares fit to the data) called a calibration curve.
Correction of all measurements by the inverse of the calibration curve.
Schematic
example of a
calibration
curve and
resulting
value
A schematic explanation is provided by the figure below for load cell calibration. The loadcell
measurements (shown as *) are plotted on the y-axis against the corresponding values of
known load shown on the x-axis.
A quadratic fit to the loadcell data produces the calibration curve that is shown as the solid
line. For a future measurement with the load cell, Y' = 1.344 on the y-axis, a dotted line is
drawn through Y' parallel to the x-axis. At the point where it intersects the calibration curve,
another dotted line is drawn parallel to the y-axis. Its point of intersection with the x-axis at X'
= 13.417 is the calibrated value.
2.3.6.1. Models for instrument calibration

2.3. Calibration
Notation The following notation is used in this chapter in discussing
models for calibration curves.
Y denotes a measurement on a reference standard
X denotes the known value of a reference standard
denotes measurement error.
a, b and c denote coefficients to be determined
Possible forms
for calibration
curves
There are several models for calibration curves that can be
considered for instrument calibration. They fall into the
following classes:
Linear:
Quadratic:
Power:
Non-linear:
Special case
of linear
model - no
calibration
required
An instrument requires no calibration if
a=0 and b=1
i.e., if measurements on the reference standards agree with
their known values given an allowance for measurement
error, the instrument is already calibrated. Guidance on
collecting data, estimating and testing the coefficients is
given on other pages.
Advantages of
the linear
The linear model ISO 11095 is widely applied to
instrument calibration because it has several advantages
model over more complicated models.
Computation of coefficients and standard deviations
is easy.
Correction for bias is easy.
There is often a theoretical basis for the model.
The analysis of uncertainty is tractable.
Warning on
excluding the
intercept term
from the
model
It is often tempting to exclude the intercept, a, from the
model because a zero stimulus on the x-axis should lead to
a zero response on the y-axis. However, the correct
procedure is to fit the full model and test for the
significance of the intercept term.
Quadratic
model and
higher order
polynomials
Responses of instruments or measurement systems which
cannot be linearized, and for which no theoretical model
exists, can sometimes be described by a quadratic model
(or higher-order polynomial). An example is a load cell
where force exerted on the cell is a non-linear function of
load.
Disadvantages
of quadratic
models
Disadvantages of quadratic and higher-order polynomials
are:
They may require more reference standards to
capture the region of curvature.
There is rarely a theoretical justification; however,
the adequacy of the model can be tested statistically.
The correction for bias is more complicated than for
the linear model.
The uncertainty analysis is difficult.
Warning A plot of the data, although always recommended, is not
sufficient for identifying the correct model for the
calibration curve. Instrument responses may not appear
non-linear over a large interval. If the response and the
known values are in the same units, differences from the
known values should be plotted versus the known values.
Power model
treated as a
linear model
The power model is appropriate when the measurement
error is proportional to the response rather than being
additive. It is frequently used for calibrating instruments
that measure dosage levels of irradiated materials.
The power model is a special case of a non-linear model
that can be linearized by a natural logarithm
transformation to
so that the model to be fit to the data is of the familiar
linear form
where W, Z and e are the transforms of the variables, Y, X
and the measurement error, respectively, and a' is the
natural logarithm of a.
Non-linear
models and
their
limitations
Instruments whose responses are not linear in the
coefficients can sometimes be described by non-linear
models. In some cases, there are theoretical foundations for
the models; in other cases, the models are developed by
trial and error. Two classes of non-linear functions that
have been shown to have practical value as calibration
functions are:
1. Exponential
2. Rational
Non-linear models are an important class of calibration
models, but they have several significant limitations.
The model itself may be difficult to ascertain and
verify.
There can be severe computational difficulties in
estimating the coefficients.
Correction for bias cannot be applied algebraically
and can only be approximated by interpolation.
Uncertainty analysis is very difficult.
Example of an
exponential
function
An exponential function is shown in the equation below.
Instruments for measuring the ultrasonic response of
reference standards with various levels of defects (holes)
that are submerged in a fluid are described by this
function.
Example of a
rational
function
A rational function is shown in the equation below.
Scanning electron microscope measurements of line widths
on semiconductors are described by this function (Kirby).

2.3. Calibration
Data
collection
The process of collecting data for creating the calibration
curve is critical to the success of the calibration program.
General rules for designing calibration experiments apply, and
guidelines that are adequate for the calibration models in this
chapter are given below.
Selection
of
reference
standards
A minimum of five reference standards is required for a linear
calibration curve, and ten reference standards should be
adequate for more complicated calibration models.
The optimal strategy in selecting the reference standards is to
space the reference standards at points corresponding to equal
increments on the y-axis, covering the range of the instrument.
Frequently, this strategy is not realistic because the person
producing the reference materials is often not the same as the
person who is creating the calibration curve. Spacing the
reference standards at equal intervals on the x-axis is a good
alternative.
Exception
to the rule
above -
bracketing
If the instrument is not to be calibrated over its entire range,
but only over a very short range for a specific application,
then it may not be necessary to develop a complete calibration
curve, and a bracketing technique (ISO 11095) will provide
satisfactory results. The bracketing technique assumes that the
instrument is linear over the interval of interest, and, in this
case, only two reference standards are required -- one at each
end of the interval.
Number of
repetitions
on each
reference
standard
A minimum of two measurements on each reference standard
is required and four is recommended. The repetitions should
be separated in time by days or weeks. These repetitions
provide the data for determining whether a candidate model is
adequate for calibrating the instrument.
2.3.6.3. Assumptions for instrument calibration

2.3. Calibration
2.3.6.3. Assumptions for instrument calibration
Assumption
regarding
reference
values
The basic assumption regarding the reference values of
artifacts that are measured in the calibration experiment is
that they are known without error. In reality, this condition
is rarely met because these values themselves usually come
from a measurement process. Systematic errors in the
reference values will always bias the results, and random
errors in the reference values can bias the results.
Rule of
thumb
It has been shown by Bruce Hoadly, in an internal NIST
publication, that the best way to mitigate the effect of
random fluctuations in the reference values is to plan for a
large spread of values on the x-axis relative to the precision
of the instrument.
Assumptions
regarding
measurement
errors
The basic assumptions regarding measurement errors
associated with the instrument are that they are:
free from outliers
independent
of equal precision
from a normal distribution.
2.3.6.4. What can go wrong with the calibration procedure

2.3. Calibration
2.3.6.4. What can go wrong with the calibration
procedure
Calibration
procedure
may fail to
eliminate
bias
There are several circumstances where the calibration curve
will not reduce or eliminate bias as intended. Some are
discussed on this page. A critical exploratory analysis of the
calibration data should expose such problems.
Lack of
precision
Poor instrument precision or unsuspected day-to-day effects
may result in standard deviations that are large enough to
jeopardize the calibration. There is nothing intrinsic to the
calibration procedure that will improve precision, and the best
strategy, before committing to a particular instrument, is to
estimate the instrument's precision in the environment of
interest to decide if it is good enough for the precision
required.
Outliers in
the
calibration
data
Outliers in the calibration data can seriously distort the
calibration curve, particularly if they lie near one of the
endpoints of the calibration interval.
Isolated outliers (single points) should be deleted from
the calibration data.
An entire day's results which are inconsistent with the
other data should be examined and rectified before
proceeding with the analysis.
Systematic
differences
among
operators
It is possible for different operators to produce measurements
with biases that differ in sign and magnitude. This is not
usually a problem for automated instrumentation, but for
instruments that depend on line of sight, results may differ
significantly by operator. To diagnose this problem,
measurements by different operators on the same artifacts are
plotted and compared. Small differences among operators can
be accepted as part of the imprecision of the measurement
process, but large systematic differences among operators
require resolution. Possible solutions are to retrain the
operators or maintain separate calibration curves by operator.
Lack of
system
The calibration procedure, once established, relies on the
instrument continuing to respond in the same way over time.
control If the system drifts or takes unpredictable excursions, the
calibrated values may not be properly corrected for bias, and
depending on the direction of change, the calibration may
further degrade the accuracy of the measurements. To assure
that future measurements are properly corrected for bias, the
calibration procedure should be coupled with a statistical
control procedure for the instrument.
Example of
differences
among
repetitions
in the
calibration
data
An important point, but one that is rarely considered, is that
there can be differences in responses from repetition to
repetition that will invalidate the analysis. A plot of the
aggregate of the calibration data may not identify changes in
the instrument response from day-to-day. What is needed is a
plot of the fine structure of the data that exposes any day to
day differences in the calibration data.
Warning -
calibration
can fail
because of
day-to-day
changes
A straight-line fit to the aggregate data will produce a
'calibration curve'. However, if straight lines fit separately to
each day's measurements show very disparate responses, the
instrument, at best, will require calibration on a daily basis
and, at worst, may be sufficiently lacking in control to be
usable.
2.3.6.4.1. Example of day-to-day changes in calibration

2.3. Calibration
Calibration
data over 4
days
Line width measurements on 10 NIST reference standards were made
with an optical imaging system on each of four days. The four data points
for each reference value appear to overlap in the plot because of the wide
spread in reference values relative to the precision. The plot suggests that
a linear calibration line is appropriate for calibrating the imaging system.
This plot
shows
measurements
made on 10
reference
materials
repeated on
four days with
the 4 points
for each day
overlapping
REFERENCE VALUES (m)
This plot
shows the
differences
between each
measurement
and the
corresponding
reference
value.
Because days
are not
identified, the
plot gives no
indication of
problems in
the control of
the imaging
system from
from day to
day.
This plot, with
linear
calibration
lines fit to
each day's
measurements
individually,
shows how
the response
of the imaging
system
changes
dramatically
from day to
day. Notice
that the slope
of the
calibration
line goes from
positive on
day 1 to
negative on
day 3.
Interpretation
of calibration
findings
Given the lack of control for this measurement process, any calibration
procedure built on the average of the calibration data will fail to properly
correct the system on some days and invalidate resulting measurements.
There is no good solution to this problem except daily calibration.
2.3.6.5. Data analysis and model validation

2.3. Calibration
First step -
plot the
calibration
data
If the model for the calibration curve is not known from
theoretical considerations or experience, it is necessary to
identify and validate a model for the calibration curve. To
begin this process, the calibration data are plotted as a
function of known values of the reference standards; this
plot should suggest a candidate model for describing the
data. A linear model should always be a consideration. If
the responses and their known values are in the same units,
a plot of differences between responses and known values
is more informative than a plot of the data for exposing
structure in the data.
Warning -
regarding
statistical
software
Once an initial model has been chosen, the coefficients in
the model are estimated from the data using a statistical
software package. It is impossible to over-emphasize the
importance of using reliable and documented software for
this analysis.
Output
required from
a software
package
The software package will use the method of least squares
for estimating the coefficients. The software package
should also be capable of performing a 'weighted' fit for
situations where errors of measurement are non-constant
over the calibration interval. The choice of weights is
usually the responsibility of the user. The software
package should, at the minimum, provide the following
information:
Coefficients of the calibration curve
Standard deviations of the coefficients
Residual standard deviation of the fit
F-ratio for goodness of fit (if there are repetitions on
the y-axis at each reference value)
Typical
analysis of a
quadratic fit
Load cell measurements are modeled as a quadratic
function of known loads as shown below. There are three
repetitions at each load level for a total of 33
measurements.
Parameter estimates for model y = a + b*x +
c*x*x + e:
Parameter Estimate Std. Error t-value
Pr(>|t|)
a -1.840e-05 2.451e-05 -0.751
0.459
b 1.001e-01 4.839e-06 20687.891
<2e-16
c 7.032e-06 2.014e-07 34.922
<2e-16
Residual standard error = 3.764e-05 (30 degrees
of freedom)
Multiple R-squared = 1
Adjusted R-squared = 1
Analysis of variance table:
Source of Degrees of Sum of Mean
Variation Freedom Squares Square
F-Ratio Pr(>F)
Model 2 12.695 6.3475
4.48e+09 <2.2e-16
Residual 30 4.2504e-08 1.4170e-
09
(Lack of fit) 8 4.7700e-09 5.9625e-
10 0.3477 0.9368
(Pure error) 22 3.7733e-08 1.7151e-
09
Total 32 12.695
The analyses shown above can be reproduced using
Note: Dataplot reports a probability associated with the F-
ratio (for example, 6.334 % for the lack-of-fit test), where
a probability greater than 95 % indicates an F-ratio that is
significant at the 5 % level. R reports a p-value that
corresponds to the probability greater than the F-ratio, so a
value less than 0.05 would indicate significance at the 5 %
level. Other software may report in other ways; therefore,
it is necessary to check the interpretation for each package.
The F-ratio is
used to test
the goodness
of the fit to
the data
The F-ratio provides information on the model as a good
descriptor of the data. The F-ratio is compared with a
critical value from the F-table. An F-ratio smaller than the
critical value indicates that all significant structure has
been captured by the model.
F-ratio < 1
always
indicates a
good fit
For the load cell analysis, a plot of the data suggests a
linear fit. However, the linear fit gives a very large F-ratio.
For the quadratic fit, the F-ratio is 0.3477 with v
1
= 8 and
v
2
= 22 degrees of freedom. The critical value of F(0.05, 8,
20) = 2.45 indicates that the quadratic function is sufficient
for describing the data. A fact to keep in mind is that an F-
ratio < 1 does not need to be checked against a critical
value; it always indicates a good fit to the data.
The t-values
are used to
test the
significance of
The t-values can be compared with critical values from a
t-table. However, for a test at the 5 % significance level, a
t-value < 2 is a good indicator of non-significance. The t-
value for the intercept term, a, is < 2 indicating that the
individual
coefficients
intercept term is not significantly different from zero. The
t-values for the linear and quadratic terms are significant
indicating that these coefficients are needed in the model.
If the intercept is dropped from the model, the analysis is
repeated to obtain new estimates for the coefficients, b and
c.
Residual
standard
deviation
The residual standard deviation estimates the standard
deviation of a single measurement with the load cell.
Further
considerations
and tests of
assumptions
The residuals (differences between the measurements and
their fitted values) from the fit should also be examined for
outliers and structure that might invalidate the calibration
curve. They are also a good indicator of whether basic
assumptions of normality and equal precision for all
measurements are valid.
If the initial model proves inappropriate for the data, a
strategy for improving the model is followed.
2.3.6.5.1. Data on load cell #32066

2.3. Calibration
2.3.6.5.1. Data on load cell #32066
Three
repetitions on
a load cell at
eleven known
loads
X Y
2. 0.20024
2. 0.20016
2. 0.20024
4. 0.40056
4. 0.40045
4. 0.40054
6. 0.60087
6. 0.60075
6. 0.60086
8. 0.80130
8. 0.80122
8. 0.80127
10. 1.00173
10. 1.00164
10. 1.00173
12. 1.20227
12. 1.20218
12. 1.20227
14. 1.40282
14. 1.40278
14. 1.40279
16. 1.60344
16. 1.60339
16. 1.60341
18. 1.80412
18. 1.80409
18. 1.80411
20. 2.00485
20. 2.00481
20. 2.00483
21. 2.10526
21. 2.10524
21. 2.10524
2.3.6.6. Calibration of future measurements

2.3. Calibration
Purpose The purpose of creating the calibration curve is to correct
future measurements made with the same instrument to the
correct units of measurement. The calibration curve can be
applied many, many times before it is discarded or reworked
as long as the instrument remains in statistical control.
Chemical measurements are an exception where frequently the
calibration curve is used only for a single batch of
measurements, and a new calibration curve is created for the
next batch.
Notation The notation for this section is as follows:
Y' denotes a future measurement.
X' denotes the associated calibrated value.
are the estimates of the coefficients, a, b, c.
are standard deviations of the coefficients, a,
b, c.
Procedure To apply a correction to a future measurement, Y*, to obtain
the calibration value X* requires the inverse of the calibration
curve.
Linear
calibration
line
The inverse of the calibration line for the linear model
gives the calibrated value
Tests for
the
intercept
and slope
of
calibration
Before correcting for the calibration line by the equation
above, the intercept and slope should be tested for a=0, and
b=1. If both
curve -- If
both
conditions
hold, no
calibration
is needed.
there is no need for calibration. If, on the other hand only the
test for a=0 fails, the error is constant; if only the test for
b=1 fails, the errors are related to the size of the reference
standards.
Table
look-up
for t-
factor
The factor, t
1-/2,
, is found in the t-table where is the
degrees of freedom for the residual standard deviation from
the calibration curve, and is chosen to be small, say, 0.05.
Quadratic
calibration
curve
The inverse of the calibration curve for the quadratic model
requires a root
The correct root (+ or -) can usually be identified from
practical considerations.
Power
curve
The inverse of the calibration curve for the power model
gives the calibrated value
where b and the natural logarithm of a are estimated from the
power model transformed to a linear function.
Non-linear
and other
calibration
curves
For more complicated models, the inverse for the calibration
curve is obtained by interpolation from a graph of the function
or from predicted values of the function.

2.3. Calibration
Purpose The purpose is to quantify the uncertainty of a 'future' result
that has been corrected by the calibration curve. In principle,
the uncertainty quantifies any possible difference between the
calibrated value and its reference base (which normally
depends on reference standards).
Explanation
in terms of
reference
artifacts
Measurements of interest are future measurements on
unknown artifacts, but one way to look at the problem is to
ask: If a measurement is made on one of the reference
standards and the calibration curve is applied to obtain the
calibrated value, how well will this value agree with the
'known' value of the reference standard?
Difficulties The answer is not easy because of the intersection of two
uncertainties associated with
1. the calibration curve itself because of limited data
2. the 'future' measurement
If the calibration experiment were to be repeated, a slightly
different calibration curve would result even for a system in
statistical control. An exposition of the intersection of the
two uncertainties is given for the calibration of proving rings
( Hockersmith and Ku).
ISO
approach to
uncertainty
can be
based on
check
standards
or
propagation
of error
General procedures for computing an uncertainty based on
ISO principles of uncertainty analysis are given in the
chapter on modeling.
Type A uncertainties for calibrated values from calibration
curves can be derived from
check standard values
An example of type A uncertainties of calibrated values from
a linear calibration curve are analyzed from measurements on
linewidth check standards. Comparison of the uncertainties
from check standards and propagation of error for the
linewidth calibration data are also illustrated.
An example of the derivation of propagation of error type A
uncertainties for calibrated values from a quadratic
calibration curve for loadcells is discussed on the next page.
2.3.6.7.1. Uncertainty for quadratic calibration using propagation of error

2.3. Calibration
2.3.6.7.1. Uncertainty for quadratic calibration using propagation of
error
Propagation
of error for
uncertainty
of
calibrated
values of
loadcells
The purpose of this page is to show the propagation of error for calibrated values of a loadcell
based on a quadratic calibration curve where the model for instrument response is
The calibration data are instrument responses at known loads (psi), and estimates of the
quadratic coefficients, a, b, c, and their associated standard deviations are shown with the
analysis.
A graph of the calibration curve showing a measurement Y' corrected to X', the proper load
(psi), is shown below.
Uncertainty of
the calibrated
value X'
The uncertainty to be evaluated is the uncertainty of the calibrated value, X', computed for any
future measurement, Y', made with the calibrated instrument where
Partial
derivatives
The partial derivatives are needed to compute uncertainty.
The variance
of the
calibrated
value from
propagation of
error
The variance of X' is defined from propagation of error as follows:
The values of the coefficients and their respective standard deviations from the quadratic fit to
the calibration curve are substituted in the equation. The standard deviation of the
measurement, Y, may not be the same as the standard deviation from the fit to the calibration
data if the measurements to be corrected are taken with a different system; here we assume
that the instrument to be calibrated has a standard deviation that is essentially the same as the
instrument used for collecting the calibration data and the residual standard deviation from the
quadratic fit is the appropriate estimate.
a = -0.183980e-04
sa = 0.2450e-04
b = 0.100102
sb = 0.4838e-05
c = 0.703186e-05
sc = 0.2013e-06
sy = 0.0000376353
Graph
showing the
standard
deviations of
calibrated
values X' for
given
instrument
responses Y'
ignoring
covariance
terms in the
propagation of
error
The standard deviation expressed above is not easily interpreted but it is easily graphed. A
graph showing standard deviations of calibrated values, X', as a function of instrument
response, Y', is shown below.
Problem with
propagation of
error
The propagation of errors shown above is not complete because it ignores the covariances
among the coefficients, a, b, c. Unfortunately, some statistical software packages do not
display these covariance terms with the other output from the analysis.
Covariance
terms for
loadcell data
The variance-covariance terms for the loadcell data set are shown below.
a b c
a 6.0049021-10
b -1.0759599-10 2.3408589-11
c 4.0191106-12 -9.5051441-13 4.0538705-14
The diagonal elements are the variances of the coefficients, a, b, c, respectively, and the off-
diagonal elements are the covariance terms.
Recomputation
of the
standard
deviation of X'
To account for the covariance terms, the variance of X' is redefined by adding the covariance
terms. Appropriate substitutions are made; the standard deviations are recomputed and
graphed as a function of instrument response.
sab = -1.0759599e-10
sac = 4.0191106e-12
sbc = -9.5051441e-13
The graph below shows the correct estimates for the standard deviation of X' and gives a
means for assessing the loss of accuracy that can be incurred by ignoring covariance terms. In
this case, the uncertainty is reduced by including covariance terms, some of which are
negative.
Graph
showing the
standard
deviations of
calibrated
values, X', for
given
instrument
responses, Y',
with
covariance
terms included
in the
propagation of
error
Sample code The results in this section can be generated using R code.
2.3.6.7.2. Uncertainty for linear calibration using check standards

2.3. Calibration
2.3.6.7.2. Uncertainty for linear calibration
using check standards
Check
standards
provide a
mechanism
for
calculating
uncertainties
The easiest method for calculating type A uncertainties for
calibrated values from a calibration curve requires periodic
measurements on check standards. The check standards, in
this case, are artifacts at the lower, mid-point and upper
ends of the calibration curve. The measurements on the
check standard are made in a way that randomly samples
the output of the calibration procedure.
Calculation of
check
standard
values
The check standard values are the raw measurements on
the artifacts corrected by the calibration curve. The
standard deviation of these values should estimate the
uncertainty associated with calibrated values. The success
of this method of estimating the uncertainties depends on
adequate sampling of the measurement process.
Measurements
corrected by a
linear
calibration
curve
As an example, consider measurements of linewidths on
photomask standards, made with an optical imaging system
and corrected by a linear calibration curve. The three
control measurements were made on reference standards
with values at the lower, mid-point, and upper end of the
calibration interval.
Compute the
calibration
standard
deviation
For the linewidth data, the regression equation from the
calibration experiment is
and the estimated regression coefficients are the following.
Next, we calculate the difference between the "predicted" X
from the regression fit and the observed X.
2.3.6.7.2. Uncertainty for linear calibration using check standards
Finally, we find the calibration standard deviation by
calculating the standard deviation of the computed
differences.
The calibration standard deviation for the linewidth data is
0.119 m.
The calculations in this section can be completed using
Comparison
with
propagation
of error
The standard deviation, 0.119 m, can be compared with a
propagation of error analysis.
Other sources
of uncertainty
In addition to the type A uncertainty, there may be other
contributors to the uncertainty such as the uncertainties of
the values of the reference materials from which the
calibration curve was derived.
2.3.6.7.3. Comparison of check standard analysis and propagation of error

2.3. Calibration
2.3.6.7.3. Comparison of check standard analysis and propagation of
error
Propagation
of error for
the linear
calibration
The analysis of uncertainty for calibrated values from a linear calibration line can be
addressed using propagation of error. On the previous page, the uncertainty was estimated
from check standard values.
Estimates
from
calibration
data
The calibration data consist of 40 measurements with an optical imaging system on 10
linewidth artifacts. A linear fit to the data gives a calibration curve with the following
estimates for the intercept, a, and the slope, b:
Parameter Estimate Std. Error t-value Pr(>|t|)
a 0.2357623 0.02430034 9.702014 7.860745e-12
b 0.9870377 0.00344058 286.881171 5.354121e-65
with the following covariance matrix.
a b
a 5.905067e-04 -7.649453e-05
b -7.649453e-05 1.183759e-05
The results shown above can be generated with R code.
Propagation
of error
The propagation of error is performed for the equation
so that the squared uncertainty of a calibrated value, X', is
where
The uncertainty of the calibrated value, X',
is dependent on the value of the instrument reponse Y'.
Graph
showing
standard
deviation of
calibrated
value X'
plotted as a
function of
instrument
response Y'
for a linear
calibration
Comparison
of check
standard
analysis and
propagation
of error
Comparison of the analysis of check standard data, which gives a standard deviation of 0.119
m, and propagation of error, which gives a maximum standard deviation of 0.068 m,
suggests that the propagation of error may underestimate the type A uncertainty. The check
standard measurements are undoubtedly sampling some sources of variability that do not
appear in the formal propagation of error formula.
2.3.7. Instrument control for linear calibration

2.3. Calibration
Purpose The purpose of the control program is to guarantee that the
calibration of an instrument does not degrade over time.
Approach This is accomplished by exercising quality control on the
instrument's output in much the same way that quality control
is exercised on components in a process using a modification
of the Shewhart control chart.
Check
standards
needed for
the control
program
For linear calibration, it is sufficient to control the end-points
and the middle of the calibration interval to ensure that the
instrument does not drift out of calibration. Therefore, check
standards are required at three points; namely,
at the lower-end of the regime
at the mid-range of the regime
at the upper-end of the regime
Data
collection
One measurement is needed on each check standard for each
checking period. It is advisable to start by making control
measurements at the start of each day or as often as
experience dictates. The time between checks can be
lengthened if the instrument continues to stay in control.
Definition
of control
value
To conform to the notation in the section on instrument
corrections, X* denotes the known value of a standard, and X
denotes the measurement on the standard.
A control value is defined as the difference
If the calibration is perfect, control values will be randomly
distributed about zero and fall within appropriate upper and
lower limits on a control chart.
Calculation
of control
limits
The upper and lower control limits (Croarkin and Varner))
are, respectively,
where s is the residual standard deviation of the fit from the
calibration experiment, and is the estimated slope of the
linear calibration curve.
Values t* The critical value, , can be found in the t* table; v is the
degrees of freedom for the residual standard deviation; and
is equal to 0.05.
Determining
t*
For the case where = 0.05 and v = 38, the critical value of
the t* statistic is 2.497575.
R code and Dataplot code can be used to determine t*
critical values using a standard t-table for the quantile and
v degrees of freedom where is computed as
where m is the number of check standards.
Sensitivity
to departure
from
linearity
If
the instrument is in statistical control. Statistical control in
this context implies not only that measurements are
repeatable within certain limits but also that instrument
response remains linear. The test is sensitive to departures
from linearity.
Control
chart for a
system
corrected
by a linear
calibration
curve
An example of measurements of line widths on photomask
standards, made with an optical imaging system and
corrected by a linear calibration curve, are shown as an
example. The three control measurements were made on
reference standards with values at the lower, mid-point, and
upper end of the calibration interval.
2.3.7.1. Control chart for a linear calibration line

2.3. Calibration
Purpose Line widths of three photomask reference standards (at the low, middle and high end of the
calibration line) were measured on six days with an optical imaging system that had been
calibrated from similar measurements on 10 reference artifacts. The control values and limits
for the control chart , which depend on the intercept and slope of the linear calibration line,
monitor the calibration and linearity of the optical imaging system.
Initial
calibration
experiment
The initial calibration experiment consisted of 40 measurements (not shown here) on 10
artifacts and produced a linear calibration line with:
Intercept = 0.2357
Slope = 0.9870
Residual standard deviation = 0.06203 micrometers
Degrees of freedom = 38
Line width
measurements
made with an
optical
imaging
system
The control measurements, Y, and known values, X, for the three artifacts at the upper, mid-
range, and lower end (U, M, L) of the calibration line are shown in the following table:
DAY POSITION X Y
1 L 0.76 1.12
1 M 3.29 3.49
1 U 8.89 9.11
2 L 0.76 0.99
2 M 3.29 3.53
2 U 8.89 8.89
3 L 0.76 1.05
3 M 3.29 3.46
3 U 8.89 9.02
4 L 0.76 0.76
4 M 3.29 3.75
4 U 8.89 9.30
5 L 0.76 0.96
5 M 3.29 3.53
5 U 8.89 9.05
6 L 0.76 1.03
6 M 3.29 3.52
6 U 8.89 9.02
Control chart The control chart shown below can be generated using both Dataplot code and R code.
Interpretation
of control
chart
The control measurements show no evidence of drift and are within the control limits except
on the fourth day when all three control values are outside the limits. The cause of the
problem on that day cannot be diagnosed from the data at hand, but all measurements made on
that day, including workload items, should be rejected and remeasured.
2.4. Gauge R & R studies

The purpose of this section is to outline the steps that can be
taken to characterize the performance of gauges and
instruments used in a production setting in terms of errors that
affect the measurements.
What are the issues for a gauge R & R study?
What are the design considerations for the study?
1. Artifacts
2. Operators
3. Gauges, parameter levels, configurations
How do we collect data for the study?
How do we quantify variability of measurements?
1. Repeatability
2. Reproducibility
3. Stability
How do we identify and analyze bias?
1. Resolution
2. Linearity
3. Hysteresis
4. Drift
5. Differences among gauges
6. Differences among geometries, configurations
Remedies and strategies
How do we quantify uncertainties of measurements made with
the gauges?
2.4.1. What are the important issues?

2.4.1. What are the important issues?
Basic
issues
The basic issue for the study is the behavior of gauges in a
particular environment with respect to:
Repeatability
Reproducibility
Stability
Bias
Strategy The strategy is to conduct and analyze a study that examines
the behavior of similar gauges to see if:
They exhibit different levels of precision;
Instruments in the same environment produce equivalent
results;
Operators in the same environment produce equivalent
results;
Responses of individual gauges are affected by
configuration or geometry changes or changes in setup
procedures.
Other
goals
Other goals are to:
Test the resolution of instruments
Test the gauges for linearity
Estimate differences among gauges (bias)
Estimate differences caused by geometries,
configurations
Estimate operator biases
Incorporate the findings in an uncertainty budget
2.4.2. Design considerations

Design
considerations
Design considerations for a gauge study are choices of:
Artifacts (check standards)
Operators
Gauges
Parameter levels
Configurations, etc.
Selection of
artifacts or
check
standards
The artifacts for the study are check standards or test items
of a type that are typically measured with the gauges under
study. It may be necessary to include check standards for
different parameter levels if the gauge is a multi-response
instrument. The discussion of check standards should be
reviewed to determine the suitability of available artifacts.
Number of
artifacts
The number of artifacts for the study should be Q (Q > 2).
Check standards for a gauge study are needed only for the
limited time period (two or three months) of the study.
Selection of
operators
Only those operators who are trained and experienced with
the gauges should be enlisted in the study, with the
following constraints:
If there is a small number of operators who are
familiar with the gauges, they should all be included
in the study.
If the study is intended to be representative of a
large pool of operators, then a random sample of L
(L > 2) operators should be chosen from the pool.
If there is only one operator for the gauge type, that
operator should make measurements on K (K > 2)
days.
Selection of
gauges
If there is only a small number of gauges in the facility,
then all gauges should be included in the study.
If the study is intended to represent a larger pool of
gauges, then a random sample of I (I > 3) gauges should
be chosen for the study.
Limit the If the gauges operate at several parameter levels (for
initial study example; frequencies), an initial study should be carried
out at 1 or 2 levels before a larger study is undertaken.
If there are differences in the way that the gauge can be
operated, an initial study should be carried out for one or
two configurations before a larger study is undertaken.
2.4.3. Data collection for time-related sources of variability

2.4.3. Data collection for time-related sources of
variability
Time-
related
analysis
The purpose of this page is to present several options for
collecting data for estimating time-dependent effects in a
Time
intervals
The following levels of time-dependent errors are considered
in this section based on the characteristics of many
measurement systems and should be adapted to a specific
measurement situation as needed.
1. Level-1 Measurements taken over a short time to
capture the precision of the gauge
2. Level-2 Measurements taken over days (of other
appropriate time increment)
3. Level-3 Measurements taken over runs separated by
months
Time
intervals
Simple design for 2 levels of random error
Nested design for 2 levels of random error
Nested design for 3 levels of random error
In all cases, data collection and analysis are straightforward,
and there is no reason to estimate interaction terms when
dealing with time-dependent errors. Two levels should be
sufficient for characterizing most measurement systems. Three
levels are recommended for measurement systems where
sources of error are not well understood and have not
previously been studied.
2.4.3.1. Simple design

Constraints
on time and
resources
In planning a gauge study, particularly for the first time, it is
advisable to start with a simple design and progress to more
complicated and/or labor intensive designs after acquiring
some experience with data collection and analysis. The
design recommended here is appropriate as a preliminary
study of variability in the measurement process that occurs
over time. It requires about two days of measurements
separated by about a month with two repetitions per day.
Relationship
to 2-level
and 3-level
nested
designs
The disadvantage of this design is that there is minimal data
for estimating variability over time. A 2-level nested design
and a 3-level nested design, both of which require
measurments over time, are discussed on other pages.
Plan of
action
Choose at least Q = 10 work pieces or check standards,
which are essentially identical insofar as their expected
responses to the measurement method. Measure each of the
check standards twice with the same gauge, being careful to
randomize the order of the check standards.
After about a month, repeat the measurement sequence,
randomizing anew the order in which the check standards are
measured.
Notation Measurements on the check standards are designated:
with the first index identifying the month of measurement
and the second index identifying the repetition number.
Analysis of
data
The level-1 standard deviation, which describes the basic
precision of the gauge, is
with v
1
= 2Q degrees of freedom.
The level-2 standard deviation, which describes the
variability of the measurement process over time, is
with v
2
= Q degrees of freedom.
Relationship
to
uncertainty
for a test
item
The standard deviation that defines the uncertainty for a
single measurement on a test item, often referred to as the
reproducibility standard deviation (ASTM), is given by
The time-dependent component is
There may be other sources of uncertainty in the
measurement process that must be accounted for in a formal
analysis of uncertainty.
2.4.3.2. 2-level nested design

Check
standard
measurements
for estimating
time-
dependent
sources of
variability
Measurements on a check standard are recommended for studying the
effect of sources of variability that manifest themselves over time. Data
collection and analysis are straightforward, and there is no reason to
estimate interaction terms when dealing with time-dependent errors. The
measurements can be made at one of two levels. Two levels should be
sufficient for characterizing most measurement systems. Three levels are
recommended for measurement systems for which sources of error are
not well understood and have not previously been studied.
Time intervals
in a nested
design
The following levels are based on the characteristics of many
measurement systems and should be adapted to a specific measurement
situation as needed.
Level-1 Measurements taken over a short term to estimate gauge
precision
Level-2 Measurements taken over days (of other appropriate time
increment)
Definition of
number of
measurements
at each level
The following symbols are defined for this chapter:
Level-1 J (J > 1) repetitions
Level-2 K (K > 2) days
Schedule for
making
measurements
A schedule for making check standard measurements over time (once a
day, twice a week, or whatever is appropriate for sampling all conditions
of measurement) should be set up and adhered to. The check standard
measurements should be structured in the same way as values reported on
the test items. For example, if the reported values are averages of two
repetitions made within 5 minutes of each other, the check standard
values should be averages of the two measurements made in the same
manner.
Exception One exception to this rule is that there should be at least J = 2 repetitions
per day, etc. Without this redundancy, there is no way to check on the
short-term precision of the measurement system.
Depiction of
schedule for
making check
standard
measurements
with 4
repetitions per
day over K
days on the
surface of a
silicon wafer
2-level design for check standard measurements
Operator
considerations
The measurements should be taken with ONE operator. Operator is not
usually a consideration with automated systems. However, systems that
require decisions regarding line edge or other feature delineations may be
operator dependent.
Case Study:
Resistivity
check
standard
Results should be recorded along with pertinent environmental readings
and identifications for significant factors. The best way to record this
information is in one file with one line or row (on a spreadsheet) of
information in fixed fields for each check standard measurement.
Data analysis
of gauge
precision
The check standard measurements are represented by
for the jth repetition on the kth day. The mean for the kth day is

and the (level-1) standard deviation for gauge precision with v = J - 1
.
Pooling
increases the
reliability of
the estimate of
the standard
deviation
The pooled level-1 standard deviation with v = K(J - 1) degrees of
freedom is
.
Data analysis
of process
The level-2 standard deviation of the check standard represents the
process variability. It is computed with v = K - 1 degrees of freedom as:
(level-2)
standard
deviation
where

Relationship
to uncertainty
for a test item
The standard deviation that defines the uncertainty for a single
measurement on a test item, often referred to as the reproducibility
standard deviation (ASTM), is given by
The time-dependent component is
There may be other sources of uncertainty in the measurement process
that must be accounted for in a formal analysis of uncertainty.

Advantages
of nested
designs
A nested design is recommended for studying the effect of
sources of variability that manifest themselves over time. Data
collection and analysis are straightforward, and there is no
reason to estimate interaction terms when dealing with time-
dependent errors. Nested designs can be run at several levels.
Three levels are recommended for measurement systems
where sources of error are not well understood and have not
previously been studied.
Time
intervals in
a nested
design
The following levels are based on the characteristics of many
measurement systems and should be adapted to a specific
measurement situation as need be. A typical design is shown
below.
Level-1 Measurements taken over a short-time to
capture the precision of the gauge
Level-2 Measurements taken over days (or other
appropriate time increment)
Level-3 Measurements taken over runs separated by
months
Definition of
number of
measurements
at each level
The following symbols are defined for this chapter:
Level-1 J (J > 1) repetitions
Level-2 K (K > 2) days
Level-3 L (L > 2) runs
For the design shown above, J = 4; K = 3 and L = 2. The
design can be repeated for:
Q (Q > 2) check standards
I (I > 3) gauges if the intent is to characterize
several similar gauges
2-level nested
design
The design can be truncated at two levels to estimate
repeatability and day-to-day variability if there is no
reason to estimate longer-term effects. The analysis
remains the same through the first two levels.
Advantages This design has advantages in ease of use and
computation. The number of repetitions at each level need
not be large because information is being gathered on
several check standards.
Operator
considerations
The measurements should be made with ONE operator.
Operator is not usually a consideration with automated
systems. However, systems that require decisions regarding
line edge or other feature delineations may be operator
dependent. If there is reason to believe that results might
differ significantly by operator, 'operators' can be
substituted for 'runs' in the design. Choose L (L > 2)
operators at random from the pool of operators who are
capable of making measurements at the same level of
precision. (Conduct a small experiment with operators
making repeatability measurements, if necessary, to verify
comparability of precision among operators.) Then
complete the data collection and analysis as outlined. In
this case, the level-3 standard deviation estimates operator
effect.
Caution Be sure that the design is truly nested; i.e., that each
operator reports results for the same set of circumstances,
particularly with regard to day of measurement so that
each operator measures every day, or every other day, and
so forth.
Randomize on
gauges
Randomize with respect to gauges for each check standard;
i.e., choose the first check standard and randomize the
gauges; choose the second check standard and randomize
gauges; and so forth.
Record results
in a file
Record the average and standard deviation from each
group of J repetitions by:
check standard
gauge
Case Study:
Resistivity
Gauges
Results should be recorded along with pertinent
environmental readings and identifications for significant
factors. The best way to record this information is in one
file with one line or row (on a spreadsheet) of information
in fixed fields for each check standard measurement. A list
of typical entries follows.
1. Month
2. Day
3. Year
5. Check standard identification
6. Gauge identification
7. Average of J repetitions
8. Short-term standard deviation from J repetitions
2.4.4. Analysis of variability

Analysis of
variability
from a nested
design
The purpose of this section is to show the effect of various
levels of time-dependent effects on the variability of the
measurement process with standard deviations for each level
of a 3-level nested design.
Level 1 - repeatability/short-term precision
Level 2 - reproducibility/day-to-day
Level 3 - stability/run-to-run
The graph below depicts possible scenarios for a 2-level
design (short-term repetitions and days) to illustrate the
concepts.
Depiction of 2
measurement
processes with
the same
short-term
variability
over 6 days
where process
1 has large
between-day
variability and
process 2 has
negligible
between-day
variability
Process 1 Process 2
Large between-day variability Small between-day
variability

Distributions of short-term measurements over 6
days where distances from centerlines illustrate
between-day variability
Hint on using
tabular
method of
analysis
An easy way to begin is with a 2-level table with J columns
and K rows for the repeatability/reproducibility measurements
and proceed as follows:
1. Compute an average for each row and put it in the J+1
column.
2. Compute the level-1 (repeatability) standard deviation
for each row and put it in the J+2 column.
3. Compute the grand average and the level-2 standard
deviation from data in the J+1 column.
4. Repeat the table for each of the L runs.
5. Compute the level-3 standard deviation from the L
grand averages.
Level-1: LK
repeatability
standard
deviations can
be computed
from the data
The measurements from the nested design are denoted by
Equations corresponding to the tabular analysis are shown
below. Level-1 repeatability standard deviations, s
1lk
, are
pooled over the K days and L runs. Individual standard
deviations with (J - 1) degrees of freedom each are computed
from J repetitions as
where
Level-2: L
reproducibility
standard
deviations can
be computed
from the data
The level-2 standard deviation, s
2l
, is pooled over the L runs.
Individual standard deviations with (K - 1) degrees of
freedom each are computed from K daily averages as
where
Level-3: A
single global
standard
deviation can
be computed
from the L-
run averages
A level-3 standard deviation with (L - 1) degrees of freedom
is computed from the L-run averages as
where
Relationship
to uncertainty
for a test item
The standard deviation that defines the uncertainty for a
single measurement on a test item is given by
where the pooled values, s
1
and s
2
, are the usual
and
There may be other sources of uncertainty in the
measurement process that must be accounted for in a formal
analysis of uncertainty.
2.4.4.1. Analysis of repeatability

Case study:
Resistivity
probes
The repeatability quantifies the basic precision for the gauge. A level-1
repeatability standard deviation is computed for each group of J
repetitions, and a graphical analysis is recommended for deciding if
repeatability is dependent on the check standard, the operator, or the
gauge. Two graphs are recommended. These should show:
Plot of repeatability standard deviations versus check standard with
day coded
Plot of repeatability standard deviations versus check standard with
gauge coded
Typically, we expect the standard deviation to be gauge dependent -- in
which case there should be a separate standard deviation for each gauge.
If the gauges are all at the same level of precision, the values can be
combined over all gauges.
Repeatability
standard
deviations
can be
pooled over
operators,
runs, and
check
standards
A repeatability standard deviation from J repetitions is not a reliable
estimate of the precision of the gauge. Fortunately, these standard
deviations can be pooled over days; runs; and check standards, if
appropriate, to produce a more reliable precision measure. The table
below shows a mechanism for pooling. The pooled repeatability standard
deviation, , has LK(J - 1) degrees of freedom for measurements taken
over:
J repetitions
K days
L runs
Basic
pooling rules
The table below gives the mechanism for pooling repeatability standard
deviations over days and runs. The pooled value is an average of
weighted variances and is shown as the last entry in the right-hand
column of the table. The pooling can also cover check standards, if
appropriate.
View of
entire
dataset from
the nested
design
To illustrate the calculations, a subset of data collected in a nested design
for one check standard (#140) and one probe (#2362) are shown below.
The measurements are resistivity (ohm.cm) readings with six repetitions
per day. The individual level-1 standard deviations from the six
repetitions and degrees of freedom are recorded in the last two columns
of the database.
Run Wafer Probe Month Day Op Temp Average Stddev
df
1 140 2362 3 15 1 23.08 96.0771 0.1024
5
1 140 2362 3 17 1 23.00 95.9976 0.0943
5
1 140 2362 3 18 1 23.01 96.0148 0.0622
5
1 140 2362 3 22 1 23.27 96.0397 0.0702
5
1 140 2362 3 23 2 23.24 96.0407 0.0627
5
1 140 2362 3 24 2 23.13 96.0445 0.0622
5

2 140 2362 4 12 1 22.88 96.0793 0.0996
5
2 140 2362 4 18 2 22.76 96.1115 0.0533
5
2 140 2362 4 19 2 22.79 96.0803 0.0364
5
2 140 2362 4 19 1 22.71 96.0411 0.0768
5
2 140 2362 4 20 2 22.84 96.0988 0.1042
5
2 140 2362 4 21 1 22.94 96.0482 0.0868
5
Pooled repeatability standard deviations over days, runs
Source of
Variability
Degrees
of
Freedom
Standard Deviations
Sum of Squares
(SS)
Probe 2362
run 1 - day 1
run 1 - day 2
run 1 - day 3
run 1 - day 4
run 1 - day 5
run 1 - day 6
run 2 - day 1
run 2 - day 2
run 2 - day 3
run 2 - day 4
run 2 - day 5
run 2 - day 6
5
5
5
5
5
5
5
5
5
5
5
5
0.1024
0.0943
0.0622
0.0702
0.0627
0.0622
0.0996
0.0533
0.0364
0.0768
0.1042
0.0868
0.05243
0.04446
0.01934
0.02464
0.01966
0.01934
0.04960
0.01420
0.00662
0.02949
0.05429
0.03767
gives the total
degrees of
freedom for s
1
60
gives the total sum of
squares for s
1
0.37176
The pooled value of s
1
is given by
0.07871
The calculations displayed in the table above can be generated using both
2.4.4.2. Analysis of reproducibility

Case study:
Resistivity
gauges
Day-to-day variability can be assessed by a graph of check standard
values (averaged over J repetitions) versus day with a separate graph
for each check standard. Graphs for all check standards should be
plotted on the same page to obtain an overall view of the
measurement situation.
Pooling
results in
more
reliable
estimates
The level-2 standard deviations with (K - 1) degrees of freedom are
computed from the check standard values for days and pooled over
runs as shown in the table below. The pooled level-2 standard
deviation has degrees of freedom
L(K - 1) for measurements made over:
K days
L runs
Mechanism
for pooling
The table below gives the mechanism for pooling level-2 standard
deviations over runs. The pooled value is an average of weighted
variances and is the last entry in the right-hand column of the table.
The pooling can be extended in the same manner to cover check
standards, if appropriate.
The table was generated using a subset of data (shown on previous
page) collected in a nested design on one check standard (#140) with
probe (#2362) over six days. The data are analyzed for between-day
effects. The level-2 standard deviations and pooled level-2 standard
deviations over runs 1 and 2 are:
Level-2 standard deviations for a single gauge pooled
over runs
Source of variability
Standard
deviations
Degrees
of
freedom
Sum of squares
Days
Run 1

Run 2

0.027280

0.027560
5

5
------
-
0.003721
0.003798
-----------
Sum
Pooled value

10
0.007519
0.02742
Relationship
to day effect
The level-2 standard deviation is related to the standard deviation for
between-day precision and gauge precision by
The size of the day effect can be calculated by subtraction using the
formula above once the other two standard deviations have been
estimated reliably.
Computation
of variance
component
for days
For our example, the variance component for between days is -
0.00028072. The negative number for the variance is interpreted as
meaning that the variance component for days is zero. However, with
only 10 degrees of freedom for the level-2 standard deviation, this
estimate is not necessarily reliable. The standard deviation for days
over the entire database shows a significant component for days.
Sample code The calculations included in this section can be implemented using
both
2.4.4.3. Analysis of stability

Case study:
Resistivity
probes
Run-to-run variability can be assessed graphically by a plot of check
standard values (averaged over J repetitions) versus time with a separate
graph for each check standard. Data on all check standards should be
plotted on one page to obtain an overall view of the measurement
situation.
Advantage
of pooling
A level-3 standard deviation with (L - 1) degrees of freedom is computed
from the run averages. Because there will rarely be more than two runs
per check standard, resulting in one degree of freedom per check
standard, it is prudent to have three or more check standards in the design
to take advantage of pooling. The mechanism for pooling over check
standards is shown in the table below. The pooled standard deviation has
Q(L - 1) degrees and is shown as the last entry in the right-hand column
of the table.
Example of
pooling
The following table shows how the level-3 standard deviations for a
single gauge (probe #2362) are pooled over check standards. The table
can be reproduced using
R code.
Level-3 standard deviations for a single gauge pooled over
check standards
Source of variability
Standard
deviation
Degrees
of
freedom
Sum of squares
Level-3
Chk std 138
Chk std 139
Chk std 140
Chk std 141
Chk std 142
Sum
Pooled value
0.0223
0.0027
0.0289
0.0133
0.0205
1
1
1
1
1
----
-
5
0.0004973
0.0000073
0.0008352
0.0001769
0.0004203
-----------
0.0019370
0.0197
Level-3
standard
deviations
A subset of data collected in a nested design on one check standard
(#140) with probe (#2362) for six days and two runs is analyzed for
between-run effects. The level-3 standard deviation, computed from the
averages of two runs, is 0.02885 with one degree of freedom. Dataplot
code and R code can be used to perform the calculations for this data.
Relationship
to long-
term
changes,
days and
gauge
precision
The size of the between-run effect can be calculated by subtraction using
the standard deviations for days and gauge precision as
2.4.4.4.4. Example of calculations

2.4.4.4.
2.4.4.4.4. Example of calculations
Example of
repeatability
calculations
Short-term standard deviations based on
J = 6 repetitions with 5 degrees of freedom
K = 6 days
L = 2 runs
were recorded with a probing instrument on Q = 5 wafers.
The standard deviations were pooled over K = 6 days and L
= 2 runs to give 60 degrees of freedom for each wafer. The
pooling of repeatability standard deviations over the 5 wafers
is demonstrated in the table below.
Pooled repeatability standard deviation for a single gauge
Source of
variability
Sum of Squares (SS)
Degrees of
freedom
(DF)
Std Devs
Repeatability
Wafer #138
Wafer #139
Wafer #140
Wafer #141
Wafer #142
SUM
0.48115
0.69209
0.48483
1.21752
0.30076
3.17635
60
60
60
60
60
300
0.10290
2.4.5. Analysis of bias

Definition
of bias
The terms 'bias' and 'systematic error' have the same meaning
in this handbook. Bias is defined (VIM) as the difference
between the measurement result and its unknown 'true value'.
It can often be estimated and/or eliminated by calibration to a
reference standard.
Potential
problem
Calibration relates output to 'true value' in an ideal
environment. However, it may not assure that the gauge reacts
properly in its working environment. Temperature, humidity,
operator, wear, and other factors can introduce bias into the
measurements. There is no single method for dealing with this
problem, but the gauge study is intended to uncover biases in
the measurement process.
Sources of
bias
Sources of bias that are discussed in this Handbook include:
Lack of gauge resolution
Lack of linearity
Drift
Hysteresis
Differences among gauges
Differences among geometries
Differences among operators
Remedial actions and strategies
2.4.5.1. Resolution

2.4.5.1. Resolution
Resolution Resolution (MSA) is the ability of the measurement
system to detect and faithfully indicate small changes in
the characteristic of the measurement result.
Definition
from (MSA)
manual
The resolution of the instrument is if there is an equal
probability that the indicated value of any artifact, which
differs from a reference standard by less than , will be
the same as the indicated value of the reference.
Good versus
poor
A small implies good resolution -- the measurement
system can discriminate between artifacts that are close
together in value.
A large implies poor resolution -- the measurement
system can only discriminate between artifacts that are far
apart in value.
Warning The number of digits displayed does not indicate the
resolution of the instrument.
Manufacturer's
statement of
resolution
Resolution as stated in the manufacturer's specifications is
usually a function of the least-significant digit (LSD) of
the instrument and other factors such as timing
mechanisms. This value should be checked in the
laboratory under actual conditions of measurement.
Experimental
determination
of resolution
To make a determination in the laboratory, select several
artifacts with known values over a range from close in
value to far apart. Start with the two artifacts that are
farthest apart and make measurements on each artifact.
Then, measure the two artifacts with the second largest
difference, and so forth, until two artifacts are found
which repeatedly give the same result. The difference
between the values of these two artifacts estimates the
resolution.
Consequence
of poor
resolution
No useful information can be gained from a study on a
gauge with poor resolution relative to measurement needs.
2.4.5.1. Resolution
2.4.5.2. Linearity of the gauge

Definition
of linearity
for gauge
studies
Linearity is given a narrow interpretation in this Handbook to
indicate that gauge response increases in equal increments to
equal increments of stimulus, or, if the gauge is biased, that
the bias remains constant throughout the course of the
Data
collection
and
repetitions
A determination of linearity requires Q (Q > 4) reference
standards that cover the range of interest in fairly equal
increments and J (J > 1) measurements on each reference
standard. One measurement is made on each of the reference
standards, and the process is repeated J times.
Plot of the
data
A test of linearity starts with a plot of the measured values
versus corresponding values of the reference standards to
obtain an indication of whether or not the points fall on a
straight line with slope equal to 1 -- indicating linearity.
Least-
squares
estimates
of bias and
slope
A least-squares fit of the data to the model
Y = a + bX + measurement error
where Y is the measurement result and X is the value of the
reference standard, produces an estimate of the intercept, a,
and the slope, b.
Output
from
software
package
The intercept and bias are estimated using a statistical
software package that should provide the following
information:
Estimates of the intercept and slope,
Standard deviations of the intercept and slope
Residual standard deviation of the fit
F-test for goodness of fit
Test for
linearity
Tests for the slope and bias are described in the section on
instrument calibration. If the slope is different from one, the
gauge is non-linear and requires calibration or repair. If the
intercept is different from zero, the gauge has a bias.
Causes of
non-
linearity
The reference manual on Measurement Systems Analysis
(MSA) lists possible causes of gauge non-linearity that should
be investigated if the gauge shows symptoms of non-linearity.
1. Gauge not properly calibrated at the lower and upper
ends of the operating range
2. Error in the value of X at the maximum or minimum
range
3. Worn gauge
4. Internal design problems (electronics)
Note - on
artifact
calibration
The requirement of linearity for artifact calibration is not so
stringent. Where the gauge is used as a comparator for
measuring small differences among test items and reference
standards of the same nominal size, as with calibration
designs, the only requirement is that the gauge be linear over
the small on-scale range needed to measure both the reference
standard and the test item.
Situation
where the
calibration
of the
gauge is
neglected
Sometimes it is not economically feasible to correct for the
calibration of the gauge (Turgel and Vecchia). In this case, the
bias that is incurred by neglecting the calibration is estimated
as a component of uncertainty.
2.4.5.3. Drift

2.4.5.3. Drift
Definition Drift can be defined (VIM) as a slow change in the response
of a gauge.
Instruments
used as
comparators
for
calibration
Short-term drift can be a problem for comparator
measurements. The cause is frequently heat build-up in the
instrument during the time of measurement. It would be
difficult, and probably unproductive, to try to pinpoint the
extent of such drift with a gauge study. The simplest solution
is to use drift-free designs for collecting calibration data.
These designs mitigate the effect of linear drift on the results.
Long-term drift should not be a problem for comparator
measurements because such drift would be constant during a
calibration design and would cancel in the difference
measurements.
Instruments
corrected by
linear
calibration
For instruments whose readings are corrected by a linear
calibration line, drift can be detected using a control chart
technique and measurements on three or more check
standards.
Drift in
direct
reading
instruments
and
uncertainty
analysis
For other instruments, measurements can be made on a daily
basis on two or more check standards over a preset time
period, say, one month. These measurements are plotted on a
time scale to determine the extent and nature of any drift.
Drift rarely continues unabated at the same rate and in the
same direction for a long time period.
Thus, the expectation from such an experiment is to
document the maximum change that is likely to occur during
a set time period and plan adjustments to the instrument
accordingly. A further impact of the findings is that
uncorrected drift is treated as a type A component in the
uncertainty analysis.
2.4.5.4. Differences among gauges

Purpose A gauge study should address whether gauges agree with
one another and whether the agreement (or disagreement) is
consistent over artifacts and time.
Data
collection
For each gauge in the study, the analysis requires
measurements on
Q (Q > 2) check standards
K (K > 2) days
The measurements should be made by a single operator.
Data
reduction
The steps in the analysis are:
1. Measurements are averaged over days by
artifact/gauge configuration.
2. For each artifact, an average is computed over
gauges.
3. Differences from this average are then computed for
each gauge.
4. If the design is run as a 3-level design, the statistics
are computed separately for each run.
Data from a
gauge study
The data in the table below come from resistivity (ohm.cm)
measurements on Q = 5 artifacts on K = 6 days. Two runs
were made which were separated by about a month's time.
The artifacts are silicon wafers and the gauges are four-
point probes specifically designed for measuring resistivity
of silicon wafers. Differences from the wafer means are
shown in the table.
Biases for 5
probes from
a gauge study
with 5
artifacts on 6
days
Table of biases for probes and silicon wafers
(ohm.cm)
Wafers

Probe 138 139 140 141
142
------------------------------------------------
---------
1 0.02476 -0.00356 0.04002 0.03938
0.00620
181 0.01076 0.03944 0.01871 -0.01072
0.03761
182 0.01926 0.00574 -0.02008 0.02458
-0.00439
2062 -0.01754 -0.03226 -0.01258 -0.02802
-0.00110
2362 -0.03725 -0.00936 -0.02608 -0.02522
-0.03830
Plot of
differences
among
probes
A graphical analysis can be more effective for detecting
differences among gauges than a table of differences. The
differences are plotted versus artifact identification with
each gauge identified by a separate plotting symbol. For
ease of interpretation, the symbols for any one gauge can
be connected by dotted lines.
Interpretation Because the plots show differences from the average by
artifact, the center line is the zero-line, and the differences
are estimates of bias. Gauges that are consistently above or
below the other gauges are biased high or low, respectively,
relative to the average. The best estimate of bias for a
particular gauge is its average bias over the Q artifacts. For
this data set, notice that probe #2362 is consistently biased
low relative to the other probes.
Strategies for
dealing with
differences
among
gauges
Given that the gauges are a random sample of like-kind
gauges, the best estimate in any situation is an average over
all gauges. In the usual production or metrology setting,
however, it may only be feasible to make the measurements
on a particular piece with one gauge. Then, there are two
methods of dealing with the differences among gauges.
1. Correct each measurement made with a particular
gauge for the bias of that gauge and report the
standard deviation of the correction as a type A
uncertainty.
2. Report each measurement as it occurs and assess a
type A uncertainty for the differences among the
gauges.
2.4.5.5. Geometry/configuration differences

How to deal
with
configuration
differences
The mechanism for identifying and/or dealing with
differences among geometries or configurations in an
instrument is basically the same as dealing with differences
among the gauges themselves.
Example of
differences
among wiring
configurations
An example is given of a study of configuration
differences for a single gauge. The gauge, a 4-point probe
for measuring resistivity of silicon wafers, can be wired in
several ways. Because it was not possible to test all wiring
configurations during the gauge study, measurements were
made in only two configurations as a way of identifying
possible problems.
Data on
wiring
configurations
and a plot of
differences
between the 2
wiring
configurations
Measurements were made on six wafers over six days
(except for 5 measurements on wafer 39) with probe #2062
wired in two configurations. This sequence of
measurements was repeated after about a month resulting
in two runs. Differences between measurements in the two
configurations on the same day are shown in the following
table.
Differences between wiring
configurations
Wafer Day Probe Run 1 Run
2
17. 1 2062. -0.0108
0.0088
17. 2 2062. -0.0111
0.0062
17. 3 2062. -0.0062
0.0074
17. 4 2062. 0.0020
0.0047
17. 5 2062. 0.0018
0.0049
17. 6 2062. 0.0002
0.0000
39. 1 2062. -0.0089
0.0075
39. 3 2062. -0.0040 -
0.0016
39. 4 2062. -0.0022
0.0052
39. 5 2062. -0.0012
0.0085
39. 6 2062. -0.0034 -
0.0018
63. 1 2062. -0.0016
0.0092
63. 2 2062. -0.0111
0.0040
63. 3 2062. -0.0059
0.0067
63. 4 2062. -0.0078
0.0016
63. 5 2062. -0.0007
0.0020
63. 6 2062. 0.0006
0.0017
103. 1 2062. -0.0050
0.0076
103. 2 2062. -0.0140
0.0002
103. 3 2062. -0.0048
0.0025
103. 4 2062. 0.0018
0.0045
103. 5 2062. 0.0016 -
0.0025
103. 6 2062. 0.0044
0.0035
125. 1 2062. -0.0056
0.0099
125. 2 2062. -0.0155
0.0123
125. 3 2062. -0.0010
0.0042
125. 4 2062. -0.0014
0.0098
125. 5 2062. 0.0003
0.0032
125. 6 2062. -0.0017
0.0115
Test of
difference
between
configurations
Because there are only two configurations, a t-test is used
to decide if there is a difference. If
the difference between the two configurations is
statistically significant.
The average and standard deviation computed from the 29
differences in each run are shown in the table below along
with the t-values which confirm that the differences are
significant for both runs.
Average differences between wiring
configurations
Run Probe Average Std dev
N t
1 2062 - 0.00383 0.00514
29 -4.0
2 2062 + 0.00489 0.00400
29 +6.6
Unexpected
result
The data reveal a wiring bias for both runs that changes
direction between runs. This is a somewhat disturbing
finding, and further study of the gauges is needed. Because
neither wiring configuration is preferred or known to give
the 'correct' result, the differences are treated as a
component of the measurement uncertainty.
2.4.5.6. Remedial actions and strategies

Variability The variability of the gauge in its normal operating mode
needs to be examined in light of measurement
requirements.
If the standard deviation is too large, relative to
requirements, the uncertainty can be reduced by making
repeated measurements and taking advantage of the
standard deviation of the average (which is reduced by a
factor of when n measurements are averaged).
Causes of
excess
variability
If multiple measurements are not economically feasible in
the workload, then the performance of the gauge must be
improved. Causes of variability which should be examined
are:
Wear
Environmental effects such as humidity
Temperature excursions
Operator technique
Resolution There is no remedy for a gauge with insufficient resolution.
The gauge will need to be replaced with a better gauge.
Lack of
linearity
Lack of linearity can be dealt with by correcting the output
of the gauge to account for bias that is dependent on the
level of the stimulus. Lack of linearity can be tolerated
(left uncorrected) if it does not increase the uncertainty of
the measurement result beyond its requirement.
Drift It would be very difficult to correct a gauge for drift unless
there is sufficient history to document the direction and
size of the drift. Drift can be tolerated if it does not
increase the uncertainty of the measurement result beyond
its requirement.
Differences
among gauges
or
configurations
Significant differences among gauges/configurations can
be treated in one of two ways:
1. By correcting each measurement for the bias of the
specific gauge/configuration.
2. By accepting the difference as part of the uncertainty
of the measurement process.
Differences
among
operators
Differences among operators can be viewed in the same
way as differences among gauges. However, an operator
who is incapable of making measurements to the required
precision because of an untreatable condition, such as a
vision problem, should be re-assigned to other tasks.
2.4.6. Quantifying uncertainties from a gauge study

2.4.6. Quantifying uncertainties from a gauge
study
Gauge
studies can
be used as
the basis for
uncertainty
assessment
One reason for conducting a gauge study is to quantify
uncertainties in the measurement process that would be
difficult to quantify under conditions of actual measurement.
This is a reasonable approach to take if the results are truly
representative of the measurement process in its working
environment. Consideration should be given to all sources of
error, particularly those sources of error which do not
exhibit themselves in the short-term run.
Potential
problem with
this
approach
The potential problem with this approach is that the
calculation of uncertainty depends totally on the gauge
study. If the measurement process changes its characteristics
over time, the standard deviation from the gauge study will
not be the correct standard deviation for the uncertainty
analysis. One way to try to avoid such a problem is to carry
out a gauge study both before and after the measurements
that are being characterized for uncertainty. The 'before' and
'after' results should indicate whether or not the
measurement process changed in the interim.
Uncertainty
analysis
requires
information
about the
specific
measurement
The computation of uncertainty depends on the particular
measurement that is of interest. The gauge study gathers the
data and estimates standard deviations for sources that
contribute to the uncertainty of the measurement result.
However, specific formulas are needed to relate these
standard deviations to the standard deviation of a
measurement result.
General
guidance
The following sections outline the general approach to
uncertainty analysis and give methods for combining the
standard deviations into a final uncertainty:
1. Approach
2. Methods for type A evaluations
3. Methods for type B evaluations
4. Propagation of error
5. Error budgets and sensitivity coefficients
6. Standard and expanded uncertainties
7. Treatment of uncorrected biases
Type A
evaluations
of random
error
Data collection methods and analyses of random sources of
uncertainty are given for the following:
1. Repeatability of the gauge
2. Reproducibility of the measurement process
3. Stability (very long-term) of the measurement process
Biases - Rule
of thumb
The approach for biases is to estimate the maximum bias
from a gauge study and compute a standard uncertainty
from the maximum bias assuming a suitable distribution.
The formulas shown below assume a uniform distribution
for each bias.
Determining
resolution
If the resolution of the gauge is , the standard uncertainty
for resolution is
Determining
non-linearity
If the maximum departure from linearity for the gauge has
been determined from a gauge study, and it is reasonable to
assume that the gauge is equally likely to be engaged at any
point within the range tested, the standard uncertainty for
linearity is
Hysteresis Hysteresis, as a performance specification, is defined (NCSL
RP-12) as the maximum difference between the upscale and
downscale readings on the same artifact during a full range
traverse in each direction. The standard uncertainty for
hysteresis is
Determining
drift
Drift in direct reading instruments is defined for a specific
time interval of interest. The standard uncertainty for drift is
where Y
0
and Y
t
are measurements at time zero and t,
respectively.
Other biases Other sources of bias are discussed as follows:
1. Differences among gauges
2. Differences among configurations
Case study: A case study on type A uncertainty analysis from a gauge
Type A
uncertainties
from a
gauge study
study is recommended as a guide for bringing together the
principles and elements discussed in this section. The study
in question characterizes the uncertainty of resistivity
measurements made on silicon wafers.
2.5. Uncertainty analysis

Uncertainty
measures
'goodness'
of a test
result
This section discusses the uncertainty of measurement results.
Uncertainty is a measure of the 'goodness' of a result.
Without such a measure, it is impossible to judge the fitness
of the value as a basis for making decisions relating to health,
safety, commerce or scientific excellence.
Contents 1. What are the issues for uncertainty analysis?
2. Approach to uncertainty analysis
1. Steps
3. Type A evaluations
1. Type A evaluations of random error
1. Time-dependent components
2. Measurement configurations
2. Type A evaluations of material inhomogeneities
3. Type A evaluations of bias
1. Treatment of inconsistent bias
2. Treatment of consistent bias
3. Treatment of bias with sparse data
4. Type B evaluations
1. Assumed distributions
5. Propagation of error considerations
1. Functions of a single variable
2. Functions of two variables
3. Functions of several variables
6. Error budgets and sensitivity coefficients
1. Sensitivity coefficients for measurements on the
test item
2. Sensitivity coefficients for measurements on a
check standard
3. Sensitivity coefficients for measurements with a
2-level design
4. Sensitivity coefficients for measurements with a
3-level design
5. Example of error budget
8. Treatment of uncorrected bias
1. Computation of revised uncertainty
2.5.1. Issues

2.5.1. Issues
Issues for
uncertainty
analysis
Evaluation of uncertainty is an ongoing process that can
consume time and resources. It can also require the
services of someone who is familiar with data analysis
techniques, particularly statistical analysis. Therefore, it
is important for laboratory personnel who are
approaching uncertainty analysis for the first time to be
aware of the resources required and to carefully lay out a
plan for data collection and analysis.
Problem areas Some laboratories, such as test laboratories, may not
have the resources to undertake detailed uncertainty
analyses even though, increasingly, quality management
standards such as the ISO 9000 series are requiring that
all measurement results be accompanied by statements of
uncertainty.
Other situations where uncertainty analyses are
problematical are:
One-of-a-kind measurements
Dynamic measurements that depend strongly on
the application for the measurement
Directions being
pursued
What can be done in these situations? There is no
definitive answer at this time. Several organizations,
such as the National Conference of Standards
Laboratories (NCSL) and the International Standards
Organization (ISO) are investigating methods for dealing
with this problem, and there is a document in draft that
will recommend a simplified approach to uncertainty
analysis based on results of interlaboratory tests.
Relationship to
interlaboratory
test results
Many laboratories or industries participate in
interlaboratory studies where the test method itself is
evaluated for:
repeatability within laboratories
reproducibility across laboratories
These evaluations do not lead to uncertainty statements
because the purpose of the interlaboratory test is to
evaluate, and then improve, the test method as it is
2.5.1. Issues
applied across the industry. The purpose of uncertainty
analysis is to evaluate the result of a particular
measurement, in a particular laboratory, at a particular
time. However, the two purposes are related.
Default
recommendation
for test
laboratories
If a test laboratory has been party to an interlaboratory
test that follows the recommendations and analyses of an
American Society for Testing Materials standard (ASTM
E691) or an ISO standard (ISO 5725), the laboratory
can, as a default, represent its standard uncertainty for a
single measurement as the reproducibility standard
deviation as defined in ASTM E691 and ISO 5725. This
standard deviation includes components for within-
laboratory repeatability common to all laboratories and
between-laboratory variation.
Drawbacks of
this procedure
The standard deviation computed in this manner
describes a future single measurement made at a
laboratory randomly drawn from the group and leads to a
prediction interval (Hahn & Meeker) rather than a
confidence interval. It is not an ideal solution and may
produce either an unrealistically small or unacceptably
large uncertainty for a particular laboratory. The
procedure can reward laboratories with poor performance
or those that do not follow the test procedures to the
letter and punish laboratories with good performance.
Further, the procedure does not take into account sources
of uncertainty other than those captured in the
interlaboratory test. Because the interlaboratory test is a
snapshot at one point in time, characteristics of the
measurement process over time cannot be accurately
evaluated. Therefore, it is a strategy to be used only
where there is no possibility of conducting a realistic
uncertainty investigation.
2.5.2. Approach

2.5.2. Approach
Procedures
in this
chapter
The procedures in this chapter are intended for test
laboratories, calibration laboratories, and scientific
laboratories that report results of measurements from
ongoing or well-documented processes.
Pertinent
sections
The following pages outline methods for estimating the
individual uncertainty components, which are consistent
with materials presented in other sections of this Handbook,
and rules and equations for combining them into a final
expanded uncertainty. The general framework is:
1. ISO Approach
2. Outline of steps to uncertainty analysis
3. Methods for type A evaluations
4. Methods for type B evaluations
5. Propagation of error considerations
6. Uncertainty budgets and sensitivity coefficients
8. Treatment of uncorrected bias
Specific
situations are
outlined in
other places
in this
chapter
Methods for calculating uncertainties for specific results are
explained in the following sections:
Calibrated values of artifacts
Calibrated values from calibration curves
From propagation of error
From check standard measurements
Comparison of check standards and
Gauge R & R studies
Type A components for resistivity measurements
Type B components for resistivity measurements
ISO
definition of
uncertainty
Uncertainty, as defined in the ISO Guide to the Expression
of Uncertainty in Measurement (GUM) and the
International Vocabulary of Basic and General Terms in
Metrology (VIM), is a
"parameter, associated with the result of a
measurement, that characterizes the dispersion
of the values that could reasonably be
2.5.2. Approach
attributed to the measurand."
Consistent
with
historical
view of
uncertainty
This definition is consistent with the well-established
concept that an uncertainty statement assigns credible limits
to the accuracy of a reported value, stating to what extent
that value may differ from its reference value (Eisenhart).
In some cases, reference values will be traceable to a
national standard, and in certain other cases, reference
values will be consensus values based on measurements
made according to a specific protocol by a group of
laboratories.
Accounts for
both random
error and
bias
The estimation of a possible discrepancy takes into account
both random error and bias in the measurement process.
The distinction to keep in mind with regard to random error
and bias is that random errors cannot be corrected, and
biases can, theoretically at least, be corrected or eliminated
from the measurement result.
Relationship
to precision
and bias
statements
Precision and bias are properties of a measurement method.
Uncertainty is a property of a specific result for a single
test item that depends on a specific measurement
configuration (laboratory/instrument/operator, etc.). It
depends on the repeatability of the instrument; the
reproducibility of the result over time; the number of
measurements in the test result; and all sources of random
and systematic error that could contribute to disagreement
between the result and its reference value.
Handbook
follows the
ISO
approach
This Handbook follows the ISO approach (GUM) to stating
and combining components of uncertainty. To this basic
structure, it adds a statistical framework for estimating
individual components, particularly those that are classified
as type A uncertainties.
Basic ISO
tenets
The ISO approach is based on the following rules:
Each uncertainty component is quantified by a
standard deviation.
All biases are assumed to be corrected and any
uncertainty is the uncertainty of the correction.
Zero corrections are allowed if the bias cannot be
corrected and an uncertainty is assessed.
All uncertainty intervals are symmetric.
ISO
approach to
classifying
sources of
error
Components are grouped into two major categories,
depending on the source of the data and not on the type of
error, and each component is quantified by a standard
deviation. The categories are:
Type A - components evaluated by statistical
2.5.2. Approach
methods
Type B - components evaluated by other means (or in
other laboratories)
Interpretation
of this
classification
One way of interpreting this classification is that it
distinguishes between information that comes from sources
local to the measurement process and information from
other sources -- although this interpretation does not always
hold. In the computation of the final uncertainty it makes no
difference how the components are classified because the
ISO guidelines treat type A and type B evaluations in the
same manner.
Rule of
quadrature
All uncertainty components (standard deviations) are
combined by root-sum-squares (quadrature) to arrive at a
'standard uncertainty', u, which is the standard deviation of
the reported value, taking into account all sources of error,
both random and systematic, that affect the measurement
result.
Expanded
uncertainty
for a high
degree of
confidence
If the purpose of the uncertainty statement is to provide
coverage with a high level of confidence, an expanded
uncertainty is computed as
U = k u
where k is chosen to be the t
1-/2,
critical value from the t-
table with degrees of freedom.
For large degrees of freedom, it is suggested to use k = 2
to approximate 95% coverage. Details for these calculations
are found under degrees of freedom.
Type B
evaluations
Type B evaluations apply to random errors and biases for
which there is little or no data from the local process, and
to random errors and biases from other measurement
processes.
2.5.2.1. Steps

2.5.2. Approach
2.5.2.1. Steps
Steps in
uncertainty
analysis -
define the
result to
be
reported
The first step in the uncertainty evaluation is the definition of
the result to be reported for the test item for which an
uncertainty is required. The computation of the standard
deviation depends on the number of repetitions on the test
item and the range of environmental and operational
conditions over which the repetitions were made, in addition
to other sources of error, such as calibration uncertainties for
reference standards, which influence the final result. If the
value for the test item cannot be measured directly, but must
be calculated from measurements on secondary quantities, the
equation for combining the various quantities must be defined.
The steps to be followed in an uncertainty analysis are
outlined for two situations:
Outline of
steps to be
followed in
the
evaluation
of
uncertainty
for a
single
quantity
A. Reported value involves measurements on one quantity.
1. Compute a type A standard deviation for random
sources of error from:
Replicated results for the test item.
Measurements on a check standard.
Measurements made according to a 2-level
designed experiment
Measurements made according to a 3-level
designed experiment
2. Make sure that the collected data and analysis cover all
sources of random error such as:
instrument imprecision
day-to-day variation
long-term variation
and bias such as:
differences among instruments
operator differences.
3. Compute a standard deviation for each type B
component of uncertainty.
4. Combine type A and type B standard deviations into a
2.5.2.1. Steps
standard uncertainty for the reported result using
sensitivity factors.
5. Compute an expanded uncertainty.
Outline of
steps to be
followed in
the
evaluation
of
uncertainty
involving
several
secondary
quantities
B. - Reported value involves more than one quantity.
1. Write down the equation showing the relationship
between the quantities.
Write-out the propagation of error equation and
do a preliminary evaluation, if possible, based on
propagation of error.
2. If the measurement result can be replicated directly,
regardless of the number of secondary quantities in the
individual repetitions, treat the uncertainty evaluation as
in (A.1) to (A.5) above, being sure to evaluate all
sources of random error in the process.
3. If the measurement result cannot be replicated
directly, treat each measurement quantity as in (A.1)
and (A.2) and:
Compute a standard deviation for each
measurement quantity.
Combine the standard deviations for the
individual quantities into a standard deviation for
the reported result via propagation of error.
4. Compute a standard deviation for each type B
5. Combine type A and type B standard deviations into a
standard uncertainty for the reported result.
6. Compute an expanded uncertainty.
7. Compare the uncerainty derived by propagation of error
with the uncertainty derived by data analysis techniques.
2.5.3. Type A evaluations

Type A
evaluations
apply to
both error
and bias
Type A evaluations can apply to both random error and bias.
The only requirement is that the calculation of the uncertainty
component be based on a statistical analysis of data. The
distinction to keep in mind with regard to random error and
bias is that:
random errors cannot be corrected
biases can, theoretically at least, be corrected or
eliminated from the result.
Caveat for
biases
The ISO guidelines are based on the assumption that all biases
are corrected and that the only uncertainty from this source is
the uncertainty of the correction. The section on type A
evaluations of bias gives guidance on how to assess, correct
and calculate uncertainties related to bias.
Random
error and
bias
require
different
types of
analyses
How the source of error affects the reported value and the
context for the uncertainty determines whether an analysis of
random error or bias is appropriate.
Consider a laboratory with several instruments that can
reasonably be assumed to be representative of all similar
instruments. Then the differences among these instruments
can be considered to be a random effect if the uncertainty
statement is intended to apply to the result of any instrument,
selected at random, from this batch.
If, on the other hand, the uncertainty statement is intended to
apply to one specific instrument, then the bias of this
instrument relative to the group is the component of interest.
The following pages outline methods for type A evaluations
of:
1. Random errors
2. Bias
2.5.3.1. Type A evaluations of random components

2.5.3.1. Type A evaluations of random
components
Type A
evaluations of
random
components
Type A sources of uncertainty fall into three main
categories:
1. Uncertainties that reveal themselves over time
2. Uncertainties caused by specific conditions of
measurement
3. Uncertainties caused by material inhomogeneities
Time-dependent
changes are a
primary source
of random
errors
One of the most important indicators of random error is
time, with the root cause perhaps being environmental
changes over time. Three levels of time-dependent
effects are discussed in this section.
Many possible
configurations
may exist in a
laboratory for
making
measurements
Other sources of uncertainty are related to measurement
configurations within the laboratory. Measurements on
test items are usually made on a single day, with a single
operator, on a single instrument, etc. If the intent of the
uncertainty is to characterize all measurements made in
the laboratory, the uncertainty should account for any
differences due to:
1. instruments
2. operators
3. geometries
4. other
Examples of
causes of
differences
within a
laboratory
Examples of causes of differences within a well-
maintained laboratory are:
1. Differences among instruments for measurements
of derived units, such as sheet resistance of silicon,
where the instruments cannot be directly calibrated
to a reference base
2. Differences among operators for optical
measurements that are not automated and depend
strongly on operator sightings
3. Differences among geometrical or electrical
configurations of the instrumentation
Calibrated
instruments do
not fall in this
class
Calibrated instruments do not normally fall in this class
because uncertainties associated with the instrument's
calibration are reported as type B evaluations, and the
instruments in the laboratory should agree within the
calibration uncertainties. Instruments whose responses are
not directly calibrated to the defined unit are candidates
for type A evaluations. This covers situations in which
the measurement is defined by a test procedure or
standard practice using a specific instrument type.
Evaluation
depends on the
context for the
uncertainty
How these differences are treated depends primarily on
the context for the uncertainty statement. The differences,
depending on the context, will be treated either as
random differences, or as bias differences.
Uncertainties
due to
inhomogeneities
Artifacts, electrical devices, and chemical substances, etc.
can be inhomogeneous relative to the quantity that is
being characterized by the measurement process. If this
fact is known beforehand, it may be possible to measure
the artifact very carefully at a specific site and then direct
the user to also measure at this site. In this case, there is
no contribution to measurement uncertainty from
inhomogeneity.
However, this is not always possible, and measurements
may be destructive. As an example, compositions of
chemical compounds may vary from bottle to bottle. If
the reported value for the lot is established from
measurements on a few bottles drawn at random from the
lot, this variability must be taken into account in the
uncertainty statement.
Methods for testing for inhomogeneity and assessing the
appropriate uncertainty are discussed on another page.
2.5.3.1.1. Type A evaluations of time-dependent effects

2.5.3.1.1. Type A evaluations of time-dependent
effects
Time-
dependent
changes are a
primary
source of
random
errors
One of the most important indicators of random error is
time. Effects not specifically studied, such as
environmental changes, exhibit themselves over time.
Three levels of time-dependent errors are discussed in this
section. These can be usefully characterized as:
1. Level-1 or short-term errors (repeatability,
imprecision)
2. Level-2 or day-to-day errors (reproducibility)
3. Level-3 or long-term errors (stability - which may
not be a concern for all processes)
Day-to-day
errors can be
the dominant
source of
uncertainty
With instrumentation that is exceedingly precise in the
short run, changes over time, often caused by small
environmental effects, are frequently the dominant source
of uncertainty in the measurement process. The uncertainty
statement is not 'true' to its purpose if it describes a
situation that cannot be reproduced over time. The
customer for the uncertainty is entitled to know the range
of possible results for the measurement result, independent
of the day or time of year when the measurement was
made.
Two levels
may be
sufficient
Two levels of time-dependent errors are probably
sufficient for describing the majority of measurement
processes. Three levels may be needed for new
measurement processes or processes whose characteristics
are not well understood.
Measurements
on test item
are used to
assess
uncertainty
only when no
other data are
available
Repeated measurements on the test item generally do not
cover a sufficient time period to capture day-to-day
changes in the measurement process. The standard
deviation of these measurements is quoted as the estimate
of uncertainty only if no other data are available for the
assessment. For J short-term measurements, this standard
deviation has v = J - 1 degrees of freedom.
2.5.3.1.1. Type A evaluations of time-dependent effects
A check
standard is
the best
device for
capturing all
sources of
random error
The best approach for capturing information on time-
dependent sources of uncertainties is to intersperse the
workload with measurements on a check standard taken at
set intervals over the life of the process. The standard
deviation of the check standard measurements estimates
the overall temporal component of uncertainty directly --
thereby obviating the estimation of individual components.
Nested design
for estimating
type A
uncertainties
Case study:
Temporal
uncertainty
from a 3-level
nested design
A less-efficient method for estimating time-dependent
sources of uncertainty is a designed experiment.
Measurements can be made specifically for estimating two
or three levels of errors. There are many ways to do this,
but the easiest method is a nested design where J short-
term measurements are replicated on K days and the entire
operation is then replicated over L runs (months, etc.). The
analysis of these data leads to:
= standard deviation with (J -1) degrees of
freedom for short-term errors
= standard deviation with (K -1) degrees of
freedom for day-to-day errors
= standard deviation with (L -1) degrees of
freedom for very long-term errors
Approaches
given in this
chapter
The computation of the uncertainty of the reported value
for a test item is outlined for situations where temporal
sources of uncertainty are estimated from:
1. measurements on the test item itself
2. measurements on a check standard
3. measurements from a 2-level nested design (gauge
study)
4. measurements from a 3-level nested design (gauge
study)
2.5.3.1.2. Measurement configuration within the laboratory

2.5.3.1.2. Measurement configuration within the
laboratory
Purpose of
this page
The purpose of this page is to outline options for estimating
uncertainties related to the specific measurement
configuration under which the test item is measured, given
other possible measurement configurations. Some of these
may be controllable and some of them may not, such as:
instrument
operator
temperature
humidity
The effect of uncontrollable environmental conditions in
the laboratory can often be estimated from check standard
data taken over a period of time, and methods for
calculating components of uncertainty are discussed on
other pages. Uncertainties resulting from controllable
factors, such as operators or instruments chosen for a
specific measurement, are discussed on this page.
First, decide
on context for
uncertainty
The approach depends primarily on the context for the
uncertainty statement. For example, if instrument effect is
the question, one approach is to regard, say, the instruments
in the laboratory as a random sample of instruments of the
same type and to compute an uncertainty that applies to all
results regardless of the particular instrument on which the
measurements are made. The other approach is to compute
an uncertainty that applies to results using a specific
instrument.
Next,
evaluate
whether or
not there are
differences
To treat instruments as a random source of uncertainty
requires that we first determine if differences due to
instruments are significant. The same can be said for
operators, etc.
Plan for
collecting
data
To evaluate the measurement process for instruments,
select a random sample of I (I > 4) instruments from those
available. Make measurements on Q (Q >2) artifacts with
each instrument.
Graph
showing
differences
among
instruments
For a graphical analysis, differences from the average for
each artifact can be plotted versus artifact, with instruments
individually identified by a special plotting symbol. The
plot is examined to determine if some instruments always
read high or low relative to the other instruments and if this
behavior is consistent across artifacts. If there are
systematic and significant differences among instruments, a
type A uncertainty for instruments is computed. Notice that
in the graph for resistivity probes, there are differences
among the probes with probes #4 and #5, for example,
consistently reading low relative to the other probes. A
standard deviation that describes the differences among the
probes is included as a component of the uncertainty.
Standard
deviation for
instruments
Given the measurements,
for each of Q artifacts and I instruments, the pooled
standard deviation that describes the differences among
instruments is:
where
Example of
resistivity
measurements
on silicon
wafers
A two-way table of resistivity measurements (ohm.cm)
with 5 probes on 5 wafers (identified as: 138, 139, 140, 141,
142) is shown below. Standard deviations for probes with 4
degrees of freedom each are shown for each wafer. The
pooled standard deviation over all wafers, with 20 degrees
of freedom, is the type A standard deviation for
instruments.
Wafers
Probe 138 139 140 141
142
------------------------------------------------
-------
1 95.1548 99.3118 96.1018 101.1248
94.2593
281 95.1408 99.3548 96.0805 101.0747
94.2907
. 283 95.1493 99.3211 96.0417 101.1100
94.2487
2062 95.1125 99.2831 96.0492 101.0574
94.2520
2362 95.0928 99.3060 96.0357 101.0602
94.2148
Std dev 0.02643 0.02612 0.02826 0.03038
0.02711
DF 4 4 4 4
4
Pooled standard deviation = 0.02770 DF =
20
2.5.3.2. Material inhomogeneity

Purpose of this
page
The purpose of this page is to outline methods for
assessing uncertainties related to material
inhomogeneities. Artifacts, electrical devices, and
chemical substances, etc. can be inhomogeneous relative
to the quantity that is being characterized by the
Effect of
inhomogeneity
on the
uncertainty
Inhomogeneity can be a factor in the uncertainty analysis
where
1. an artifact is characterized by a single value and
the artifact is inhomogeneous over its surface, etc.
2. a lot of items is assigned a single value from a few
samples from the lot and the lot is inhomogeneous
from sample to sample.
An unfortunate aspect of this situation is that the
uncertainty from inhomogeneity may dominate the
uncertainty. If the measurement process itself is very
precise and in statistical control, the total uncertainty may
still be unacceptable for practical purposes because of
material inhomogeneities.
Targeted
measurements
can eliminate
the effect of
inhomogeneity
It may be possible to measure an artifact very carefully at
a specific site and direct the user to also measure at this
site. In this case there is no contribution to measurement
uncertainty from inhomogeneity.
Example Silicon wafers are doped with boron to produce desired
levels of resistivity (ohm.cm). Manufacturing processes
for semiconductors are not yet capable (at least at the
time this was originally written) of producing 2" diameter
wafers with constant resistivity over the surfaces.
However, because measurements made at the center of a
wafer by a certification laboratory can be reproduced in
the industrial setting, the inhomogeneity is not a factor in
the uncertainty analysis -- as long as only the center-
point of the wafer is used for future measurements.
Random Random inhomogeneities are assessed using statistical
inhomogeneities methods for quantifying random errors. An example of
inhomogeneity is a chemical compound which cannot be
sufficiently homogenized with respect to isotopes of
interest. Isotopic ratio determinations, which are
destructive, must be determined from measurements on a
few bottles drawn at random from the lot.
Best strategy The best strategy is to draw a sample of bottles from the
lot for the purpose of identifying and quantifying
between-bottle variability. These measurements can be
made with a method that lacks the accuracy required to
certify isotopic ratios, but is precise enough to allow
between-bottle comparisons. A second sample is drawn
from the lot and measured with an accurate method for
determining isotopic ratios, and the reported value for the
lot is taken to be the average of these determinations.
There are therefore two components of uncertainty
assessed:
1. component that quantifies the imprecision of the
average
2. component that quantifies how much an individual
bottle can deviate from the average.
Systematic
inhomogeneities
Systematic inhomogeneities require a somewhat different
approach. Roughness can vary systematically over the
surface of a 2" square metal piece lathed to have a
specific roughness profile. The certification laboratory
can measure the piece at several sites, but unless it is
possible to characterize roughness as a mathematical
function of position on the piece, inhomogeneity must be
assessed as a source of uncertainty.
Best strategy In this situation, the best strategy is to compute the
reported value as the average of measurements made over
the surface of the piece and assess an uncertainty for
departures from the average. The component of
uncertainty can be assessed by one of several methods
for evaluating bias -- depending on the type of
inhomogeneity.
Standard
method
The simplest approach to the computation of uncertainty
for systematic inhomogeneity is to compute the
maximum deviation from the reported value and,
assuming a uniform, normal or triangular distribution for
the distribution of inhomogeneity, compute the
appropriate standard deviation. Sometimes the
approximate shape of the distribution can be inferred
from the inhomogeneity measurements. The standard
deviation for inhomogeneity assuming a uniform
distribution is:
2.5.3.2.1. Data collection and analysis

Purpose of
this page
The purpose of this page is to outline methods for:
collecting data
testing for inhomogeneity
quantifying the component of uncertainty
Balanced
measurements
at 2-levels
The simplest scheme for identifying and quantifying the effect of
inhomogeneity of a measurement result is a balanced (equal number of
measurements per cell) 2-level nested design. For example, K bottles
of a chemical compound are drawn at random from a lot and J (J > 1)
measurements are made per bottle. The measurements are denoted by
where the k index runs over bottles and the j index runs over
repetitions within a bottle.
Analysis of
measurements
The between (bottle) variance is calculated using an analysis of
variance technique that is repeated here for convenience.
where
and
Between
bottle
variance may
be negative
If this variance is negative, there is no contribution to uncertainty, and
the bottles are equivalent with regard to their chemical compositions.
Even if the variance is positive, inhomogeneity still may not be
statistically significant, in which case it is not required to be included
as a component of the uncertainty.
If the between-bottle variance is statistically significantly (i.e., judged
to be greater than zero), then inhomogeneity contributes to the
uncertainty of the reported value.
Certification,
reported
value and
associated
uncertainty
The purpose of assessing inhomogeneity is to be able to assign a value
to the entire batch based on the average of a few bottles, and the
determination of inhomogeneity is usually made by a less accurate
method than the certification method. The reported value for the batch
would be the average of N repetitions on Q bottles using the
certification method.
The uncertainty calculation is summarized below for the case where
the only contribution to uncertainty from the measurement method
itself is the repeatability standard deviation, s
1
associated with the
certification method. For more complicated scenarios, see the pages on
uncertainty budgets.
If s
reported value

If , we need to distinguish two cases and their interpretations:
1. The standard deviation
leads to an interval that covers the difference between the
reported value and the average for a bottle selected at random
from the batch.
2. The standard deviation
allows one to test the instrument using a single measurement.
The prediction interval for the difference between the reported
value and a single measurement, made with the same precision
as the certification measurements, on a bottle selected at random
from the batch. This is appropriate when the instrument under
test is similar to the certification instrument. If the difference is
not within the interval, the user's instrument is in need of
calibration.
Relationship
to prediction
intervals
When the standard deviation for inhomogeneity is included in the
calculation, as in the last two cases above, the uncertainty interval
becomes a prediction interval ( Hahn & Meeker) and is interpreted as
characterizing a future measurement on a bottle drawn at random from
the lot.
2.5.3.3. Type A evaluations of bias

Sources of
bias relate to
the specific
measurement
environment
The sources of bias discussed on this page cover specific
measurement configurations. Measurements on test items
are usually made on a single day, with a single operator,
with a single instrument, etc. Even if the intent of the
uncertainty is to characterize only those measurements made
in one specific configuration, the uncertainty must account
for any significant differences due to:
1. instruments
2. operators
3. geometries
4. other
Calibrated
instruments
do not fall in
this class
Calibrated instruments do not normally fall in this class
because uncertainties associated with the instrument's
calibration are reported as type B evaluations, and the
instruments in the laboratory should agree within the
calibration uncertainties. Instruments whose responses are
not directly calibrated to the defined unit are candidates for
type A evaluations. This covers situations where the
measurement is defined by a test procedure or standard
practice using a specific instrument type.
The best
strategy is to
correct for
bias and
compute the
uncertainty
of the
correction
This problem was treated on the foregoing page as an
analysis of random error for the case where the uncertainty
was intended to apply to all measurements for all
configurations. If measurements for only one configuration
are of interest, such as measurements made with a specific
instrument, or if a smaller uncertainty is required, the
differences among, say, instruments are treated as biases.
The best strategy in this situation is to correct all
measurements made with a specific instrument to the
average for the instruments in the laboratory and compute a
type A uncertainty for the correction. This strategy, of
course, relies on the assumption that the instruments in the
laboratory represent a random sample of all instruments of a
specific type.
Only limited
comparisons
can be made
However, suppose that it is possible to make comparisons
among, say, only two instruments and neither is known to
be 'unbiased'. This scenario requires a different strategy
among
sources of
possible bias
because the average will not necessarily be an unbiased
result. The best strategy if there is a significant difference
between the instruments, and this should be tested, is to
apply a 'zero' correction and assess a type A uncertainty of
the correction.
Guidelines
for treatment
of biases
The discussion above is intended to point out that there are
many possible scenarios for biases and that they should be
treated on a case-by-case basis. A plan is needed for:
gathering data
testing for bias (graphically and/or statistically)
estimating biases
assessing uncertainties associated with significant
biases.
caused by:
instruments
operators
configurations, geometries, etc.
inhomogeneities
Plan for
testing for
assessing
bias
Measurements needed for assessing biases among
instruments, say, requires a random sample of I (I > 1)
instruments from those available and measurements on Q (Q
>2) artifacts with each instrument. The same can be said for
the other sources of possible bias. General strategies for
dealing with significant biases are given in the table below.
Data collection and analysis for assessing biases related to:
lack of resolution of instrument
non-linearity of instrument
drift
are addressed in the section on gauge studies.
Sources of
data for
evaluating
this type of
bias
Databases for evaluating bias may be available from:
check standards
gauge R and R studies
control measurements
Strategies for assessing corrections and uncertainties associated with significant biases
Type of bias Examples Type of correction Uncertainty
1. Inconsistent
Sign change (+ to -)
Varying magnitude
Zero
Based on
maximum
bias
2. Consistent
Instrument bias ~ same
magnitude over many
artifacts
Bias (for a single
instrument) =
difference from
average over several
instruments
Standard
deviation of
correction
3. Not correctable
because of sparse data
- consistent or
inconsistent
Limited testing; e.g.,
only 2 instruments,
operators,
configurations, etc.
Zero
Standard
deviation of
correction
4. Not correctable -
consistent
Lack of resolution,
non-linearity, drift,
material inhomogeneity
Zero
Based on
maximum
bias
Strategy
for no
significant
bias
If there is no significant bias over time, there is no correction
and no contribution to uncertainty.
2.5.3.3.1. Inconsistent bias

Strategy for
inconsistent
bias -- apply
a zero
correction
If there is significant bias but it changes direction over time,
a zero correction is assumed and the standard deviation of
the correction is reported as a type A uncertainty; namely,
Computations
based on
uniform or
normal
distribution
The equation for estimating the standard deviation of the
correction assumes that biases are uniformly distributed
between {-max |bias|, + max |bias|}. This assumption is
quite conservative. It gives a larger uncertainty than the
assumption that the biases are normally distributed. If
normality is a more reasonable assumption, substitute the
number '3' for the 'square root of 3' in the equation above.
Example of
change in
bias over
time
The results of resistivity measurements with five probes on
five silicon wafers are shown below for probe #283, which
is the probe of interest at this level with the artifacts being
1 ohm.cm wafers. The bias for probe #283 is negative for
run 1 and positive for run 2 with the runs separated by a
two-month time period. The correction is taken to be zero.
Table of biases (ohm.cm) for probe 283
Wafer Probe Run 1 Run 2
-----------------------------------
11 283 0.0000340 -0.0001841
26 283 -0.0001000 0.0000861
42 283 0.0000181 0.0000781
131 283 -0.0000701 0.0001580
208 283 -0.0000240 0.0001879
Average 283 -0.0000284 0.0000652
A conservative assumption is that the bias could fall
somewhere within the limits a, with a = maximum bias or
0.0000652 ohm.cm. The standard deviation of the
correction is included as a type A systematic component of
the uncertainty.
2.5.3.3.2. Consistent bias

Consistent
bias
Bias that is significant and persists consistently over time for a
specific instrument, operator, or configuration should be corrected if it
can be reliably estimated from repeated measurements. Results with
the instrument of interest are then corrected to:
Corrected result = Measurement - Estimate of bias
The example below shows how bias can be identified graphically
from measurements on five artifacts with five instruments and
estimated from the differences among the instruments.
Graph
showing
consistent
bias for
probe #5
An analysis of bias for five instruments based on measurements on
five artifacts shows differences from the average for each artifact
plotted versus artifact with instruments individually identified by a
special plotting symbol. The plot is examined to determine if some
instruments always read high or low relative to the other instruments,
and if this behavior is consistent across artifacts. Notice that on the
graph for resistivity probes, probe #2362, (#5 on the graph), which is
the instrument of interest for this measurement process, consistently
reads low relative to the other probes. This behavior is consistent over
2 runs that are separated by a two-month time period.
Strategy -
correct for
bias
Because there is significant and consistent bias for the instrument of
interest, the measurements made with that instrument should be
corrected for its average bias relative to the other instruments.
Computation
of bias
Given the measurements,
on Q artifacts with I instruments, the average bias for instrument, I'
say, is
where
Computation
of correction
The correction that should be made to measurements made with
instrument I' is
Type A
uncertainty
of the
correction
The type A uncertainty of the correction is the standard deviation of
the average bias or
Example of
consistent
bias for
probe #2362
used to
measure
resistivity of
silicon
wafers
The table below comes from the table of resistivity measurements
from a type A analysis of random effects with the average for each
wafer subtracted from each measurement. The differences, as shown,
represent the biases for each probe with respect to the other probes.
Probe #2362 has an average bias, over the five wafers, of -0.02724
ohm.cm. If measurements made with this probe are corrected for this
bias, the standard deviation of the correction is a type A uncertainty.
Table of biases for probes and silicon wafers (ohm.cm)
Wafers
Probe 138 139 140 141 142
-------------------------------------------------------
1 0.02476 -0.00356 0.04002 0.03938 0.00620
181 0.01076 0.03944 0.01871 -0.01072 0.03761
182 0.01926 0.00574 -0.02008 0.02458 -0.00439
2062 -0.01754 -0.03226 -0.01258 -0.02802 -0.00110
2362 -0.03725 -0.00936 -0.02608 -0.02522 -0.03830
Average bias for probe #2362 = - 0.02724
Standard deviation of bias = 0.01171 with
4 degrees of freedom
Standard deviation of correction =
0.01171/sqrt(5) = 0.00523
Note on
different
approaches
to
instrument
bias
The analysis on this page considers the case where only one
instrument is used to make the certification measurements; namely
probe #2362, and the certified values are corrected for bias due to this
probe. The analysis in the section on type A analysis of random effects
considers the case where any one of the probes could be used to make
the certification measurements.
2.5.3.3.3. Bias with sparse data

Strategy for
dealing with
limited data
The purpose of this discussion is to outline methods for dealing with biases that may be
real but which cannot be estimated reliably because of the sparsity of the data. For
example, a test between two, of many possible, configurations of the measurement
process cannot produce a reliable enough estimate of bias to permit a correction, but it
can reveal problems with the measurement process. The strategy for a significant bias is
to apply a 'zero' correction. The type A uncertainty component is the standard deviation
of the correction, and the calculation depends on whether the bias is
inconsistent
consistent
The analyses in this section can be produced using both Dataplot code and R code.
Example of
differences
among wiring
settings
An example is given of a study of wiring settings for a single gauge. The gauge, a 4-
point probe for measuring resistivity of silicon wafers, can be wired in several ways.
Because it was not possible to test all wiring configurations during the gauge study,
measurements were made in only two configurations as a way of identifying possible
problems.
Data on
wiring
configurations
Measurements were made on six wafers over six days (except for 5 measurements on
wafer 39) with probe #2062 wired in two configurations. This sequence of
measurements was repeated after about a month resulting in two runs. A database of
differences between measurements in the two configurations on the same day are
analyzed for significance.
Plot the
differences
between the
two wiring
configurations
A plot of the differences between the two configurations shows that the differences for
run 1 are, for the most part, less than zero, and the differences for run 2 are greater than
zero.
Statistical test
for difference
between two
configurations
A t-statistic is used as an approximate test where we are
assuming the differences are approximately normal. The
average difference and standard deviation of the difference
are required for this test. If
the difference between the two configurations is statistically
significant.
The average and standard deviation computed from the N =
29 differences in each run from the table above are shown
along with corresponding t-values which confirm that the
differences are significant, but in opposite directions, for
both runs.
Average differences between wiring
configurations
Run Probe Average Std dev N
t
1 2062 - 0.00383 0.00514 29
- 4.0
2 2062 + 0.00489 0.00400 29
+ 6.6
Case of
inconsistent
bias
The data reveal a significant wiring bias for both runs that
changes direction between runs. Because of this
inconsistency, a 'zero' correction is applied to the results,
and the type A uncertainty is taken to be
For this study, the type A uncertainty for wiring bias is
Case of
consistent
bias
Even if the bias is consistent over time, a 'zero' correction is
applied to the results, and for a single run, the estimated
standard deviation of the correction is
For two runs (1 and 2), the estimated standard deviation of
the correction is
2.5.4. Type B evaluations

Type B
evaluations
apply to both
error and
bias
Type B evaluations can apply to both random error and bias.
The distinguishing feature is that the calculation of the
uncertainty component is not based on a statistical analysis
of data. The distinction to keep in mind with regard to
random error and bias is that:
random errors cannot be corrected
biases can, theoretically at least, be corrected or
eliminated from the result.
Sources of
type B
evaluations
Some examples of sources of uncertainty that lead to type B
evaluations are:
Reference standards calibrated by another laboratory
Physical constants used in the calculation of the
reported value
Environmental effects that cannot be sampled
Possible configuration/geometry misalignment in the
instrument
Lack of resolution of the instrument
Documented
sources of
uncertainty
from other
processes
Documented sources of uncertainty, such as calibration
reports for reference standards or published reports of
uncertainties for physical constants, pose no difficulties in
the analysis. The uncertainty will usually be reported as an
expanded uncertainty, U, which is converted to the standard
uncertainty,
u = U/k
If the k factor is not known or documented, it is probably
conservative to assume that k = 2.
Sources of
uncertainty
that are
local to the
measurement
process
Sources of uncertainty that are local to the measurement
process but which cannot be adequately sampled to allow a
statistical analysis require type B evaluations. One
technique, which is widely used, is to estimate the worst-
case effect, a, for the source of interest, from
experience
scientific judgment
scant data
A standard deviation, assuming that the effect is two-sided,
can then be computed based on a uniform, triangular, or
normal distribution of possible effects.
Following the Guide to the Expression of Uncertainty of
Measurement (GUM), the convention is to assign infinite
degrees of freedom to standard deviations derived in this
manner.
2.5.4.1. Standard deviations from assumed distributions

2.5.4.1. Standard deviations from assumed
distributions
Difficulty
of
obtaining
reliable
uncertainty
estimates
The methods described on this page attempt to avoid the
difficulty of allowing for sources of error for which reliable
estimates of uncertainty do not exist. The methods are based
on assumptions that may, or may not, be valid and require the
experimenter to consider the effect of the assumptions on the
final uncertainty.
Difficulty
of
obtaining
reliable
uncertainty
estimates
The ISO guidelines do not allow worst-case estimates of bias
to be added to the other components, but require they in some
way be converted to equivalent standard deviations. The
approach is to consider that any error or bias, for the situation
at hand, is a random draw from a known statistical
distribution. Then the standard deviation is calculated from
known (or assumed) characteristics of the distribution.
Distributions that can be considered are:
Uniform
Triangular
Normal (Gaussian)
Standard
deviation
for a
uniform
distribution
The uniform distribution leads to the most conservative
estimate of uncertainty; i.e., it gives the largest standard
deviation. The calculation of the standard deviation is based
on the assumption that the end-points, a, of the distribution
are known. It also embodies the assumption that all effects on
the reported value, between -a and +a, are equally likely for
the particular source of uncertainty.
Standard
deviation
for a
triangular
The triangular distribution leads to a less conservative
estimate of uncertainty; i.e., it gives a smaller standard
deviation than the uniform distribution. The calculation of the
standard deviation is based on the assumption that the end-
2.5.4.1. Standard deviations from assumed distributions
distribution points, a, of the distribution are known and the mode of the
triangular distribution occurs at zero.
Standard
deviation
for a
normal
distribution
The normal distribution leads to the least conservative
estimate of uncertainty; i.e., it gives the smallest standard
deviation. The calculation of the standard deviation is based
on the assumption that the end-points, a, encompass 99.7
percent of the distribution.
Degrees of
freedom
In the context of using the Welch-Saitterthwaite formula with
the above distributions, the degrees of freedom is assumed to
be infinite.
2.5.5. Propagation of error considerations

Top-down
approach
consists of
estimating the
uncertainty
from direct
repetitions of
the
measurement
result
The approach to uncertainty analysis that has been followed up to this point
in the discussion has been what is called a top-down approach. Uncertainty
components are estimated from direct repetitions of the measurement result.
To contrast this with a propagation of error approach, consider the simple
example where we estimate the area of a rectangle from replicate
measurements of length and width. The area
area = length x width
can be computed from each replicate. The standard deviation of the reported
area is estimated directly from the replicates of area.
Advantages of
top-down
approach
This approach has the following advantages:
proper treatment of covariances between measurements of length and
width
proper treatment of unsuspected sources of error that would emerge if
measurements covered a range of operating conditions and a
sufficiently long time period
independence from propagation of error model
Propagation
of error
approach
combines
estimates from
individual
auxiliary
measurements
The formal propagation of error approach is to compute:
1. standard deviation from the length measurements
2. standard deviation from the width measurements
and combine the two into a standard deviation for area using the
approximation for products of two variables (ignoring a possible covariance
between length and width),
Exact formula Goodman (1960) derived an exact formula for the variance between two
products. Given two random variables, x and y (correspond to width and
length in the above approximate formula), the exact formula for the variance
is:
with
X = E(x) and Y = E(y) (corresponds to width and length, respectively,
in the approximate formula)
V(x) = variance of x and V(y) = variance Y (corresponds to s
2
for
width and length, respectively, in the approximate formula)
E
ij
= {( x)
i
, ( y)
j
} where x = x - X and y = y - Y
To obtain the standard deviation, simply take the square root of the above
formula. Also, an estimate of the statistic is obtained by substituting sample
estimates for the corresponding population values on the right hand side of
the equation.
Approximate
formula
assumes
indpendence
The approximate formula assumes that length and width are independent.
The exact formula assumes that length and width are not independent.
Disadvantages
of
propagation
of error
approach
In the ideal case, the propagation of error estimate above will not differ from
the estimate made directly from the area measurements. However, in
complicated scenarios, they may differ because of:
unsuspected covariances
disturbances that affect the reported value and not the elementary
measurements (usually a result of mis-specification of the model)
mistakes in propagating the error through the defining formulas
Propagation
of error
formula
Sometimes the measurement of interest cannot be replicated directly and it is
necessary to estimate its uncertainty via propagation of error formulas (Ku).
The propagation of error formula for
Y = f(X, Z, ... )
a function of one or more variables with measurements, X, Z, ... gives the
following estimate for the standard deviation of Y:
where
is the standard deviation of the X measurements
is the standard deviation of Z measurements
is the standard deviation of Y measurements
is the partial derivative of the function Y with respect to X,
etc.
is the estimated covariance between the X,Z measurements
Treatment of
covariance
terms
Covariance terms can be difficult to estimate if measurements are not made
in pairs. Sometimes, these terms are omitted from the formula. Guidance on
when this is acceptable practice is given below:
1. If the measurements of X, Z are independent, the associated covariance
term is zero.
2. Generally, reported values of test items from calibration designs have
non-zero covariances that must be taken into account if Y is a
summation such as the mass of two weights, or the length of two gage
blocks end-to-end, etc.
3. Practically speaking, covariance terms should be included in the
computation only if they have been estimated from sufficient data. See
Ku (1966) for guidance on what constitutes sufficient data.
Sensitivity
coefficients
The partial derivatives are the sensitivity coefficients for the associated
components.
Examples of
propagation
of error
analyses
Examples of propagation of error that are shown in this chapter are:
Case study of propagation of error for resistivity measurements
Comparison of check standard analysis and propagation of error for
linear calibration
Propagation of error for quadratic calibration showing effect of
covariance terms
Specific
formulas
Formulas for specific functions can be found in the following sections:
functions of a single variable
functions of two variables
functions of many variables
2.5.5.1. Formulas for functions of one variable

Case:
Y=f(X,Z)
Standard deviations of reported values that are functions of a
single variable are reproduced from a paper by H. Ku (Ku).
The reported value, Y, is a function of the average of N
measurements on a single variable.
Notes
Function of
is an average of N
measurements
Standard deviation of
= standard deviation of X.

Approximation
could be
seriously in
error if n is
small--
Not directly
derived from
the formulas Note: we need to assume that the
original data follow an
approximately normal distribution.
2.5.5.2. Formulas for functions of two variables

Case:
Y=f(X,Z)
Standard deviations of reported values that are functions of
measurements on two variables are reproduced from a paper
by H. Ku (Ku).
The reported value, Y is a function of averages of N
measurements on two variables.
Function of ,
and are averages of N
measurements
Standard deviation of
= standard dev of X;
= standard dev of Z;
= covariance of X,Z
Note: Covariance term is to be included only if
there is a reliable estimate
Note: this is an approximation. The exact result
could be obtained starting from the exact formula
for the standard deviation of a product derived by
Goodman (1960).
2.5.5.3. Propagation of error for many variables

Example
from fluid
flow with a
nonlinear
function
Computing uncertainty for measurands based on more complicated functions
can be done using basic propagation of errors principles. For example,
suppose we want to compute the uncertainty of the discharge coefficient for
fluid flow (Whetstone et al.). The measurement equation is
where
Assuming the variables in the equation are uncorrelated, the squared
uncertainty of the discharge coefficient is
and the partial derivatives are the following.
Software can
simplify
propagation
of error
Propagation of error for more complicated functions can be done reliably with
software capable of symbolic computations or algebraic representations.
Symbolic computation software can also be used to combine the partial
derivatives with the appropriate standard deviations, and then the standard
deviation for the discharge coefficient can be evaluated and plotted for
specific values of the secondary variables, as shown in the comparison of
check standard analysis and propagation of error.
Simplification
for dealing
with
multiplicative
variables
Propagation of error for several variables can be simplified considerably for
the special case where:
the function, Y, is a simple multiplicative function of secondary
variables, and
uncertainty is evaluated as a percentage.
For three variables, X, Z, W, the function
has a standard deviation in absolute units of
In percent units, the standard deviation can be written as
if all covariances are negligible. These formulas are easily extended to more
than three variables.
2.5.6. Uncertainty budgets and sensitivity coefficients

2.5.6. Uncertainty budgets and sensitivity
coefficients
Case study
showing
uncertainty
budget
Uncertainty components are listed in a table along with their
corresponding sensitivity coefficients, standard deviations and
degrees of freedom. A table of typical entries illustrates the
concept.
Typical budget of type A and type B uncertainty components
Type A components Sensitivity coefficient
Standard
deviation
Degrees
freedom
1. Time (repeatability) v1
2. Time (reproducibility) v2
3. Time (long-term) v3
Type B components
5. Reference standard (nominal test / nominal ref) v4
Sensitivity
coefficients
show how
components
are related
to result
The sensitivity coefficient shows the relationship of the
individual uncertainty component to the standard deviation
of the reported value for a test item. The sensitivity
coefficient relates to the result that is being reported and not
to the method of estimating uncertainty components where
the uncertainty, u, is
Sensitivity
coefficients
for type A
components
of
uncertainty
This section defines sensitivity coefficients that are
appropriate for type A components estimated from repeated
measurements. The pages on type A evaluations, particularly
the pages related to estimation of repeatability and
reproducibility components, should be reviewed before
continuing on this page. The convention for the notation for
sensitivity coefficients for this section is that:
1. refers to the sensitivity coefficient for the
repeatability standard deviation,
2. refers to the sensitivity coefficient for the
reproducibility standard deviation,
3. refers to the sensitivity coefficient for the stability
standard deviation,
with some of the coefficients possibly equal to zero.
Note on
long-term
errors
Even if no day-to-day nor run-to-run measurements were
made in determining the reported value, the sensitivity
coefficient is non-zero if that standard deviation proved to
be significant in the analysis of data.
Sensitivity
coefficients
for other
type A
components
of random
error
Procedures for estimating differences among instruments,
operators, etc., which are treated as random components of
uncertainty in the laboratory, show how to estimate the
standard deviations so that the sensitivity coefficients = 1.
Sensitivity
coefficients
for type A
components
for bias
This Handbook follows the ISO guidelines in that biases are
corrected (correction may be zero), and the uncertainty
component is the standard deviation of the correction.
Procedures for dealing with biases show how to estimate the
standard deviation of the correction so that the sensitivity
coefficients are equal to one.
Sensitivity
coefficients
for specific
applications
The following pages outline methods for computing
sensitivity coefficients where the components of uncertainty
are derived in the following manner:
1. From measurements on the test item itself
2. From measurements on a check standard
3. From measurements in a 2-level design
4. From measurements in a 3-level design
and give an example of an uncertainty budget with
sensitivity coefficients from a 3-level design.
Sensitivity
coefficients
for type B
evaluations
The majority of sensitivity coefficients for type B
evaluations will be one with a few exceptions. The
sensitivity coefficient for the uncertainty of a reference
standard is the nominal value of the test item divided by the
nominal value of the reference standard.
Case study-
sensitivity
coefficients
for
propagation
of error
If the uncertainty of the reported value is calculated from
propagation of error, the sensitivity coefficients are the
multipliers of the individual variance terms in the
propagation of error formula. Formulas are given for
selected functions of:
1. functions of a single variable
2. functions of two variables
3. several variables
2.5.6.1. Sensitivity coefficients for measurements on the test item

2.5.6.1. Sensitivity coefficients for
measurements on the test item
From data
on the test
item itself
If the temporal component is estimated from N short-term
readings on the test item itself
Y
1
, Y
2
, ..., Y
N
and
and the reported value is the average, the standard deviation of
the reported value is
with degrees of freedom .
Sensitivity
coefficients
The sensitivity coefficient is . The risk in using this
method is that it may seriously underestimate the uncertainty.
To
improve
the
reliability
of the
uncertainty
calculation
If possible, the measurements on the test item should be
repeated over M days and averaged to estimate the reported
value. The standard deviation for the reported value is
computed from the daily averages>, and the standard
deviation for the temporal component is:
with degrees of freedom where are the daily
averages and is the grand average.
The sensitivity coefficients are: a
1
= 0; a
2
= .
Note on Even if no day-to-day nor run-to-run measurements were
2.5.6.1. Sensitivity coefficients for measurements on the test item
long-term
errors
made in determining the reported value, the sensitivity
coefficient is non-zero if that standard deviation proved to be
significant in the analysis of data.
2.5.6.2. Sensitivity coefficients for measurements on a check standard

measurements on a check standard
From
measurements
on check
standards
If the temporal component of the measurement process is
evaluated from measurements on a check standard and
there are M days (M = 1 is permissible) of measurements
on the test item that are structured in the same manner as
the measurements on the check standard, the standard
deviation for the reported value is
with degrees of freedom from the K entries in
the check standard database.
Standard
deviation
from check
standard
measurements
The computation of the standard deviation from the check
standard values and its relationship to components of
instrument precision and day-to-day variability of the
process are explained in the section on two-level nested
designs using check standards.
Sensitivity
coefficients
1
; a
2
= .
2.5.6.3. Sensitivity coefficients for measurements from a 2-level design

2.5.6.3. Sensitivity coefficients for measurements
from a 2-level design
Sensitivity
coefficients
from a 2-
level
design
If the temporal components are estimated from a 2-level
nested design, and the reported value for a test item is an
average over
N short-term repetitions
M (M = 1 is permissible) days
of measurements on the test item, the standard deviation for
the reported value is:
See the relationships in the section on 2-level nested design
for definitions of the standard deviations and their respective
degrees of freedom.
Problem
with
estimating
degrees of
freedom
If degrees of freedom are required for the uncertainty of the
reported value, the formula above cannot be used directly and
must be rewritten in terms of the standard deviations, and
.
Sensitivity
coefficients
1
= ;
a
2
= .
Specific sensitivity coefficients are shown in the table below
for selections of N, M.
Sensitivity coefficients for two
components
of uncertainty
Number Number
Short-term
Day-to-
short-
term
N
day-to-
day
M
sensitivity
coefficient
day
sensitivity
coefficient
1 1 1
N 1 1
N M

measurements from a 3-level design
Sensitivity
coefficients
from a 3-
level
design
Case study
showing
sensitivity
coefficients
for 3-level
design
If the temporal components are estimated from a 3-level
nested design and the reported value is an average over
N short-term repetitions
M days
P runs
of measurements on the test item, the standard deviation for
the reported value is:
See the section on analysis of variability for definitions and
relationships among the standard deviations shown in the
equation above.
Problem
with
estimating
degrees of
freedom
If degrees of freedom are required for the uncertainty, the
formula above cannot be used directly and must be rewritten
in terms of the standard deviations , , and .
Sensitivity
coefficients
The sensitivity coefficients are:
a
1
= ; a
2
= ;
a
3
= .
Specific sensitivity coefficients are shown in the table below
for selections of N, M, P. In addition, the following
constraints must be observed:
J must be > or = N and K must be > or = M
Sensitivity coefficients for three components of uncertainty
Number
short-
term
N
Number
day-to-
day
M
Number
run-to-
run
P
Short-term
sensitivity coefficient
Day-to-day
sensitivity coefficient
Run-to-
run
sensitivity
coefficient
1 1 1 1
N 1 1 1
N M 1 1
N M P
2.5.6.5. Example of uncertainty budget

2.5.6.5. Example of uncertainty budget
Example of
uncertainty
budget for
three
components
of temporal
uncertainty
An uncertainty budget that illustrates several principles of
uncertainty analysis is shown below. The reported value for a
test item is the average of N short-term measurements where
the temporal components of uncertainty were estimated from
a 3-level nested design with J short-term repetitions over K
days.
The number of measurements made on the test item is the
same as the number of short-term measurements in the
design; i.e., N = J. Because there were no repetitions over
days or runs on the test item, M = 1; P = 1. The sensitivity
coefficients for this design are shown on the foregoing page.
Example of
instrument
bias
This example also illustrates the case where the measuring
instrument is biased relative to the other instruments in the
laboratory, with a bias correction applied accordingly. The
sensitivity coefficient, given that the bias correction is based
on measurements on Q artifacts, is defined as a
4
= 1, and the
4
, is the standard deviation of the
correction.
Example of error budget for type A and type B uncertainties
Type A components Sensitivity coefficient
Standard
deviation
Degrees
freedom
1. Repeatability
= 0
J - 1
2. Reproducibility
=
K - 1
2. Stability
= 1
L - 1
3. Instrument bias
= 1
Q - 1
2.5.7. Standard and expanded uncertainties

Definition of
standard
uncertainty
The sensitivity coefficients and standard deviations are
combined by root sum of squares to obtain a 'standard
uncertainty'. Given R components, the standard uncertainty
is:
Expanded
uncertainty
assures a
high level of
confidence
If the purpose of the uncertainty statement is to provide
coverage with a high level of confidence, an expanded
uncertainty is computed as
where k is chosen to be the t
1-/2,
critical value from the t-
table with degrees of freedom. For large degrees of
freedom, k = 2 approximates 95 % coverage.
Interpretation
of uncertainty
statement
The expanded uncertainty defined above is assumed to
provide a high level of coverage for the unknown true value
of the measurement of interest so that for any measurement
result, Y,
2.5.7.1. Degrees of freedom

2.5.7.1. Degrees of freedom
Degrees of
freedom for
individual
components
of
uncertainty
Degrees of freedom for type A uncertainties are the degrees
of freedom for the respective standard deviations. Degrees of
freedom for Type B evaluations may be available from
published reports or calibration certificates. Special cases
where the standard deviation must be estimated from
fragmentary data or scientific judgment are assumed to have
infinite degrees of freedom; for example,
Worst-case estimate based on a robustness study or
other evidence
Estimate based on an assumed distribution of possible
errors
Type B uncertainty component for which degrees of
freedom are not documented
Degrees of
freedom for
the
standard
uncertainty
Degrees of freedom for the standard uncertainty, u, which
may be a combination of many standard deviations, is not
generally known. This is particularly troublesome if there are
large components of uncertainty with small degrees of
freedom. In this case, the degrees of freedom is approximated
by the Welch-Satterthwaite formula (Brownlee).
Case study:
Uncertainty
and
degrees of
freedom
A case study of type A uncertainty analysis shows the
computations of temporal components of uncertainty;
instrument bias; geometrical bias; standard uncertainty;
degrees of freedom; and expanded uncertainty.
2.5.8. Treatment of uncorrected bias

Background The ISO Guide ( ISO) for expressing measurement
uncertainties assumes that all biases are corrected and that the
uncertainty applies to the corrected result. For measurements
at the factory floor level, this approach has several
disadvantages. It may not be practical, may be expensive and
may not be economically sound to correct for biases that do
not impact the commercial value of the product (Turgel and
Vecchia).
Reasons for
not
correcting
for bias
Corrections may be expensive to implement if they require
modifications to existing software and "paper and pencil"
corrections can be both time consuming and prone to error.
In the scientific or metrology laboratory, biases may be
documented in certain situations, but the mechanism that
causes the bias may not be fully understood, or repeatable,
which makes it difficult to argue for correction. In these
cases, the best course of action is to report the measurement
as taken and adjust the uncertainty to account for the "bias".
The
question is
how to
adjust the
uncertainty
A method needs to be developed which assures that the
resulting uncertainty has the following properties (Phillips
and Eberhardt):
1. The final uncertainty must be greater than or equal to
the uncertainty that would be quoted if the bias were
corrected.
2. The final uncertainty must reduce to the same
uncertainty given that the bias correction is applied.
3. The level of coverage that is achieved by the final
uncertainty statement should be at least the level
obtained for the case of corrected bias.
4. The method should be transferable so that both the
uncertainty and the bias can be used as components of
uncertainty in another uncertainty statement.
5. The method should be easy to implement.
2.5.8.1. Computation of revised uncertainty

Definition of
the bias and
corrected
measurement
If the bias is and the corrected measurement is defined by
,
the corrected value of Y has the usual expanded uncertainty
interval which is symmetric around the unknown true value
for the measurement process and is of the following type:
Definition of
asymmetric
uncertainty
interval to
account for
uncorrected
measurement
If no correction is made for the bias, the uncertainty interval
is contaminated by the effect of the bias term as follows:
and can be rewritten in terms of upper and lower endpoints
that are asymmetric around the true value; namely,
Conditions
on the
relationship
between the
bias and U
The definition above can lead to a negative uncertainty
limit; e.g., if the bias is positive and greater than U, the
upper endpoint becomes negative. The requirement that the
uncertainty limits be greater than or equal to zero for all
values of the bias guarantees non-negative uncertainty
limits and is accepted at the cost of somewhat wider
uncertainty intervals. This leads to the following set of
restrictions on the uncertainty limits:
Situation
where bias is
not known
exactly but
must be
If the bias is not known exactly, its magnitude is estimated
from repeated measurements, from sparse data or from
theoretical considerations, and the standard deviation is
estimated from repeated measurements or from an assumed
distribution. The standard deviation of the bias becomes a
estimated component in the uncertainty analysis with the standard
uncertainty restructured to be:
and the expanded uncertainty limits become:
.
Interpretation The uncertainty intervals described above have the
desirable properties outlined on a previous page. For more
information on theory and industrial examples, the reader
should consult the paper by the authors of this technique
(Phillips and Eberhardt).
2.6. Case studies

2.6. Case studies
Contents The purpose of this section is to illustrate the planning,
procedures, and analyses outlined in the various sections of
this chapter with data taken from measurement processes at
the National Institute of Standards and Technology.
1. Gauge study of resistivity probes
2. Check standard study for resistivity measurements
3. Type A uncertainty analysis
4. Type B uncertainty analysis and propagation of
error
2.6.1. Gauge study of resistivity probes

2.6. Case studies
Purpose The purpose of this case study is to outline the analysis of a
gauge study that was undertaken to identify the sources of
uncertainty in resistivity measurements of silicon wafers.
Outline 1. Background and data
2. Analysis and interpretation
3. Graphs showing repeatability standard deviations
4. Graphs showing day-to-day variability
5. Graphs showing differences among gauges
6. Run this example yourself with Dataplot
7. Dataplot macros
2.6.1.1. Background and data

2.6. Case studies
Description
of
measurements
Measurements of resistivity on 100 ohm.cm wafers were
made according to an ASTM Standard Test Method
(ASTM F84) to assess the sources of uncertainty in the
measurement system. Resistivity measurements have been
studied over the years, and it is clear from those data that
there are sources of variability affecting the process beyond
the basic imprecision of the gauges. Changes in
measurement results have been noted over days and over
months and the data in this study are structured to quantify
these time-dependent changes in the measurement process.
Gauges The gauges for the study were five probes used to measure
resistivity of silicon wafers. The five gauges are assumed to
represent a random sample of typical 4-point gauges for
making resistivity measurements. There is a question of
whether or not the gauges are essentially equivalent or
whether biases among them are possible.
Check
standards
The check standards for the study were five wafers selected
at random from the batch of 100 ohm.cm wafers.
Operators The effect of operator was not considered to be significant
for this study.
Database of
measurements
The 3-level nested design consisted of:
J = 6 measurements at the center of each wafer per
day
K = 6 days
L = 2 runs
To characterize the probes and the influence of wafers on
the measurements, the design was repeated over:
Q = 5 wafers (check standards 138, 139, 140, 141,
142)
I = 5 probes (1, 281, 283, 2062, 2362)
The runs were separated by about one month in time. The J
= 6 measurements at the center of each wafer are reduced
to an average and repeatability standard deviation and
recorded in a database with identifications for wafer, probe,
and day.
2.6.1.1.1. Database of resistivity measurements

2.6. Case studies
The check
standards are
five wafers
chosen at
random from
a batch of
wafers
Measurements of resistivity (ohm.cm) were made
according to an ASTM Standard Test Method (F4) at NIST
to assess the sources of uncertainty in the measurement
system. The gauges for the study were five probes owned
by NIST; the check standards for the study were five
wafers selected at random from a batch of wafers cut from
one silicon crystal doped with phosphorous to give a
nominal resistivity of 100 ohm.cm.
Measurements
on the check
standards are
used to
estimate
repeatability,
day effect,
and run effect
The effect of operator was not considered to be significant
for this study; therefore, 'day' replaces 'operator' as a factor
in the nested design. Averages and standard deviations
from J = 6 measurements at the center of each wafer are
shown in the table.
J = 6 measurements at the center of the wafer per
day
K = 6 days (one operator) per repetition
L = 2 runs (complete)
142)
R = 5 probes (1, 281, 283, 2062, 2362)
Run Wafer Probe Month Day Op Temp Average
Std Dev
1 138. 1. 3. 15. 1. 22.98 95.1772
0.1191
1 138. 1. 3. 17. 1. 23.02 95.1567
0.0183
1 138. 1. 3. 18. 1. 22.79 95.1937
0.1282
1 138. 1. 3. 21. 1. 23.17 95.1959
0.0398
1 138. 1. 3. 23. 2. 23.25 95.1442
0.0346
1 138. 1. 3. 23. 1. 23.20 95.0610
0.1539
1 138. 281. 3. 16. 1. 22.99 95.1591
0.0963
1 138. 281. 3. 17. 1. 22.97 95.1195
0.0606
1 138. 281. 3. 18. 1. 22.83 95.1065
0.0842
1 138. 281. 3. 21. 1. 23.28 95.0925
0.0973
1 138. 281. 3. 23. 2. 23.14 95.1990
0.1062
1 138. 281. 3. 23. 1. 23.16 95.1682
0.1090
1 138. 283. 3. 16. 1. 22.95 95.1252
0.0531
1 138. 283. 3. 17. 1. 23.08 95.1600
0.0998
1 138. 283. 3. 18. 1. 23.13 95.0818
0.1108
1 138. 283. 3. 21. 1. 23.28 95.1620
0.0408
1 138. 283. 3. 22. 1. 23.36 95.1735
0.0501
1 138. 283. 3. 24. 2. 22.97 95.1932
0.0287
1 138. 2062. 3. 16. 1. 22.97 95.1311
0.1066
1 138. 2062. 3. 17. 1. 22.98 95.1132
0.0415
1 138. 2062. 3. 18. 1. 23.16 95.0432
0.0491
1 138. 2062. 3. 21. 1. 23.16 95.1254
0.0603
1 138. 2062. 3. 22. 1. 23.28 95.1322
0.0561
1 138. 2062. 3. 24. 2. 23.19 95.1299
0.0349
1 138. 2362. 3. 15. 1. 23.08 95.1162
0.0480
1 138. 2362. 3. 17. 1. 23.01 95.0569
0.0577
1 138. 2362. 3. 18. 1. 22.97 95.0598
0.0516
1 138. 2362. 3. 22. 1. 23.23 95.1487
0.0386
1 138. 2362. 3. 23. 2. 23.28 95.0743
0.0256
1 138. 2362. 3. 24. 2. 23.10 95.1010
0.0420
1 139. 1. 3. 15. 1. 23.01 99.3528
0.1424
1 139. 1. 3. 17. 1. 23.00 99.2940
0.0660
1 139. 1. 3. 17. 1. 23.01 99.2340
0.1179
1 139. 1. 3. 21. 1. 23.20 99.3489
0.0506
1 139. 1. 3. 23. 2. 23.22 99.2625
0.1111
1 139. 1. 3. 23. 1. 23.22 99.3787
0.1103
1 139. 281. 3. 16. 1. 22.95 99.3244
0.1134
1 139. 281. 3. 17. 1. 22.98 99.3378
0.0949
1 139. 281. 3. 18. 1. 22.86 99.3424
0.0847
1 139. 281. 3. 22. 1. 23.17 99.4033
0.0801
1 139. 281. 3. 23. 2. 23.10 99.3717
0.0630
1 139. 281. 3. 23. 1. 23.14 99.3493
0.1157
1 139. 283. 3. 16. 1. 22.94 99.3065
0.0381
1 139. 283. 3. 17. 1. 23.09 99.3280
0.1153
1 139. 283. 3. 18. 1. 23.11 99.3000
0.0818
1 139. 283. 3. 21. 1. 23.25 99.3347
0.0972
1 139. 283. 3. 22. 1. 23.36 99.3929
0.1189
1 139. 283. 3. 23. 1. 23.18 99.2644
0.0622
1 139. 2062. 3. 16. 1. 22.94 99.3324
0.1531
1 139. 2062. 3. 17. 1. 23.08 99.3254
0.0543
1 139. 2062. 3. 18. 1. 23.15 99.2555
0.1024
1 139. 2062. 3. 18. 1. 23.18 99.1946
0.0851
1 139. 2062. 3. 22. 1. 23.27 99.3542
0.1227
1 139. 2062. 3. 24. 2. 23.23 99.2365
0.1218
1 139. 2362. 3. 15. 1. 23.08 99.2939
0.0818
1 139. 2362. 3. 17. 1. 23.02 99.3234
0.0723
1 139. 2362. 3. 18. 1. 22.93 99.2748
0.0756
1 139. 2362. 3. 22. 1. 23.29 99.3512
0.0475
1 139. 2362. 3. 23. 2. 23.25 99.2350
0.0517
1 139. 2362. 3. 24. 2. 23.05 99.3574
0.0485
1 140. 1. 3. 15. 1. 23.07 96.1334
0.1052
1 140. 1. 3. 17. 1. 23.08 96.1250
0.0916
1 140. 1. 3. 18. 1. 22.77 96.0665
0.0836
1 140. 1. 3. 21. 1. 23.18 96.0725
0.0620
1 140. 1. 3. 23. 2. 23.20 96.1006
0.0582
1 140. 1. 3. 23. 1. 23.21 96.1131
0.1757
1 140. 281. 3. 16. 1. 22.94 96.0467
0.0565
1 140. 281. 3. 17. 1. 22.99 96.1081
0.1293
1 140. 281. 3. 18. 1. 22.91 96.0578
0.1148
1 140. 281. 3. 22. 1. 23.15 96.0700
0.0495
1 140. 281. 3. 22. 1. 23.33 96.1052
0.1722
1 140. 281. 3. 23. 1. 23.19 96.0952
0.1786
1 140. 283. 3. 16. 1. 22.89 96.0650
0.1301
1 140. 283. 3. 17. 1. 23.07 96.0870
0.0881
1 140. 283. 3. 18. 1. 23.07 95.8906
0.1842
1 140. 283. 3. 21. 1. 23.24 96.0842
0.1008
1 140. 283. 3. 22. 1. 23.34 96.0189
0.0865
1 140. 283. 3. 23. 1. 23.19 96.1047
0.0923
1 140. 2062. 3. 16. 1. 22.95 96.0379
0.2190
1 140. 2062. 3. 17. 1. 22.97 96.0671
0.0991
1 140. 2062. 3. 18. 1. 23.15 96.0206
0.0648
1 140. 2062. 3. 21. 1. 23.14 96.0207
0.1410
1 140. 2062. 3. 22. 1. 23.32 96.0587
0.1634
1 140. 2062. 3. 24. 2. 23.17 96.0903
0.0406
1 140. 2362. 3. 15. 1. 23.08 96.0771
0.1024
1 140. 2362. 3. 17. 1. 23.00 95.9976
0.0943
1 140. 2362. 3. 18. 1. 23.01 96.0148
0.0622
1 140. 2362. 3. 22. 1. 23.27 96.0397
0.0702
1 140. 2362. 3. 23. 2. 23.24 96.0407
0.0627
1 140. 2362. 3. 24. 2. 23.13 96.0445
0.0622
1 141. 1. 3. 15. 1. 23.01 101.2124
0.0900
1 141. 1. 3. 17. 1. 23.08 101.1018
0.0820
1 141. 1. 3. 18. 1. 22.75 101.1119
0.0500
1 141. 1. 3. 21. 1. 23.21 101.1072
0.0641
1 141. 1. 3. 23. 2. 23.25 101.0802
0.0704
1 141. 1. 3. 23. 1. 23.19 101.1350
0.0699
1 141. 281. 3. 16. 1. 22.93 101.0287
0.0520
1 141. 281. 3. 17. 1. 23.00 101.0131
0.0710
1 141. 281. 3. 18. 1. 22.90 101.1329
0.0800
1 141. 281. 3. 22. 1. 23.19 101.0562
0.1594
1 141. 281. 3. 23. 2. 23.18 101.0891
0.1252
1 141. 281. 3. 23. 1. 23.17 101.1283
0.1151
1 141. 283. 3. 16. 1. 22.85 101.1597
0.0990
1 141. 283. 3. 17. 1. 23.09 101.0784
0.0810
1 141. 283. 3. 18. 1. 23.08 101.0715
0.0460
1 141. 283. 3. 21. 1. 23.27 101.0910
0.0880
1 141. 283. 3. 22. 1. 23.34 101.0967
0.0901
1 141. 283. 3. 24. 2. 23.00 101.1627
0.0888
1 141. 2062. 3. 16. 1. 22.97 101.1077
0.0970
1 141. 2062. 3. 17. 1. 22.96 101.0245
0.1210
1 141. 2062. 3. 18. 1. 23.19 100.9650
0.0700
1 141. 2062. 3. 18. 1. 23.18 101.0319
0.1070
1 141. 2062. 3. 22. 1. 23.34 101.0849
0.0960
1 141. 2062. 3. 24. 2. 23.21 101.1302
0.0505
1 141. 2362. 3. 15. 1. 23.08 101.0471
0.0320
1 141. 2362. 3. 17. 1. 23.01 101.0224
0.1020
1 141. 2362. 3. 18. 1. 23.05 101.0702
0.0580
1 141. 2362. 3. 22. 1. 23.22 101.0904
0.1049
1 141. 2362. 3. 23. 2. 23.29 101.0626
0.0702
1 141. 2362. 3. 24. 2. 23.15 101.0686
0.0661
1 142. 1. 3. 15. 1. 23.02 94.3160
0.1372
1 142. 1. 3. 17. 1. 23.04 94.2808
0.0999
1 142. 1. 3. 18. 1. 22.73 94.2478
0.0803
1 142. 1. 3. 21. 1. 23.19 94.2862
0.0700
1 142. 1. 3. 23. 2. 23.25 94.1859
0.0899
1 142. 1. 3. 23. 1. 23.21 94.2389
0.0686
1 142. 281. 3. 16. 1. 22.98 94.2640
0.0862
1 142. 281. 3. 17. 1. 23.00 94.3333
0.1330
1 142. 281. 3. 18. 1. 22.88 94.2994
0.0908
1 142. 281. 3. 21. 1. 23.28 94.2873
0.0846
1 142. 281. 3. 23. 2. 23.07 94.2576
0.0795
1 142. 281. 3. 23. 1. 23.12 94.3027
0.0389
1 142. 283. 3. 16. 1. 22.92 94.2846
0.1021
1 142. 283. 3. 17. 1. 23.08 94.2197
0.0627
1 142. 283. 3. 18. 1. 23.09 94.2119
0.0785
1 142. 283. 3. 21. 1. 23.29 94.2536
0.0712
1 142. 283. 3. 22. 1. 23.34 94.2280
0.0692
1 142. 283. 3. 24. 2. 22.92 94.2944
0.0958
1 142. 2062. 3. 16. 1. 22.96 94.2238
0.0492
1 142. 2062. 3. 17. 1. 22.95 94.3061
0.2194
1 142. 2062. 3. 18. 1. 23.16 94.1868
0.0474
1 142. 2062. 3. 21. 1. 23.11 94.2645
0.0697
1 142. 2062. 3. 22. 1. 23.31 94.3101
0.0532
1 142. 2062. 3. 24. 2. 23.24 94.2204
0.1023
1 142. 2362. 3. 15. 1. 23.08 94.2437
0.0503
1 142. 2362. 3. 17. 1. 23.00 94.2115
0.0919
1 142. 2362. 3. 18. 1. 22.99 94.2348
0.0282
1 142. 2362. 3. 22. 1. 23.26 94.2124
0.0513
1 142. 2362. 3. 23. 2. 23.27 94.2214
0.0627
1 142. 2362. 3. 24. 2. 23.08 94.1651
0.1010
2 138. 1. 4. 13. 1. 23.12 95.1996
0.0645
2 138. 1. 4. 15. 1. 22.73 95.1315
0.1192
2 138. 1. 4. 18. 2. 22.76 95.1845
0.0452
2 138. 1. 4. 19. 1. 22.73 95.1359
0.1498
2 138. 1. 4. 20. 2. 22.73 95.1435
0.0629
2 138. 1. 4. 21. 2. 22.93 95.1839
0.0563
2 138. 281. 4. 14. 2. 22.46 95.2106
0.1049
2 138. 281. 4. 18. 2. 22.80 95.2505
0.0771
2 138. 281. 4. 18. 2. 22.77 95.2648
0.1046
2 138. 281. 4. 20. 2. 22.80 95.2197
0.1779
2 138. 281. 4. 20. 2. 22.87 95.2003
0.1376
2 138. 281. 4. 21. 2. 22.95 95.0982
0.1611
2 138. 283. 4. 18. 2. 22.83 95.1211
0.0794
2 138. 283. 4. 13. 1. 23.17 95.1327
0.0409
2 138. 283. 4. 18. 1. 22.67 95.2053
0.1525
2 138. 283. 4. 19. 2. 23.00 95.1292
0.0655
2 138. 283. 4. 21. 2. 22.91 95.1669
0.0619
2 138. 283. 4. 21. 2. 22.96 95.1401
0.0831
2 138. 2062. 4. 15. 1. 22.64 95.2479
0.2867
2 138. 2062. 4. 15. 1. 22.67 95.2224
0.1945
2 138. 2062. 4. 19. 2. 22.99 95.2810
0.1960
2 138. 2062. 4. 19. 1. 22.75 95.1869
0.1571
2 138. 2062. 4. 21. 2. 22.84 95.3053
0.2012
2 138. 2062. 4. 21. 2. 22.92 95.1432
0.1532
2 138. 2362. 4. 12. 1. 22.74 95.1687
0.0785
2 138. 2362. 4. 18. 2. 22.75 95.1564
0.0430
2 138. 2362. 4. 19. 2. 22.88 95.1354
0.0983
2 138. 2362. 4. 19. 1. 22.73 95.0422
0.0773
2 138. 2362. 4. 20. 2. 22.86 95.1354
0.0587
2 138. 2362. 4. 21. 2. 22.94 95.1075
0.0776
2 139. 1. 4. 13. 2. 23.14 99.3274
0.0220
2 139. 1. 4. 15. 2. 22.77 99.5020
0.0997
2 139. 1. 4. 18. 2. 22.80 99.4016
0.0704
2 139. 1. 4. 19. 1. 22.68 99.3181
0.1245
2 139. 1. 4. 20. 2. 22.78 99.3858
0.0903
2 139. 1. 4. 21. 2. 22.93 99.3141
0.0255
2 139. 281. 4. 14. 2. 23.05 99.2915
0.0859
2 139. 281. 4. 15. 2. 22.71 99.4032
0.1322
2 139. 281. 4. 18. 2. 22.79 99.4612
0.1765
2 139. 281. 4. 20. 2. 22.74 99.4001
0.0889
2 139. 281. 4. 20. 2. 22.91 99.3765
0.1041
2 139. 281. 4. 21. 2. 22.92 99.3507
0.0717
2 139. 283. 4. 13. 2. 23.11 99.3848
0.0792
2 139. 283. 4. 18. 2. 22.84 99.4952
0.1122
2 139. 283. 4. 18. 2. 22.76 99.3220
0.0915
2 139. 283. 4. 19. 2. 23.03 99.4165
0.0503
2 139. 283. 4. 21. 2. 22.87 99.3791
0.1138
2 139. 283. 4. 21. 2. 22.98 99.3985
0.0661
2 139. 2062. 4. 14. 2. 22.43 99.4283
0.0891
2 139. 2062. 4. 15. 2. 22.70 99.4139
0.2147
2 139. 2062. 4. 19. 2. 22.97 99.3813
0.1143
2 139. 2062. 4. 19. 1. 22.77 99.4314
0.1685
2 139. 2062. 4. 21. 2. 22.79 99.4166
0.2080
2 139. 2062. 4. 21. 2. 22.94 99.4052
0.2400
2 139. 2362. 4. 12. 1. 22.82 99.3408
0.1279
2 139. 2362. 4. 18. 2. 22.77 99.3116
0.1131
2 139. 2362. 4. 19. 2. 22.82 99.3241
0.0519
2 139. 2362. 4. 19. 1. 22.74 99.2991
0.0903
2 139. 2362. 4. 20. 2. 22.88 99.3049
0.0783
2 139. 2362. 4. 21. 2. 22.94 99.2782
0.0718
2 140. 1. 4. 13. 1. 23.10 96.0811
0.0463
2 140. 1. 4. 15. 2. 22.75 96.1460
0.0725
2 140. 1. 4. 18. 2. 22.78 96.1582
0.1428
2 140. 1. 4. 19. 1. 22.70 96.1039
0.1056
2 140. 1. 4. 20. 2. 22.75 96.1262
0.0672
2 140. 1. 4. 21. 2. 22.93 96.1478
0.0562
2 140. 281. 4. 15. 2. 22.71 96.1153
0.1097
2 140. 281. 4. 14. 2. 22.49 96.1297
0.1202
2 140. 281. 4. 18. 2. 22.81 96.1233
0.1331
2 140. 281. 4. 20. 2. 22.78 96.1731
0.1484
2 140. 281. 4. 20. 2. 22.89 96.0872
0.0857
2 140. 281. 4. 21. 2. 22.91 96.1331
0.0944
2 140. 283. 4. 13. 2. 23.22 96.1135
0.0983
2 140. 283. 4. 18. 2. 22.85 96.1111
0.1210
2 140. 283. 4. 18. 2. 22.78 96.1221
0.0644
2 140. 283. 4. 19. 2. 23.01 96.1063
0.0921
2 140. 283. 4. 21. 2. 22.91 96.1155
0.0704
2 140. 283. 4. 21. 2. 22.94 96.1308
0.0258
2 140. 2062. 4. 15. 2. 22.60 95.9767
0.2225
2 140. 2062. 4. 15. 2. 22.66 96.1277
0.1792
2 140. 2062. 4. 19. 2. 22.96 96.1858
0.1312
2 140. 2062. 4. 19. 1. 22.75 96.1912
0.1936
2 140. 2062. 4. 21. 2. 22.82 96.1650
0.1902
2 140. 2062. 4. 21. 2. 22.92 96.1603
0.1777
2 140. 2362. 4. 12. 1. 22.88 96.0793
0.0996
2 140. 2362. 4. 18. 2. 22.76 96.1115
0.0533
2 140. 2362. 4. 19. 2. 22.79 96.0803
0.0364
2 140. 2362. 4. 19. 1. 22.71 96.0411
0.0768
2 140. 2362. 4. 20. 2. 22.84 96.0988
0.1042
2 140. 2362. 4. 21. 1. 22.94 96.0482
0.0868
2 141. 1. 4. 13. 1. 23.07 101.1984
0.0803
2 141. 1. 4. 15. 2. 22.72 101.1645
0.0914
2 141. 1. 4. 18. 2. 22.75 101.2454
0.1109
2 141. 1. 4. 19. 1. 22.69 101.1096
0.1376
2 141. 1. 4. 20. 2. 22.83 101.2066
0.0717
2 141. 1. 4. 21. 2. 22.93 101.0645
0.1205
2 141. 281. 4. 15. 2. 22.72 101.1615
0.1272
2 141. 281. 4. 14. 2. 22.40 101.1650
0.0595
2 141. 281. 4. 18. 2. 22.78 101.1815
0.1393
2 141. 281. 4. 20. 2. 22.73 101.1106
0.1189
2 141. 281. 4. 20. 2. 22.86 101.1420
0.0713
2 141. 281. 4. 21. 2. 22.94 101.0116
0.1088
2 141. 283. 4. 13. 2. 23.26 101.1554
0.0429
2 141. 283. 4. 18. 2. 22.85 101.1267
0.0751
2 141. 283. 4. 18. 2. 22.76 101.1227
0.0826
2 141. 283. 4. 19. 2. 22.82 101.0635
0.1715
2 141. 283. 4. 21. 2. 22.89 101.1264
0.1447
2 141. 283. 4. 21. 2. 22.96 101.0853
0.1189
2 141. 2062. 4. 15. 2. 22.65 101.1332
0.2532
2 141. 2062. 4. 15. 1. 22.68 101.1487
0.1413
2 141. 2062. 4. 19. 2. 22.95 101.1778
0.1772
2 141. 2062. 4. 19. 1. 22.77 101.0988
0.0884
2 141. 2062. 4. 21. 2. 22.87 101.1686
0.2940
2 141. 2062. 4. 21. 2. 22.94 101.3289
0.2072
2 141. 2362. 4. 12. 1. 22.83 101.1353
0.0585
2 141. 2362. 4. 18. 2. 22.83 101.1201
0.0868
2 141. 2362. 4. 19. 2. 22.91 101.0946
0.0855
2 141. 2362. 4. 19. 1. 22.71 100.9977
0.0645
2 141. 2362. 4. 20. 2. 22.87 101.0963
0.0638
2 141. 2362. 4. 21. 2. 22.94 101.0300
0.0549
2 142. 1. 4. 13. 1. 23.07 94.3049
0.1197
2 142. 1. 4. 15. 2. 22.73 94.3153
0.0566
2 142. 1. 4. 18. 2. 22.77 94.3073
0.0875
2 142. 1. 4. 19. 1. 22.67 94.2803
0.0376
2 142. 1. 4. 20. 2. 22.80 94.3008
0.0703
2 142. 1. 4. 21. 2. 22.93 94.2916
0.0604
2 142. 281. 4. 14. 2. 22.90 94.2557
0.0619
2 142. 281. 4. 18. 2. 22.83 94.3542
0.1027
2 142. 281. 4. 18. 2. 22.80 94.3007
0.1492
2 142. 281. 4. 20. 2. 22.76 94.3351
0.1059
2 142. 281. 4. 20. 2. 22.88 94.3406
0.1508
2 142. 281. 4. 21. 2. 22.92 94.2621
0.0946
2 142. 283. 4. 13. 2. 23.25 94.3124
0.0534
2 142. 283. 4. 18. 2. 22.85 94.3680
0.1643
2 142. 283. 4. 18. 1. 22.67 94.3442
0.0346
2 142. 283. 4. 19. 2. 22.80 94.3391
0.0616
2 142. 283. 4. 21. 2. 22.91 94.2238
0.0721
2 142. 283. 4. 21. 2. 22.95 94.2721
0.0998
2 142. 2062. 4. 14. 2. 22.49 94.2915
0.2189
2 142. 2062. 4. 15. 2. 22.69 94.2803
0.0690
2 142. 2062. 4. 19. 2. 22.94 94.2818
0.0987
2 142. 2062. 4. 19. 1. 22.76 94.2227
0.2628
2 142. 2062. 4. 21. 2. 22.74 94.4109
0.1230
2 142. 2062. 4. 21. 2. 22.94 94.2616
0.0929
2 142. 2362. 4. 12. 1. 22.86 94.2052
0.0813
2 142. 2362. 4. 18. 2. 22.83 94.2824
0.0605
2 142. 2362. 4. 19. 2. 22.85 94.2396
0.0882
2 142. 2362. 4. 19. 1. 22.75 94.2087
0.0702
2 142. 2362. 4. 20. 2. 22.86 94.2937
0.0591
2 142. 2362. 4. 21. 1. 22.93 94.2330
0.0556
2.6.1.2. Analysis and interpretation

2.6. Case studies
Graphs of
probe effect
on
repeatability
A graphical analysis shows repeatability standard deviations
plotted by wafer and probe. Probes are coded by numbers
with probe #2362 coded as #5. The plots show that for both
runs the precision of this probe is better than for the other
probes.
Probe #2362, because of its superior precision, was chosen
as the tool for measuring all 100 ohm.cm resistivity wafers at
NIST. Therefore, the remainder of the analysis focuses on
this probe.
Plot of
repeatability
standard
deviations
for probe
#2362 from
the nested
design over
days,
wafers, runs
The precision of probe #2362 is first checked for consistency
by plotting the repeatability standard deviations over days,
wafers and runs. Days are coded by letter. The plots verify
that, for both runs, probe repeatability is not dependent on
wafers or days although the standard deviations on days D,
E, and F of run 2 are larger in some instances than for the
other days. This is not surprising because repeated probing
on the wafer surfaces can cause slight degradation. Then the
repeatability standard deviations are pooled over:
K = 6 days for K(J - 1) = 30 degrees of freedom
L = 2 runs for LK(J - 1) = 60 degrees of freedom
Q = 5 wafers for QLK(J - 1) = 300 degrees of freedom
The results of pooling are shown below. Intermediate steps
are not shown, but the section on repeatability standard
deviations shows an example of pooling over wafers.
Pooled level-1 standard deviations (ohm.cm)
Probe Run 1 DF Run 2 DF Pooled
DF
2362. 0.0658 150 0.0758 150 0.0710
300
Graphs of
reproducibility
and stability for
Averages of the 6 center measurements on each wafer
are plotted on a single graph for each wafer. The points
(connected by lines) on the left side of each graph are
probe #2362 averages at the wafer center plotted over 5 days; the
points on the right are the same measurements repeated
after one month as a check on the stability of the
measurement process. The plots show day-to-day
variability as well as slight variability from run-to-run.
Earlier work discounts long-term drift in the gauge as the
cause of these changes. A reasonable conclusion is that
day-to-day and run-to-run variations come from random
fluctuations in the measurement process.
Level-2
(reproducibility)
standard
deviations
computed from
day averages
and pooled over
wafers and runs
Level-2 standard deviations (with K - 1 = 5 degrees of
freedom each) are computed from the daily averages that
are recorded in the database. Then the level-2 standard
deviations are pooled over:
L = 2 runs for L(K - 1) = 10 degrees of freedom
Q = 5 wafers for QL(K - 1) = 50 degrees of
freedom
as shown in the table below. The table shows that the
level-2 standard deviations are consistent over wafers
and runs.
Level-2 standard deviations (ohm.cm) for 5 wafers
Run 1 Run 2
Wafer Probe Average Stddev DF Average
Stddev DF
138. 2362. 95.0928 0.0359 5 95.1243
0.0453 5
139. 2362. 99.3060 0.0472 5 99.3098
0.0215 5
140. 2362. 96.0357 0.0273 5 96.0765
0.0276 5
141. 2362. 101.0602 0.0232 5 101.0790
0.0537 5
142. 2362. 94.2148 0.0274 5 94.2438
0.0370 5
2362. Pooled 0.0333 25
0.0388 25
(over 2 runs)
0.0362 50
Level-3
(stability)
standard
deviations
computed
from run
averages
and pooled
Level-3 standard deviations are computed from the averages
of the two runs. Then the level-3 standard deviations are
pooled over the five wafers to obtain a standard deviation with
5 degrees of freedom as shown in the table below.
over
wafers
Level-3 standard deviations (ohm.cm) for 5 wafers
Run 1 Run 2
Wafer Probe Average Average Diff
Stddev DF
138. 2362. 95.0928 95.1243 -0.0315
0.0223 1
139. 2362. 99.3060 99.3098 -0.0038
0.0027 1
140. 2362. 96.0357 96.0765 -0.0408
0.0289 1
141. 2362. 101.0602 101.0790 -0.0188
0.0133 1
142. 2362. 94.2148 94.2438 -0.0290
0.0205 1
2362. Pooled
0.0197 5
Graphs of
probe
biases
A graphical analysis shows the relative biases among the 5
probes. For each wafer, differences from the wafer average
by probe are plotted versus wafer number. The graphs verify
that probe #2362 (coded as 5) is biased low relative to the
other probes. The bias shows up more strongly after the
probes have been in use (run 2).
Formulas
for
computation
of biases for
probe
#2362
Biases by probe are shown in the following table.
Differences from the mean for each wafer
Wafer Probe Run 1 Run 2
138. 1. 0.0248 -0.0119
138. 281. 0.0108 0.0323
138. 283. 0.0193 -0.0258
138. 2062. -0.0175 0.0561
138. 2362. -0.0372 -0.0507
139. 1. -0.0036 -0.0007
139. 281. 0.0394 0.0050
139. 283. 0.0057 0.0239
139. 2062. -0.0323 0.0373
139. 2362. -0.0094 -0.0657
140. 1. 0.0400 0.0109
140. 281. 0.0187 0.0106
140. 283. -0.0201 0.0003
140. 2062. -0.0126 0.0182
140. 2362. -0.0261 -0.0398
141. 1. 0.0394 0.0324
141. 281. -0.0107 -0.0037
141. 283. 0.0246 -0.0191
141. 2062. -0.0280 0.0436
141. 2362. -0.0252 -0.0534
142. 1. 0.0062 0.0093
142. 281. 0.0376 0.0174
142. 283. -0.0044 0.0192
142. 2062. -0.0011 0.0008
142. 2362. -0.0383 -0.0469
How to
deal with
bias due to
the probe
Probe #2362 was chosen for the certification process because
of its superior precision, but its bias relative to the other
probes creates a problem. There are two possibilities for
handling this problem:
1. Correct all measurements made with probe #2362 to the
average of the probes.
2. Include the standard deviation for the difference among
probes in the uncertainty budget.
The better choice is (1) if we can assume that the probes in the
study represent a random sample of probes of this type. This is
particularly true when the unit (resistivity) is defined by a test
method.
2.6.1.3. Repeatability standard deviations

2.6. Case studies
Run 1 -
Graph of
repeatability
standard
deviations
for probe
#2362 -- 6
days and 5
wafers
showing
that
repeatability
is constant
across
wafers and
days
Run 2 -
Graph of
repeatability
standard
deviations
for probe
#2362 -- 6
days and 5
wafers
showing
that
repeatability
is constant
across
wafers and
days
Run 1 -
Graph
showing
repeatability
standard
deviations
for five
probes as a
function of
wafers and
probes
Symbols for codes: 1 = #1; 2 = #281; 3 = #283; 4 = #2062; 5 =
#2362
Run 2 -
Graph
showing
repeatability
standard
deviations
for 5 probes
as a
function of
wafers and
probes
Symbols for probes: 1 = #1; 2 = #281; 3 = #283; 4 = #2062; 5 =
#2362
2.6.1.4. Effects of days and long-term stability

2.6. Case studies
Effects of
days and
long-term
stability on
the
measurements
The data points that are plotted in the five graphs shown below are averages of
resistivity measurements at the center of each wafer for wafers #138, 139, 140, 141,
142. Data for each of two runs are shown on each graph. The six days of
measurements for each run are separated by approximately one month and show,
with the exception of wafer #139, that there is a very slight shift upwards between
run 1 and run 2. The size of the effect is estimated as a level-3 standard deviation in
the analysis of the data.
Wafer 138
Wafer 139
Wafer 140
Wafer 141
Wafer 142
2.6.1.5. Differences among 5 probes

2.6. Case studies
Run 1 -
Graph of
differences
from
wafer
averages
for each of
5 probes
showing
that
probes
#2062 and
#2362 are
biased low
relative to
the other
probes
#2362
Run 2 -
Graph of
differences
from
wafer
averages
for each of
5 probes
showing
that probe
#2362
continues
to be
biased low
relative to
the other
probes
#2362
2.6.1.6. Run gauge study example using Dataplot

2.6. Case studies
2.6.1.6. Run gauge study example using
Dataplot
View of
Dataplot
macros for
this case
study
Output Window, the Graphics window, the Command History
Data Analysis Steps Results and Conclusions
Click on the links below to start Dataplot
and run this case study yourself. Each
step may use results from previous steps,
so please be patient. Wait until the
software verifies that the current step is
The links in this column will connect you
with more detailed information about
each analysis step from the case study
description.
Graphical analyses of variability
Graphs to test for:
1. Wafer/day effect on repeatability
(run 1)
2. Wafer/day effect on repeatability
(run 2)
3. Probe effect on repeatability (run 1)
4. Probe effect on repeatability (run 2)
5. Reproducibility and stability
1. and 2. Interpretation: The plots verify
that, for both runs, the repeatability of
probe #2362 is not dependent on wafers
or days, although the standard deviations
on days D, E, and F of run 2 are larger in
some instances than for the other days.
3. and 4. Interpretation: Probe #2362
appears as #5 in the plots which show
that, for both runs, the precision of this
probe is better than for the other probes.
5. Interpretation: There is a separate plot
for each wafer. The points on the left side
of each plot are averages at the wafer
center plotted over 5 days; the points on
2.6.1.6. Run gauge study example using Dataplot
the right are the same measurements
repeated after one month to check on the
stability of the measurement process. The
plots show day-to-day variability as well
as slight variability from run-to-run.
Table of estimates for probe #2362
1. Level-1 (repeatability)
2. Level-2 (reproducibility)
3. Level-3 (stability)
1., 2. and 3.: Interpretation: The
repeatability of the gauge (level-1
standard deviation) dominates the
imprecision associated with
measurements and days and runs are less
important contributors. Of course, even if
the gauge has high precision, biases may
contribute substantially to the uncertainty
of measurement.
Bias estimates
1. Differences among probes - run 1
2. Differences among probes - run 2
1. and 2. Interpretation: The graphs show
the relative biases among the 5 probes.
For each wafer, differences from the
wafer average by probe are plotted versus
wafer number. The graphs verify that
probe #2362 (coded as 5) is biased low
relative to the other probes. The bias
shows up more strongly after the probes
have been in use (run 2).
2.6.1.7. Dataplot macros

2.6. Case studies
Plot of wafer
and day effect
on
repeatability
standard
deviations for
run 1
reset data
reset plot control
reset i/o
dimension 500 30
label size 3
read mpc61.dat run wafer probe mo day op hum y
sw
y1label ohm.cm
title GAUGE STUDY
lines blank all
let z = pattern 1 2 3 4 5 6 for I = 1 1 300
let z2 = wafer + z/10 -0.25
characters a b c d e f
X1LABEL WAFERS
X2LABEL REPEATABILITY STANDARD DEVIATIONS BY
WAFER AND DAY
X3LABEL CODE FOR DAYS: A, B, C, D, E, F
TITLE RUN 1
plot sw z2 day subset run 1
Plot of wafer
and day effect
on
repeatability
standard
deviations for
run 2
reset data
reset plot control
reset i/o
dimension 500 30
label size 3
sw
y1label ohm.cm
title GAUGE STUDY
lines blank all
let z2 = wafer + z/10 -0.25
characters a b c d e f
X1LABEL WAFERS
WAFER AND DAY
X3LABEL CODE FOR DAYS: A, B, C, D, E, F
TITLE RUN 2
plot sw z2 day subset run 2
Plot of
repeatability
standard
deviations for
5 probes - run
1
reset data
reset plot control
reset i/o
dimension 500 30
label size 3
sw
y1label ohm.cm
title GAUGE STUDY
lines blank all
let z2 = wafer + z/10 -0.25
characters 1 2 3 4 5
X1LABEL WAFERS
WAFER AND PROBE
X3LABEL CODE FOR PROBES: 1= SRM1; 2= 281; 3=283;
4=2062; 5=2362
TITLE RUN 1
plot sw z2 probe subset run 1
Plot of
repeatability
standard
deviations for
5 probes - run
2
reset data
reset plot control
reset i/o
dimension 500 30
label size 3
sw
y1label ohm.cm
title GAUGE STUDY
lines blank all
let z2 = wafer + z/10 -0.25
X1LABEL WAFERS
WAFER AND PROBE
X3LABEL CODE FOR PROBES: 1= SRM1; 2= 281; 3=283;
4=2062; 5=2362
TITLE RUN 2
plot sw z2 probe subset run 2
Plot of
differences
from the wafer
mean for 5
probes - run 1
reset data
reset plot control
reset i/o
dimension 500 30
read mpc61a.dat wafer probe d1 d2
let biasrun1 = mean d1 subset probe 2362
print biasrun1
title GAUGE STUDY FOR 5 PROBES
Y1LABEL OHM.CM
lines dotted dotted dotted dotted dotted solid
characters 1 2 3 4 5 blank
xlimits 137 143
let zero = pattern 0 for I = 1 1 30
x1label DIFFERENCES AMONG PROBES VS WAFER (RUN
1)
plot d1 wafer probe and
plot zero wafer
Plot of
differences
from the wafer
mean for 5
probes - run 2
reset data
reset plot control
reset i/o
dimension 500 30
print biasrun2
Y1LABEL OHM.CM
xlimits 137 143
2)
plot zero wafer
Plot of
averages by
day showing
reproducibility
and stability
for
measurements
made with
probe #2362
on 5 wafers
reset data
reset plot control
reset i/o
dimension 300 50
label size 3
read mcp61b.dat wafer probe mo1 day1 y1 mo2
day2 y2 diff
let t = mo1+(day1-1)/31.
let t2= mo2+(day2-1)/31.
x3label WAFER 138
multiplot 3 2
plot y1 t subset wafer 138 and
plot y2 t2 subset wafer 138
x3label wafer 139
x3label WAFER 140
x3label WAFER 140
x3label WAFER 142
2.6.2. Check standard for resistivity measurements

2.6. Case studies
2.6.2. Check standard for resistivity
measurements
Purpose The purpose of this page is to outline the analysis of check
standard data with respect to controlling the precision and
long-term variability of the process.
Outline 1. Background and data
2. Analysis and interpretation
3. Run this example yourself using Dataplot

2.6. Case studies
Explanation
of check
standard
measurements
The process involves the measurement of resistivity
(ohm.cm) of individual silicon wafers cut from a single
crystal (# 51939). The wafers were doped with
phosphorous to give a nominal resistivity of 100 ohm.cm.
A single wafer (#137), chosen at random from a batch of
130 wafers, was designated as the check standard for this
process.
Design of
data
collection and
Database
The measurements were carried out according to an ASTM
Test Method (F84) with NIST probe #2362. The
measurements on the check standard duplicate certification
measurements that were being made, during the same time
period, on individual wafers from crystal #51939. For the
check standard there were:
J = 6 repetitions at the center of the wafer on each
day
K = 25 days
The K = 25 days cover the time during which the
individual wafers were being certified at the National
Institute of Standards and Technology.
2.6.2.1.1. Database for resistivity check standard

2.6. Case studies
2.6.2.1.1. Database for resistivity check
standard
Description
of check
standard
A single wafer (#137), chosen at random from a batch of
130 wafers, is the check standard for resistivity
measurements at the 100 ohm.cm level at the National
Institute of Standards and Technology. The average of six
measurements at the center of the wafer is the check
standard value for one occasion, and the standard deviation
of the six measurements is the short-term standard
deviation. The columns of the database contain the
following:
1. Crystal ID
2. Check standard ID
3. Month
4. Day
5. Hour
6. Minute
7. Operator
8. Humidity
9. Probe ID
10. Temperature
12. Short-term standard deviation
Database of
measurements
on check
standard
Crystal Waf Mo Da Hr Mn Op Hum Probe Temp Avg
Stddev DF
51939 137 03 24 18 01 drr 42 2362 23.003 97.070
0.085 5
51939 137 03 25 12 41 drr 35 2362 23.115 97.049
0.052 5
51939 137 03 25 15 57 drr 33 2362 23.196 97.048
0.038 5
51939 137 03 28 10 10 JMT 47 2362 23.383 97.084
0.036 5
51939 137 03 28 13 31 JMT 44 2362 23.491 97.106
0.049 5
51939 137 03 28 17 33 drr 43 2362 23.352 97.014
0.036 5
2.6.2.1.1. Database for resistivity check standard
51939 137 03 29 14 40 drr 36 2362 23.202 97.047
0.052 5
51939 137 03 29 16 33 drr 35 2362 23.222 97.078
0.117 5
51939 137 03 30 05 45 JMT 32 2362 23.337 97.065
0.085 5
51939 137 03 30 09 26 JMT 33 2362 23.321 97.061
0.052 5
51939 137 03 25 14 59 drr 34 2362 22.993 97.060
0.060 5
51939 137 03 31 10 10 JMT 37 2362 23.164 97.102
0.048 5
51939 137 03 31 13 00 JMT 37 2362 23.169 97.096
0.026 5
51939 137 03 31 15 32 JMT 35 2362 23.156 97.035
0.088 5
51939 137 04 01 13 05 JMT 34 2362 23.097 97.114
0.031 5
51939 137 04 01 15 32 JMT 34 2362 23.127 97.069
0.037 5
51939 137 04 01 10 32 JMT 48 2362 22.963 97.095
0.032 5
51939 137 04 06 14 38 JMT 49 2362 23.454 97.088
0.056 5
51939 137 04 07 10 50 JMT 34 2362 23.285 97.079
0.067 5
51939 137 04 07 15 46 JMT 33 2362 23.123 97.016
0.116 5
51939 137 04 08 09 37 JMT 33 2362 23.373 97.051
0.046 5
51939 137 04 08 12 53 JMT 33 2362 23.296 97.070
0.078 5
51939 137 04 08 15 03 JMT 33 2362 23.218 97.065
0.040 5
51939 137 04 11 09 30 JMT 36 2362 23.415 97.111
0.038 5
51939 137 04 11 11 34 JMT 35 2362 23.395 97.073
0.039 5

2.6. Case studies
Estimates of
the
repeatability
standard
deviation and
level-2
standard
deviation
The level-1 standard deviations (with J - 1 = 5 degrees of
freedom each) from the database are pooled over the K = 25
days to obtain a reliable estimate of repeatability. This
pooled value is
s
1
= 0.06139 ohm.cm
with K(J - 1) = 125 degrees of freedom. The level-2
standard deviation is computed from the daily averages to
be
s
2
= 0.02680 ohm.cm
with K - 1 = 24 degrees of freedom.
Relationship
to uncertainty
calculations
These standard deviations are appropriate for estimating the
uncertainty of the average of six measurements on a wafer
that is of the same material and construction as the check
standard. The computations are explained in the section on
sensitivity coefficients for check standard measurements.
For other numbers of measurements on the test wafer, the
computations are explained in the section on sensitivity
coefficients for level-2 designs.
Illustrative
table showing
computations
of
repeatability
and level-2
standard
deviations
A tabular presentation of a subset of check standard data (J
= 6 repetitions and K = 6 days) illustrates the computations.
The pooled repeatability standard deviation with K(J - 1) =
30 degrees of freedom from this limited database is shown
in the next to last row of the table. A level-2 standard
deviation with K - 1= 5 degrees of freedom is computed
from the center averages and is shown in the last row of the
table.
Control chart
for probe
#2362
The control chart for monitoring the precision of probe
#2362 is constructed as discussed in the section on control
charts for standard deviations. The upper control limit
(UCL) for testing for degradation of the probe is computed
using the critical value from the F table with numerator
degrees of freedom J - 1 = 5 and denominator degrees of
freedom K(J - 1) = 125. For a 0.05 significance level,
Interpretation
of control
chart for
probe #2362
The control chart shows two points exceeding the upper
control limit. We expect 5 % of the standard deviations to
exceed the UCL for a measurement process that is in-
control. Two outliers are not indicative of significant
problems with the repeatability for the probe, but the probe
should be monitored closely in the future.
Control chart
for bias and
variability
The control limits for monitoring the bias and long-term
variability of resistivity with a Shewhart control chart are
given by
UCL = Average + 2*s
2
= 97.1234 ohm.cm
Centerline = Average = 97.0698 ohm.cm
LCL = Average - 2*s
2
= 97.0162 ohm.cm
Interpretation
of control
chart for bias
The control chart shows that the points scatter randomly
about the center line with no serious problems, although one
point exceeds the upper control limit and one point exceeds
the lower control limit by a small amount. The conclusion is
that there is:
No evidence of bias, change or drift in the
No evidence of long-term lack of control.
Future measurements that exceed the control limits must be
evaluated for long-term changes in bias and/or variability.

2.6. Case studies
2.6.2.2.1. Repeatability and level-2 standard
deviations
Example The table below illustrates the computation of repeatability and
level-2 standard deviations from measurements on a check standard.
The check standard measurements are resistivities at the center of a
100 ohm.cm wafer. There are J = 6 repetitions per day and K = 5
days for this example.
Table of
data,
averages,
and
repeatability
standard
deviations
Measurements on check standard #137
Repetitions per day
Days 1 2 3 4 5 6
1 96.920 97.054 97.057 97.035 97.189 96.965
2 97.118 96.947 97.110 97.047 96.945 97.013
3 97.034 97.084 97.023 97.045 97.061 97.074
4 97.047 97.099 97.087 97.076 97.117 97.070
5 97.127 97.067 97.106 96.995 97.052 97.121
6 96.995 96.984 97.053 97.065 96.976 96.997
Averages 97.040 97.039 97.073 97.044 97.057 97.037
Repeatability
Standard
Deviations
0.0777 0.0602 0.0341 0.0281 0.0896 0.0614
Pooled
Repeatability
Standard
Deviation
0.0625
30 df
Level-2
Standard
Deviation
0.0139
5 df
2.6.2.3. Control chart for probe precision

2.6. Case studies
2.6.2.3. Control chart for probe precision
Control
chart for
probe
#2362
showing
violations
of the
control
limits --
all
standard
deviations
are based
on 6
repetitions
and the
control
limits are
95%
limits
2.6.2.4. Control chart for bias and long-term variability

2.6. Case studies
2.6.2.4. Control chart for bias and long-term variability
Shewhart
control chart
for
measurements
on a
resistivity
check
standard
showing that
the process is
in-control --
all
measurements
are averages
of 6
repetitions
2.6.2.5. Run check standard example yourself

2.6. Case studies
View of
Dataplot
macros for
this case
study
case study description on the previous page using Dataplot. It
Dataplot and configured your browser to run Dataplot. Output
description.
Graphical tests of assumptions
Histogram
Normal probability plot
The histogram and normal probability
plots show no evidence of non-normality.
Control chart for precision
Control chart for probe #2362
Computations:
1. Pooled repeatability standard
deviation
2. Control limit
The precision control chart shows two
points exceeding the upper control limit.
We expect 5% of the standard deviations
to exceed the UCL even when the
measurement process is in-control.
Control chart for check standard
Control chart for check standard #137
The Shewhart control chart shows that the
points scatter randomly about the center
Computations:
1. Average check standard value
2. Process standard deviation
3. Upper and lower control limits
line with no serious problems, although
one point exceeds the upper control limit
and one point exceeds the lower control
limit by a small amount. The conclusion
is that there is no evidence of bias or lack
of long-term control.

2.6. Case studies
Histogram
for check
standard
#137 to test
assumption
of normality
reset data
reset plot control
reset i/o
dimension 500 30
skip 14
read mpc62.dat crystal wafer mo day hour min op
hum probe temp y sw df
histogram y
Normal
probability
plot for
check
standard
#137 to test
assumption
of normality
reset data
reset plot control
reset i/o
dimension 500 30
skip 14
normal probabilty plot y
Control
chart for
precision of
probe
#2372 and
computation
of control
parameter
estimates
reset data
reset plot control
reset i/o
dimension 500 30
skip 14
let time = mo +(day-1)/31.
let s = sw*sw
let spool = mean s
let spool = spool**.5
print spool
let f = fppf(.95, 5, 125)
let ucl = spool*(f)**.5
print ucl
title Control chart for precision
characters blank blank O
lines solid dashed blank
y1label ohm.cm
x1label Time in days
x2label Standard deviations with probe #2362
x3label 5% upper control limit
let center = sw - sw + spool
let cl = sw - sw + ucl
plot center cl sw vs time
Shewhart
control
chart for
check
standard
#137 with
computation
of control
reset data
reset plot control
reset i/o
dimension 500 30
skip 14
let time = mo +(day-1)/31.
let avg = mean y
let sprocess = standard deviation y
let ucl = avg + 2*sprocess
let lcl = avg - 2*sprocess
chart
parameters
print avg
print sprocess
print ucl lcl
title Shewhart control chart
characters O blank blank blank
lines blank dashed solid dashed
y1label ohm.cm
x1label Time in days
x2label Check standard 137 with probe 2362
x3label 2-sigma control limits
let ybar = y - y + avg
let lc1 = y - y + lcl
let lc2 = y - y + ucl
plot y lc1 ybar lc2 vs time
2.6.3. Evaluation of type A uncertainty

2.6. Case studies
Purpose The purpose of this case study is to demonstrate the
computation of uncertainty for a measurement process with
several sources of uncertainty from data taken during a gauge
study.
Outline 1. Background and data for the study
2. Graphical and quantitative analyses and interpretations
3. Run this example yourself with Dataplot

2.6. Case studies
Description
of
measurements
The measurements in question are resistivities (ohm.cm) of
silicon wafers. The intent is to calculate an uncertainty
associated with the resistivity measurements of
approximately 100 silicon wafers that were certified with
probe #2362 in wiring configuration A, according to
ASTM Method F84 (ASTM F84) which is the defined
reference for this measurement. The reported value for each
wafer is the average of six measurements made at the
center of the wafer on a single day. Probe #2362 is one of
five probes owned by the National Institute of Standards
and Technology that is capable of making the
measurements.
Sources of
uncertainty in
NIST
measurements
The uncertainty analysis takes into account the following
sources of variability:
Repeatability of measurements at the center of the
wafer
Day-to-day effects
Run-to-run effects
Bias due to probe #2362
Bias due to wiring configuration
Database of
3-level nested
design -- for
estimating
time-
dependent
sources of
uncertainty
The certification measurements themselves are not the
primary source for estimating uncertainty components
because they do not yield information on day-to-day effects
and long-term effects. The standard deviations for the three
time-dependent sources of uncertainty are estimated from a
3-level nested design. The design was replicated on each of
Q = 5 wafers which were chosen at random, for this
purpose, from the lot of wafers. The certification
measurements were made between the two runs in order to
check on the long-term stability of the process. The data
consist of repeatability standard deviations (with J - 1 = 5
degrees of freedom each) from measurements at the wafer
center.

2.6. Case studies
Check
standards are
five wafers
chosen at
random from
a batch of
wafers
according to an ASTM Standard Test Method (F4) at the
National Institute of Standards and Technology to assess
the sources of uncertainty in the measurement system. The
gauges for the study were five probes owned by NIST; the
check standards for the study were five wafers selected at
random from a batch of wafers cut from one silicon crystal
doped with phosphorous to give a nominal resistivity of
100 ohm.cm.
Measurements
on the check
standards are
used to
estimate
repeatability,
day effect, run
effect
The effect of operator was not considered to be significant
for this study. Averages and standard deviations from J =
6 measurements at the center of each wafer are shown in
the table.
J = 6 measurements at the center of the wafer per
day
K = 6 days (one operator) per repetition
L = 2 runs (complete)
142)
I = 5 probes (1, 281, 283, 2062, 2362)

Standard
Run Wafer Probe Month Day Operator Temp Average
Deviation
1 138. 1. 3. 15. 1. 22.98 95.1772
0.1191
1 138. 1. 3. 17. 1. 23.02 95.1567
0.0183
1 138. 1. 3. 18. 1. 22.79 95.1937
0.1282
1 138. 1. 3. 21. 1. 23.17 95.1959
0.0398
1 138. 1. 3. 23. 2. 23.25 95.1442
0.0346
1 138. 1. 3. 23. 1. 23.20 95.0610
0.1539
1 138. 281. 3. 16. 1. 22.99 95.1591
0.0963
1 138. 281. 3. 17. 1. 22.97 95.1195
0.0606
1 138. 281. 3. 18. 1. 22.83 95.1065
0.0842
1 138. 281. 3. 21. 1. 23.28 95.0925
0.0973
1 138. 281. 3. 23. 2. 23.14 95.1990
0.1062
1 138. 281. 3. 23. 1. 23.16 95.1682
0.1090
1 138. 283. 3. 16. 1. 22.95 95.1252
0.0531
1 138. 283. 3. 17. 1. 23.08 95.1600
0.0998
1 138. 283. 3. 18. 1. 23.13 95.0818
0.1108
1 138. 283. 3. 21. 1. 23.28 95.1620
0.0408
1 138. 283. 3. 22. 1. 23.36 95.1735
0.0501
1 138. 283. 3. 24. 2. 22.97 95.1932
0.0287
1 138. 2062. 3. 16. 1. 22.97 95.1311
0.1066
1 138. 2062. 3. 17. 1. 22.98 95.1132
0.0415
1 138. 2062. 3. 18. 1. 23.16 95.0432
0.0491
1 138. 2062. 3. 21. 1. 23.16 95.1254
0.0603
1 138. 2062. 3. 22. 1. 23.28 95.1322
0.0561
1 138. 2062. 3. 24. 2. 23.19 95.1299
0.0349
1 138. 2362. 3. 15. 1. 23.08 95.1162
0.0480
1 138. 2362. 3. 17. 1. 23.01 95.0569
0.0577
1 138. 2362. 3. 18. 1. 22.97 95.0598
0.0516
1 138. 2362. 3. 22. 1. 23.23 95.1487
0.0386
1 138. 2362. 3. 23. 2. 23.28 95.0743
0.0256
1 138. 2362. 3. 24. 2. 23.10 95.1010
0.0420
1 139. 1. 3. 15. 1. 23.01 99.3528
0.1424
1 139. 1. 3. 17. 1. 23.00 99.2940
0.0660
1 139. 1. 3. 17. 1. 23.01 99.2340
0.1179
1 139. 1. 3. 21. 1. 23.20 99.3489
0.0506
1 139. 1. 3. 23. 2. 23.22 99.2625
0.1111
1 139. 1. 3. 23. 1. 23.22 99.3787
0.1103
1 139. 281. 3. 16. 1. 22.95 99.3244
0.1134
1 139. 281. 3. 17. 1. 22.98 99.3378
0.0949
1 139. 281. 3. 18. 1. 22.86 99.3424
0.0847
1 139. 281. 3. 22. 1. 23.17 99.4033
0.0801
1 139. 281. 3. 23. 2. 23.10 99.3717
0.0630
1 139. 281. 3. 23. 1. 23.14 99.3493
0.1157
1 139. 283. 3. 16. 1. 22.94 99.3065
0.0381
1 139. 283. 3. 17. 1. 23.09 99.3280
0.1153
1 139. 283. 3. 18. 1. 23.11 99.3000
0.0818
1 139. 283. 3. 21. 1. 23.25 99.3347
0.0972
1 139. 283. 3. 22. 1. 23.36 99.3929
0.1189
1 139. 283. 3. 23. 1. 23.18 99.2644
0.0622
1 139. 2062. 3. 16. 1. 22.94 99.3324
0.1531
1 139. 2062. 3. 17. 1. 23.08 99.3254
0.0543
1 139. 2062. 3. 18. 1. 23.15 99.2555
0.1024
1 139. 2062. 3. 18. 1. 23.18 99.1946
0.0851
1 139. 2062. 3. 22. 1. 23.27 99.3542
0.1227
1 139. 2062. 3. 24. 2. 23.23 99.2365
0.1218
1 139. 2362. 3. 15. 1. 23.08 99.2939
0.0818
1 139. 2362. 3. 17. 1. 23.02 99.3234
0.0723
1 139. 2362. 3. 18. 1. 22.93 99.2748
0.0756
1 139. 2362. 3. 22. 1. 23.29 99.3512
0.0475
1 139. 2362. 3. 23. 2. 23.25 99.2350
0.0517
1 139. 2362. 3. 24. 2. 23.05 99.3574
0.0485
1 140. 1. 3. 15. 1. 23.07 96.1334
0.1052
1 140. 1. 3. 17. 1. 23.08 96.1250
0.0916
1 140. 1. 3. 18. 1. 22.77 96.0665
0.0836
1 140. 1. 3. 21. 1. 23.18 96.0725
0.0620
1 140. 1. 3. 23. 2. 23.20 96.1006
0.0582
1 140. 1. 3. 23. 1. 23.21 96.1131
0.1757
1 140. 281. 3. 16. 1. 22.94 96.0467
0.0565
1 140. 281. 3. 17. 1. 22.99 96.1081
0.1293
1 140. 281. 3. 18. 1. 22.91 96.0578
0.1148
1 140. 281. 3. 22. 1. 23.15 96.0700
0.0495
1 140. 281. 3. 22. 1. 23.33 96.1052
0.1722
1 140. 281. 3. 23. 1. 23.19 96.0952
0.1786
1 140. 283. 3. 16. 1. 22.89 96.0650
0.1301
1 140. 283. 3. 17. 1. 23.07 96.0870
0.0881
1 140. 283. 3. 18. 1. 23.07 95.8906
0.1842
1 140. 283. 3. 21. 1. 23.24 96.0842
0.1008
1 140. 283. 3. 22. 1. 23.34 96.0189
0.0865
1 140. 283. 3. 23. 1. 23.19 96.1047
0.0923
1 140. 2062. 3. 16. 1. 22.95 96.0379
0.2190
1 140. 2062. 3. 17. 1. 22.97 96.0671
0.0991
1 140. 2062. 3. 18. 1. 23.15 96.0206
0.0648
1 140. 2062. 3. 21. 1. 23.14 96.0207
0.1410
1 140. 2062. 3. 22. 1. 23.32 96.0587
0.1634
1 140. 2062. 3. 24. 2. 23.17 96.0903
0.0406
1 140. 2362. 3. 15. 1. 23.08 96.0771
0.1024
1 140. 2362. 3. 17. 1. 23.00 95.9976
0.0943
1 140. 2362. 3. 18. 1. 23.01 96.0148
0.0622
1 140. 2362. 3. 22. 1. 23.27 96.0397
0.0702
1 140. 2362. 3. 23. 2. 23.24 96.0407
0.0627
1 140. 2362. 3. 24. 2. 23.13 96.0445
0.0622
1 141. 1. 3. 15. 1. 23.01 101.2124
0.0900
1 141. 1. 3. 17. 1. 23.08 101.1018
0.0820
1 141. 1. 3. 18. 1. 22.75 101.1119
0.0500
1 141. 1. 3. 21. 1. 23.21 101.1072
0.0641
1 141. 1. 3. 23. 2. 23.25 101.0802
0.0704
1 141. 1. 3. 23. 1. 23.19 101.1350
0.0699
1 141. 281. 3. 16. 1. 22.93 101.0287
0.0520
1 141. 281. 3. 17. 1. 23.00 101.0131
0.0710
1 141. 281. 3. 18. 1. 22.90 101.1329
0.0800
1 141. 281. 3. 22. 1. 23.19 101.0562
0.1594
1 141. 281. 3. 23. 2. 23.18 101.0891
0.1252
1 141. 281. 3. 23. 1. 23.17 101.1283
0.1151
1 141. 283. 3. 16. 1. 22.85 101.1597
0.0990
1 141. 283. 3. 17. 1. 23.09 101.0784
0.0810
1 141. 283. 3. 18. 1. 23.08 101.0715
0.0460
1 141. 283. 3. 21. 1. 23.27 101.0910
0.0880
1 141. 283. 3. 22. 1. 23.34 101.0967
0.0901
1 141. 283. 3. 24. 2. 23.00 101.1627
0.0888
1 141. 2062. 3. 16. 1. 22.97 101.1077
0.0970
1 141. 2062. 3. 17. 1. 22.96 101.0245
0.1210
1 141. 2062. 3. 18. 1. 23.19 100.9650
0.0700
1 141. 2062. 3. 18. 1. 23.18 101.0319
0.1070
1 141. 2062. 3. 22. 1. 23.34 101.0849
0.0960
1 141. 2062. 3. 24. 2. 23.21 101.1302
0.0505
1 141. 2362. 3. 15. 1. 23.08 101.0471
0.0320
1 141. 2362. 3. 17. 1. 23.01 101.0224
0.1020
1 141. 2362. 3. 18. 1. 23.05 101.0702
0.0580
1 141. 2362. 3. 22. 1. 23.22 101.0904
0.1049
1 141. 2362. 3. 23. 2. 23.29 101.0626
0.0702
1 141. 2362. 3. 24. 2. 23.15 101.0686
0.0661
1 142. 1. 3. 15. 1. 23.02 94.3160
0.1372
1 142. 1. 3. 17. 1. 23.04 94.2808
0.0999
1 142. 1. 3. 18. 1. 22.73 94.2478
0.0803
1 142. 1. 3. 21. 1. 23.19 94.2862
0.0700
1 142. 1. 3. 23. 2. 23.25 94.1859
0.0899
1 142. 1. 3. 23. 1. 23.21 94.2389
0.0686
1 142. 281. 3. 16. 1. 22.98 94.2640
0.0862
1 142. 281. 3. 17. 1. 23.00 94.3333
0.1330
1 142. 281. 3. 18. 1. 22.88 94.2994
0.0908
1 142. 281. 3. 21. 1. 23.28 94.2873
0.0846
1 142. 281. 3. 23. 2. 23.07 94.2576
0.0795
1 142. 281. 3. 23. 1. 23.12 94.3027
0.0389
1 142. 283. 3. 16. 1. 22.92 94.2846
0.1021
1 142. 283. 3. 17. 1. 23.08 94.2197
0.0627
1 142. 283. 3. 18. 1. 23.09 94.2119
0.0785
1 142. 283. 3. 21. 1. 23.29 94.2536
0.0712
1 142. 283. 3. 22. 1. 23.34 94.2280
0.0692
1 142. 283. 3. 24. 2. 22.92 94.2944
0.0958
1 142. 2062. 3. 16. 1. 22.96 94.2238
0.0492
1 142. 2062. 3. 17. 1. 22.95 94.3061
0.2194
1 142. 2062. 3. 18. 1. 23.16 94.1868
0.0474
1 142. 2062. 3. 21. 1. 23.11 94.2645
0.0697
1 142. 2062. 3. 22. 1. 23.31 94.3101
0.0532
1 142. 2062. 3. 24. 2. 23.24 94.2204
0.1023
1 142. 2362. 3. 15. 1. 23.08 94.2437
0.0503
1 142. 2362. 3. 17. 1. 23.00 94.2115
0.0919
1 142. 2362. 3. 18. 1. 22.99 94.2348
0.0282
1 142. 2362. 3. 22. 1. 23.26 94.2124
0.0513
1 142. 2362. 3. 23. 2. 23.27 94.2214
0.0627
1 142. 2362. 3. 24. 2. 23.08 94.1651
0.1010
2 138. 1. 4. 13. 1. 23.12 95.1996
0.0645
2 138. 1. 4. 15. 1. 22.73 95.1315
0.1192
2 138. 1. 4. 18. 2. 22.76 95.1845
0.0452
2 138. 1. 4. 19. 1. 22.73 95.1359
0.1498
2 138. 1. 4. 20. 2. 22.73 95.1435
0.0629
2 138. 1. 4. 21. 2. 22.93 95.1839
0.0563
2 138. 281. 4. 14. 2. 22.46 95.2106
0.1049
2 138. 281. 4. 18. 2. 22.80 95.2505
0.0771
2 138. 281. 4. 18. 2. 22.77 95.2648
0.1046
2 138. 281. 4. 20. 2. 22.80 95.2197
0.1779
2 138. 281. 4. 20. 2. 22.87 95.2003
0.1376
2 138. 281. 4. 21. 2. 22.95 95.0982
0.1611
2 138. 283. 4. 18. 2. 22.83 95.1211
0.0794
2 138. 283. 4. 13. 1. 23.17 95.1327
0.0409
2 138. 283. 4. 18. 1. 22.67 95.2053
0.1525
2 138. 283. 4. 19. 2. 23.00 95.1292
0.0655
2 138. 283. 4. 21. 2. 22.91 95.1669
0.0619
2 138. 283. 4. 21. 2. 22.96 95.1401
0.0831
2 138. 2062. 4. 15. 1. 22.64 95.2479
0.2867
2 138. 2062. 4. 15. 1. 22.67 95.2224
0.1945
2 138. 2062. 4. 19. 2. 22.99 95.2810
0.1960
2 138. 2062. 4. 19. 1. 22.75 95.1869
0.1571
2 138. 2062. 4. 21. 2. 22.84 95.3053
0.2012
2 138. 2062. 4. 21. 2. 22.92 95.1432
0.1532
2 138. 2362. 4. 12. 1. 22.74 95.1687
0.0785
2 138. 2362. 4. 18. 2. 22.75 95.1564
0.0430
2 138. 2362. 4. 19. 2. 22.88 95.1354
0.0983
2 138. 2362. 4. 19. 1. 22.73 95.0422
0.0773
2 138. 2362. 4. 20. 2. 22.86 95.1354
0.0587
2 138. 2362. 4. 21. 2. 22.94 95.1075
0.0776
2 139. 1. 4. 13. 2. 23.14 99.3274
0.0220
2 139. 1. 4. 15. 2. 22.77 99.5020
0.0997
2 139. 1. 4. 18. 2. 22.80 99.4016
0.0704
2 139. 1. 4. 19. 1. 22.68 99.3181
0.1245
2 139. 1. 4. 20. 2. 22.78 99.3858
0.0903
2 139. 1. 4. 21. 2. 22.93 99.3141
0.0255
2 139. 281. 4. 14. 2. 23.05 99.2915
0.0859
2 139. 281. 4. 15. 2. 22.71 99.4032
0.1322
2 139. 281. 4. 18. 2. 22.79 99.4612
0.1765
2 139. 281. 4. 20. 2. 22.74 99.4001
0.0889
2 139. 281. 4. 20. 2. 22.91 99.3765
0.1041
2 139. 281. 4. 21. 2. 22.92 99.3507
0.0717
2 139. 283. 4. 13. 2. 23.11 99.3848
0.0792
2 139. 283. 4. 18. 2. 22.84 99.4952
0.1122
2 139. 283. 4. 18. 2. 22.76 99.3220
0.0915
2 139. 283. 4. 19. 2. 23.03 99.4165
0.0503
2 139. 283. 4. 21. 2. 22.87 99.3791
0.1138
2 139. 283. 4. 21. 2. 22.98 99.3985
0.0661
2 139. 2062. 4. 14. 2. 22.43 99.4283
0.0891
2 139. 2062. 4. 15. 2. 22.70 99.4139
0.2147
2 139. 2062. 4. 19. 2. 22.97 99.3813
0.1143
2 139. 2062. 4. 19. 1. 22.77 99.4314
0.1685
2 139. 2062. 4. 21. 2. 22.79 99.4166
0.2080
2 139. 2062. 4. 21. 2. 22.94 99.4052
0.2400
2 139. 2362. 4. 12. 1. 22.82 99.3408
0.1279
2 139. 2362. 4. 18. 2. 22.77 99.3116
0.1131
2 139. 2362. 4. 19. 2. 22.82 99.3241
0.0519
2 139. 2362. 4. 19. 1. 22.74 99.2991
0.0903
2 139. 2362. 4. 20. 2. 22.88 99.3049
0.0783
2 139. 2362. 4. 21. 2. 22.94 99.2782
0.0718
2 140. 1. 4. 13. 1. 23.10 96.0811
0.0463
2 140. 1. 4. 15. 2. 22.75 96.1460
0.0725
2 140. 1. 4. 18. 2. 22.78 96.1582
0.1428
2 140. 1. 4. 19. 1. 22.70 96.1039
0.1056
2 140. 1. 4. 20. 2. 22.75 96.1262
0.0672
2 140. 1. 4. 21. 2. 22.93 96.1478
0.0562
2 140. 281. 4. 15. 2. 22.71 96.1153
0.1097
2 140. 281. 4. 14. 2. 22.49 96.1297
0.1202
2 140. 281. 4. 18. 2. 22.81 96.1233
0.1331
2 140. 281. 4. 20. 2. 22.78 96.1731
0.1484
2 140. 281. 4. 20. 2. 22.89 96.0872
0.0857
2 140. 281. 4. 21. 2. 22.91 96.1331
0.0944
2 140. 283. 4. 13. 2. 23.22 96.1135
0.0983
2 140. 283. 4. 18. 2. 22.85 96.1111
0.1210
2 140. 283. 4. 18. 2. 22.78 96.1221
0.0644
2 140. 283. 4. 19. 2. 23.01 96.1063
0.0921
2 140. 283. 4. 21. 2. 22.91 96.1155
0.0704
2 140. 283. 4. 21. 2. 22.94 96.1308
0.0258
2 140. 2062. 4. 15. 2. 22.60 95.9767
0.2225
2 140. 2062. 4. 15. 2. 22.66 96.1277
0.1792
2 140. 2062. 4. 19. 2. 22.96 96.1858
0.1312
2 140. 2062. 4. 19. 1. 22.75 96.1912
0.1936
2 140. 2062. 4. 21. 2. 22.82 96.1650
0.1902
2 140. 2062. 4. 21. 2. 22.92 96.1603
0.1777
2 140. 2362. 4. 12. 1. 22.88 96.0793
0.0996
2 140. 2362. 4. 18. 2. 22.76 96.1115
0.0533
2 140. 2362. 4. 19. 2. 22.79 96.0803
0.0364
2 140. 2362. 4. 19. 1. 22.71 96.0411
0.0768
2 140. 2362. 4. 20. 2. 22.84 96.0988
0.1042
2 140. 2362. 4. 21. 1. 22.94 96.0482
0.0868
2 141. 1. 4. 13. 1. 23.07 101.1984
0.0803
2 141. 1. 4. 15. 2. 22.72 101.1645
0.0914
2 141. 1. 4. 18. 2. 22.75 101.2454
0.1109
2 141. 1. 4. 19. 1. 22.69 101.1096
0.1376
2 141. 1. 4. 20. 2. 22.83 101.2066
0.0717
2 141. 1. 4. 21. 2. 22.93 101.0645
0.1205
2 141. 281. 4. 15. 2. 22.72 101.1615
0.1272
2 141. 281. 4. 14. 2. 22.40 101.1650
0.0595
2 141. 281. 4. 18. 2. 22.78 101.1815
0.1393
2 141. 281. 4. 20. 2. 22.73 101.1106
0.1189
2 141. 281. 4. 20. 2. 22.86 101.1420
0.0713
2 141. 281. 4. 21. 2. 22.94 101.0116
0.1088
2 141. 283. 4. 13. 2. 23.26 101.1554
0.0429
2 141. 283. 4. 18. 2. 22.85 101.1267
0.0751
2 141. 283. 4. 18. 2. 22.76 101.1227
0.0826
2 141. 283. 4. 19. 2. 22.82 101.0635
0.1715
2 141. 283. 4. 21. 2. 22.89 101.1264
0.1447
2 141. 283. 4. 21. 2. 22.96 101.0853
0.1189
2 141. 2062. 4. 15. 2. 22.65 101.1332
0.2532
2 141. 2062. 4. 15. 1. 22.68 101.1487
0.1413
2 141. 2062. 4. 19. 2. 22.95 101.1778
0.1772
2 141. 2062. 4. 19. 1. 22.77 101.0988
0.0884
2 141. 2062. 4. 21. 2. 22.87 101.1686
0.2940
2 141. 2062. 4. 21. 2. 22.94 101.3289
0.2072
2 141. 2362. 4. 12. 1. 22.83 101.1353
0.0585
2 141. 2362. 4. 18. 2. 22.83 101.1201
0.0868
2 141. 2362. 4. 19. 2. 22.91 101.0946
0.0855
2 141. 2362. 4. 19. 1. 22.71 100.9977
0.0645
2 141. 2362. 4. 20. 2. 22.87 101.0963
0.0638
2 141. 2362. 4. 21. 2. 22.94 101.0300
0.0549
2 142. 1. 4. 13. 1. 23.07 94.3049
0.1197
2 142. 1. 4. 15. 2. 22.73 94.3153
0.0566
2 142. 1. 4. 18. 2. 22.77 94.3073
0.0875
2 142. 1. 4. 19. 1. 22.67 94.2803
0.0376
2 142. 1. 4. 20. 2. 22.80 94.3008
0.0703
2 142. 1. 4. 21. 2. 22.93 94.2916
0.0604
2 142. 281. 4. 14. 2. 22.90 94.2557
0.0619
2 142. 281. 4. 18. 2. 22.83 94.3542
0.1027
2 142. 281. 4. 18. 2. 22.80 94.3007
0.1492
2 142. 281. 4. 20. 2. 22.76 94.3351
0.1059
2 142. 281. 4. 20. 2. 22.88 94.3406
0.1508
2 142. 281. 4. 21. 2. 22.92 94.2621
0.0946
2 142. 283. 4. 13. 2. 23.25 94.3124
0.0534
2 142. 283. 4. 18. 2. 22.85 94.3680
0.1643
2 142. 283. 4. 18. 1. 22.67 94.3442
0.0346
2 142. 283. 4. 19. 2. 22.80 94.3391
0.0616
2 142. 283. 4. 21. 2. 22.91 94.2238
0.0721
2 142. 283. 4. 21. 2. 22.95 94.2721
0.0998
2 142. 2062. 4. 14. 2. 22.49 94.2915
0.2189
2 142. 2062. 4. 15. 2. 22.69 94.2803
0.0690
2 142. 2062. 4. 19. 2. 22.94 94.2818
0.0987
2 142. 2062. 4. 19. 1. 22.76 94.2227
0.2628
2 142. 2062. 4. 21. 2. 22.74 94.4109
0.1230
2 142. 2062. 4. 21. 2. 22.94 94.2616
0.0929
2 142. 2362. 4. 12. 1. 22.86 94.2052
0.0813
2 142. 2362. 4. 18. 2. 22.83 94.2824
0.0605
2 142. 2362. 4. 19. 2. 22.85 94.2396
0.0882
2 142. 2362. 4. 19. 1. 22.75 94.2087
0.0702
2 142. 2362. 4. 20. 2. 22.86 94.2937
0.0591
2 142. 2362. 4. 21. 1. 22.93 94.2330
0.0556
2.6.3.1.2. Measurements on wiring configurations

2.6. Case studies
2.6.3.1.2. Measurements on wiring
configurations
Check wafers
were
measured
with the probe
wired in two
configurations
according to an ASTM Standard Test Method (F4) to
identify differences between 2 wiring configurations for
probe #2362. The check standards for the study were five
wafers selected at random from a batch of wafers cut from
one silicon crystal doped with phosphorous to give a
nominal resistivity of 100 ohm.cm.
Description of
database
The data are averages of K = 6 days' measurements and J
= 6 repetitions at the center of each wafer. There are L = 2
complete runs, separated by two months time, on each
wafer.
The data recorded in the 10 columns are:
1. Wafer
2. Probe
3. Average - configuration A; run 1
4. Standard deviation - configuration A; run 1
5. Average - configuration B; run 1
6. Standard deviation - configuration B; run 1
7. Average - configuration A; run 2
8. Standard deviation - configuration A; run 2
9. Average - configuration B; run 2
10. Standard deviation - configuration B; run 2
Wafer Probe Config A-run1 Config B-run1 Config A-run2
Config B-run2.
138. 2362. 95.1162 0.0480 95.0993 0.0466 95.1687 0.0785
95.1589 0.0642
138. 2362. 95.0569 0.0577 95.0657 0.0450 95.1564 0.0430
95.1705 0.0730
138. 2362. 95.0598 0.0516 95.0622 0.0664 95.1354 0.0983
95.1221 0.0695
138. 2362. 95.1487 0.0386 95.1625 0.0311 95.0422 0.0773
95.0513 0.0840
138. 2362. 95.0743 0.0256 95.0599 0.0488 95.1354 0.0587
95.1531 0.0482
138. 2362. 95.1010 0.0420 95.0944 0.0393 95.1075 0.0776
95.1537 0.0230
139. 2362. 99.2939 0.0818 99.3018 0.0905 99.3408 0.1279
99.3637 0.1025
139. 2362. 99.3234 0.0723 99.3488 0.0350 99.3116 0.1131
99.3881 0.0451
2.6.3.1.2. Measurements on wiring configurations
139. 2362. 99.2748 0.0756 99.3571 0.1993 99.3241 0.0519
99.3737 0.0699
139. 2362. 99.3512 0.0475 99.3512 0.1286 99.2991 0.0903
99.3066 0.0709
139. 2362. 99.2350 0.0517 99.2255 0.0738 99.3049 0.0783
99.3040 0.0744
139. 2362. 99.3574 0.0485 99.3605 0.0459 99.2782 0.0718
99.3680 0.0470
140. 2362. 96.0771 0.1024 96.0915 0.1257 96.0793 0.0996
96.1041 0.0890
140. 2362. 95.9976 0.0943 96.0057 0.0806 96.1115 0.0533
96.0774 0.0983
140. 2362. 96.0148 0.0622 96.0244 0.0833 96.0803 0.0364
96.1004 0.0758
140. 2362. 96.0397 0.0702 96.0422 0.0738 96.0411 0.0768
96.0677 0.0663
140. 2362. 96.0407 0.0627 96.0738 0.0800 96.0988 0.1042
96.0585 0.0960
140. 2362. 96.0445 0.0622 96.0557 0.1129 96.0482 0.0868
96.0062 0.0895
141. 2362. 101.0471 0.0320 101.0241 0.0670 101.1353 0.0585
101.1156 0.1027
141. 2362. 101.0224 0.1020 101.0660 0.1030 101.1201 0.0868
101.1077 0.1141
141. 2362. 101.0702 0.0580 101.0509 0.0710 101.0946 0.0855
101.0455 0.1070
141. 2362. 101.0904 0.1049 101.0983 0.0894 100.9977 0.0645
101.0274 0.0666
141. 2362. 101.0626 0.0702 101.0614 0.0849 101.0963 0.0638
101.1106 0.0788
141. 2362. 101.0686 0.0661 101.0811 0.0490 101.0300 0.0549
101.1073 0.0663
142. 2362. 94.2437 0.0503 94.2088 0.0815 94.2052 0.0813
94.2487 0.0719
142. 2362. 94.2115 0.0919 94.2043 0.1176 94.2824 0.0605
94.2886 0.0499
142. 2362. 94.2348 0.0282 94.2324 0.0519 94.2396 0.0882
94.2739 0.1075
142. 2362. 94.2124 0.0513 94.2347 0.0694 94.2087 0.0702
94.2023 0.0416
142. 2362. 94.2214 0.0627 94.2416 0.0757 94.2937 0.0591
94.2600 0.0731
142. 2362. 94.1651 0.1010 94.2287 0.0919 94.2330 0.0556
94.2406 0.0651

2.6. Case studies
Purpose of
this page
The purpose of this page is to outline an analysis of data taken
during a gauge study to quantify the type A uncertainty
component for resistivity (ohm.cm) measurements on silicon
wafers made with a gauge that was part of the initial study.
Summary of
standard
deviations at
three levels
The level-1, level-2, and level-3 standard deviations for the
uncertainty analysis are summarized in the table below from the
gauge case study.
Standard deviations for probe #2362
Level Symbol Estimate DF
Level-1 s
1
0.0729 300
Level-2 s
2
0.0362 50
Level-3 s
3
0.0197 5
Calculation of
individual
components
for days and
runs
The standard deviation that estimates the day effect is
The standard deviation that estimates the run effect is
Calculation of
the standard
deviation of
the certified
value showing
sensitivity
coefficients
The certified value for each wafer is the average of N = 6
repeatability measurements at the center of the wafer on M = 1
days and over P = 1 runs. Notice that N, M and P are not
necessarily the same as the number of measurements in the
gauge study per wafer; namely, J, K and L. The standard
deviation of a certified value (for time-dependent sources of
error), is
Standard deviations for days and runs are included in this
calculation, even though there were no replications over days or
runs for the certification measurements. These factors contribute
to the overall uncertainty of the measurement process even
though they are not sampled for the particular measurements of
interest.
The equation
must be
rewritten to
calculate
degrees of
freedom
Degrees of freedom cannot be calculated from the equation
above because the calculations for the individual components
involve differences among variances. The table of sensitivity
coefficients for a 3-level design shows that for
N = J, M = 1, P = 1
the equation above can be rewritten in the form
Then the degrees of freedom can be approximated using the
Welch-Satterthwaite method.
Probe bias -
Graphs of
probe biases
A graphical analysis shows the relative biases among the 5
probes. For each wafer, differences from the wafer average by
probe are plotted versus wafer number. The graphs verify that
probe #2362 (coded as 5) is biased low relative to the other
probes. The bias shows up more strongly after the probes have
been in use (run 2).
How to deal
with bias due
to the probe
Probe #2362 was chosen for the certification process because of
its superior precision, but its bias relative to the other probes
creates a problem. There are two possibilities for handling this
problem:
1. Correct all measurements made with probe #2362 to the
average of the probes.
2. Include the standard deviation for the difference among
probes in the uncertainty budget.
The best strategy, as followed in the certification process, is to
correct all measurements for the average bias of probe #2362 and
take the standard deviation of the correction as a type A
Correction for
bias or probe
#2362 and
uncertainty
Biases by probe and wafer are shown in the gauge case study.
Biases for probe #2362 are summarized in table below for the
two runs. The correction is taken to be the negative of the
average bias. The standard deviation of the correction is the
standard deviation of the average of the ten biases.
Estimated biases for probe #2362

Wafer Probe Run 1 Run 2 All
138 2362 -0.0372 -0.0507
139 2362 -0.0094 -0.0657
140 2362 -0.0261 -0.0398
141 2362 -0.0252 -0.0534
142 2362 -0.0383 -0.0469
Average -0.0272 -0.0513 -0.0393
Standard deviation 0.0162
(10 values)
Configurations
Database and
plot of
differences
Measurements on the check wafers were made with the probe
wired in two different configurations (A, B). A plot of
differences between configuration A and configuration B shows
no bias between the two configurations.
Test for
difference
between
configurations
This finding is consistent over runs 1 and 2 and is confirmed by
the t-statistics in the table below where the average differences
and standard deviations are computed from 6 days of
measurements on 5 wafers. A t-statistic < 2 indicates no
significant difference. The conclusion is that there is no bias due
to wiring configuration and no contribution to uncertainty from
this source.
Differences between configurations
Status Average Std dev DF t

Pre -0.00858 0.0242 29 1.9
Post -0.0110 0.0354 29 1.7
Error budget
showing
sensitivity
coefficients,
standard
deviations and
degrees of
freedom
The error budget showing sensitivity coefficients for computing
the standard uncertainty and degrees of freedom is outlined
below.
Error budget for resistivity (ohm.cm)
Source Type Sensitivity
Standard
Deviation DF
Repeatability A a
1
= 0 0.0729 300
Reproducibility A
a
2
=
0.0362 50
Run-to-run A a
3
= 1 0.0197 5
Probe #2362 A
a
4
=
0.0162 5
Wiring
Configuration A
A a
5
= 1 0 --
Standard
uncertainty
includes
components
for
repeatability,
days, runs and
probe
The standard uncertainty is computed from the error budget as
Approximate The degrees of freedom associated with u are approximated by
degrees of
freedom and
expanded
uncertainty
the Welch-Satterthwaite formula as:
where the
i
are the degrees of freedom given in the rightmost
column of the table.
The critical value at the 0.05 significance level with 42 degrees
of freedom, from the t-table, is 2.018 so the expanded
uncertainty is
U = 2.018 u = 0.078 ohm.cm
2.6.3.2.1. Difference between 2 wiring configurations

2.6. Case studies
2.6.3.2.1. Difference between 2 wiring
configurations
Measurements
with the probe
configured in
two ways
The graphs below are constructed from resistivity
measurements (ohm.cm) on five wafers where the probe
(#2362) was wired in two different configurations, A and
B. The probe is a 4-point probe with many possible wiring
configurations. For this experiment, only two
configurations were tested as a means of identifying large
discrepancies.
Artifacts for
the study
The five wafers; namely, #138, #139, #140, #141, and #142
are coded 1, 2, 3, 4, 5, respectively, in the graphs. These
wafers were chosen at random from a batch of
approximately 100 wafers that were being certified for
resistivity.
Interpretation Differences between measurements in configurations A
and B, made on the same day, are plotted over six days for
each wafer. The two graphs represent two runs separated
by approximately two months time. The dotted line in the
center is the zero line. The pattern of data points scatters
fairly randomly above and below the zero line -- indicating
no difference between configurations for probe #2362. The
conclusion applies to probe #2362 and cannot be extended
to all probes of this type.
2.6.3.3. Run the type A uncertainty analysis using Dataplot

2.6. Case studies
2.6.3.3. Run the type A uncertainty analysis
using Dataplot
View of
Dataplot
macros for
this case
study
description.
Time-dependent components from 3-
level nested design
Pool repeatability standard deviations for:
1. Run 1
2. Run 2
Compute level-2 standard
deviations for:
3. Run 1
4. Run 2
5. Pool level-2 standard deviations
Database of measurements with probe
#2362
1. The repeatability standard deviation
is 0.0658 ohm.cm for run 1 and
0.0758 ohm.cm for run 2. This
represents the basic precision of the
measuring instrument.
2. The level-2 standard deviation
pooled over 5 wafers and 2 runs is
0.0362 ohm.cm. This is significant
in the calculation of uncertainty.
3. The level-3 standard deviation
pooled over 5 wafers is 0.0197
ohm.cm. This is small compared to
the other components but is
2.6.3.3. Run the type A uncertainty analysis using Dataplot
6. Compute level-3 standard
deviations
included in the uncertainty
calculation for completeness.
1. Plot biases for 5 NIST probes
2. Compute wafer bias and average
bias for probe #2362
3. Correction for bias and standard
deviation
Database of measurements with 5 probes
1. The plot shows that probe #2362 is
biased low relative to the other
probes and that this bias is
consistent over 5 wafers.
2. The bias correction is the average
bias = 0.0393 ohm.cm over the 5
wafers. The correction is to be
subtracted from all measurements
made with probe #2362.
3. The uncertainty of the bias
correction = 0.0051 ohm.cm is
computed from the standard
deviation of the biases for the 5
wafers.
Bias due to wiring configuration A
1. Plot differences between wiring
configurations
2. Averages, standard deviations and
t-statistics
Database of wiring configurations A and
B
1. The plot of measurements in wiring
configurations A and B shows no
difference between A and B.
2. The statistical test confirms that
there is no difference between the
wiring configurations.
Uncertainty
1. Standard uncertainty, df, t-value
and expanded uncertainty
Elements of error budget
1. The uncertainty is computed from
the error budget. The uncertainty
for an average of 6 measurements
on one day with probe #2362 is
0.078 with 42 degrees of freedom.

2.6. Case studies
Reads data
and plots the
repeatability
standard
deviations for
probe #2362
and pools
standard
deviations
over days,
wafers -- run
1
reset data
reset plot control
reset i/o
dimension 500 rows
label size 3
set read format f1.0,f6.0,f8.0,32x,f10.4,f10.4
read mpc633a.dat run wafer probe y sr
retain run wafer probe y sr subset probe =
2362
let df = sr - sr + 5.
y1label ohm.cm
characters * all
lines blank all
x2label Repeatability standard deviations for
probe 2362 - run 1
plot sr subset run 1
let var = sr*sr
let df11 = sum df subset run 1
let s11 = sum var subset run 1
. repeatability standard deviation for run 1
let s11 = (5.*s11/df11)**(1/2)
print s11 df11
. end of calculations
Reads data
and plots
repeatability
standard
deviations for
probe #2362
and pools
standard
deviations
over days,
wafers -- run
2
reset data
reset plot control
reset i/o
dimension 500 30
label size 3
retain run wafer probe y sr subset probe 2362
let df = sr - sr + 5.
y1label ohm.cm
characters * all
lines blank all
x2label Repeatability standard deviations for
probe 2362 - run 2
plot sr subset run 2
let var = sr*sr
let s11 = (5.*s11/df11)**(1/2)
let s12 = (5.*s12/df12)**(1/2)
print s11 df11
print s12 df12
let s1 = ((s11**2 + s12**2)/2.)**(1/2)
let df1=df11+df12
. repeatability standard deviation and df for
run 2
print s1 df1
Computes
level-2
standard
deviations
reset data
reset plot control
reset i/o
dimension 500 rows
label size 3
from daily
averages and
pools over
wafers -- run
1
sd plot y wafer subset run 1
let s21 = yplot
let wafer1 = xplot
retain s21 wafer1 subset tagplot = 1
let nwaf = size s21
let df21 = 5 for i = 1 1 nwaf
. level-2 standard deviations and df for 5
wafers - run 1
print wafer1 s21 df21
Computes
level-2
standard
deviations
from daily
averages and
pools over
wafers -- run
2
reset data
reset plot control
reset i/o
dimension 500 rows
label size 3
let s22 = yplot
let wafer1 = xplot
retain s22 wafer1 subset tagplot = 1
let nwaf = size s22
wafers - run 1
Pools level-2
standard
deviations
over wafers
and runs
reset data
reset plot control
reset i/o
dimension 500 30
label size 3
let s21 = yplot
let wafer1 = xplot
let s22 = yplot
retain s21 s22 wafer1 subset tagplot = 1
let nwaf = size wafer1
let s2a = (s21**2)/5 + (s22**2)/5
let s2 = sum s2a
let s2 = sqrt(s2/2)
let df2a = df21 + df22
let df2 = sum df2a
. pooled level-2 standard deviation and df
across wafers and runs
print s2 df2
Computes
level-
3standard
deviations
from run
averages and
pools over
wafers
reset data
reset plot control
reset i/o
dimension 500 rows
label size 3
.
mean plot y wafer subset run 1
let m31 = yplot
let wafer1 = xplot
mean plot y wafer subset run 2
let m32 = yplot
retain m31 m32 wafer1 subset tagplot = 1
let nwaf = size m31
let s31 =(((m31-m32)**2)/2.)**(1/2)
wafers
let s31 = (s31**2)/5
let s3 = sum s31
let s3 = sqrt(s3)
let df3=sum df31
. pooled level-3 std deviation and df over 5
wafers
print s3 df3
Plot
differences
from the
average wafer
value for each
probe
showing bias
for probe
#2362
reset data
reset plot control
reset i/o
dimension 500 30
print biasrun1 biasrun2
Y1LABEL OHM.CM
xlimits 137 143
1)
plot zero wafer
print biasrun2
Y1LABEL OHM.CM
xlimits 137 143
2)
plot zero wafer
Compute bias
for probe
#2362 by
wafer
reset data
reset plot control
reset i/o
dimension 500 30
label size 3
set read format
.
cross tabulate mean y run wafer
skip 1
read dpst1f.dat runid wafid ybar
print runid wafid ybar
let ngroups = size ybar
skip 0
.
let m3 = y - y
feedback off
loop for k = 1 1 ngroups
let runa = runid(k)
let wafera = wafid(k)
let ytemp = ybar(k)
let m3 = ytemp subset run = runa subset
wafer = wafera
end of loop
feedback on
.
let d = y - m3
let bias1 = average d subset run 1
.
mean plot d wafer subset run 1
let b1 = yplot
let wafer1 = xplot
let b2 = yplot
retain b1 b2 wafer1 subset tagplot = 1
let nwaf = size b1
. biases for run 1 and run 2 by wafers
print wafer1 b1 b2
. average biases over wafers for run 1 and run 2
print bias1 bias2
Compute
correction for
bias for
measurements
with probe
#2362 and the
standard
deviation of
the correction
reset data
reset plot control
reset i/o
dimension 500 30
label size 3
set read format
.
cross tabulate mean y run wafer
skip 1
read dpst1f.dat runid wafid ybar
let ngroups = size ybar
skip 0
.
let m3 = y - y
feedback off
loop for k = 1 1 ngroups
let runa = runid(k)
let wafera = wafid(k)
let ytemp = ybar(k)
let m3 = ytemp subset run = runa subset
wafer = wafera
end of loop
feedback on
.
let d = y - m3
.
let b1 = yplot
let wafer1 = xplot
let b2 = yplot
retain b1 b2 wafer1 subset tagplot = 1
.
extend b1 b2
let sd = standard deviation b1
let sdcorr = sd/(10**(1/2))
let correct = -(bias1+bias2)/2.
. correction for probe #2362, standard dev, and
standard dev of corr
print correct sd sdcorr
Plot
differences
between
wiring
configurations
A and B
reset data
reset plot control
reset i/o
dimension 500 30
label size 3
read mpc633k.dat wafer probe a1 s1 b1 s2 a2 s3
b2 s4
let diff1 = a1 - b1
let diff2 = a2 - b2
let t = sequence 1 1 30
lines blank all
y1label ohm.cm
x1label Config A - Config B -- Run 1
x2label over 6 days and 5 wafers
x3label legend for wafers 138, 139, 140, 141,
142: 1, 2, 3, 4, 5
plot diff1 t wafer
x1label Config A - Config B -- Run 2
plot diff2 t wafer
reset data
Compute
average
differences
between
configuration
A and B;
standard
deviations and
t-statistics for
testing
significance
reset plot control
reset i/o
separator character @
dimension 500 rows
label size 3
read mpc633k.dat wafer probe a1 s1 b1 s2 a2 s3
b2 s4
let diff1 = a1 - b1
let diff2 = a2 - b2
let d1 = average diff1
let d2 = average diff2
let s1 = standard deviation diff1
let s2 = standard deviation diff2
let t1 = (30.)**(1/2)*(d1/s1)
let t2 = (30.)**(1/2)*(d2/s2)
. Average config A-config B; std dev difference;
t-statistic for run 1
print d1 s1 t1
. Average config A-config B; std dev difference;
t-statistic for run 2
print d2 s2 t2
separator character ;
Compute
standard
uncertainty,
effective
degrees of
freedom, t
value and
expanded
uncertainty
reset data
reset plot control
reset i/o
dimension 500 rows
label size 3
read mpc633m.dat sz a df
let c = a*sz*sz
let d = c*c
let e = d/(df)
let sume = sum e
let u = sum c
let u = u**(1/2)
let effdf=(u**4)/sume
let tvalue=tppf(.975,effdf)
let expu=tvalue*u
.
. uncertainty, effective degrees of freedom,
tvalue and
. expanded uncertainty
print u effdf tvalue expu
2.6.4. Evaluation of type B uncertainty and propagation of error

2.6. Case studies
2.6.4. Evaluation of type B uncertainty and
Focus of this
case study
The purpose of this case study is to demonstrate uncertainty
analysis using statistical techniques coupled with type B analyses
and propagation of error. It is a continuation of the case study of
type A uncertainties.
Background -
description of
measurements
and
constraints
The measurements in question are volume resistivities (ohm.cm)
of silicon wafers which have the following definition:
= Xo
.
K
a
.
F
t

.
t
.
F
t/s
with explanations of the quantities and their nominal values
shown below:
= resistivity = 0.00128 ohm.cm
X = voltage/current (ohm)
t = thickness
wafer
(cm) = 0.628 cm
K
a
= factor
electrical
= 4.50 ohm.cm
F
F
= correction
temp

F
t/s
= factor
thickness/separation
1.0
Type A
evaluations
The resistivity measurements, discussed in the case study of type
A evaluations, were replicated to cover the following sources of
uncertainty in the measurement process, and the associated
uncertainties are reported in units of resistivity (ohm.cm).
Repeatability of measurements at the center of the wafer
Day-to-day effects
Run-to-run effects
Bias due to wiring configuration
Need for
propagation
of error
Not all factors could be replicated during the gauge experiment.
Wafer thickness and measurements required for the scale
corrections were measured off-line. Thus, the type B evaluation
of uncertainty is computed using propagation of error. The
propagation of error formula in units of resistivity is as follows:
Standard
deviations for
type B
evaluations
Standard deviations for the type B components are summarized
here. For a complete explanation, see the publication (Ehrstein
and Croarkin).
Electrical
measurements
There are two basic sources of uncertainty for the electrical
measurements. The first is the least-count of the digital volt
meter in the measurement of X with a maximum bound of
a = 0.0000534 ohm
which is assumed to be the half-width of a uniform distribution.
The second is the uncertainty of the electrical scale factor. This
has two sources of uncertainty:
1. error in the solution of the transcendental equation for
determining the factor
2. errors in measured voltages
The maximum bounds to these errors are assumed to be half-
widths of
a = 0.0001 ohm.cm and a = 0.00038 ohm.cm
respectively, from uniform distributions. The corresponding
standard deviations are shown below.
s
x
= 0.0000534/ = 0.0000308 ohm
Thickness The standard deviation for thickness, t, accounts for two sources
of uncertainty:
1. calibration of the thickness measuring tool with precision
gauge blocks
2. variation in thicknesses of the silicon wafers
The maximum bounds to these errors are assumed to be half-
widths of
a = 0.000015 cm and a = 0.000001 cm
respectively, from uniform distributions. Thus, the standard
deviation for thickness is
Temperature
correction
The standard deviation for the temperature correction is
calculated from its defining equation as shown below. Thus, the
standard deviation for the correction is the standard deviation
associated with the measurement of temperature multiplied by
the temperature coefficient, C(t) = 0.0083. The maximum
bound to the error of the temperature measurement is assumed to
be the half-width
a = 0.13 C
of a triangular distribution. Thus the standard deviation of the
correction for
is
Thickness
scale factor
The standard deviation for the thickness scale factor is negligible.
Associated
sensitivity
coefficients
Sensitivity coefficients for translating the standard deviations for
the type B components into units of resistivity (ohm.cm) from
the propagation of error equation are listed below and in the
error budget. The sensitivity coefficient for a source is the
multiplicative factor associated with the standard deviation in the
formula above; i.e., the partial derivative with respect to that
variable from the propagation of error equation.
a
6
= ( /X) = 100/0.111 = 900.901
a
7
= ( /K
a
) = 100/4.50 = 22.222
a
8
= ( /t) = 100/0.628 = 159.24
a
9
= ( /F
T
) = 100
a
10
= ( /F
t/S
) = 100
Sensitivity
coefficients
and degrees
of freedom
Sensitivity coefficients for the type A components are shown in
the case study of type A uncertainty analysis and repeated below.
Degrees of freedom for type B uncertainties based on assumed
distributions, according to the convention, are assumed to be
infinite.
Error budget
showing
The error budget showing sensitivity coefficients for computing
the relative standard uncertainty of volume resistivity (ohm.cm)
sensitivity
coefficients,
standard
deviations
and degrees
of freedom
with degrees of freedom is outlined below.
Error budget for volume resistivity (ohm.cm)
Source Type Sensitivity
Standard
Deviation DF
Repeatability A a
1
= 0 0.0729 300
Reproducibility A
a
2
=
0.0362 50
Run-to-run A a
3
= 1 0.0197 5
Probe #2362 A
a
4
=
0.0162 5
Wiring
Configuration A
A a
5
= 1 0 --
Resistance
ratio
B a
6
= 900.901 0.0000308
Electrical
scale
B a
7
= 22.222 0.000227
Thickness B a
8
= 159.20 0.00000868
Temperature
correction
B a
9
= 100 0.000441
Thickness
scale
B a
10
= 100 0 --
Standard
uncertainty
The standard uncertainty is computed as:
Approximate
degrees of
freedom and
expanded
uncertainty
The degrees of freedom associated with u are approximated by
the Welch-Satterthwaite formula as:
This calculation is not affected by components with infinite
degrees of freedom, and therefore, the degrees of freedom for the
standard uncertainty is the same as the degrees of freedom for the
type A uncertainty. The critical value at the 0.05 significance
level with 42 degrees of freedom, from the t-table, is 2.018 so
the expanded uncertainty is
U = 2.018 u = 0.13 ohm.cm
2.7. References

2.7. References
Degrees of
freedom
K. A. Brownlee (1960). Statistical Theory and
Methodology in Science and Engineering, John Wiley &
Sons, Inc., New York, p. 236.
Calibration
designs
J. M. Cameron, M. C. Croarkin and R. C. Raybold (1977).
Designs for the Calibration of Standards of Mass, NBS
Technical Note 952, U.S. Dept. Commerce, 58 pages.
Calibration
designs for
eliminating
drift
J. M. Cameron and G. E. Hailes (1974). Designs for the
Calibration of Small Groups of Standards in the
Presence of Drift, Technical Note 844, U.S. Dept.
Commerce, 31 pages.
Measurement
assurance for
measurements
on ICs
Carroll Croarkin and Ruth Varner (1982). Measurement
Assurance for Dimensional Measurements on
I ntegrated-circuit Photomasks, NBS Technical Note
1164, U.S. Dept. Commerce, 44 pages.
Calibration
designs for
gauge blocks
Ted Doiron (1993). Drift Eliminating Designs for Non-
Simultaneous Comparison Calibrations, J Research
National Institute of Standards and Technology, 98, pp.
217-224.
Type A & B
uncertainty
analyses for
resistivities
J. R. Ehrstein and M. C. Croarkin (1998). Standard
Reference Materials: The Certification of 100 mm
Diameter Silicon Resistivity SRMs 2541 through 2547
Using Dual-Configuration Four-Point Probe
Measurements, NIST Special Publication 260-131,
Revised, 84 pages.
Calibration
designs for
electrical
standards
W. G. Eicke and J. M. Cameron (1967). Designs for
Surveillance of the Volt Maintained By a Group of
Saturated Standard Cells, NBS Technical Note 430, U.S.
Dept. Commerce 19 pages.
Theory of
uncertainty
analysis
Churchill Eisenhart (1962). Realistic Evaluation of the
Precision and Accuracy of I nstrument Calibration
SystemsJ Research National Bureau of Standards-C.
Engineering and Instrumentation, Vol. 67C, No.2, p. 161-
187.
Confidence,
prediction, and
Gerald J. Hahn and William Q. Meeker (1991). Statistical
I ntervals: A Guide for Practitioners, John Wiley & Sons,
2.7. References
tolerance
intervals
Inc., New York.
Original
calibration
designs for
weighings
J. A. Hayford (1893). On the Least Square Adjustment of
Weighings, U.S. Coast and Geodetic Survey Appendix 10,
Report for 1892.
Uncertainties
for values from
a calibration
curve
Thomas E. Hockersmith and Harry H. Ku (1993).
Uncertainties associated with proving ring calibrations,
NBS Special Publication 300: Precision Measurement and
Calibration, Statistical Concepts and Procedures, Vol. 1,
pp. 257-263, H. H. Ku, editor.
EWMA control
charts
J. Stuart Hunter (1986). The Exponentially Weighted
Moving Average, J Quality Technology, Vol. 18, No. 4,
pp. 203-207.
Fundamentals
of mass
metrology
K. B. Jaeger and R. S. Davis (1984). A Primer for Mass
Metrology, NBS Special Publication 700-1, 85 pages.
Fundamentals
of propagation
of error
Harry Ku (1966). Notes on the Use of Propagation of
Error Formulas, J Research of National Bureau of
Standards-C. Engineering and Instrumentation, Vol. 70C,
No.4, pp. 263-273.
Handbook of
statistical
methods
Mary Gibbons Natrella (1963). Experimental Statistics,
NBS Handbook 91, US Deptartment of Commerce.
Omnitab Sally T. Peavy, Shirley G. Bremer, Ruth N. Varner, David
Hogben (1986). OMNI TAB 80: An I nterpretive System
for Statistical and Numerical Data Analysis, NBS
Special Publication 701, US Deptartment of Commerce.
Uncertainties
for
uncorrected
bias
Steve D. Phillips and Keith R. Eberhardt (1997).
Guidelines for Expressing the Uncertainty of
Measurement Results Containing Uncorrected Bias,
NIST Journal of Research, Vol. 102, No. 5.
Calibration of
roundness
artifacts
Charles P. Reeve (1979). Calibration designs for
roundness standards, NBSIR 79-1758, 21 pages.
Calibration
designs for
angle blocks
Charles P. Reeve (1967). The Calibration of Angle
Blocks by Comparison, NBSIR 80-19767, 24 pages.
SI units Barry N. Taylor (1991). I nterpretation of the SI for the
United States and Metric Conversion Policy for Federal
Agencies, NIST Special Publication 841, U.S.
2.7. References
Deptartment of Commerce.
Uncertainties
for calibrated
values
Raymond Turgel and Dominic Vecchia (1987). Precision
Calibration of Phase Meters, IEEE Transactions on
Instrumentation and Measurement, Vol. IM-36, No. 4., pp.
918-922.
Example of
propagation of
error for flow
measurements
James R. Whetstone et al. (1989). Measurements of
Coefficients of Discharge for Concentric Flange-Tapped
Square-Edged Orifice Meters in Water Over the
Reynolds Number Range 600 to 2,700,000, NIST
Technical Note 1264. pp. 97.
Mathematica
software
Stephen Wolfram (1993). Mathematica, A System of
Doing Mathematics by Computer, 2nd edition, Addison-
Wesley Publishing Co., New York.
Restrained
least squares
Marvin Zelen (1962). "Linear Estimation and Related
Topics" in Survey of Numerical Analysis edited by John
Todd, McGraw-Hill Book Co. Inc., New York, pp. 558-
577.
ASTM F84 for
resistivity
ASTM Method F84-93, Standard Test Method for
Measuring Resistivity of Silicon Wafers With an I n-line
Four-Point Probe. Annual Book of ASTM Standards,
10.05, West Conshohocken, PA 19428.
ASTM E691
for
interlaboratory
testing
ASTM Method E691-92, Standard Practice for
Conducting an I nterlaboratory Study to Determine the
Precision of a Test Method. Annual Book of ASTM
Standards, 10.05, West Conshohocken, PA 19428.
Guide to
uncertainty
analysis
Guide to the Expression of Uncertainty of Measurement
(1993). ISBN 91-67-10188-9, 1st ed. ISO, Case postale
56, CH-1211, Genve 20, Switzerland, 101 pages.
ISO 5725 for
interlaboratory
testing
I SO 5725: 1997. Accuracy (trueness and precision) of
measurement results, Part 2: Basic method for
repeatability and reproducibility of a standard
measurement method, ISO, Case postale 56, CH-1211,
Genve 20, Switzerland.
ISO 11095 on
linear
calibration
I SO 11095: 1997. Linear Calibration using Reference
Materials, ISO, Case postale 56, CH-1211, Genve 20,
Switzerland.
MSA gauge
studies manual
Measurement Systems Analysis Reference Manual, 2nd
ed., (1995). Chrysler Corp., Ford Motor Corp., General
Motors Corp., 120 pages.
NCSL RP on
uncertainty
Determining and Reporting Measurement Uncertainties,
National Conference of Standards Laboratories RP-12,
2.7. References
analysis (1994), Suite 305B, 1800 30th St., Boulder, CO 80301.
ISO
Vocabulary for
metrology
I nternational Vocabulary of Basic and General Terms in
Metrology, 2nd ed., (1993). ISO, Case postale 56, CH-
1211, Genve 20, Switzerland, 59 pages.
Exact variance
for length and
width
Leo Goodman (1960). "On the Exact Variance of
Products" in Journal of the American Statistical
Association, December, 1960, pp. 708-713.
3. Production Process Characterization
http://www.itl.nist.gov/div898/handbook/ppc/ppc.htm[6/27/2012 2:11:14 PM]

The goal of this chapter is to learn how to plan and conduct a Production
Process Characterization Study (PPC) on manufacturing processes. We will
learn how to model manufacturing processes and use these models to design
a data collection scheme and to guide data analysis activities. We will look
in detail at how to analyze the data collected in characterization studies and
how to interpret and report the results. The accompanying Case Studies
provide detailed examples of several process characterization studies.
1. Introduction
1. Definition
2. Uses
3. Terminology/Concepts
4. PPC Steps
2. Assumptions
1. General Assumptions
2. Specific PPC Models
3. Data Collection
1. Set Goals
2. Model the Process
3. Define Sampling Plan
4. Analysis
1. First Steps
2. Exploring Relationships
3. Model Building
4. Variance Components
5. Process Stability
6. Process Capability
7. Checking Assumptions
5. Case Studies
1. Furnace Case Study
2. Machine Case Study
Detailed Chapter Table of Contents
References
http://www.itl.nist.gov/div898/handbook/ppc/ppc_d.htm[6/27/2012 2:10:10 PM]

3. Production Process Characterization - Detailed Table of
Contents [3.]
1. Introduction to Production Process Characterization [3.1.]
1. What is PPC? [3.1.1.]
2. What are PPC Studies Used For? [3.1.2.]
3. Terminology/Concepts [3.1.3.]
1. Distribution (Location, Spread and Shape) [3.1.3.1.]
2. Process Variability [3.1.3.2.]
1. Controlled/Uncontrolled Variation [3.1.3.2.1.]
3. Propagating Error [3.1.3.3.]
4. Populations and Sampling [3.1.3.4.]
5. Process Models [3.1.3.5.]
6. Experiments and Experimental Design [3.1.3.6.]
4. PPC Steps [3.1.4.]
2. Assumptions / Prerequisites [3.2.]
1. General Assumptions [3.2.1.]
2. Continuous Linear Model [3.2.2.]
3. Analysis of Variance Models (ANOVA) [3.2.3.]
1. One-Way ANOVA [3.2.3.1.]
1. One-Way Value-Splitting [3.2.3.1.1.]
2. Two-Way Crossed ANOVA [3.2.3.2.]
1. Two-way Crossed Value-Splitting Example [3.2.3.2.1.]
3. Two-Way Nested ANOVA [3.2.3.3.]
1. Two-Way Nested Value-Splitting Example [3.2.3.3.1.]
4. Discrete Models [3.2.4.]
3. Data Collection for PPC [3.3.]
1. Define Goals [3.3.1.]
2. Process Modeling [3.3.2.]
3. Define Sampling Plan [3.3.3.]
1. Identifying Parameters, Ranges and Resolution [3.3.3.1.]
2. Choosing a Sampling Scheme [3.3.3.2.]
3. Selecting Sample Sizes [3.3.3.3.]
4. Data Storage and Retrieval [3.3.3.4.]
5. Assign Roles and Responsibilities [3.3.3.5.]
4. Data Analysis for PPC [3.4.]
1. First Steps [3.4.1.]
2. Exploring Relationships [3.4.2.]
1. Response Correlations [3.4.2.1.]
2. Exploring Main Effects [3.4.2.2.]
http://www.itl.nist.gov/div898/handbook/ppc/ppc_d.htm[6/27/2012 2:10:10 PM]
3. Exploring First Order Interactions [3.4.2.3.]
3. Building Models [3.4.3.]
1. Fitting Polynomial Models [3.4.3.1.]
2. Fitting Physical Models [3.4.3.2.]
4. Analyzing Variance Structure [3.4.4.]
5. Assessing Process Stability [3.4.5.]
6. Assessing Process Capability [3.4.6.]
7. Checking Assumptions [3.4.7.]
5. Case Studies [3.5.]
1. Furnace Case Study [3.5.1.]
1. Background and Data [3.5.1.1.]
2. Initial Analysis of Response Variable [3.5.1.2.]
3. Identify Sources of Variation [3.5.1.3.]
4. Analysis of Variance [3.5.1.4.]
5. Final Conclusions [3.5.1.5.]
6. Work This Example Yourself [3.5.1.6.]
2. Machine Screw Case Study [3.5.2.]
2. Box Plots by Factors [3.5.2.2.]
3. Analysis of Variance [3.5.2.3.]
4. Throughput [3.5.2.4.]
5. Final Conclusions [3.5.2.5.]
3.1. Introduction to Production Process Characterization
http://www.itl.nist.gov/div898/handbook/ppc/section1/ppc1.htm[6/27/2012 2:10:17 PM]

3.1. Introduction to Production Process
Characterization
Overview
Section
The goal of this section is to provide an introduction to PPC.
We will define PPC and the terminology used and discuss
some of the possible uses of a PPC study. Finally, we will
look at the steps involved in designing and executing a PPC
study.
Contents:
Section 1
1. What is PPC?
2. What are PPC studies used for?
3. What terminology is used in PPC?
1. Location, Spread and Shape
2. Process Variability
3. Propagating Error
4. Populations and Sampling
5. Process Models
6. Experiments and Experimental Design
4. What are the steps of a PPC?
1. Plan PPC
2. Collect Data
3. Analyze and Interpret Data
4. Report Conclusions
3.1.1. What is PPC?

3.1.1. What is PPC?
In PPC,
we build
data-
based
models
Process characterization is an activity in which we:
identify the key inputs and outputs of a process
collect data on their behavior over the entire operating
range
estimate the steady-state behavior at optimal operating
conditions
and build models describing the parameter relationships
across the operating range
The result of this activity is a set of mathematical process
models that we can use to monitor and improve the process.
This is a
three-step
process
This activity is typically a three-step process.
The Screening Step
In this phase we identify all possible significant process
inputs and outputs and conduct a series of screening
experiments in order to reduce that list to the key inputs
and outputs. These experiments will also allow us to
develop initial models of the relationships between those
inputs and outputs.
The Mapping Step
In this step we map the behavior of the key outputs over
their expected operating ranges. We do this through a
series of more detailed experiments called Response
Surface experiments.
The Passive Step
In this step we allow the process to run at nominal
conditions and estimate the process stability and
capability.
Not all of
the steps
need to be
performed
The first two steps are only needed for new processes or when
the process has undergone some significant engineering
change. There are, however, many times throughout the life
of a process when the third step is needed. Examples might
be: initial process qualification, control chart development,
after minor process adjustments, after schedule equipment
maintenance, etc.
3.1.1. What is PPC?
3.1.2. What are PPC Studies Used For?

3.1.2. What are PPC Studies Used For?
PPC is the core
of any CI
program
Process characterization is an integral part of any
continuous improvement program. There are many steps
in that program for which process characterization is
required. These might include:
When process
characterization
is required
when we are bringing a new process or tool into
use.
when we are bringing a tool or process back up
after scheduled/unscheduled maintenance.
when we want to compare tools or processes.
when we want to check the health of our process
during the monitoring phase.
when we are troubleshooting a bad process.
The techniques described in this chapter are equally
applicable to the other chapters covered in this
Handbook. These include:
Process
characterization
techniques are
applicable in
other areas
calibration
process monitoring
process improvement
process/product comparison
reliability
3.1.3. Terminology/Concepts

There are just a few fundamental concepts needed
for PPC. This section will review these ideas briefly
and provide links to other sections in the Handbook
where they are covered in more detail.
Distribution(location,
spread, shape)
For basic data analysis, we will need to understand
how to estimate location, spread and shape from the
data. These three measures comprise what is known
as the distribution of the data. We will look at both
graphical and numerical techniques.
Process variability We need to thoroughly understand the concept of
process variability. This includes how variation
explains the possible range of expected data values,
the various classifications of variability, and the role
that variability plays in process stability and
capability.
Error propagation We also need to understand how variation
propagates through our manufacturing processes
and how to decompose the total observed variation
into components attributable to the contributing
sources.
Populations and
sampling
It is important to have an understanding of the
various issues related to sampling. We will define a
population and discuss how to acquire
representative random samples from the population
of interest. We will also discuss a useful formula
for estimating the number of observations required
to answer specific questions.
Modeling For modeling, we will need to know how to identify
important factors and responses. We will also need
to know how to graphically and quantitatively build
models of the relationships between the factors and
responses.
Experiments Finally, we will need to know about the basics of
designed experiments including screening designs
and response surface designs so that we can
quantify these relationships. This topic will receive
only a cursory treatment in this chapter. It is
covered in detail in the process improvement
chapter. However, examples of its use are in the
case studies.
3.1.3.1. Distribution (Location, Spread and Shape)

3.1.3.1. Distribution (Location, Spread and
Shape)
Distributions
are
characterized
by location,
spread and
shape
A fundamental concept in representing any of the outputs
from a production process is that of a distribution.
Distributions arise because any manufacturing process
output will not yield the same value every time it is
measured. There will be a natural scattering of the
measured values about some central tendency value. This
scattering about a central value is known as a distribution.
A distribution is characterized by three values:
Location
The location is the expected value of the output being
measured. For a stable process, this is the value
around which the process has stabilized.
Spread
The spread is the expected amount of variation
associated with the output. This tells us the range of
possible values that we would expect to see.
Shape
The shape shows how the variation is distributed
about the location. This tells us if our variation is
symmetric about the mean or if it is skewed or
possibly multimodal.
A primary
goal of PPC
is to estimate
the
distributions
of the
process
outputs
One of the primary goals of a PPC study is to characterize
our process outputs in terms of these three measurements. If
we can demonstrate that our process is stabilized about a
constant location, with a constant variance and a known
stable shape, then we have a process that is both predictable
and controllable. This is required before we can set up
control charts or conduct experiments.
Click on
each item to
read more
detail
The table below shows the most common numerical and
graphical measures of location, spread and shape.
Parameter Numerical Graphical
Location
mean
median
scatter plot
boxplot
3.1.3.1. Distribution (Location, Spread and Shape)
histogram
Spread
variance
range
inter-quartile range
boxplot
histogram
Shape
skewness
kurtosis
boxplot
histogram
probability plot
3.1.3.2. Process Variability

Variability
is present
everywhere
All manufacturing and measurement processes exhibit variation. For example, when
we take sample data on the output of a process, such as critical dimensions, oxide
thickness, or resistivity, we observe that all the values are NOT the same. This results
in a collection of observed values distributed about some location value. This is what
we call spread or variability. We represent variability numerically with the variance
calculation and graphically with a histogram.
How does
the
standard
deviation
describe the
spread of
the data?
The standard deviation (square root of the variance) gives insight into the spread of the
data through the use of what is known as the Empirical Rule. This rule (shown in the
graph below) is:
Approximately 60-78% of the data are within a distance of one standard deviation
from the average .
Approximately 90-98% of the data are within a distance of two standard deviations
from the average .
More than 99% of the data are within a distance of three standard deviations from the
average
Variability
accumulates
from many
sources
This observed variability is an accumulation of many different sources of variation that
have occurred throughout the manufacturing process. One of the more important
activities of process characterization is to identify and quantify these various sources
of variation so that they may be minimized.
There are
also
different
types
There are not only different sources of variation, but there are also different types of
variation. Two important classifications of variation for the purposes of PPC are
controlled variation and uncontrolled variation.
Click here
to see
examples
CONTROLLED VARIATION
Variation that is characterized by a stable and consistent pattern of variation
over time. This type of variation will be random in nature and will be exhibited
by a uniform fluctuation about a constant level.
UNCONTROLLED VARIATION
Variation that is characterized by a pattern of variation that changes over time
and hence is unpredictable. This type of variation will typically contain some
structure.
Stable
processes
only exhibit
controlled
variation
This concept of controlled/uncontrolled variation is important in determining if a
process is stable. A process is deemed stable if it runs in a consistent and predictable
manner. This means that the average process value is constant and the variability is
controlled. If the variation is uncontrolled, then either the process average is changing
or the process variation is changing or both. The first process in the example above is
stable; the second is not.
In the course of process characterization we should endeavor to eliminate all sources
of uncontrolled variation.
3.1.3.2.1. Controlled/Uncontrolled Variation

Two trend
plots
The two figures below are two trend plots from two different oxide
growth processes. Thirty wafers were sampled from each process: one
per day over 30 days. Thickness at the center was measured on each
wafer. The x-axis of each graph is the wafer number and the y-axis is the
film thickness in angstroms.
Examples
of"in
control" and
"out of
control"
processes
The first process is an example of a process that is "in control" with
random fluctuation about a process location of approximately 990. The
second process is an example of a process that is "out of control" with a
process location trending upward after observation 20.
This process
exhibits
controlled
variation.
Note the
random
fluctuation
about a
constant
mean.
This process
exhibits
uncontrolled
variation.
Note the
structure in
the
variation in
the form of
a linear
trend.
3.1.3.3. Propagating Error

3.1.3.3. Propagating Error
The
variation we
see can
come from
many
sources
When we estimate the variance at a particular process step, this
variance is typically not just a result of the current step, but rather is
an accumulation of variation from previous steps and from
measurement error. Therefore, an important question that we need
to answer in PPC is how the variation from the different sources
accumulates. This will allow us to partition the total variation and
assign the parts to the various sources. Then we can attack the
sources that contribute the most.
How do I
partition the
error?
Usually we can model the contribution of the various sources of
error to the total error through a simple linear relationship. If we
have a simple linear relationship between two variables, say,
then the variance associated with, y, is given by,
.
If the variables are not correlated, then there is no covariance and
the last term in the above equation drops off. A good example of
this is the case in which we have both process error and
measurement error. Since these are usually independent of each
other, the total observed variance is just the sum of the variances for
process and measurement. Remember to never add standard
deviations, we must add variances.
How do I
calculate the
individual
components?
Of course, we rarely have the individual components of variation
and wish to know the total variation. Usually, we have an estimate
of the overall variance and wish to break that variance down into its
individual components. This is known as components of variance
estimation and is dealt with in detail in the analysis of variance
page later in this chapter.
3.1.3.4. Populations and Sampling

3.1.3.4. Populations and Sampling
We take
samples
from a
target
population
and make
inferences
In survey sampling, if you want to know what everyone thinks
about a particular topic, you can just ask everyone and record
their answers. Depending on how you define the term,
everyone (all the adults in a town, all the males in the USA,
etc.), it may be impossible or impractical to survey everyone.
The other option is to survey a small group (Sample) of the
people whose opinions you are interested in (Target
Population) , record their opinions and use that information to
make inferences about what everyone thinks. Opinion pollsters
have developed a whole body of tools for doing just that and
many of those tools apply to manufacturing as well. We can
use these sampling techniques to take a few measurements
from a process and make statements about the behavior of that
process.
Facts
about a
sample are
not
necessarily
facts about
a
population
If it weren't for process variation we could just take one
sample and everything would be known about the target
population. Unfortunately this is never the case. We cannot
take facts about the sample to be facts about the population.
Our job is to reach appropriate conclusions about the
population despite this variation. The more observations we
take from a population, the more our sample data resembles
the population. When we have reached the point at which facts
about the sample are reasonable approximations of facts about
the population, then we say the sample is adequate.
Four
attributes
of samples
Adequacy of a sample depends on the following four
attributes:
Representativeness of the sample (is it random?)
Size of the sample
Variability in the population
Desired precision of the estimates
We will learn about choosing representative samples of
adequate size in the section on defining sampling plans.
3.1.3.5. Process Models

Black box
model and
fishbone
diagram
As we will see in Section 3 of this chapter, one of the first steps in PPC is to
model the process that is under investigation. Two very useful tools for
doing this are the black-box model and the fishbone diagram.
We use the
black-box
model to
describe
our
processes
We can use the simple black-box model, shown below, to describe most of
the tools and processes we will encounter in PPC. The process will be
stimulated by inputs. These inputs can either be controlled (such as recipe or
machine settings) or uncontrolled (such as humidity, operators, power
fluctuations, etc.). These inputs interact with our process and produce
outputs. These outputs are usually some characteristic of our process that we
can measure. The measurable inputs and outputs can be sampled in order to
observe and understand how they behave and relate to each other.
Diagram
of the
black box
model
These inputs and outputs are also known as Factors and Responses,
respectively.
Factors
Observed inputs used to explain response behavior (also called
explanatory variables). Factors may be fixed-level controlled inputs or
sampled uncontrolled inputs.
Responses
Sampled process outputs. Responses may also be functions of sampled
outputs such as average thickness or uniformity.
Factors
and
Responses
are further
classified
by
variable
type
We further categorize factors and responses according to their Variable
Type, which indicates the amount of information they contain. As the name
implies, this classification is useful for data modeling activities and is
critical for selecting the proper analysis technique. The table below
summarizes this categorization. The types are listed in order of the amount
of information they contain with Measurement containing the most
information and Nominal containing the least.
Table
describing
the
different
variable
types
Type Description Example
Measurement
discrete/continuous, order is
important, infinite range
particle count, oxide
thickness, pressure,
temperature
Ordinal
discrete, order is important,
finite range
run #, wafer #, site, bin
Nominal
discrete, no order, very few
possible values
good/bad, bin,
high/medium/low, shift,
operator

Fishbone
diagrams
help to
decompose
complexity
We can use the fishbone diagram to further refine the modeling process.
Fishbone diagrams are very useful for decomposing the complexity of our
manufacturing processes. Typically, we choose a process characteristic
(either Factors or Responses) and list out the general categories that may
influence the characteristic (such as material, machine method, environment,
etc.), and then provide more specific detail within each category. Examples
of how to do this are given in the section on Case Studies.
Sample
fishbone
diagram
3.1.3.6. Experiments and Experimental Design

Factors and
responses
Besides just observing our processes for evidence of stability
and capability, we quite often want to know about the
relationships between the various Factors and Responses.
We look for
correlations
and causal
relationships
There are generally two types of relationships that we are
interested in for purposes of PPC. They are:
Correlation
Two variables are said to be correlated if an observed
change in the level of one variable is accompanied by
a change in the level of another variable. The change
may be in the same direction (positive correlation) or
in the opposite direction (negative correlation).
Causality
There is a causal relationship between two variables if
a change in the level of one variable causes a change
in the other variable.
Note that correlation does not imply causality. It is possible
for two variables to be associated with each other without
one of them causing the observed behavior in the other.
When this is the case it is usually because there is a third
(possibly unknown) causal factor.
Our goal is
to find
causal
relationships
Generally, our ultimate goal in PPC is to find and quantify
causal relationships. Once this is done, we can then take
advantage of these relationships to improve and control our
processes.
Find
correlations
and then try
to establish
causal
relationships
Generally, we first need to find and explore correlations and
then try to establish causal relationships. It is much easier to
find correlations as these are just properties of the data. It is
much more difficult to prove causality as this additionally
requires sound engineering judgment. There is a systematic
procedure we can use to accomplish this in an efficient
manner. We do this through the use of designed
experiments.
First we
screen, then
we build
When we have many potential factors and we want to see
which ones are correlated and have the potential to be
involved in causal relationships with the responses, we use
models screening designs to reduce the number of candidates. Once
we have a reduced set of influential factors, we can use
response surface designs to model the causal relationships
with the responses across the operating range of the process
factors.
Techniques
discussed in
process
improvement
chapter
The techniques are covered in detail in the process
improvement section and will not be discussed much in this
chapter. Examples of how the techniques are used in PPC
are given in the Case Studies.
3.1.4. PPC Steps

3.1.4. PPC Steps
Follow
these 4
steps to
ensure
efficient
use of
resources
The primary activity of a PPC is to collect and analyze data
so that we may draw conclusions about and ultimately
improve our production processes. In many industrial
applications, access to production facilities for the purposes of
conducting experiments is very limited. Thus we must be
very careful in how we go about these activities so that we
can be sure of doing them in a cost-effective manner.
Step 1:
Plan
The most important step by far is the planning step. By
faithfully executing this step, we will ensure that we only
collect data in the most efficient manner possible and still
support the goals of the PPC. Planning should generate the
following:
a statement of the goals
a descriptive process model (a list of process inputs and
outputs)
a description of the sampling plan (including a
description of the procedure and settings to be used to
run the process during the study with clear assignments
for each person involved)
a description of the method of data collection, tasks and
responsibilities, formatting, and storage
an outline of the data analysis
All decisions that affect how the characterization will be
conducted should be made during the planning phase. The
process characterization should be conducted according to
this plan, with all exceptions noted.
Step 2:
Collect
Data collection is essentially just the execution of the
sampling plan part of the previous step. If a good job were
done in the planning step, then this step should be pretty
straightforward. It is important to execute to the plan as
closely as possible and to note any exceptions.
Step 3:
Analyze
and
interpret
This is the combination of quantitative (regression, ANOVA,
correlation, etc.) and graphical (histograms, scatter plots, box
plots, etc.) analysis techniques that are applied to the collected
data in order to accomplish the goals of the PPC.
Step 4: Reporting is an important step that should not be overlooked.
3.1.4. PPC Steps
Report By creating an informative report and archiving it in an
accessible place, we can ensure that others have access to the
information generated by the PPC. Often, the work involved
in a PPC can be minimized by using the results of other,
similar studies. Examples of PPC reports can be found in the
Case Studies section.
Further
information
The planning and data collection steps are described in detail
in the data collection section. The analysis and interpretation
steps are covered in detail in the analysis section. Examples
of the reporting step can be seen in the Case Studies.
3.2. Assumptions / Prerequisites

Primary
goal is to
identify
and
quantify
sources of
variation
The primary goal of PPC is to identify and quantify sources of
variation. Only by doing this will we be able to define an
effective plan for variation reduction and process
improvement. Sometimes, in order to achieve this goal, we
must first build mathematical/statistical models of our
processes. In these models we will identify influential factors
and the responses on which they have an effect. We will use
these models to understand how the sources of variation are
influenced by the important factors. This subsection will
review many of the modeling tools we have at our disposal to
accomplish these tasks. In particular, the models covered in
this section are linear models, Analysis of Variance (ANOVA)
models and discrete models.
Contents:
Section 2
1. General Assumptions
2. Continuous Linear
3. Analysis of Variance
1. One-Way
2. Crossed
3. Nested
4. Discrete
3.2.1. General Assumptions

3.2.1. General Assumptions
Assumption:
process is
sum of a
systematic
component
and a random
component
In order to employ the modeling techniques described in
this section, there are a few assumptions about the process
under study that must be made. First, we must assume that
the process can adequately be modeled as the sum of a
systematic component and a random component. The
systematic component is the mathematical model part and
the random component is the error or noise present in the
system. We also assume that the systematic component is
fixed over the range of operating conditions and that the
random component has a constant location, spread and
distributional form.
Assumption:
data used to
fit these
models are
representative
of the process
being
modeled
Finally, we assume that the data used to fit these models
are representative of the process being modeled. As a
result, we must additionally assume that the measurement
system used to collect the data has been studied and proven
to be capable of making measurements to the desired
precision and accuracy. If this is not the case, refer to the
Measurement Capability Section of this Handbook.
3.2.2. Continuous Linear Model

Description The continuous linear model (CLM) is probably the most
commonly used model in PPC. It is applicable in many instances
ranging from simple control charts to response surface models.
The CLM is a mathematical function that relates explanatory
variables (either discrete or continuous) to a single continuous
response variable. It is called linear because the coefficients of
the terms are expressed as a linear sum. The terms themselves do
not have to be linear.
Model The general form of the CLM is:
This equation just says that if we have p explanatory variables
then the response is modeled by a constant term plus a sum of
functions of those explanatory variables, plus some random error
term. This will become clear as we look at some examples below.
Estimation The coefficients for the parameters in the CLM are estimated by
the method of least squares. This is a method that gives estimates
which minimize the sum of the squared distances from the
observations to the fitted line or plane. See the chapter on Process
Modeling for a more complete discussion on estimating the
coefficients for these models.
Testing The tests for the CLM involve testing that the model as a whole is
a good representation of the process and whether any of the
coefficients in the model are zero or have no effect on the overall
fit. Again, the details for testing are given in the chapter on
Process Modeling.
Assumptions For estimation purposes, there are no additional assumptions
necessary for the CLM beyond those stated in the assumptions
section. For testing purposes, however, it is necessary to assume
that the error term is adequately modeled by a Gaussian
distribution.
Uses The CLM has many uses such as building predictive process
models over a range of process settings that exhibit linear
behavior, control charts, process capability, building models from
the data produced by designed experiments, and building response
surface models for automated process control applications.
Examples Shewhart Control Chart - The simplest example of a very
common usage of the CLM is the underlying model used for
Shewhart control charts. This model assumes that the process
parameter being measured is a constant with additive Gaussian
noise and is given by:
Diffusion Furnace - Suppose we want to model the average wafer
sheet resistance as a function of the location or zone in a furnace
tube, the temperature, and the anneal time. In this case, let there
be 3 distinct zones (front, center, back) and temperature and time
are continuous explanatory variables. This model is given by the
CLM:
Diffusion Furnace (cont.) - Usually, the fitted line for the average
wafer sheet resistance is not straight but has some curvature to it.
This can be accommodated by adding a quadratic term for the
time parameter as follows:
3.2.3. Analysis of Variance Models (ANOVA)

ANOVA
allows us
to compare
the effects
of multiple
levels of
multiple
factors
One of the most common analysis activities in PPC is
comparison. We often compare the performance of similar
tools or processes. We also compare the effect of different
treatments such as recipe settings. When we compare two
things, such as two tools running the same operation, we use
comparison techniques. When we want to compare multiple
things, like multiple tools running the same operation or
multiple tools with multiple operators running the same
operation, we turn to ANOVA techniques to perform the
analysis.
ANOVA
splits the
data into
components
The easiest way to understand ANOVA is through a concept
known as value splitting. ANOVA splits the observed data
values into components that are attributable to the different
levels of the factors. Value splitting is best explained by
example.
Example:
Turned
Pins
The simplest example of value splitting is when we just have
one level of one factor. Suppose we have a turning operation
in a machine shop where we are turning pins to a diameter of
.125 +/- .005 inches. Throughout the course of a day we take
five samples of pins and obtain the following measurements:
.125, .127, .124, .126, .128.
We can split these data values into a common value (mean)
and residuals (what's left over) as follows:
.125 .127 .124 .126 .128
=
.126 .126 .126 .126 .126
+
-.001 .001 -.002 .000 .002
From these tables, also called overlays, we can easily
calculate the location and spread of the data as follows:
mean = .126
std. deviation = .0016.
Other
layouts
While the above example is a trivial structural layout, it
illustrates how we can split data values into its components.
In the next sections, we will look at more complicated
structural layouts for the data. In particular we will look at
multiple levels of one factor (One-Way ANOVA) and
multiple levels of two factors (Two-Way ANOVA) where the
factors are crossed and nested.
3.2.3.1. One-Way ANOVA

Description A one-way layout consists of a single factor with several levels and multiple
observations at each level. With this kind of layout we can calculate the mean of the
observations within each level of our factor. The residuals will tell us about the
variation within each level. We can also average the means of each level to obtain a
grand mean. We can then look at the deviation of the mean of each level from the
grand mean to understand something about the level effects. Finally, we can compare
the variation within levels to the variation across levels. Hence the name analysis of
variance.
Model It is easy to model all of this with an equation of the form:
The equation indicates that the jth data value, from level i, is the sum of three
components: the common value (grand mean), the level effect (the deviation of each
level mean from the grand mean), and the residual (what's left over).
Estimation
click here to
see details
of one-way
value
splitting
Estimation for the one-way layout can be performed one of two ways. First, we can
calculate the total variation, within-level variation and across-level variation. These can
be summarized in a table as shown below and tests can be made to determine if the
factor levels are significant. The value splitting example illustrates the calculations
involved.
ANOVA
table for
one-way
case
In general, the ANOVA table for the one-way case is given by:
where
and
.
The row labeled, "Corr. Total", in the ANOVA table contains the corrected total sum of
squares and the associated degrees of freedom (DoF).
Level effects
must sum to
zero
The second way to estimate effects is through the use of CLM techniques. If you look at
the model above you will notice that it is in the form of a CLM. The only problem is
that the model is saturated and no unique solution exists. We overcome this problem by
applying a constraint to the model. Since the level effects are just deviations from the
grand mean, they must sum to zero. By applying the constraint that the level effects
must sum to zero, we can now obtain a unique solution to the CLM equations. Most
analysis programs will handle this for you automatically. See the chapter on Process
Modeling for a more complete discussion on estimating the coefficients for these
models.
Testing We are testing to see if the observed data support the hypothesis that the levels of the
factor are significantly different from each other. The way we do this is by comparing
the within-level variancs to the between-level variance.
If we assume that the observations within each level have the same variance, we can
calculate the variance within each level and pool these together to obtain an estimate of
the overall population variance. This works out to be the mean square of the residuals.
Similarly, if there really were no level effect, the mean square across levels would be an
estimate of the overall variance. Therefore, if there really were no level effect, these
two estimates would be just two different ways to estimate the same parameter and
should be close numerically. However, if there is a level effect, the level mean square
will be higher than the residual mean square.
It can be shown that given the assumptions about the data stated below, the ratio of the
level mean square and the residual mean square follows an F distribution with degrees
of freedom as shown in the ANOVA table. If the F
0
value is significant at a given
significance level (greater than the cut-off value in a F table), then there is a level effect
present in the data.
Assumptions For estimation purposes, we assume the data can adequately be modeled as the sum of
a deterministic component and a random component. We further assume that the fixed
(deterministic) component can be modeled as the sum of an overall mean and some
contribution from the factor level. Finally, it is assumed that the random component can
be modeled with a Gaussian distribution with fixed location and spread.
Uses The one-way ANOVA is useful when we want to compare the effect of multiple levels
of one factor and we have multiple observations at each level. The factor can be either
discrete (different machine, different plants, different shifts, etc.) or continuous
(different gas flows, temperatures, etc.).
Example Let's extend the machining example by assuming that we have five different machines
making the same part and we take five random samples from each machine to obtain the
following diameter data:
Machine
1 2 3 4 5
0.125 0.118 0.123 0.126 0.118
0.127 0.122 0.125 0.128 0.129
0.125 0.120 0.125 0.126 0.127
0.126 0.124 0.124 0.127 0.120
0.128 0.119 0.126 0.129 0.121
Analyze Using ANOVA software or the techniques of the value-splitting example, we
summarize the data in an ANOVA table as follows:
Source Sum of Squares Deg. of Freedom Mean Square
F
0
Factor 0.000137 4 0.000034 4.86
Residual 0.000132 20 0.000007
Corrected Total 0.000269 24
Test By dividing the factor-level mean square by the residual mean square, we obtain an F
0
value of 4.86 which is greater than the cut-off value of 2.87 from the F distribution with
4 and 20 degrees of freedom and a significance level of 0.05. Therefore, there is
sufficient evidence to reject the hypothesis that the levels are all the same.
Conclusion From the analysis of these data we can conclude that the factor "machine" has an effect.
There is a statistically significant difference in the pin diameters across the machines on
which they were manufactured.
3.2.3.1.1. One-Way Value-Splitting

Example Let's use the data from the machining example to illustrate
how to use the techniques of value-splitting to break each data
value into its component parts. Once we have the component
parts, it is then a trivial matter to calculate the sums of squares
and form the F-value for the test.

Machine
1 2 3 4 5
.125 .118 .123 .126 .118
.127 .122 .125 .128 .129
.125 .120 .125 .126 .127
.126 .124 .124 .127 .120
.128 .119 .126 .129 .121
Calculate
level-
means
Remember from our model, , we say each
observation is the sum of a common value, a level effect and a
residual value. Value-splitting just breaks each observation
into its component parts. The first step in value-splitting is to
calculate the mean values (rounding to the nearest thousandth)
within each machine to get the level means.
Machine
1 2 3 4 5
.1262 .1206 .1246 .1272 .123
Sweep
level
means
We can then sweep (subtract the level mean from each
associated data value) the means through the original data
table to get the residuals:
Machine
1 2 3 4 5
-
.0012
-
.0026
-
.0016
-
.0012
-
.005
.0008 .0014 .0004 .0008 .006
- - -
.0012 .0006
.0004
.0012
.004
-
.0002
.0034
-
.0006
-
.0002
-
.003
.0018
-
.0016
.0014 .0018
-
.002
Calculate
the grand
mean
The next step is to calculate the grand mean from the
individual machine means as:
Grand
Mean
.12432
Sweep the
grand
mean
through
the level
means
Finally, we can sweep the grand mean through the individual
level means to obtain the level effects:
Machine
1 2 3 4 5
.00188
-
.00372
.00028 .00288
-
.00132
It is easy to verify that the original data table can be
constructed by adding the overall mean, the machine effect
and the appropriate residual.
Calculate
ANOVA
values
Now that we have the data values split and the overlays
created, the next step is to calculate the various values in the
One-Way ANOVA table. We have three values to calculate
for each overlay. They are the sums of squares, the degrees of
freedom, and the mean squares.
Total sum
of squares
The total sum of squares is calculated by summing the squares
of all the data values and subtracting from this number the
square of the grand mean times the total number of data
values. We usually don't calculate the mean square for the
total sum of squares because we don't use this value in any
statistical test.
Residual
sum of
squares,
degrees of
freedom
and mean
The residual sum of squares is calculated by summing the
squares of the residual values. This is equal to .000132. The
degrees of freedom is the number of unconstrained values.
Since the residuals for each level of the factor must sum to
zero, once we know four of them, the last one is determined.
This means we have four unconstrained values for each level,
square or 20 degrees of freedom. This gives a mean square of
.000007.
Level sum
of squares,
degrees of
freedom
and mean
square
Finally, to obtain the sum of squares for the levels, we sum the
squares of each value in the level effect overlay and multiply
the sum by the number of observations for each level (in this
case 5) to obtain a value of .000137. Since the deviations from
the level means must sum to zero, we have only four
unconstrained values so the degrees of freedom for level
effects is 4. This produces a mean square of .000034.
Calculate
F-value
The last step is to calculate the F-value and perform the test of
equal level means. The F- value is just the level mean square
divided by the residual mean square. In this case the F-
value=4.86. If we look in an F-table for 4 and 20 degrees of
freedom at 95% confidence, we see that the critical value is
2.87, which means that we have a significant result and that
there is thus evidence of a strong machine effect. By looking
at the level-effect overlay we see that this is driven by
machines 2 and 4.
3.2.3.2. Two-Way Crossed ANOVA

Description When we have two factors with at least two levels and one or more observations at each level, we say we have a
two-way layout. We say that the two-way layout is crossed when every level of Factor A occurs with every level
of Factor B. With this kind of layout we can estimate the effect of each factor (Main Effects) as well as any
interaction between the factors.
Model If we assume that we have K observations at each combination of I levels of Factor A and J levels of Factor B,
then we can model the two-way layout with an equation of the form:
This equation just says that the kth data value for the jth level of Factor B and the ith level of Factor A is the sum
of five components: the common value (grand mean), the level effect for Factor A, the level effect for Factor B,
the interaction effect, and the residual. Note that (ab) does not mean multiplication; rather that there is interaction
between the two factors.
Estimation Like the one-way case, the estimation for the two-way layout can be done either by calculating the variance
components or by using CLM techniques.
Click here
for the
value
splitting
example
For the two-way ANOVA, we display the data in a two-dimensional table with the levels of Factor A in columns
and the levels of Factor B in rows. The replicate observations fill each cell. We can sweep out the common
value, the row effects, the column effects, the interaction effects and the residuals using value-splitting
techniques. Sums of squares can be calculated and summarized in an ANOVA table as shown below.
.
The row labeled, "Corr. Total", in the ANOVA table contains the corrected total sum of squares and the
associated degrees of freedom (DoF).
We can use CLM techniques to do the estimation. We still have the problem that the model is saturated and no
unique solution exists. We overcome this problem by applying the constraints to the model that the two main
effects and interaction effects each sum to zero.
Testing Like testing in the one-way case, we are testing that two main effects and the interaction are zero. Again we just
form a ratio of each main effect mean square and the interaction mean square to the residual mean square. If the
assumptions stated below are true then those ratios follow an F distribution and the test is performed by
comparing the F
0
ratios to values in an F table with the appropriate degrees of freedom and confidence level.
Assumptions For estimation purposes, we assume the data can be adequately modeled as described in the model above. It is
assumed that the random component can be modeled with a Gaussian distribution with fixed location and spread.
Uses The two-way crossed ANOVA is useful when we want to compare the effect of multiple levels of two factors
and we can combine every level of one factor with every level of the other factor. If we have multiple
observations at each level, then we can also estimate the effects of interaction between the two factors.
Example Let's extend the one-way machining example by assuming that we want to test if there are any differences in pin
diameters due to different types of coolant. We still have five different machines making the same part and we
take five samples from each machine for each coolant type to obtain the following data:
Machine
Coolant
A
1 2 3 4 5
0.125 0.118 0.123 0.126 0.118
0.127 0.122 0.125 0.128 0.129
0.125 0.120 0.125 0.126 0.127
0.126 0.124 0.124 0.127 0.120
0.128 0.119 0.126 0.129 0.121
Coolant
B
0.124 0.116 0.122 0.126 0.125
0.128 0.125 0.121 0.129 0.123
0.127 0.119 0.124 0.125 0.114
0.126 0.125 0.126 0.130 0.124
0.129 0.120 0.125 0.124 0.117
Analyze For analysis details see the crossed two-way value splitting example. We can summarize the analysis results in
an ANOVA table as follows:
F
0
machine 0.000303 4 0.000076 8.8
coolant 0.00000392 1 0.00000392 0.45
interaction 0.00001468 4 0.00000367 0.42
residuals 0.000346 40 0.0000087
corrected total 0.000668 49
Test By dividing the mean square for machine by the mean square for residuals we obtain an F
0
value of 8.8 which is
greater than the critical value of 2.61 based on 4 and 40 degrees of freedom and a 0.05 significance level.
Likewise the F
0
values for Coolant and Interaction, obtained by dividing their mean squares by the residual mean
square, are less than their respective critical values of 4.08 and 2.61 (0.05 significance level).
Conclusion From the ANOVA table we can conclude that machine is the most important factor and is statistically
significant. Coolant is not significant and neither is the interaction. These results would lead us to believe that
some tool-matching efforts would be useful for improving this process.
3.2.3.2.1. Two-way Crossed Value-Splitting Example

3.2.3.2.1. Two-way Crossed Value-Splitting
Example
Example:
Coolant is
completely
crossed
with
machine
The data table below is five samples each collected from five
different lathes each running two different types of coolant.
The measurement is the diameter of a turned pin.
Machine
Coolant
A
1 2 3 4 5
.125 .118 .123 .126 .118
.127 .122 .125 .128 .129
.125 .120 .125 .126 .127
.126 .124 .124 .127 .120
.128 .119 .126 .129 .121
Coolant
B
.124 .116 .122 .126 .125
.128 .125 .121 .129 .123
.127 .119 .124 .125 .114
.126 .125 .126 .130 .124
.129 .120 .125 .124 .117
For the crossed two-way case, the first thing we need to do is
to sweep the cell means from the data table to obtain the
residual values. This is shown in the tables below.
The first
step is to
sweep out
the cell
means to
obtain the
residuals
and means
Machine
1 2 3 4 5
A .1262 .1206 .1246 .1272 .123
B .1268 .121 .1236 .1268 .1206
Coolant
A
-
.0012
-
.0026
-
.0016
-
.0012
-.005
.0008 .0014 .0004 .0008 .006
-
.0012
-
.0006
.0004
-
.0012
.004
-
.0002
.0034
-
.0006
-
.0002
-.003
.0018
-
.0016
.0014 .0018 -.002
Coolant
B
-
.0028
-.005
-
.0016
-
.0008
.0044
.0012 .004
-
.0026
.0022 .0024
.0002 -.002 .0004
-
.0018
-
.0066
-
.0008
.004 .0024 .0032 .0034
.0022 -.001 .0014
-
.0028
-
.0036
Sweep the
row means
The next step is to sweep out the row means. This gives the
table below.
Machine
1 2 3 4 5
A .1243 .0019
-
.0037
.0003 .0029
-
.0013
B .1238 .003
-
.0028
-
.0002
.003
-
.0032
Sweep the
column
means
Finally, we sweep the column means to obtain the grand mean,
row (coolant) effects, column (machine) effects and the
interaction effects.
Machine
1 2 3 4 5
.1241 .0025
-
.0033
.00005 .003
-
.0023
A .0003
-
.0006
-
.0005
.00025 .0000 .001
B
-
.0003
.0006 .0005
-
.00025
.0000 -.001
What do
these
tables tell
us?
By looking at the table of residuals, we see that the residuals
for coolant B tend to be a little higher than for coolant A. This
implies that there may be more variability in diameter when
we use coolant B. From the effects table above, we see that
machines 2 and 5 produce smaller pin diameters than the other
machines. There is also a very slight coolant effect but the
machine effect is larger. Finally, there also appears to be slight
interaction effects. For instance, machines 1 and 2 had smaller
diameters with coolant A but the opposite was true for
machines 3,4 and 5.
Calculate We can calculate the values for the ANOVA table according
sums of
squares
and mean
squares
to the formulae in the table on the crossed two-way page. This
gives the table below. From the F-values we see that the
machine effect is significant but the coolant and the
interaction are not.
Source
Sums of
Squares
Degrees of
Freedom
Mean
Square
F-
value
Machine .000303 4 .000076
8.8 >
2.61
Coolant .00000392 1 .00000392
.45 <
4.08
Interaction .00001468 4 .00000367
.42 <
2.61
Residual .000346 40 .0000087
Corrected
Total
.000668 49
3.2.3.3. Two-Way Nested ANOVA

Description Sometimes, constraints prevent us from crossing every level of one factor with every level of the
other factor. In these cases we are forced into what is known as a nested layout. We say we have
a nested layout when fewer than all levels of one factor occur within each level of the other
factor. An example of this might be if we want to study the effects of different machines and
different operators on some output characteristic, but we can't have the operators change the
machines they run. In this case, each operator is not crossed with each machine but rather only
runs one machine.
Model If Factor B is nested within Factor A, then a level of Factor B can only occur within one level of
Factor A and there can be no interaction. This gives the following model:
This equation indicates that each data value is the sum of a common value (grand mean), the
level effect for Factor A, the level effect of Factor B nested within Factor A, and the residual.
Estimation For a nested design we typically use variance components methods to perform the analysis. We
can sweep out the common value, the Factor A effects, the Factor B within A effects and the
residuals using value-splitting techniques. Sums of squares can be calculated and summarized in
an ANOVA table as shown below.
Click here
for nested
value-
splitting
example
It is important to note that with this type of layout, since each level of one factor is only present
with one level of the other factor, we can't estimate interaction between the two.
ANOVA
table for
nested case
The row labeled, "Corr. Total", in the ANOVA table contains the corrected total sum of squares
and the associated degrees of freedom (DoF).
As with the crossed layout, we can also use CLM techniques. We still have the problem that the
model is saturated and no unique solution exists. We overcome this problem by applying to the
model the constraints that the two main effects sum to zero.
Testing We are testing that two main effects are zero. Again we just form a ratio (F
0
) of each main effect
mean square to the appropriate mean-squared error term. (Note that the error term for Factor A is
not MSE, but is MSB.) If the assumptions stated below are true then those ratios follow an F
distribution and the test is performed by comparing the F
0
ratios to values in an F table with the
appropriate degrees of freedom and confidence level.
Assumptions For estimation purposes, we assume the data can be adequately modeled by the model above and
that there is more than one variance component. It is assumed that the random component can be
modeled with a Gaussian distribution with fixed location and spread.
Uses The two-way nested ANOVA is useful when we are constrained from combining all the levels of
one factor with all of the levels of the other factor. These designs are most useful when we have
what is called a random effects situation. When the levels of a factor are chosen at random rather
than selected intentionally, we say we have a random effects model. An example of this is when
we select lots from a production run, then select units from the lot. Here the units are nested
within lots and the effect of each factor is random.
Example Let's change the two-way machining example slightly by assuming that we have five different
machines making the same part and each machine has two operators, one for the day shift and
one for the night shift. We take five samples from each machine for each operator to obtain the
following data:
Machine
Operator
Day
1 2 3 4 5
0.125 0.118 0.123 0.126 0.118
0.127 0.122 0.125 0.128 0.129
0.125 0.120 0.125 0.126 0.127
0.126 0.124 0.124 0.127 0.120
0.128 0.119 0.126 0.129 0.121
Operator
Night
0.124 0.116 0.122 0.126 0.125
0.128 0.125 0.121 0.129 0.123
0.127 0.119 0.124 0.125 0.114
0.126 0.125 0.126 0.130 0.124
0.129 0.120 0.125 0.124 0.117
Analyze For analysis details see the nested two-way value splitting example. We can summarize the
analysis results in an ANOVA table as follows:
F
0
Machine 3.03e-4 4 7.58e-5 20.38
Operator(Machine) 1.86e-5 5 3.72e-6 0.428
Residuals 3.46e-4 40 8.70e-6
Corrected Total 6.68e-4 49
Test By dividing the mean square for Machine by the mean square for Operator within Machine, or
Operator(Machine), we obtain an F
0
value of 20.38 which is greater than the critical value of
5.19 for 4 and 5 degrees of freedom at the 0.05 significance level. The F
0
value for
Operator(Machine), obtained by dividing its mean square by the residual mean square, is less than
the critical value of 2.45 for 5 and 40 degrees of freedom at the 0.05 significance level.
Conclusion From the ANOVA table we can conclude that the Machine is the most important factor and is
statistically significant. The effect of Operator nested within Machine is not statistically
significant. Again, any improvement activities should be focused on the tools.
3.2.3.3.1. Two-Way Nested Value-Splitting Example

Example:
Operator
is nested
within
machine.
The data table below contains data collected from five different lathes, each run by two
different operators. Note we are concerned here with the effect of operators, so the
layout is nested. If we were concerned with shift instead of operator, the layout would
be crossed. The measurement is the diameter of a turned pin.
Machine Operator
Sample
1 2 3 4 5
1
Day .125 .127 .125 .126 .128
Night .124 .128 .127 .126 .129
2
Day .118 .122 .120 .124 .119
Night .116 .125 .119 .125 .120
3
Day .123 .125 .125 .124 .126
Night .122 .121 .124 .126 .125
4
Day .126 .128 .126 .127 .129
Night .126 .129 .125 .130 .124
5
Day .118 .129 .127 .120 .121
Night .125 .123 .114 .124 .117
For the nested two-way case, just as in the crossed case, the first thing we need to do is
to sweep the cell means from the data table to obtain the residual values. We then
sweep the nested factor (Operator) and the top level factor (Machine) to obtain the
table below.
Machine Operator
Common Machine Operator
Sample
1 2 3 4 5
1
Day
.12404
.00246
-.0003
-
.0012
.0008
-
.0012
-
.0002
.0018
Night .0003
-
.0028
.0012 .002
-
.0008
.0022
2
Day
-.00324
-.0002
-
.0026
.0014
-
.0006
.0034
-
.0016
Night .0002 -.005 .004 -.002 .004 -.001
3
Day
.00006
.0005
-
.0016
.0004 .0004
-
.0006
.0014
Night -.0005
-
.0016
-
.0026
.0004 .0024 .0014
4
Day
.00296
.0002
-
.0012
.0008
-
.0012
-.002 .0018
Night -.0002
-
.0008
.0022
-
.0018
.0032
-
.0028
5
Day
-.00224
.0012 -.005 .006 .004 -.003 -.002
Night -.0012 .0044 .0024
-
.0066
.0034
-
.0036
What
does this
table tell
us?
By looking at the residuals we see that machines 2 and 5 have the greatest variability.
There does not appear to be much of an operator effect but there is clearly a strong
machine effect.
Calculate
sums of
squares
and mean
squares
We can calculate the values for the ANOVA table according to the formulae in the
table on the nested two-way page. This produces the table below. From the F-values
we see that the machine effect is significant but the operator effect is not. (Here it is
assumed that both factors are fixed).
Source
Sums of
Squares
Degrees of
Freedom
Mean
Square
F-value
Machine .000303 4 .0000758
8.77 >
2.61
Operator(Machine) .0000186 5 .00000372
.428 <
2.45
Residual .000346 40 .0000087
Corrected Total .000668 49
3.2.4. Discrete Models

Description There are many instances when we are faced with the
analysis of discrete data rather than continuous data.
Examples of this are yield (good/bad), speed bins
(slow/fast/faster/fastest), survey results (favor/oppose), etc.
We then try to explain the discrete outcomes with some
combination of discrete and/or continuous explanatory
variables. In this situation the modeling techniques we have
learned so far (CLM and ANOVA) are no longer appropriate.
Contingency
table
analysis and
log-linear
model
There are two primary methods available for the analysis of
discrete response data. The first one applies to situations in
which we have discrete explanatory variables and discrete
responses and is known as Contingency Table Analysis. The
model for this is covered in detail in this section. The second
model applies when we have both discrete and continuous
explanatory variables and is referred to as a Log-Linear
Model. That model is beyond the scope of this Handbook,
but interested readers should refer to the reference section of
this chapter for a list of useful books on the topic.
Model Suppose we have n individuals that we classify according to
two criteria, A and B. Suppose there are r levels of criterion
A and s levels of criterion B. These responses can be
displayed in an r x s table. For example, suppose we have a
box of manufactured parts that we classify as good or bad
and whether they came from supplier 1, 2 or 3.
Now, each cell of this table will have a count of the
individuals who fall into its particular combination of
classification levels. Let's call this count N
ij
. The sum of all
of these counts will be equal to the total number of
individuals, N. Also, each row of the table will sum to N
i.
and each column will sum to N
.j
.
Under the assumption that there is no interaction between the
two classifying variables (like the number of good or bad
parts does not depend on which supplier they came from),
we can calculate the counts we would expect to see in each
cell. Let's call the expected count for any cell E
ij
. Then the
expected value for a cell is E
ij
= N
i.
* N
.j
/N . All we need to
do then is to compare the expected counts to the observed
counts. If there is a consderable difference between the
observed counts and the expected values, then the two
variables interact in some way.
Estimation The estimation is very simple. All we do is make a table of
the observed counts and then calculate the expected counts as
described above.
Testing The test is performed using a Chi-Square goodness-of-fit
test according to the following formula:
where the summation is across all of the cells in the table.
Given the assumptions stated below, this statistic has
approximately a chi-square distribution and is therefore
compared against a chi-square table with (r-1)(s-1) degrees
of freedom, with r and s as previously defined. If the value
of the test statistic is less than the chi-square value for a
given level of confidence, then the classifying variables are
declared independent, otherwise they are judged to be
dependent.
Assumptions The estimation and testing results above hold regardless of
whether the sample model is Poisson, multinomial, or
product-multinomial. The chi-square results start to break
down if the counts in any cell are small, say < 5.
Uses The contingency table method is really just a test of
interaction between discrete explanatory variables for
discrete responses. The example given below is for two
factors. The methods are equally applicable to more factors,
but as with any interaction, as you add more factors the
interpretation of the results becomes more difficult.
Example Suppose we are comparing the yield from two manufacturing
processes. We want want to know if one process has a higher
yield.
Make table
of counts
Good Bad Totals
Process A 86 14 100
Process B 80 20 100
Totals 166 34 200
Table 1. Yields for two production processes
We obtain the expected values by the formula given above.
This gives the table below.
Calculate
expected
counts
Good Bad Totals
Process A 83 17 100
Process B 83 17 100
Totals 166 34 200
Table 2. Expected values for two production processes
Calculate
chi-square
statistic and
compare to
table value
The chi-square statistic is 1.276. This is below the chi-square
value for 1 degree of freedom and 90% confidence of 2.71 .
Therefore, we conclude that there is not a (significant)
difference in process yield.
Conclusion Therefore, we conclude that there is no statistically
significant difference between the two processes.
3.3. Data Collection for PPC

Start with
careful
planning
The data collection process for PPC starts with careful
planning. The planning consists of the definition of clear and
concise goals, developing process models and devising a
sampling plan.
Many
things can
go wrong
in the data
collection
This activity of course ends without the actual collection of
the data which is usually not as straightforward as it might
appear. Many things can go wrong in the execution of the
sampling plan. The problems can be mitigated with the use of
check lists and by carefully documenting all exceptions to the
original sampling plan.
Table of
Contents
1. Set Goals
2. Modeling Processes
1. Black-Box Models
2. Fishbone Diagrams
3. Relationships and Sensitivities
3. Define the Sampling Plan
1. Identify the parameters, ranges and
resolution
2. Design sampling scheme
3. Select sample sizes
4. Design data storage formats
5. Assign roles and responsibilities
3.3.1. Define Goals

3.3.1. Define Goals
State concise
goals
The goal statement is one of the most important parts of the
characterization plan. With clearly and concisely stated
goals, the rest of the planning process falls naturally into
place.
Goals
usually
defined in
terms of key
specifications
The goals are usually defined in terms of key specifications
or manufacturing indices. We typically want to characterize
a process and compare the results against these
specifications. However, this is not always the case. We
may, for instance, just want to quantify key process
parameters and use our estimates of those parameters in
some other activity like controller design or process
improvement.
Example
goal
statements
Click on each of the links below to see Goal Statements for
each of the case studies.
1. Furnace Case Study (Goal)
2. Machine Case Study (Goal)
3.3.2. Process Modeling

Identify
influential
parameters
Process modeling begins by identifying all of the important
factors and responses. This is usually best done as a team
effort and is limited to the scope set by the goal statement.
Document
with black-
box
models
This activity is best documented in the form of a black-box
model as seen in the figure below. In this figure all of the
outputs are shown on the right and all of the controllable
inputs are shown on the left. Any inputs or factors that may be
observable but not controllable are shown on the top or
bottom.
Model
relationships
using
fishbone
diagrams
The next step is to model relationships of the previously
identified factors and responses. In this step we choose a
parameter and identify all of the other parameters that may
have an influence on it. This process is easily documented
with fishbone diagrams as illustrated in the figure below.
The influenced parameter is put on the center line and the
influential factors are listed off of the centerline and can be
grouped into major categories like Tool, Material, Work
Methods and Environment.
Document
relationships
and
sensitivities
The final step is to document all known information about
the relationships and sensitivities between the inputs and
outputs. Some of the inputs may be correlated with each
other as well as the outputs. There may be detailed
mathematical models available from other studies or the
information available may be vague such as for a machining
process we know that as the feed rate increases, the quality
of the finish decreases.
It is best to document this kind of information in a table
with all of the inputs and outputs listed both on the left
column and on the top row. Then, correlation information
can be filled in for each of the appropriate cells. See the case
studies for an example.
Examples Click on each of the links below to see the process models
for each of the case studies.
1. Case Study 1 (Process Model)
2. Case Study 2 (Process Model)
3.3.3. Define Sampling Plan

Sampling
plan is
detailed
outline of
measurements
to be taken
A sampling plan is a detailed outline of which
measurements will be taken at what times, on which
material, in what manner, and by whom. Sampling plans
should be designed in such a way that the resulting data
will contain a representative sample of the parameters of
interest and allow for all questions, as stated in the goals, to
be answered.
Steps in the
sampling plan
The steps involved in developing a sampling plan are:
1. identify the parameters to be measured, the range of
possible values, and the required resolution
2. design a sampling scheme that details how and when
samples will be taken
3. select sample sizes
4. design data storage formats
5. assign roles and responsibilities
Verify and
execute
Once the sampling plan has been developed, it can be
verified and then passed on to the responsible parties for
execution.
3.3.3.1. Identifying Parameters, Ranges and Resolution

3.3.3.1. Identifying Parameters, Ranges and
Resolution
Our goals and the models we built in the previous steps
should provide all of the information needed for selecting
parameters and determining the expected ranges and the
required measurement resolution.
Goals will
tell us what
to measure
and how
The first step is to carefully examine the goals. This will tell
you which response variables need to be sampled and how.
For instance, if our goal states that we want to determine if
an oxide film can be grown on a wafer to within 10
Angstroms of the target value with a uniformity of <2%,
then we know we have to measure the film thickness on the
wafers to an accuracy of at least +/- 3 Angstroms and we
must measure at multiple sites on the wafer in order to
calculate uniformity.
The goals and the models we build will also indicate which
explanatory variables need to be sampled and how. Since
the fishbone diagrams define the known important
relationships, these will be our best guide as to which
explanatory variables are candidates for measurement.
Ranges help
screen
outliers
Defining the expected ranges of values is useful for
screening outliers. In the machining example , we would not
expect to see many values that vary more than +/- .005"
from nominal. Therefore we know that any values that are
much beyond this interval are highly suspect and should be
remeasured.
Resolution
helps choose
measurement
equipment
Finally, the required resolution for the measurements should
be specified. This specification will help guide the choice of
metrology equipment and help define the measurement
procedures. As a rule of thumb, we would like our
measurement resolution to be at least 1/10 of our tolerance.
For the oxide growth example, this means that we want to
measure with an accuracy of 2 Angstroms. Similarly, for the
turning operation we would need to measure the diameter
within .001". This means that vernier calipers would be
adequate as the measurement device for this application.
Examples Click on each of the links below to see the parameter
3.3.3.1. Identifying Parameters, Ranges and Resolution
descriptions for each of the case studies.
1. Case Study 1 (Sampling Plan)
3.3.3.2. Choosing a Sampling Scheme

A sampling
scheme
defines what
data will be
obtained and
how
A sampling scheme is a detailed description of what data
will be obtained and how this will be done. In PPC we are
faced with two different situations for developing
sampling schemes. The first is when we are conducting a
controlled experiment. There are very efficient and exact
methods for developing sampling schemes for designed
experiments and the reader is referred to the Process
Improvement chapter for details.
Passive data
collection
The second situation is when we are conducting a passive
data collection (PDC) study to learn about the inherent
properties of a process. These types of studies are usually
for comparison purposes when we wish to compare
properties of processes against each other or against some
hypothesis. This is the situation that we will focus on here.
There are two
principles that
guide our
choice of
sampling
scheme
Once we have selected our response parameters, it would
seem to be a rather straightforward exercise to take some
measurements, calculate some statistics and draw
conclusions. There are, however, many things which can
go wrong along the way that can be avoided with careful
planning and knowing what to watch for. There are two
overriding principles that will guide the design of our
sampling scheme.
The first is
precision
The first principle is that of precision. If the sampling
scheme is properly laid out, the difference between our
estimate of some parameter of interest and its true value
will be due only to random variation. The size of this
random variation is measured by a quantity called
standard error. The magnitude of the standard error is
known as precision. The smaller the standard error, the
more precise are our estimates.
Precision of
an estimate
depends on
several factors
The precision of any estimate will depend on:
the inherent variability of the process estimator
the measurement error
the number of independent replications (sample size)
the efficiency of the sampling scheme.
The second is
systematic
sampling error
(or
confounded
effects)
The second principle is the avoidance of systematic errors.
Systematic sampling error occurs when the levels of one
explanatory variable are the same as some other
unaccounted for explanatory variable. This is also referred
to as confounded effects. Systematic sampling error is best
seen by example.
Example 1: We want to compare the effect of
two different coolants on the resulting surface
finish from a turning operation. It is decided
to run one lot, change the coolant and then
run another lot. With this sampling scheme,
there is no way to distinguish the coolant
effect from the lot effect or from tool wear
considerations. There is systematic sampling
error in this sampling scheme.
Example 2: We wish to examine the effect of
two pre-clean procedures on the uniformity of
an oxide growth process. We clean one
cassette of wafers with one method and
another cassette with the other method. We
load one cassette in the front of the furnace
tube and the other cassette in the middle. To
complete the run, we fill the rest of the tube
with other lots. With this sampling scheme,
there is no way to distinguish between the
effect of the different pre-clean methods and
the cassette effect or the tube location effect.
Again, we have systematic sampling errors.
Stratification
helps to
overcome
systematic
error
The way to combat systematic sampling errors (and at the
same time increase precision) is through stratification and
randomization. Stratification is the process of segmenting
our population across levels of some factor so as to
minimize variability within those segments or strata. For
instance, if we want to try several different process recipes
to see which one is best, we may want to be sure to apply
each of the recipes to each of the three work shifts. This
will ensure that we eliminate any systematic errors caused
by a shift effect. This is where the ANOVA designs are
particularly useful.
Randomization
helps too
Randomization is the process of randomly applying the
various treatment combinations. In the above example, we
would not want to apply recipe 1, 2 and 3 in the same
order for each of the three shifts but would instead
randomize the order of the three recipes in each shift. This
will avoid any systematic errors caused by the order of the
recipes.
Examples The issues here are many and complicated. Click on each
of the links below to see the sampling schemes for each of
the case studies.
3.3.3.3. Selecting Sample Sizes

Consider
these things
when
selecting a
sample size
When choosing a sample size, we must consider the
following issues:
What population parameters we want to estimate
Cost of sampling (importance of information)
How much is already known
Spread (variability) of the population
Practicality: how hard is it to collect data
How precise we want the final estimates to be
Cost of
taking
samples
The cost of sampling issue helps us determine how precise
our estimates should be. As we will see below, when
choosing sample sizes we need to select risk values. If the
decisions we will make from the sampling activity are very
valuable, then we will want low risk values and hence
larger sample sizes.
Prior
information
If our process has been studied before, we can use that prior
information to reduce sample sizes. This can be done by
using prior mean and variance estimates and by stratifying
the population to reduce variation within groups.
Inherent
variability
We take samples to form estimates of some characteristic
of the population of interest. The variance of that estimate
is proportional to the inherent variability of the population
divided by the sample size:
.
with denoting the parameter we are trying to estimate.
This means that if the variability of the population is large,
then we must take many samples. Conversely, a small
population variance means we don't have to take as many
samples.
Practicality Of course the sample size you select must make sense. This
is where the trade-offs usually occur. We want to take
enough observations to obtain reasonably precise estimates
of the parameters of interest but we also want to do this
within a practical resource budget. The important thing is to
quantify the risks associated with the chosen sample size.
Sample size
determination
In summary, the steps involved in estimating a sample size
are:
1. There must be a statement about what is expected of
the sample. We must determine what is it we are
trying to estimate, how precise we want the estimate
to be, and what are we going to do with the estimate
once we have it. This should easily be derived from
the goals.
2. We must find some equation that connects the desired
precision of the estimate with the sample size. This is
a probability statement. A couple are given below;
see your statistician if these are not appropriate for
your situation.
3. This equation may contain unknown properties of the
population such as the mean or variance. This is
where prior information can help.
4. If you are stratifying the population in order to reduce
variation, sample size determination must be
performed for each stratum.
5. The final sample size should be scrutinized for
practicality. If it is unacceptable, the only way to
reduce it is to accept less precision in the sample
estimate.
Sampling
proportions
When we are sampling proportions we start with a
probability statement about the desired precision. This is
given by:
where
is the estimated proportion
P is the unknown population parameter
is the specified precision of the estimate
is the probability value (usually low)
This equation simply shows that we want the probability
that the precision of our estimate being less than we want is
. Of course we like to set low, usually .1 or less.
Using some assumptions about the proportion being
approximately normally distributed we can obtain an
estimate of the required sample size as:
where z is the ordinate on the Normal curve corresponding
to .
Example Let's say we have a new process we want to try. We plan to
run the new process and sample the output for yield
(good/bad). Our current process has been yielding 65%
(p=.65, q=.35). We decide that we want the estimate of the
new process yield to be accurate to within = .10 at 95%
confidence ( = .05, z
= -2). Using the formula above we

get a sample size estimate of n=91. Thus, if we draw 91
random parts from the output of the new process and
estimate the yield, then we are 95% sure the yield estimate
is within .10 of the true process yield.
Estimating
location:
relative error
If we are sampling continuous normally distributed
variables, quite often we are concerned about the relative
error of our estimates rather than the absolute error. The
probability statement connecting the desired precision to
the sample size is given by:
where is the (unknown) population mean and is the
sample mean.
Again, using the normality assumptions we obtain the
estimated sample size to be:
with
2
denoting the population variance.
Estimating
location:
absolute
error
If instead of relative error, we wish to use absolute error,
the equation for sample size looks alot like the one for the
case of proportions:
where is the population standard deviation (but in
practice is usually replaced by an engineering guesstimate).
Example Suppose we want to sample a stable process that deposits a
500 Angstrom film on a semiconductor wafer in order to
determine the process mean so that we can set up a control
chart on the process. We want to estimate the mean within
10 Angstroms ( = 10) of the true mean with 95%
confidence ( = .05, z
= -2). Our initial guess regarding

the variation in the process is that one standard deviation is
about 20 Angstroms. This gives a sample size estimate of n
= 16. Thus, if we take at least 16 samples from this process
and estimate the mean film thickness, we can be 95% sure
that the estimate is within 10 angstroms of the true mean
value.
3.3.3.4. Data Storage and Retrieval

3.3.3.4. Data Storage and Retrieval
Data
control
depends
on facility
size
If you are in a small manufacturing facility or a lab, you can
simply design a sampling plan, run the material, take the
measurements, fill in the run sheet and go back to your
computer to analyze the results. There really is not much to be
concerned with regarding data storage and retrieval.
In most larger facilities, however, the people handling the
material usually have nothing to do with the design. Quite
often the measurements are taken automatically and may not
even be made in the same country where the material was
produced. Your data go through a long chain of automatic
acquisition, storage, reformatting, and retrieval before you are
ever able to see it. All of these steps are fraught with peril and
should be examined closely to ensure that valuable data are
not lost or accidentally altered.
Know the
process
involved
In the planning phase of the PPC, be sure to understand the
entire data collection process. Things to watch out for include:
automatic measurement machines rejecting outliers
only summary statistics (mean and standard deviation)
being saved
values for explanatory variables (location, operator, etc.)
are not being saved
how missing values are handled
Consult
with
support
staff early
on
It is important to consult with someone from the organization
responsible for maintaining the data system early in the
planning phase of the PPC. It can also be worthwhile to
perform some "dry runs" of the data collection to ensure you
will be able to actually acquire the data in the format as
defined in the plan.
3.3.3.5. Assign Roles and Responsibilities

PPC is a team
effort, get
everyone
involved early
In today's manufacturing environment, it is unusual when
an investigative study is conducted by a single individual.
Most PPC studies will be a team effort. It is important that
all individuals who will be involved in the study become a
part of the team from the beginning. Many of the various
collateral activities will need approvals and sign-offs. Be
sure to account for that cycle time in your plan.
Table showing
roles and
potential
responsibilities
A partial list of these individuals along with their roles
and potential responsibilities is given in the table below.
There may be multiple occurrences of each of these
individuals across shifts or process steps, so be sure to
include everyone.
Tool
Owner
Controls
Tool
Operations
Schedules tool time
Ensures tool state
Advises on
experimental design
Process
Owner
Controls
Process
Recipe
Advises on
experimental design
Controls recipe settings
Tool
Operator
Executes
Experimental
Plan
Executes experimental
runs
May take
measurements
Metrology Own
Measurement
Tools
Maintains metrology
equipment
Conducts gauge studies
May take
measurements
CIM Owns
Enterprise
Information
System
Maintains data
collection system
Maintains equipment
interfaces and data
formatters
Maintains databases
and information access
Statistician Consultant Consults on
experimental design
Consults on data
analysis
Quality
Control
Controls
Material
Ensures quality of
incoming material
Must approve shipment
of outgoing material
(especially for recipe
changes)
3.4. Data Analysis for PPC

In this section we will learn how to analyze and interpret the
data we collected in accordance with our data collection plan.
Click on
desired
topic to
read more
This section discusses the following topics:
1. Initial Data Analysis
1. Gather Data
2. Quality Checking the Data
3. Summary Analysis (Location, Spread and Shape)
2. Exploring Relationships
1. Response Correlations
2. Exploring Main Effects
3. Exploring First-Order Interactions
3. Building Models
1. Fitting Polynomial Models
2. Fitting Physical Models
4. Analyzing Variance Structure
5. Assessing Process Stablility
6. Assessing Process Capability
7. Checking Assumptions
3.4.1. First Steps

3.4.1. First Steps
Gather all
of the data
into one
place
After executing the data collection plan for the
characterization study, the data must be gathered up for
analysis. Depending on the scope of the study, the data may
reside in one place or in many different places. It may be in
common factory databases, flat files on individual computers,
or handwritten on run sheets. Whatever the case, the first step
will be to collect all of the data from the various sources and
enter it into a single data file. The most convenient format for
most data analyses is the variables-in-columns format. This
format has the variable names in column headings and the
values for the variables in the rows.
Perform a
quality
check on
the data
using
graphical
and
numerical
techniques
The next step is to perform a quality check on the data. Here
we are typically looking for data entry problems, unusual data
values, missing data, etc. The two most useful tools for this
step are the scatter plot and the histogram. By constructing
scatter plots of all of the response variables, any data entry
problems will be easily identified. Histograms of response
variables are also quite useful for identifying data entry
problems. Histograms of explanatory variables help identify
problems with the execution of the sampling plan. If the
counts for each level of the explanatory variables are not the
same as called for in the sampling plan, you know you may
have an execution problem. Running numerical summary
statistics on all of the variables (both response and
explanatory) also helps to identify data problems.
Summarize
data by
estimating
location,
spread and
shape
Once the data quality problems are identified and fixed, we
should estimate the location, spread and shape for all of the
response variables. This is easily done with a combination of
histograms and numerical summary statistics.
3.4.2. Exploring Relationships

The first
analysis of
our data is
exploration
Once we have a data file created in the desired format,
checked the data integrity, and have estimated the summary
statistics on the response variables, the next step is to start
exploring the data and to try to understand the underlying
structure. The most useful tools will be various forms of the
basic scatter plot and box plot.
These techniques will allow pairwise explorations for
examining relationships between any pair of response
variables, any pair of explanatory and response variables, or a
response variable as a function of any two explanatory
variables. Beyond three dimensions we are pretty much
limited by our human frailties at visualization.
Graph
everything
that makes
sense
In this exploratory phase, the key is to graph everything that
makes sense to graph. These pictures will not only reveal any
additional quality problems with the data but will also reveal
influential data points and will guide the subsequent modeling
activities.
Graph
responses,
then
explanatory
versus
response,
then
conditional
plots
The order that generally proves most effective for data
analysis is to first graph all of the responses against each
other in a pairwise fashion. Then we graph responses against
the explanatory variables. This will give an indication of the
main factors that have an effect on response variables.
Finally, we graph response variables, conditioned on the
levels of explanatory factors. This is what reveals interactions
between explanatory variables. We will use nested boxplots
and block plots to visualize interactions.
3.4.2.1. Response Correlations

Make
scatter
plots of
all of the
response
variables
In this first phase of exploring our data, we plot all of the response variables in a
pairwise fashion. The individual scatter plots are displayed in a matrix form with the
y-axis scaling the same for all plots in a row of the matrix.
Check the
slope of
the data
on the
scatter
plots
The scatterplot matrix shows how the response variables are related to each other. If
there is a linear trend with a positive slope, this indicates that the responses are
positively correlated. If there is a linear trend with a negative slope, then the variables
are negatively correlated. If the data appear random with no slope, the variables are
probably not correlated. This will be important information for subsequent model
building steps.
This
scatterplot
matrix
shows
examples
of both
negatively
and
positively
correlated
variables
An example of a scatterplot matrix is given below. In this semiconductor
manufacturing example, three responses, yield (Bin1), N-channel Id effective
(NIDEFF), and P-channel Id effective (PIDEFF) are plotted against each other in a
scatterplot matrix. We can see that Bin1 is positively correlated with NIDEFF and
negatively correlated with PIDEFF. Also, as expected, NIDEFF is negatively
correlated with PIDEFF. This kind of information will prove to be useful when we
build models for yield improvement.
3.4.2.2. Exploring Main Effects

The next
step is to
look for
main effects
The next step in the exploratory analysis of our data is to see which factors have an
effect on which response variables and to quantify that effect. Scatter plots and box
plots will be the tools of choice here.
Watch out
for varying
sample
sizes across
levels
This step is relatively self explanatory. However there are two points of caution. First,
be cognizant of not only the trends in these graphs but also the amount of data
represented in those trends. This is especially true for categorical explanatory
variables. There may be many more observations in some levels of the categorical
variable than in others. In any event, take unequal sample sizes into account when
making inferences.
Graph
implicit as
well as
explicit
explanatory
variables
The second point is to be sure to graph the responses against implicit explanatory
variables (such as observation order) as well as the explicit explanatory variables.
There may be interesting insights in these hidden explanatory variables.
Example:
wafer
processing
In the example below, we have collected data on the particles added to a wafer during
a particular processing step. We ran a number of cassettes through the process and
sampled wafers from certain slots in the cassette. We also kept track of which load
lock the wafers passed through. This was done for two different process temperatures.
We measured both small particles (< 2 microns) and large particles (> 2 microns). We
plot the responses (particle counts) against each of the explanatory variables.
Cassette
does not
appear to
be an
important
factor for
small or
large
particles
This first graph is a box plot of the number of small particles added for each cassette
type. The "X"'s in the plot represent the maximum, median, and minimum number of
particles.
The second graph is a box plot of the number of large particles added for each cassette
type.
We conclude from these two box plots that cassette does not appear to be an important
factor for small or large particles.
There is a
difference
between
slots for
small
particles,
one slot is
different for
large
particles
We next generate box plots of small and large particles for the slot variable. First, the
box plot for small particles.
Next, the box plot for large particles.
We conclude that there is a difference between slots for small particles. We also
conclude that one slot appears to be different for large particles.
Load lock
may have a
slight effect
for small
and large
particles
We next generate box plots of small and large particles for the load lock variable.
First, the box plot for small particles.
We conclude that there may be a slight effect for load lock for small and large
particles.
For small
particles,
temperature
has a
strong
effect on
both
location
and spread.
For large
particles,
there may
be a slight
temperature
effect but
this may
just be due
to the
outliers
We next generate box plots of small and large particles for the temperature variable.
First, the box plot for small particles.
'
We conclude that temperature has a strong effect on both location and spread for small
particles. We conclude that there might be a small temperature effect for large
particles, but this may just be due to outliers.
3.4.2.3. Exploring First Order Interactions

It is
important
to identify
interactions
The final step (and perhaps the most important one) in the exploration phase is to find
any first order interactions. When the difference in the response between the levels of
one factor is not the same for all of the levels of another factor we say we have an
interaction between those two factors. When we are trying to optimize responses based
on factor settings, interactions provide for compromise.
The eyes
can be
deceiving -
be careful
Interactions can be seen visually by using nested box plots. However, caution should
be exercised when identifying interactions through graphical means alone. Any
graphically identified interactions should be verified by numerical methods as well.
Previous
example
continued
To continue the previous example, given below are nested box plots of the small and
large particles. The load lock is nested within the two temperature values. There is
some evidence of possible interaction between these two factors. The effect of load
lock is stronger at the lower temperature than at the higher one. This effect is stronger
for the smaller particles than for the larger ones. As this example illustrates, when you
have significant interactions the main effects must be interpreted conditionally. That
is, the main effects do not tell the whole story by themselves.
For small
particles,
the load
lock effect
is not as
strong for
high
temperature
as it is for
low
temperature
The following is the box plot of small particles for load lock nested within
temperature.
We conclude from this plot that for small particles, the load lock effect is not as strong
for high temperature as it is for low temperature.
The same
may be true
for large
particles
but not as
strongly
The following is the box plot of large particles for load lock nested within temperature.
We conclude from this plot that for large particles, the load lock effect may not be as
strong for high temperature as it is for low temperature. However, this effect is not as
strong as it is for small particles.
3.4.3. Building Models

Black box
models
When we develop a data collection plan we build black box models of
the process we are studying like the one below:
In our data
collection plan
we drew
process model
pictures
Numerical
models are
explicit
representations
of our process
model pictures
In the Exploring Relationships section, we looked at how to identify
the input/output relationships through graphical methods. However, if
we want to quantify the relationships and test them for statistical
significance, we must resort to building mathematical models.
Polynomial
models are
generic
descriptors of
our output
surface
There are two cases that we will cover for building mathematical
models. If our goal is to develop an empirical prediction equation or
to identify statistically significant explanatory variables and quantify
their influence on output responses, we typically build polynomial
models. As the name implies, these are polynomial functions
(typically linear or quadratic functions) that describe the relationships
between the explanatory variables and the response variable.
Physical
models
describe the
underlying
physics of our
processes
On the other hand, if our goal is to fit an existing theoretical equation,
then we want to build physical models. Again, as the name implies,
this pertains to the case when we already have equations representing
the physics involved in the process and we want to estimate specific
parameter values.
3.4.3.1. Fitting Polynomial Models

3.4.3.1. Fitting Polynomial Models
Polynomial
models are
a great tool
for
determining
which input
factors
drive
responses
and in what
direction
We use polynomial models to estimate and predict the shape
of response values over a range of input parameter values.
Polynomial models are a great tool for determining which
input factors drive responses and in what direction. These are
also the most common models used for analysis of designed
experiments. A quadratic (second-order) polynomial model
for two explanatory variables has the form of the equation
below. The single x-terms are called the main effects. The
squared terms are called the quadratic effects and are used to
model curvature in the response surface. The cross-product
terms are used to model interactions between the explanatory
variables.
We
generally
don't need
more than
second-
order
equations
In most engineering and manufacturing applications we are
concerned with at most second-order polynomial models.
Polynomial equations obviously could become much more
complicated as we increase the number of explanatory
variables and hence the number of cross-product terms.
Fortunately, we rarely see significant interaction terms above
the two-factor level. This helps to keep the equations at a
manageable level.
Use
multiple
regression
to fit
polynomial
models
When the number of factors is small (less than 5), the
complete polynomial equation can be fitted using the
technique known as multiple regression. When the number of
factors is large, we should use a technique known as stepwise
regression. Most statistical analysis programs have a stepwise
regression capability. We just enter all of the terms of the
polynomial models and let the software choose which terms
best describe the data. For a more thorough discussion of this
topic and some examples, refer to the process improvement
chapter.
3.4.3.2. Fitting Physical Models

Sometimes
we want
to use a
physical
model
Sometimes, rather than approximating response behavior with
polynomial models, we know and can model the physics behind the
underlying process. In these cases we would want to fit physical
models to our data. This kind of modeling allows for better prediction
and is less subject to variation than polynomial models (as long as the
underlying process doesn't change).
We will
use a
CMP
process to
illustrate
We will illustrate this concept with an example. We have collected
data on a chemical/mechanical planarization process (CMP) at a
particular semiconductor processing step. In this process, wafers are
polished using a combination of chemicals in a polishing slurry using
polishing pads. We polished a number of wafers for differing periods
of time in order to calculate material removal rates.
CMP
removal
rate can
be
modeled
with a
non-linear
equation
From first principles we know that removal rate changes with time.
Early on, removal rate is high and as the wafer becomes more planar
the removal rate declines. This is easily modeled with an exponential
function of the form:
removal rate = p1 + p2 x exp
p3 x time
where p1, p2, and p3 are the parameters we want to estimate.
A non-
linear
regression
routine
was used
to fit the
data to
the
equation
The equation was fit to the data using a non-linear regression routine.
A plot of the original data and the fitted line are given in the image
below. The fit is quite good. This fitted equation was subsequently
used in process optimization work.
3.4.4. Analyzing Variance Structure

Studying
variation is
important
in PPC
One of the most common activities in process characterization work is to study the
variation associated with the process and to try to determine the important sources
of that variation. This is called analysis of variance. Refer to the section of this
chapter on ANOVA models for a discussion of the theory behind this kind of
analysis.
The key is
to know the
structure
The key to performing an analysis of variance is identifying the structure
represented by the data. In the ANOVA models section we discussed one-way
layouts and two-way layouts where the factors are either crossed or nested. Review
these sections if you want to learn more about ANOVA structural layouts.
To perform the analysis, we just identify the structure, enter the data for each of
the factors and levels into a statistical analysis program and then interpret the
ANOVA table and other output. This is all illustrated in the example below.
Example:
furnace
oxide
thickness
with a 1-
way layout
The example is a furnace operation in semiconductor manufacture where we are
growing an oxide layer on a wafer. Each lot of wafers is placed on quartz
containers (boats) and then placed in a long tube-furnace. They are then raised to a
certain temperature and held for a period of time in a gas flow. We want to
understand the important factors in this operation. The furnace is broken down into
four sections (zones) and two wafers from each lot in each zone are measured for
the thickness of the oxide layer.
Look at
effect of
zone
location on
oxide
thickness
The first thing to look at is the effect of zone location on the oxide thickness. This
is a classic one-way layout. The factor is furnace zone and we have four levels. A
plot of the data and an ANOVA table are given below.
The zone
effect is
masked by
the lot-to-
lot variation
ANOVA
table

Source DF SS Mean Square F Ratio Prob > F
Zone 3 912.6905 304.23 0.467612 0.70527
Within 164 106699.1 650.604
Let's
account for
lot with a
nested
layout
From the graph there does not appear to be much of a zone effect; in fact, the
ANOVA table indicates that it is not significant. The problem is that variation due
to lots is so large that it is masking the zone effect. We can fix this by adding a
factor for lot. By treating this as a nested two-way layout, we obtain the ANOVA
table below.
Now both
lot and zone
are
revealed as
important

Source DF SS Mean Square F Ratio Prob > F
Lot 20 61442.29 3072.11 5.37404 1.39e-7
Zone[lot] 63 36014.5 571.659 4.72864 3.9e-11
Within 84 10155 120.893
Conclusions Since the "Prob > F" is less than 0.05, for both lot and zone, we know that these
factors are statistically significant at the 0.05 significance level.
3.4.5. Assessing Process Stability

3.4.5. Assessing Process Stability
A process is
stable if it has
a constant
mean and a
constant
variance over
time
A manufacturing process cannot be released to production
until it has been proven to be stable. Also, we cannot
begin to talk about process capability until we have
demonstrated stability in our process. A process is said to
be stable when all of the response parameters that we use
to measure the process have both constant means and
constant variances over time, and also have a constant
distribution. This is equivalent to our earlier definition of
controlled variation.
The graphical
tool we use to
assess stability
is the scatter
plot or the
control chart
The graphical tool we use to assess process stability is the
scatter plot. We collect a sufficient number of
independent samples (greater than 100) from our process
over a sufficiently long period of time (this can be
specified in days, hours of processing time or number of
parts processed) and plot them on a scatter plot with
sample order on the x-axis and the sample value on the y-
axis. The plot should look like constant random variation
about a constant mean. Sometimes it is helpful to
calculate control limits and plot them on the scatter plot
along with the data. The two plots in the controlled
variation example are good illustrations of stable and
unstable processes.
Numerically,
we assess its
stationarity
using the
autocorrelation
function
Numerically, we evaluate process stability through a times
series analysis concept know as stationarity. This is just
another way of saying that the process has a constant
mean and a constant variance. The numerical technique
used to assess stationarity is the autocovariance function.
Graphical
methods
usually good
enough
Typically, graphical methods are good enough for
evaluating process stability. The numerical methods are
generally only used for modeling purposes.
3.4.6. Assessing Process Capability

Capability
compares a
process
against its
specification
Process capability analysis entails comparing the performance of a process against its
specifications. We say that a process is capable if virtually all of the possible variable
values fall within the specification limits.
Use a
capability
chart
Graphically, we assess process capability by plotting the process specification limits on
a histogram of the observations. If the histogram falls within the specification limits,
then the process is capable. This is illustrated in the graph below. Note how the
process is shifted below target and the process variation is too large. This is an
example of an incapable process.
Notice how
the process is
off target and
has too much
variation
Numerically, we measure capability with a capability index. The general equation for
the capability index, C
p
, is:
Numerically,
we use the C
p
index
Interpretation
of the C
p
index
This equation just says that the measure of our process capability is how much of our
observed process variation is covered by the process specifications. In this case the
process variation is measured by 6 standard deviations (+/- 3 on each side of the
mean). Clearly, if C
p
> 1.0, then the process specification covers almost all of our
process observations.
C
p
does not
account for
process that
is off center
The only problem with with the C
p
index is that it does not account for a process that
is off-center. We can modify this equation slightly to account for off-center processes
to obtain the C
pk
index as follows:
Or the C
pk
index
C
pk
accounts
for a process
being off
center
This equation just says to take the minimum distance between our specification limits
and the process mean and divide it by 3 standard deviations to arrive at the measure of
process capability. This is all covered in more detail in the process capability section
of the process monitoring chapter. For the example above, note how the C
pk
value is
less than the C
p
value. This is because the process distribution is not centered between
the specification limits.
3.4.7. Checking Assumptions

Check the
normality of
the data
Many of the techniques discussed in this chapter, such as hypothesis tests, control
charts and capability indices, assume that the underlying structure of the data can be
adequately modeled by a normal distribution. Many times we encounter data where
this is not the case.
Some causes
of non-
normality
There are several things that could cause the data to appear non-normal, such as:
The data come from two or more different sources. This type of data will often
have a multi-modal distribution. This can be solved by identifying the reason for
the multiple sets of data and analyzing the data separately.
The data come from an unstable process. This type of data is nearly impossible
to analyze because the results of the analysis will have no credibility due to the
changing nature of the process.
The data were generated by a stable, yet fundamentally non-normal mechanism.
For example, particle counts are non-normal by the very nature of the particle
generation process. Data of this type can be handled using transformations.
We can
sometimes
transform the
data to make it
look normal
For the last case, we could try transforming the data using what is known as a power
transformation. The power transformation is given by the equation:
where Y represents the data and lambda is the transformation value. Lambda is
typically any value between -2 and 2. Some of the more common values for lambda
are 0, 1/2, and -1, which give the following transformations:
General
algorithm for
trying to make
non-normal
data
approximately
normal
The general algorithm for trying to make non-normal data appear to be approximately
normal is to:
1. Determine if the data are non-normal. (Use normal probability plot and
histogram).
2. Find a transformation that makes the data look approximately normal, if
possible. Some data sets may include zeros (i.e., particle data). If the data set
does include zeros, you must first add a constant value to the data and then
transform the results.
Example:
particle count
data
As an example, let's look at some particle count data from a semiconductor processing
step. Count data are inherently non-normal. Below are histograms and normal
probability plots for the original data and the ln, sqrt and inverse of the data. You can
see that the log transform does the best job of making the data appear as if it is normal.
All analyses can be performed on the log-transformed data and the assumptions will
be approximately satisfied.
The original
data is non-
normal, the
log transform
looks fairly
normal
Neither the
square root
nor the
inverse
transformation
looks normal
3.5. Case Studies

3.5. Case Studies
Summary This section presents several case studies that demonstrate the
application of production process characterizations to specific
problems.
Table of
Contents
The following case studies are available.
1. Furnace Case Study
2. Machine Case Study
3.5.1. Furnace Case Study

3.5. Case Studies
Introduction This case study analyzes a furnace oxide growth process.
Table of
Contents
The case study is broken down into the following steps.
2. Initial Analysis of Response Variable
3. Identify Sources of Variation
5. Final Conclusions
3.5.1.1. Background and Data

3.5. Case Studies
Introduction In a semiconductor manufacturing process flow, we have a
step whereby we grow an oxide film on the silicon wafer
using a furnace. In this step, a cassette of wafers is placed in
a quartz "boat" and the boats are placed in the furnace. The
furnace can hold four boats. A gas flow is created in the
furnace and it is brought up to temperature and held there for
a specified period of time (which corresponds to the desired
oxide thickness). This study was conducted to determine if
the process was stable and to characterize sources of
variation so that a process control strategy could be
developed.
Goal The goal of this study is to determine if this process is
capable of consistently growing oxide films with a thickness
of 560 Angstroms +/- 100 Angstroms. An additional goal is
to determine important sources of variation for use in the
development of a process control strategy.
Process
Model
In the picture below we are modeling this process with one
output (film thickness) that is influenced by four controlled
factors (gas flow, pressure, temperature and time) and two
uncontrolled factors (run and zone). The four controlled
factors are part of our recipe and will remain constant
throughout this study. We know that there is run-to-run
variation that is due to many different factors (input material
variation, variation in consumables, etc.). We also know that
the different zones in the furnace have an effect. A zone is a
region of the furnace tube that holds one boat. There are four
zones in these tubes. The zones in the middle of the tube
grow oxide a little bit differently from the ones on the ends.
In fact, there are temperature offsets in the recipe to help
minimize this problem.
Sensitivity
Model
The sensitivity model for this process is fairly straightforward
and is given in the figure below. The effects of the machin are
mostly related to the preventative maintenance (PM) cycle.
We want to make sure the quartz tube has been cleaned
recently, the mass flow controllers are in good shape and the
temperature controller has been calibrated recently. The same
is true of the measurement equipment where the thickness
readings will be taken. We want to make sure a gauge study
has been performed. For material, the incoming wafers will
certainly have an effect on the outgoing thickness as well as
the quality of the gases used. Finally, the recipe will have an
effect including gas flow, temperature offset for the different
zones, and temperature profile (how quickly we raise the
temperature, how long we hold it and how quickly we cool it
off).
Sampling
Plan
Given our goal statement and process modeling, we can now
define a sampling plan. The primary goal is to determine if the
process is capable. This just means that we need to monitor the
process over some period of time and compare the estimates
of process location and spread to the specifications. An
additional goal is to identify sources of variation to aid in
setting up a process control strategy. Some obvious sources of
variation are incoming wafers, run-to-run variability, variation
due to operators or shift, and variation due to zones within a
furnace tube. One additional constraint that we must work
under is that this study should not have a significant impact on
normal production operations.
Given these constraints, the following sampling plan was
selected. It was decided to monitor the process for one day
(three shifts). Because this process is operator independent, we
will not keep shift or operator information but just record run
number. For each run, we will randomly assign cassettes of
wafers to a zone. We will select two wafers from each zone
after processing and measure two sites on each wafer. This
plan should give reasonable estimates of run-to-run variation
and within zone variability as well as good overall estimates
of process location and spread.
We are expecting readings around 560 Angstroms. We would
not expect many readings above 700 or below 400. The
measurement equipment is accurate to within 0.5 Angstroms
which is well within the accuracy needed for this study.
Data The following are the data that were collected for this study.
RUN ZONE WAFER THICKNESS
--------------------------------
1 1 1 546
1 1 2 540
1 2 1 566
1 2 2 564
1 3 1 577
1 3 2 546
1 4 1 543
1 4 2 529
2 1 1 561
2 1 2 556
2 2 1 577
2 2 2 553
2 3 1 563
2 3 2 577
2 4 1 556
2 4 2 540
3 1 1 515
3 1 2 520
3 2 1 548
3 2 2 542
3 3 1 505
3 3 2 487
3 4 1 506
3 4 2 514
4 1 1 568
4 1 2 584
4 2 1 570
4 2 2 545
4 3 1 589
4 3 2 562
4 4 1 569
4 4 2 571
5 1 1 550
5 1 2 550
5 2 1 562
5 2 2 580
5 3 1 560
5 3 2 554
5 4 1 545
5 4 2 546
6 1 1 584
6 1 2 581
6 2 1 567
6 2 2 558
6 3 1 556
6 3 2 560
6 4 1 591
6 4 2 599
7 1 1 593
7 1 2 626
7 2 1 584
7 2 2 559
7 3 1 634
7 3 2 598
7 4 1 569
7 4 2 592
8 1 1 522
8 1 2 535
8 2 1 535
8 2 2 581
8 3 1 527
8 3 2 520
8 4 1 532
8 4 2 539
9 1 1 562
9 1 2 568
9 2 1 548
9 2 2 548
9 3 1 533
9 3 2 553
9 4 1 533
9 4 2 521
10 1 1 555
10 1 2 545
10 2 1 584
10 2 2 572
10 3 1 546
10 3 2 552
10 4 1 586
10 4 2 584
11 1 1 565
11 1 2 557
11 2 1 583
11 2 2 585
11 3 1 582
11 3 2 567
11 4 1 549
11 4 2 533
12 1 1 548
12 1 2 528
12 2 1 563
12 2 2 588
12 3 1 543
12 3 2 540
12 4 1 585
12 4 2 586
13 1 1 580
13 1 2 570
13 2 1 556
13 2 2 569
13 3 1 609
13 3 2 625
13 4 1 570
13 4 2 595
14 1 1 564
14 1 2 555
14 2 1 585
14 2 2 588
14 3 1 564
14 3 2 583
14 4 1 563
14 4 2 558
15 1 1 550
15 1 2 557
15 2 1 538
15 2 2 525
15 3 1 556
15 3 2 547
15 4 1 534
15 4 2 542
16 1 1 552
16 1 2 547
16 2 1 563
16 2 2 578
16 3 1 571
16 3 2 572
16 4 1 575
16 4 2 584
17 1 1 549
17 1 2 546
17 2 1 584
17 2 2 593
17 3 1 567
17 3 2 548
17 4 1 606
17 4 2 607
18 1 1 539
18 1 2 554
18 2 1 533
18 2 2 535
18 3 1 522
18 3 2 521
18 4 1 547
18 4 2 550
19 1 1 610
19 1 2 592
19 2 1 587
19 2 2 587
19 3 1 572
19 3 2 612
19 4 1 566
19 4 2 563
20 1 1 569
20 1 2 609
20 2 1 558
20 2 2 555
20 3 1 577
20 3 2 579
20 4 1 552
20 4 2 558
21 1 1 595
21 1 2 583
21 2 1 599
21 2 2 602
21 3 1 598
21 3 2 616
21 4 1 580
21 4 2 575
3.5.1.2. Initial Analysis of Response Variable

3.5. Case Studies
Initial Plots
of Response
Variable
The initial step is to assess data quality and to look for anomalies. This is done by
generating a normal probability plot, a histogram, and a boxplot. For convenience,
these are generated on a single page.
Conclusions
From the
Plots
We can make the following conclusions based on these initial plots.
The box plot indicates one outlier. However, this outlier is only slightly
smaller than the other numbers.
The normal probability plot and the histogram (with an overlaid normal
density) indicate that this data set is reasonably approximated by a normal
distribution.
Parameter
Estimates
Parameter estimates for the film thickness are summarized in
the following table.
Parameter Estimates
Type Parameter Estimate
Lower
(95%)
Confidence
Bound
Upper
(95%)
Confidence
Bound
Location Mean 563.0357 559.1692 566.9023
Dispersion
Standard
Deviation
25.3847 22.9297 28.4331
Quantiles Quantiles for the film thickness are summarized in the following table.
Quantiles for Film Thickness
100.0% Maximum 634.00
99.5% 634.00
97.5% 615.10
90.0% 595.00
75.0% Upper Quartile 582.75
50.0% Median 562.50
25.0% Lower Quartile 546.25
10.0% 532.90
2.5% 514.23
0.5% 487.00
0.0% Minimum 487.00
Capability
Analysis
From the above preliminary analysis, it looks reasonable to proceed with the
capability analysis.
The lower specification limit is 460, the upper specification limit is 660, and
the target specification is 560.
Percent
Defective
We summarize the percent defective (i.e., the number of items outside the
specification limits) in the following table.
Percentage Outside Specification Limits
Specification Value Percent Actual
Theoretical
(% Based On
Normal)
Lower
Specification
Limit
460
Percent Below LSL =
100* ((LSL - )/s)
0.0000 0.0025%
Upper
Specification
Limit
660
Percent Above USL =
100*(1 - ((USL -
)/s))
0.0000 0.0067%
Specification
Target
560
Combined Percent
Below LSL and Above
USL
0.0000 0.0091%
Standard
Deviation
25.38468
The value denotes the normal cumulative distribution function, the
sample mean, and s the sample standard deviation.
Capability
Index
Statistics
We summarize various capability index statistics in the following table.
Capability Index Statistics
Capability Statistic Index Lower CI Upper CI
CP 1.313 1.172 1.454
CPK 1.273 1.128 1.419
CPM 1.304 1.165 1.442
CPL 1.353 1.218 1.488
CPU 1.273 1.142 1.404
Conclusions The above capability analysis indicates that the process is capable and we
can proceed with the analysis.
3.5.1.3. Identify Sources of Variation

3.5. Case Studies
The next part of the analysis is to break down the sources of variation.
Box Plot by
Run
The following is a box plot of the thickness by run number.
Conclusions
From Box
Plot
We can make the following conclusions from this box plot.
1. There is significant run-to-run variation.
2. Although the means of the runs are different, there is no discernable trend due
to run.
3. In addition to the run-to-run variation, there is significant within-run variation
as well. This suggests that a box plot by furnace location may be useful as
well.
Box Plot by The following is a box plot of the thickness by furnace location.
Furnace
Location
Conclusions
From Box
Plot
1. There is considerable variation within a given furnace location.
2. The variation between furnace locations is small. That is, the locations and
scales of each of the four furnace locations are fairly comparable (although
furnace location 3 seems to have a few mild outliers).
Box Plot by
Wafer
The following is a box plot of the thickness by wafer.
Conclusion
From Box
Plot
From this box plot, we conclude that wafer does not seem to be a significant factor.
Block Plot In order to show the combined effects of run, furnace location, and wafer, we draw a
block plot of the thickness. Note that for aesthetic reasons, we have used connecting
lines rather than enclosing boxes.
Conclusions
From Block
Plot
We can draw the following conclusions from this block plot.
1. There is significant variation both between runs and between furnace
locations. The between-run variation appears to be greater.
2. Run 3 seems to be an outlier.
3.5.1.4. Analysis of Variance

3.5. Case Studies
Analysis of
Variance
The next step is to confirm our interpretation of the plots in
the previous section by running a nested analysis of variance.
Source Degrees
of
Freedom
Sum of
Squares
Mean
Square
Error
F Ratio Prob > F
Run 20 61,442.29 3,072.11 5.37404 0.0000001
Furnace
Location
[Run]
63 36,014.5 571.659 4.72864 3.85e-11
Within 84 10,155 120.893
Total 167 107,611.8 644.382
Components
of Variance
From the above analysis of variance table, we can compute
the components of variance. Recall that for this data set we
have 2 wafers measured at 4 furnace locations for 21 runs.
This leads to the following set of equations.
3072.11 = (4*2)*Var(Run) + 2*Var(Furnace Location)
+ Var(Within)
571.659 = 2*Var(Furnace Location) + Var(Within)
120.893 = Var(Within)
Solving these equations yields the following components of
variance.
Components of Variance
Component Variance
Component
Percent
of Total
Sqrt(Variance
Component)
Run 312.55694 47.44 17.679
Furnace
Location[Run]
225.38294 34.21 15.013
Within 120.89286 18.35 10.995
3.5.1.5. Final Conclusions

3.5. Case Studies
Final
Conclusions
This simple study of a furnace oxide growth process
indicated that the process is capable and showed that both
run-to-run and zone-within-run are significant sources of
variation. We should take this into account when designing
the control strategy for this process. The results also pointed
to where we should look when we perform process
improvement activities.
3.5.1.6. Work This Example Yourself

3.5. Case Studies
View
Dataplot
Macro for
this Case
Study
case study description on the previous page using Dataplot, if
you have downloaded and installed it. Output from each
analysis step below will be displayed in one or more of the
Dataplot windows. The four main windows are the Output
window, the Graphics window, the Command History window
and the Data Sheet window. Across the top of the main
windows there are menus for executing Dataplot commands.
Across the bottom is a command entry window where
commands can be typed in.
Data Analysis Steps
Results and
Conclusions
more detailed
1. Get set up and started.

1. You have read 4
columns of numbers
into Dataplot,
variables run, zone,
wafer, and
filmthic.
2. Analyze the response variable.
1. Normal probability plot,
box plot, and histogram of
film thickness.
2. Compute summary statistics
and quantiles of film
thickness.
1. Initial plots
indicate that the
film thickness is
reasonably
approximated by a
normal
distribution with
no significant
outliers.
2. Mean is 563.04
and standard
3. Perform a capability analysis. deviation is
25.38. Data range
from 487 to 634.
3. Capability
analysis indicates
that the process
is capable.
3. Identify Sources of Variation.
1. Generate a box plot by run.
2. Generate a box plot by furnace
location.
3. Generate a box plot by wafer.
4. Generate a block plot.
1. The box plot
shows significant
variation both
between runs and
within runs.
2. The box plot
shows significant
variation within
furnace location
but not between
furnace location.
3. The box plot
shows no significant
effect for wafer.
4. The block plot
shows both run
and furnace
location are
significant.
4. Perform an Analysis of Variance
1. Perform the analysis of
variance and compute the
components of variance.
1. The results of
the ANOVA are
summarized in an
ANOVA table
and a components
of variance
table.
3.5.2. Machine Screw Case Study

3.5. Case Studies
Introduction This case study analyzes three automatic screw machines
with the intent of replacing one of them.
Table of
Contents
The case study is broken down into the following steps.
2. Box Plots by Factor
4. Throughput
5. Final Conclusions

3.5. Case Studies
Introduction A machine shop has three automatic screw machines that
produce various parts. The shop has enough capital to
replace one of the machines. The quality control department
has been asked to conduct a study and make a
recommendation as to which machine should be replaced. It
was decided to monitor one of the most commonly produced
parts (an 1/8
th
inch diameter pin) on each of the machines
and see which machine is the least stable.
Goal The goal of this study is to determine which machine is least
stable in manufacturing a steel pin with a diameter of .125
+/- .003 inches. Stability will be measured in terms of a
constant variance about a constant mean. If all machines are
stable, the decision will be based on process variability and
throughput. Namely, the machine with the highest variability
and lowest throughput will be selected for replacement.
Process
Model
The process model for this operation is trivial and need not
be addressed.
Sensitivity
Model
The sensitivity model, however, is important and is given in
the figure below. The material is not very important. All
machines will receive barstock from the same source and the
coolant will be the same. The method is important. Each
machine is slightly different and the operator must make
adjustments to the speed (how fast the part rotates), feed
(how quickly the cut is made) and stops (where cuts are
finished) for each machine. The same operator will be
running all three machines simultaneously. Measurement is
not too important. An experienced QC engineer will be
collecting the samples and making the measurements.
Finally, the machine condition is really what this study is all
about. The wear on the ways and the lead screws will largely
determine the stability of the machining process. Also, tool
wear is important. The same type of tool inserts will be used
on all three machines. The tool insert wear will be monitored
by the operator and they will be changed as needed.
Sampling
Plan
Given our goal statement and process modeling, we can now
define a sampling plan. The primary goal is to determine if the
process is stable and to compare the variances of the three
machines. We also need to monitor throughput so that we can
compare the productivity of the three machines.
There is an upcoming three-day run of the particular part of
interest, so this study will be conducted on that run. There is a
suspected time-of-day effect that we must account for. It is
sometimes the case that the machines do not perform as well
in the morning, when they are first started up, as they do later
in the day. To account for this we will sample parts in the
morning and in the afternoon. So as not to impact other QC
operations too severely, it was decided to sample 10 parts,
twice a day, for three days from each of the three machines.
Daily throughput will be recorded as well.
We are expecting readings around .125 +/- .003 inches. The
parts will be measured using a standard micrometer with
readings recorded to 0.0001 of an inch. Throughput will be
measured by reading the part counters on the machines at the
end of each day.
Data The following are the data that were collected for this study.
MACHINE DAY TIME SAMPLE
DIAMETER
(1-3) (1-3) 1 = AM (1-10)
(inches)
2 = PM
---------------------------------------------------
---
1 1 1 1
0.1247
1 1 1 2
0.1264
1 1 1 3
0.1252
1 1 1 4
0.1253
1 1 1 5
0.1263
1 1 1 6
0.1251
1 1 1 7
0.1254
1 1 1 8
0.1239
1 1 1 9
0.1235
1 1 1 10
0.1257
1 1 2 1
0.1271
1 1 2 2
0.1253
1 1 2 3
0.1265
1 1 2 4
0.1254
1 1 2 5
0.1243
1 1 2 6
0.124
1 1 2 7
0.1246
1 1 2 8
0.1244
1 1 2 9
0.1271
1 1 2 10
0.1241
1 2 1 1
0.1251
1 2 1 2
0.1238
1 2 1 3
0.1255
1 2 1 4
0.1234
1 2 1 5
0.1235
1 2 1 6
0.1266
1 2 1 7
0.125
1 2 1 8
0.1246
1 2 1 9
0.1243
1 2 1 10
0.1248
1 2 2 1
0.1248
1 2 2 2
0.1235
1 2 2 3
0.1243
1 2 2 4
0.1265
1 2 2 5
0.127
1 2 2 6
0.1229
1 2 2 7
0.125
1 2 2 8
0.1248
1 2 2 9
0.1252
1 2 2 10
0.1243
1 3 1 1
0.1255
1 3 1 2
0.1237
1 3 1 3
0.1235
1 3 1 4
0.1264
1 3 1 5
0.1239
1 3 1 6
0.1266
1 3 1 7
0.1242
1 3 1 8
0.1231
1 3 1 9
0.1232
1 3 1 10
0.1244
1 3 2 1
0.1233
1 3 2 2
0.1237
1 3 2 3
0.1244
1 3 2 4
0.1254
1 3 2 5
0.1247
1 3 2 6
0.1254
1 3 2 7
0.1258
1 3 2 8
0.126
1 3 2 9
0.1235
1 3 2 10
0.1273
2 1 1 1
0.1239
2 1 1 2
0.1239
2 1 1 3
0.1239
2 1 1 4
0.1231
2 1 1 5
0.1221
2 1 1 6
0.1216
2 1 1 7
0.1233
2 1 1 8
0.1228
2 1 1 9
0.1227
2 1 1 10
0.1229
2 1 2 1
0.122
2 1 2 2
0.1239
2 1 2 3
0.1237
2 1 2 4
0.1216
2 1 2 5
0.1235
2 1 2 6
0.124
2 1 2 7
0.1224
2 1 2 8
0.1236
2 1 2 9
0.1236
2 1 2 10
0.1217
2 2 1 1
0.1247
2 2 1 2
0.122
2 2 1 3
0.1218
2 2 1 4
0.1237
2 2 1 5
0.1234
2 2 1 6
0.1229
2 2 1 7
0.1235
2 2 1 8
0.1237
2 2 1 9
0.1224
2 2 1 10
0.1224
2 2 2 1
0.1239
2 2 2 2
0.1226
2 2 2 3
0.1224
2 2 2 4
0.1239
2 2 2 5
0.1237
2 2 2 6
0.1227
2 2 2 7
0.1218
2 2 2 8
0.122
2 2 2 9
0.1231
2 2 2 10
0.1244
2 3 1 1
0.1219
2 3 1 2
0.1243
2 3 1 3
0.1231
2 3 1 4
0.1223
2 3 1 5
0.1218
2 3 1 6
0.1218
2 3 1 7
0.1225
2 3 1 8
0.1238
2 3 1 9
0.1244
2 3 1 10
0.1236
2 3 2 1
0.1231
2 3 2 2
0.1223
2 3 2 3
0.1241
2 3 2 4
0.1215
2 3 2 5
0.1221
2 3 2 6
0.1236
2 3 2 7
0.1229
2 3 2 8
0.1205
2 3 2 9
0.1241
2 3 2 10
0.1232
3 1 1 1
0.1255
3 1 1 2
0.1215
3 1 1 3
0.1219
3 1 1 4
0.1253
3 1 1 5
0.1232
3 1 1 6
0.1266
3 1 1 7
0.1271
3 1 1 8
0.1209
3 1 1 9
0.1212
3 1 1 10
0.1249
3 1 2 1
0.1228
3 1 2 2
0.126
3 1 2 3
0.1242
3 1 2 4
0.1236
3 1 2 5
0.1248
3 1 2 6
0.1243
3 1 2 7
0.126
3 1 2 8
0.1231
3 1 2 9
0.1234
3 1 2 10
0.1246
3 2 1 1
0.1207
3 2 1 2
0.1279
3 2 1 3
0.1268
3 2 1 4
0.1222
3 2 1 5
0.1244
3 2 1 6
0.1225
3 2 1 7
0.1234
3 2 1 8
0.1244
3 2 1 9
0.1207
3 2 1 10
0.1264
3 2 2 1
0.1224
3 2 2 2
0.1254
3 2 2 3
0.1237
3 2 2 4
0.1254
3 2 2 5
0.1269
3 2 2 6
0.1236
3 2 2 7
0.1248
3 2 2 8
0.1253
3 2 2 9
0.1252
3 2 2 10
0.1237
3 3 1 1
0.1217
3 3 1 2
0.122
3 3 1 3
0.1227
3 3 1 4
0.1202
3 3 1 5
0.127
3 3 1 6
0.1224
3 3 1 7
0.1219
3 3 1 8
0.1266
3 3 1 9
0.1254
3 3 1 10
0.1258
3 3 2 1
0.1236
3 3 2 2
0.1247
3 3 2 3
0.124
3 3 2 4
0.1235
3 3 2 5
0.124
3 3 2 6
0.1217
3 3 2 7
0.1235
3 3 2 8
0.1242
3 3 2 9
0.1247
3 3 2 10
0.125
3.5.2.2. Box Plots by Factors

3.5. Case Studies
Initial Steps The initial step is to plot box plots of the measured diameter for each of the
explanatory variables.
Box Plot by
Machine
The following is a box plot of the diameter by machine.
Conclusions
From Box
Plot
1. The location appears to be significantly different for the three machines, with
machine 2 having the smallest median diameter and machine 1 having the
largest median diameter.
2. Machines 1 and 2 have comparable variability while machine 3 has somewhat
larger variability.
Box Plot by
Day
The following is a box plot of the diameter by day.
Conclusions
From Box
Plot
We can draw the following conclusion from this box plot. Neither the location nor
the spread seem to differ significantly by day.
Box Plot by
Time of Day
The following is a box plot of the time of day.
Conclusion
From Box
Plot
We can draw the following conclusion from this box plot. Neither the location nor
the spread seem to differ significantly by time of day.
Box Plot by
Sample
Number
The following is a box plot of the sample number.
Conclusion
From Box
Plot
We can draw the following conclusion from this box plot. Although there are some
minor differences in location and spread between the samples, these differences do
not show a noticeable pattern and do not seem significant.

3.5. Case Studies
Analysis of
Variance
Using All
Factors
We can confirm our interpretation of the box plots by running an analysis of
variance when all four factors are included.
Source DF Sum of Mean F Statistic Prob > F
Squares Square
------------------------------------------------------------------
Machine 2 0.000111 0.000055 29.3159 1.3e-11
Day 2 0.000004 0.000002 0.9884 0.37
Time 1 0.000002 0.000002 1.2478 0.27
Sample 9 0.000009 0.000001 0.5205 0.86
Residual 165 0.000312 0.000002
------------------------------------------------------------------
Corrected Total 179 0.000437 0.000002
Interpretation
of ANOVA
Output
We fit the model
which has an overall mean, as opposed to the model
These models are mathematically equivalent. The effect estimates in the first model
are relative to the overall mean. The effect estimates for the second model can be
obtained by simply adding the overall mean to effect estimates from the first model.
Only the machine factor is statistically significant. This confirms what the box plots
in the previous section had indicated graphically.
Analysis of
Variance
Using Only
Machine
The previous analysis of variance indicated that only the machine factor was
statistically significant. The following table displays the ANOVA results using only
the machine factor.
Squares Square
------------------------------------------------------------------
Machine 2 0.000111 0.000055 30.0094 6.0E-12
Residual 177 0.000327 0.000002
------------------------------------------------------------------
Corrected Total 179 0.000437 0.000002
Interpretation
of ANOVA
Output
At this stage, we are interested in the level means for the machine variable. These
can be summarized in the following table.
Machine Means for One-way ANOVA
Level Number Mean Standard Error Lower 95% CI Upper 95% CI
1 60 0.124887 0.00018 0.12454 0.12523
2 60 0.122968 0.00018 0.12262 0.12331
3 60 0.124022 0.00018 0.12368 0.12437
Model
Validation
As a final step, we validate the model by generating a 4-plot of the residuals.
The 4-plot does not indicate any significant problems with the ANOVA model.
3.5.2.4. Throughput

3.5. Case Studies
3.5.2.4. Throughput
Summary of
Throughput
The throughput is summarized in the following table (this was part of the original
data collection, not the result of analysis).
Machine Day 1 Day 2 Day 3
1 576 604 583
2 657 604 586
3 510 546 571
This table shows that machine 3 had significantly lower throughput.
Graphical
Representation
of Throughput
We can show the throughput graphically.
The graph clearly shows the lower throughput for machine 3.
Analysis of
Variance for
Throughput
We can confirm the statistical significance of the lower throughput of machine 3 by
running an analysis of variance.
3.5.2.4. Throughput
Squares Square
-------------------------------------------------------------------
Machine 2 8216.89 4108.45 4.9007 0.0547
Residual 6 5030.00 838.33
-------------------------------------------------------------------
Corrected Total 8 13246.89 1655.86
Interpretation
of ANOVA
Output
We summarize the level means for machine 3 in the following table.
Machine 3 Level Means for One-way ANOVA
Level Number Mean Standard Error Lower 95% CI Upper 95% CI
1 3 587.667 16.717 546.76 628.57
2 3 615.667 16.717 574.76 656.57
3 3 542.33 16.717 501.43 583.24

3.5. Case Studies
Final
Conclusions
The analysis shows that machines 1 and 2 had about the same
variablity but significantly different locations. The
throughput for machine 2 was also higher with greater
variability than for machine 1. An interview with the operator
revealed that he realized the second machine was not set
correctly. However, he did not want to change the settings
because he knew a study was being conducted and was afraid
he might impact the results by making changes. Machine 3
had significantly more variation and lower throughput. The
operator indicated that the machine had to be taken down
several times for minor repairs. Given the preceeding
analysis results, the team recommended replacing machine 3.

3.5. Case Studies
View
Dataplot
Macro for
this Case
Study
case study description on the previous page using Dataplot, if
you have downloaded and installed it. Output from each
analysis step below will be displayed in one or more of the
Dataplot windows. The four main windows are the Output
window, the Graphics window, the Command History window
and the Data Sheet window. Across the top of the main
windows there are menus for executing Dataplot commands.
Data Analysis Steps
Results and
Conclusions
more detailed

1. You have read 5
columns of numbers
into Dataplot,
variables machine,
day, time,
sample, and diameter.
2. Box Plots by Factor Variables
1. Generate a box plot by machine.
2. Generate a box plot by day.
3. Generate a box plot by time of
day.
1. The box plot
shows significant
variation for
both location and
spread.
2. The box plot
location or
spread effects for
day.
4. Generate a box plot by
sample.
3. The box plot
location or
spread effects for
time of day.
4. The box plot
location or
spread effects for
sample.
1. Perform an analysis of variance
with all factors.
2. Perform an analysis of variance
with only the machine factor.
3. Perform model validation by
generating a 4-plot of the
residuals.
1. The analysis of
variance shows
that only the
machine factor
is statistically
significant.
2. The analysis of
variance shows
the overall mean
and the
effect estimates
for the levels
of the machine
variable.
3. The 4-plot of
the residuals does
not indicate any
significant
problems with the
model.
4. Graph of Throughput
1. Generate a graph of the
throughput.
2. Perform an analysis of
variance of the throughput.
1. The graph shows
the throughput
for machine 3 is
lower than
the other
machines.
2. The effect
estimates from the
ANIVA are given.
3.6. References

3.6. References
Box, G.E.P., Hunter, W.G., and Hunter, J.S. (1978), Statistics for
Experimenters, John Wiley and Sons, New York.
Cleveland, W.S. (1993), Visualizing Data, Hobart Press, New Jersey.
Hoaglin, D.C., Mosteller, F., and Tukey, J.W. (1985), Exploring Data
Tables, Trends, and Shapes, John Wiley and Sons, New York.
Hoaglin, D.C., Mosteller, F., and Tukey, J.W. (1991), Fundamentals of
Exploratory Analysis of Variance, John Wiley and Sons, New York.
4. Process Modeling
http://www.itl.nist.gov/div898/handbook/pmd/pmd.htm[6/27/2012 2:16:03 PM]

4. Process Modeling
The goal for this chapter is to present the background and specific analysis
techniques needed to construct a statistical model that describes a particular
scientific or engineering process. The types of models discussed in this
chapter are limited to those based on an explicit mathematical function.
These types of models can be used for prediction of process outputs, for
calibration, or for process optimization.
1. Introduction
1. Definition
2. Terminology
3. Uses
4. Methods
2. Assumptions
1. Assumptions
3. Design
1. Definition
2. Importance
3. Design Principles
4. Optimal Designs
5. Assessment
4. Analysis
1. Modeling Steps
2. Model Selection
3. Model Fitting
4. Model Validation
5. Model Improvement
5. Interpretation & Use
1. Prediction
2. Calibration
3. Optimization
6. Case Studies
1. Load Cell Output
2. Alaska Pipeline
3. Ultrasonic Reference Block
4. Thermal Expansion of Copper
Detailed Table of Contents: Process Modeling
References: Process Modeling
Appendix: Some Useful Functions for Process Modeling
4. Process Modeling
http://www.itl.nist.gov/div898/handbook/pmd/pmd_d.htm[6/27/2012 2:16:30 PM]

4. Process Modeling - Detailed Table of Contents [4.]
The goal for this chapter is to present the background and specific analysis techniques needed to construct a
statistical model that describes a particular scientific or engineering process. The types of models discussed in
this chapter are limited to those based on an explicit mathematical function. These types of models can be used
for prediction of process outputs, for calibration, or for process optimization.
1. Introduction to Process Modeling [4.1.]
1. What is process modeling? [4.1.1.]
2. What terminology do statisticians use to describe process models? [4.1.2.]
3. What are process models used for? [4.1.3.]
1. Estimation [4.1.3.1.]
2. Prediction [4.1.3.2.]
3. Calibration [4.1.3.3.]
4. Optimization [4.1.3.4.]
4. What are some of the different statistical methods for model building? [4.1.4.]
1. Linear Least Squares Regression [4.1.4.1.]
2. Nonlinear Least Squares Regression [4.1.4.2.]
3. Weighted Least Squares Regression [4.1.4.3.]
4. LOESS (aka LOWESS) [4.1.4.4.]
2. Underlying Assumptions for Process Modeling [4.2.]
1. What are the typical underlying assumptions in process modeling? [4.2.1.]
1. The process is a statistical process. [4.2.1.1.]
2. The means of the random errors are zero. [4.2.1.2.]
3. The random errors have a constant standard deviation. [4.2.1.3.]
4. The random errors follow a normal distribution. [4.2.1.4.]
5. The data are randomly sampled from the process. [4.2.1.5.]
6. The explanatory variables are observed without error. [4.2.1.6.]
3. Data Collection for Process Modeling [4.3.]
1. What is design of experiments (DOE)? [4.3.1.]
2. Why is experimental design important for process modeling? [4.3.2.]
3. What are some general design principles for process modeling? [4.3.3.]
4. I've heard some people refer to "optimal" designs, shouldn't I use those? [4.3.4.]
5. How can I tell if a particular experimental design is good for my application? [4.3.5.]
4. Data Analysis for Process Modeling [4.4.]
1. What are the basic steps for developing an effective process model? [4.4.1.]
2. How do I select a function to describe my process? [4.4.2.]
1. Incorporating Scientific Knowledge into Function Selection [4.4.2.1.]
2. Using the Data to Select an Appropriate Function [4.4.2.2.]
3. Using Methods that Do Not Require Function Specification [4.4.2.3.]
3. How are estimates of the unknown parameters obtained? [4.4.3.]
1. Least Squares [4.4.3.1.]
2. Weighted Least Squares [4.4.3.2.]
4. Process Modeling
4. How can I tell if a model fits my data? [4.4.4.]
1. How can I assess the sufficiency of the functional part of the model? [4.4.4.1.]
2. How can I detect non-constant variation across the data? [4.4.4.2.]
3. How can I tell if there was drift in the measurement process? [4.4.4.3.]
4. How can I assess whether the random errors are independent from one to the next? [4.4.4.4.]
5. How can I test whether or not the random errors are distributed normally? [4.4.4.5.]
6. How can I test whether any significant terms are missing or misspecified in the functional part
of the model? [4.4.4.6.]
7. How can I test whether all of the terms in the functional part of the model are
necessary? [4.4.4.7.]
5. If my current model does not fit the data well, how can I improve it? [4.4.5.]
1. Updating the Function Based on Residual Plots [4.4.5.1.]
2. Accounting for Non-Constant Variation Across the Data [4.4.5.2.]
3. Accounting for Errors with a Non-Normal Distribution [4.4.5.3.]
5. Use and Interpretation of Process Models [4.5.]
1. What types of predictions can I make using the model? [4.5.1.]
1. How do I estimate the average response for a particular set of predictor variable
values? [4.5.1.1.]
2. How can I predict the value and and estimate the uncertainty of a single response? [4.5.1.2.]
2. How can I use my process model for calibration? [4.5.2.]
1. Single-Use Calibration Intervals [4.5.2.1.]
3. How can I optimize my process using the process model? [4.5.3.]
6. Case Studies in Process Modeling [4.6.]
1. Load Cell Calibration [4.6.1.]
1. Background & Data [4.6.1.1.]
2. Selection of Initial Model [4.6.1.2.]
3. Model Fitting - Initial Model [4.6.1.3.]
4. Graphical Residual Analysis - Initial Model [4.6.1.4.]
5. Interpretation of Numerical Output - Initial Model [4.6.1.5.]
6. Model Refinement [4.6.1.6.]
7. Model Fitting - Model #2 [4.6.1.7.]
8. Graphical Residual Analysis - Model #2 [4.6.1.8.]
9. Interpretation of Numerical Output - Model #2 [4.6.1.9.]
10. Use of the Model for Calibration [4.6.1.10.]
2. Alaska Pipeline [4.6.2.]
2. Check for Batch Effect [4.6.2.2.]
3. Initial Linear Fit [4.6.2.3.]
4. Transformations to Improve Fit and Equalize Variances [4.6.2.4.]
5. Weighting to Improve Fit [4.6.2.5.]
6. Compare the Fits [4.6.2.6.]
3. Ultrasonic Reference Block Study [4.6.3.]
2. Initial Non-Linear Fit [4.6.3.2.]
3. Transformations to Improve Fit [4.6.3.3.]
4. Weighting to Improve Fit [4.6.3.4.]
5. Compare the Fits [4.6.3.5.]
4. Thermal Expansion of Copper Case Study [4.6.4.]
4. Process Modeling
2. Rational Function Models [4.6.4.2.]
3. Initial Plot of Data [4.6.4.3.]
4. Quadratic/Quadratic Rational Function Model [4.6.4.4.]
5. Cubic/Cubic Rational Function Model [4.6.4.5.]
7. References For Chapter 4: Process Modeling [4.7.]
8. Some Useful Functions for Process Modeling [4.8.]
1. Univariate Functions [4.8.1.]
1. Polynomial Functions [4.8.1.1.]
1. Straight Line [4.8.1.1.1.]
2. Quadratic Polynomial [4.8.1.1.2.]
3. Cubic Polynomial [4.8.1.1.3.]
2. Rational Functions [4.8.1.2.]
1. Constant / Linear Rational Function [4.8.1.2.1.]
2. Linear / Linear Rational Function [4.8.1.2.2.]
3. Linear / Quadratic Rational Function [4.8.1.2.3.]
4. Quadratic / Linear Rational Function [4.8.1.2.4.]
5. Quadratic / Quadratic Rational Function [4.8.1.2.5.]
6. Cubic / Linear Rational Function [4.8.1.2.6.]
7. Cubic / Quadratic Rational Function [4.8.1.2.7.]
8. Linear / Cubic Rational Function [4.8.1.2.8.]
9. Quadratic / Cubic Rational Function [4.8.1.2.9.]
10. Cubic / Cubic Rational Function [4.8.1.2.10.]
11. Determining m and n for Rational Function Models [4.8.1.2.11.]
4.1. Introduction to Process Modeling
http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd1.htm[6/27/2012 2:16:09 PM]

4. Process Modeling
Overview
of Section
4.1
The goal for this section is to give the big picture of function-
based process modeling. This includes a discussion of what
process modeling is, the goals of process modeling, and a
comparison of the different statistical methods used for model
building. Detailed information on how to collect data,
construct appropriate models, interpret output, and use process
models is covered in the following sections. The final section
of the chapter contains case studies that illustrate the general
information presented in the first five sections using data from
a variety of scientific and engineering applications.
Contents
of Section
4.1
1. What is process modeling?
2. What terminology do statisticians use to describe
process models?
3. What are process models used for?
1. Estimation
2. Prediction
3. Calibration
4. Optimization
4. What are some of the statistical methods for model
building?
1. Linear Least Squares Regression
2. Nonlinear Least Squares Regression
3. Weighted Least Squares Regression
4. LOESS (aka LOWESS)
4.1.1. What is process modeling?

4. Process Modeling
Basic
Definition
Process modeling is the concise description of the total variation in one quantity, ,
by partitioning it into
1. a deterministic component given by a mathematical function of one or more
other quantities, , plus
2. a random component that follows a particular probability distribution.
Example For example, the total variation of the measured pressure of a fixed amount of a gas
in a tank can be described by partitioning the variability into its deterministic part,
which is a function of the temperature of the gas, plus some left-over random error.
Charles' Law states that the pressure of a gas is proportional to its temperature under
the conditions described here, and in this case most of the variation will be
deterministic. However, due to measurement error in the pressure gauge, the
relationship will not be purely deterministic. The random errors cannot be
characterized individually, but will follow some probability distribution that will
describe the relative frequencies of occurrence of different-sized errors.
Graphical
Interpretation
Using the example above, the definition of process modeling can be graphically
depicted like this:
Click Figure
for Full-Sized
Copy
The top left plot in the figure shows pressure data that vary deterministically with
temperature except for a small amount of random error. The relationship between
pressure and temperature is a straight line, but not a perfect straight line. The top
row plots on the right-hand side of the equals sign show a partitioning of the data
into a perfect straight line and the remaining "unexplained" random variation in the
data (note the different vertical scales of these plots). The plots in the middle row of
the figure show the deterministic structure in the data again and a histogram of the
random variation. The histogram shows the relative frequencies of observing
different-sized random errors. The bottom row of the figure shows how the relative
frequencies of the random errors can be summarized by a (normal) probability
distribution.
An Example
from a More
Complex
Process
Of course, the straight-line example is one of the simplest functions used for process
modeling. Another example is shown below. The concept is identical to the straight-
line example, but the structure in the data is more complex. The variation in is
partitioned into a deterministic part, which is a function of another variable, , plus
some left-over random variation. (Again note the difference in the vertical axis
scales of the two plots in the top right of the figure.) A probability distribution
describes the leftover random variation.
An Example
with Multiple
Explanatory
Variables
The examples of process modeling shown above have only one explanatory variable
but the concept easily extends to cases with more than one explanatory variable. The
three-dimensional perspective plots below show an example with two explanatory
variables. Examples with three or more explanatory variables are exactly analogous,
but are difficult to show graphically.
4.1.2. What terminology do statisticians use to describe process models?

4. Process Modeling
4.1.2. What terminology do statisticians use to
describe process models?
Model
Components
There are three main parts to every process model. These are
1. the response variable, usually denoted by ,
2. the mathematical function, usually denoted as ,
and
3. the random errors, usually denoted by .
Form of
Model
The general form of the model is
.
All process models discussed in this chapter have this general
form. As alluded to earlier, the random errors that are
included in the model make the relationship between the
response variable and the predictor variables a "statistical"
one, rather than a perfect deterministic one. This is because
the functional relationship between the response and
predictors holds only on average, not for each data point.
Some of the details about the different parts of the model are
discussed below, along with alternate terminology for the
different components of the model.
Response
Variable
The response variable, , is a quantity that varies in a way that
we hope to be able to summarize and exploit via the modeling
process. Generally it is known that the variation of the
response variable is systematically related to the values of one
or more other variables before the modeling process is begun,
although testing the existence and nature of this dependence is
part of the modeling process itself.
Mathematical
Function
The mathematical function consists of two parts. These parts
are the predictor variables, , and the parameters,
. The predictor variables are observed along with
the response variable. They are the quantities described on the
previous page as inputs to the mathematical function, .
The collection of all of the predictor variables is denoted by
for short.
The parameters are the quantities that will be estimated during
the modeling process. Their true values are unknown and
unknowable, except in simulation experiments. As for the
predictor variables, the collection of all of the parameters is
denoted by for short.
The parameters and predictor variables are combined in
different forms to give the function used to describe the
deterministic variation in the response variable. For a straight
line with an unknown intercept and slope, for example, there
are two parameters and one predictor variable
.
For a straight line with a known slope of one, but an unknown
intercept, there would only be one parameter
.
For a quadratic surface with two predictor variables, there are
six parameters for the full model.
.
Random
Error
Like the parameters in the mathematical function, the random
errors are unknown. They are simply the difference between
the data and the mathematical function. They are assumed to
follow a particular probability distribution, however, which is
used to describe their aggregate behavior. The probability
distribution that describes the errors has a mean of zero and
an unknown standard deviation, denoted by , that is another
parameter in the model, like the 's.
Alternate
Terminology
Unfortunately, there are no completely standardardized names
for the parts of the model discussed above. Other publications
or software may use different terminology. For example,
another common name for the response variable is "dependent
variable". The response variable is also simply called "the
response" for short. Other names for the predictor variables
include "explanatory variables", "independent variables",
"predictors" and "regressors". The mathematical function used
to describe the deterministic variation in the response variable
is sometimes called the "regression function", the "regression
equation", the "smoothing function", or the "smooth".
Scope of
"Model"
In its correct usage, the term "model" refers to the equation
above and also includes the underlying assumptions made
about the probability distribution used to describe the
variation of the random errors. Often, however, people will
also use the term "model" when referring specifically to the
mathematical function describing the deterministic variation in
the data. Since the function is part of the model, the more
limited usage is not wrong, but it is important to remember
that the term "model" might refer to more than just the
mathematical function.
4.1.3. What are process models used for?

4. Process Modeling
Three Main
Purposes
Process models are used for four main purposes:
1. estimation,
2. prediction,
3. calibration, and
4. optimization.
The rest of this page lists brief explanations of the different
uses of process models. More detailed explanations of the
uses for process models are given in the subsections of this
section listed at the bottom of this page.
Estimation The goal of estimation is to determine the value of the
regression function (i.e., the average value of the response
variable), for a particular combination of the values of the
predictor variables. Regression function values can be
estimated for any combination of predictor variable values,
including values for which no data have been measured or
observed. Function values estimated for points within the
observed space of predictor variable values are sometimes
called interpolations. Estimation of regression function
values for points outside the observed space of predictor
variable values, called extrapolations, are sometimes
necessary, but require caution.
Prediction The goal of prediction is to determine either
1. the value of a new observation of the response
variable, or
2. the values of a specified proportion of all future
observations of the response variable
for a particular combination of the values of the predictor
variables. Predictions can be made for any combination of
predictor variable values, including values for which no data
have been measured or observed. As in the case of
estimation, predictions made outside the observed space of
predictor variable values are sometimes necessary, but
require caution.
Calibration The goal of calibration is to quantitatively relate
measurements made using one measurement system to those
of another measurement system. This is done so that
measurements can be compared in common units or to tie
results from a relative measurement method to absolute
units.
Optimization Optimization is performed to determine the values of process
inputs that should be used to obtain the desired process
output. Typical optimization goals might be to maximize the
yield of a process, to minimize the processing time required
to fabricate a product, or to hit a target product specification
with minimum variation in order to maintain specified
tolerances.
Further
Details
1. Estimation
2. Prediction
3. Calibration
4. Optimization
4.1.4. What are some of the different statistical methods for model building?

4. Process Modeling
4.1.4. What are some of the different statistical
methods for model building?
Selecting an
Appropriate
Stat
Method:
General
Case
For many types of data analysis problems there are no more
than a couple of general approaches to be considered on the
route to the problem's solution. For example, there is often a
dichotomy between highly-efficient methods appropriate for
data with noise from a normal distribution and more general
methods for data with other types of noise. Within the
different approaches for a specific problem type, there are
usually at most a few competing statistical tools that can be
used to obtain an appropriate solution. The bottom line for
most types of data analysis problems is that selection of the
best statistical method to solve the problem is largely
determined by the goal of the analysis and the nature of the
data.
Selecting an
Appropriate
Stat
Method:
Modeling
Model building, however, is different from most other areas
of statistics with regard to method selection. There are more
general approaches and more competing techniques available
for model building than for most other types of problems.
There is often more than one statistical tool that can be
effectively applied to a given modeling application. The large
menu of methods applicable to modeling problems means
that there is both more opportunity for effective and efficient
solutions and more potential to spend time doing different
analyses, comparing different solutions and mastering the use
of different tools. The remainder of this section will
introduce and briefly discuss some of the most popular and
well-established statistical techniques that are useful for
different model building situations.
Process
Modeling
Methods
1. Linear Least Squares Regression
2. Nonlinear Least Squares Regression
3. Weighted Least Squares Regression
4. LOESS (aka LOWESS)
4.2. Underlying Assumptions for Process Modeling

4. Process Modeling
4.2. Underlying Assumptions for Process
Modeling
Implicit
Assumptions
Underlie
Most
Actions
Most, if not all, thoughtful actions that people take are based
on ideas, or assumptions, about how those actions will affect
the goals they want to achieve. The actual assumptions used
to decide on a particular course of action are rarely laid out
explicitly, however. Instead, they are only implied by the
nature of the action itself. Implicit assumptions are inherent
to process modeling actions, just as they are to most other
types of action. It is important to understand what the
implicit assumptions are for any process modeling method
because the validity of these assumptions affect whether or
not the goals of the analysis will be met.
Checking
Assumptions
Provides
Feedback
on Actions
If the implicit assumptions that underlie a particular action
are not true, then that action is not likely to meet
expectations either. Sometimes it is abundantly clear when a
goal has been met, but unfortunately that is not always the
case. In particular, it is usually not possible to obtain
immediate feedback on the attainment of goals in most
process modeling applications. The goals of process
modeling, sucha as answering a scientific or engineering
question, depend on the correctness of a process model,
which can often only be directly and absolutely determined
over time. In lieu of immediate, direct feedback, however,
indirect information on the effectiveness of a process
modeling analysis can be obtained by checking the validity
of the underlying assumptions. Confirming that the
underlying assumptions are valid helps ensure that the
methods of analysis were appropriate and that the results will
be consistent with the goals.
Overview of
Section 4.2
This section discusses the specific underlying assumptions
associated with most model-fitting methods. In discussing
the underlying assumptions, some background is also
provided on the consequences of stopping the modeling
process short of completion and leaving the results of an
analysis at odds with the underlying assumptions. Specific
data analysis methods that can be used to check whether or
not the assumptions hold in a particular case are discussed in
Section 4.4.4.
Contents of
Section 4.2
1. What are the typical underlying assumptions in process
modeling?
1. The process is a statistical process.
2. The means of the random errors are zero.
3. The random errors have a constant standard
deviation.
4. The random errors follow a normal distribution.
5. The data are randomly sampled from the
process.
6. The explanatory variables are observed without
error.
4.2.1. What are the typical underlying assumptions in process modeling?

4. Process Modeling
4.2.1. What are the typical underlying
assumptions in process modeling?
Overview of
Section
4.2.1
This section lists the typical assumptions underlying most
process modeling methods. On each of the following pages,
one of the six major assumptions is described individually;
the reasons for it's importance are also briefly discussed; and
any methods that are not subject to that particular assumption
are noted. As discussed on the previous page, these are
implicit assumptions based on properties inherent to the
process modeling methods themselves. Successful use of
these methods in any particular application hinges on the
validity of the underlying assumptions, whether their
existence is acknowledged or not. Section 4.4.4 discusses
methods for checking the validity of these assumptions.
Typical
Assumptions
for Process
Modeling
1. The process is a statistical process.
2. The means of the random errors are zero.
3. The random errors have a constant standard deviation.
4. The random errors follow a normal distribution.
5. The data are randomly sampled from the process.
6. The explanatory variables are observed without error.
4.3. Data Collection for Process Modeling

4. Process Modeling
Collecting
Good Data
This section lays out some general principles for collecting
data for construction of process models. Using well-planned
data collection procedures is often the difference between
successful and unsuccessful experiments. In addition, well-
designed experiments are often less expensive than those that
are less well thought-out, regardless of overall success or
failure.
Specifically, this section will answer the question:
What can the analyst do even prior to collecting the data
(that is, at the experimental design stage) that would
allow the analyst to do an optimal job of modeling the
process?
Contents:
Section 3
This section deals with the following five questions:
1. What is design of experiments (DOE)?
2. Why is experimental design important for process
modeling?
3. What are some general design principles for process
modeling?
4. I've heard some people refer to "optimal" designs,
shouldn't I use those?
5. How can I tell if a particular experimental design is
good for my application?
4.3.1. What is design of experiments (DOE)?

4. Process Modeling
Systematic
Approach to
Data Collection
Design of experiments (DOE) is a systematic, rigorous
approach to engineering problem-solving that applies
principles and techniques at the data collection stage so
as to ensure the generation of valid, defensible, and
supportable engineering conclusions. In addition, all of
this is carried out under the constraint of a minimal
expenditure of engineering runs, time, and money.
DOE Problem
Areas
There are four general engineering problem areas in
which DOE may be applied:
1. Comparative
2. Screening/Characterizing
3. Modeling
4. Optimizing
Comparative In the first case, the engineer is interested in assessing
whether a change in a single factor has in fact resulted
in a change/improvement to the process as a whole.
Screening
Characterization
In the second case, the engineer is interested in
"understanding" the process as a whole in the sense that
he/she wishes (after design and analysis) to have in hand
a ranked list of important through unimportant factors
(most important to least important) that affect the
process.
Modeling In the third case, the engineer is interested in
functionally modeling the process with the output being
a good-fitting (= high predictive power) mathematical
function, and to have good (= maximal accuracy)
estimates of the coefficients in that function.
Optimizing In the fourth case, the engineer is interested in
determining optimal settings of the process factors; that
is, to determine for each factor the level of the factor
that optimizes the process response.
In this section, we focus on case 3: modeling.
4.3.2. Why is experimental design important for process modeling?

4. Process Modeling
4.3.2. Why is experimental design important for
process modeling?
Output from
Process
Model is
Fitted
Mathematical
Function
The output from process modeling is a fitted mathematical
function with estimated coefficients. For example, in
modeling resistivity, , as a function of dopant density, ,
an analyst may suggest the function
in which the coefficients to be estimated are , , and
. Even for a given functional form, there is an infinite
number of potential coefficient values that potentially may
be used. Each of these coefficient values will in turn yield
predicted values.
What are
Good
Coefficient
Values?
Poor values of the coefficients are those for which the
resulting predicted values are considerably different from
the observed raw data . Good values of the coefficients are
those for which the resulting predicted values are close to
the observed raw data . The best values of the coefficients
are those for which the resulting predicted values are close
to the observed raw data , and the statistical uncertainty
connected with each coefficient is small.
There are two considerations that are useful for the
generation of "best" coefficients:
1. Least squares criterion
2. Design of experiment principles
Least
Squares
Criterion
For a given data set (e.g., 10 ( , ) pairs), the most common
procedure for obtaining the coefficients for
is the least squares estimation criterion. This criterion yields
coefficients with predicted values that are closest to the raw
data in the sense that the sum of the squared differences
between the raw data and the predicted values is as small as
possible.
The overwhelming majority of regression programs today
use the least squares criterion for estimating the model
coefficients. Least squares estimates are popular because
1. the estimators are statistically optimal (BLUEs: Best
Linear Unbiased Estimators);
2. the estimation algorithm is mathematically tractable,
in closed form, and therefore easily programmable.
How then can this be improved? For a given set of values
it cannot be; but frequently the choice of the values is
under our control. If we can select the values, the
coefficients will have less variability than if the are not
controlled.
Design of
Experiment
Principles
As to what values should be used for the 's, we look to
established experimental design principles for guidance.
Principle 1:
Minimize
Coefficient
Estimation
Variation
The first principle of experimental design is to control the
values within the vector such that after the data are
collected, the subsequent model coefficients are as good, in
the sense of having the smallest variation, as possible.
The key underlying point with respect to design of
experiments and process modeling is that even though (for
simple ( , ) fitting, for example) the least squares criterion
may yield optimal (minimal variation) estimators for a
given distribution of values, some distributions of data in
the vector may yield better (smaller variation) coefficient
estimates than other vectors. If the analyst can specify the
values in the vector, then he or she may be able to
drastically change and reduce the noisiness of the
subsequent least squares coefficient estimates.
Five Designs To see the effect of experimental design on process
modeling, consider the following simplest case of fitting a
line:
Suppose the analyst can afford 10 observations (that is, 10 (
, ) pairs) for the purpose of determining optimal (that is,
minimal variation) estimators of and . What 10
values should be used for the purpose of collecting the
corresponding 10 values? Colloquially, where should the
10 values be sprinkled along the horizontal axis so as to
minimize the variation of the least squares estimated
coefficients for and ? Should the 10 values be:
1. ten equi-spaced values across the range of interest?
2. five replicated equi-spaced values across the range of
interest?
3. five values at the minimum of the range and five
values at the maximum of the range?
4. one value at the minimum, eight values at the mid-
range, and one value at the maximum?
5. four values at the minimum, two values at mid-range,
and four values at the maximum?
or (in terms of "quality" of the resulting estimates for
and ) perhaps it doesn't make any difference?
For each of the above five experimental designs, there will
of course be data collected, followed by the generation of
least squares estimates for and , and so each design
will in turn yield a fitted line.
Are the
Fitted Lines
Better for
Some
Designs?
But are the fitted lines, i.e., the fitted process models, better
for some designs than for others? Are the coefficient
estimator variances smaller for some designs than for
others? For given estimates, are the resulting predicted
values better (that is, closer to the observed values) than
for other designs? The answer to all of the above is YES. It
DOES make a difference.
The most popular answer to the above question about which
design to use for linear modeling is design #1 with ten equi-
spaced points. It can be shown, however, that the variance
of the estimated slope parameter depends on the design
according to the relationship
.
Therefore to obtain minimum variance estimators, one
maximizes the denominator on the right. To maximize the
denominator, it is (for an arbitrarily fixed ), best to position
the 's as far away from as possible. This is done by
positioning half of the 's at the lower extreme and the
other half at the upper extreme. This is design #3 above,
and this "dumbbell" design (half low and half high) is in
fact the best possible design for fitting a line. Upon
reflection, this is intuitively arrived at by the adage that "2
points define a line", and so it makes the most sense to
determine those 2 points as far apart as possible (at the
extremes) and as well as possible (having half the data at
each extreme). Hence the design of experiment solution to
model processing when the model is a line is the
"dumbbell" design--half the X's at each extreme.
What is the
Worst
What is the worst design in the above case? Of the five
designs, the worst design is the one that has maximum
Design? variation. In the mathematical expression above, it is the
one that minimizes the denominator, and so this is design
#4 above, for which almost all of the data are located at the
mid-range. Clearly the estimated line in this case is going to
chase the solitary point at each end and so the resulting
linear fit is intuitively inferior.
Designs 1, 2,
and 5
What about the other 3 designs? Designs 1, 2, and 5 are
useful only for the case when we think the model may be
linear, but we are not sure, and so we allow additional
points that permit fitting a line if appropriate, but build into
the design the "capacity" to fit beyond a line (e.g.,
quadratic, cubic, etc.) if necessary. In this regard, the
ordering of the designs would be
design 5 (if our worst-case model is quadratic),
design 2 (if our worst-case model is quartic)
design 1 (if our worst-case model is quintic and
beyond)
4.3.3. What are some general design principles for process modeling?

4. Process Modeling
4.3.3. What are some general design principles
for process modeling?
Experimental
Design
Principles
Applied to
Process
Modeling
There are six principles of experimental design as applied
to process modeling:
1. Capacity for Primary Model
2. Capacity for Alternative Model
3. Minimum Variance of Coefficient Estimators
4. Sample where the Variation Is
5. Replication
6. Randomization
We discuss each in detail below.
Capacity for
Primary
Model
For your best-guess model, make sure that the design has
the capacity for estimating the coefficients of that model.
For a simple example of this, if you are fitting a quadratic
model, then make sure you have at least three distinct
horixontal axis points.
Capacity for
Alternative
Model
If your best-guess model happens to be inadequate, make
sure that the design has the capacity to estimate the
coefficients of your best-guess back-up alternative model
(which means implicitly that you should have already
identified such a model). For a simple example, if you
suspect (but are not positive) that a linear model is
appropriate, then it is best to employ a globally robust
design (say, four points at each extreme and three points in
the middle, for a ten-point design) as opposed to the
locally optimal design (such as five points at each
extreme). The locally optimal design will provide a best fit
to the line, but have no capacity to fit a quadratic. The
globally robust design will provide a good (though not
optimal) fit to the line and additionally provide a good
(though not optimal) fit to the quadratic.
Minimum
Variance of
Coefficient
Estimators
For a given model, make sure the design has the property
of minimizing the variation of the least squares estimated
coefficients. This is a general principle that is always in
effect but which in practice is hard to implement for many
models beyond the simpler 1-factor
models. For more complicated 1-factor models, and for
most multi-factor
models, the expressions for the variance of the least
squares estimators, although available, are complicated and
assume more than the analyst typically knows. The net
result is that this principle, though important, is harder to
apply beyond the simple cases.
Sample Where
the Variation
Is (Non
Constant
Variance
Case)
Regardless of the simplicity or complexity of the model,
there are situations in which certain regions of the curve
are noisier than others. A simple case is when there is a
linear relationship between and but the recording
device is proportional rather than absolute and so larger
values of are intrinsically noisier than smaller values of
. In such cases, sampling where the variation is means to
have more replicated points in those regions that are
noisier. The practical answer to how many such replicated
points there should be is
with denoting the theoretical standard deviation for that
given region of the curve. Usually is estimated by a-
priori guesses for what the local standard deviations are.
Sample Where
the Variation
Is (Steep
Curve Case)
A common occurence for non-linear models is for some
regions of the curve to be steeper than others. For
example, in fitting an exponential model (small
corresponding to large , and large corresponding to
small ) it is often the case that the data in the steep
region are intrinsically noisier than the data in the
relatively flat regions. The reason for this is that
commonly the values themselves have a bit of noise and
this -noise gets translated into larger -noise in the steep
sections than in the shallow sections. In such cases, when
we know the shape of the response curve well enough to
identify steep-versus-shallow regions, it is often a good
idea to sample more heavily in the steep regions than in
the shallow regions. A practical rule-of-thumb for where
to position the values in such situations is to
1. sketch out your best guess for what the resulting
curve will be;
2. partition the vertical (that is the ) axis into equi-
spaced points (with denoting the total number of
data points that you can afford);
3. draw horizontal lines from each vertical axis point to
where it hits the sketched-in curve.
4. drop a vertical projection line from the curve
intersection point to the horizontal axis.
These will be the recommended values to use in the
design.
The above rough procedure for an exponentially
decreasing curve would thus yield a logarithmic
preponderance of points in the steep region of the curve
and relatively few points in the flatter part of the curve.
Replication If affordable, replication should be part of every design.
Replication allows us to compute a model-independent
estimate of the process standard deviation. Such an
estimate may then be used as a criterion in an objective
lack-of-fit test to assess whether a given model is
adequate. Such an objective lack-of-fit F-test can be
employed only if the design has built-in replication. Some
replication is essential; replication at every point is ideal.
Randomization Just because the 's have some natural ordering does not
mean that the data should be collected in the same order as
the 's. Some aspect of randomization should enter into
every experiment, and experiments for process modeling
are no exception. Thus if your are sampling ten points on a
curve, the ten values should not be collected by
sequentially stepping through the values from the
smallest to the largest. If you do so, and if some
extraneous drifting or wear occurs in the machine, the
operator, the environment, the measuring device, etc., then
that drift will unwittingly contaminate the values and in
turn contaminate the final fit. To minimize the effect of
such potential drift, it is best to randomize (use random
number tables) the sequence of the values. This will not
make the drift go away, but it will spread the drift effect
evenly over the entire curve, realistically inflating the
variation of the fitted values, and providing some
mechanism after the fact (at the residual analysis model
validation stage) for uncovering or discovering such a
drift. If you do not randomize the run sequence, you give
up your ability to detect such a drift if it occurs.
4.3.4. I've heard some people refer to "optimal" designs, shouldn't I use those?

4. Process Modeling
4.3.4. I've heard some people refer to "optimal"
designs, shouldn't I use those?
Classical
Designs Heavily
Used in Industry
The most heavily used designs in industry are the
"classical designs" (full factorial designs, fractional
factorial designs, Latin square designs, Box-Behnken
designs, etc.). They are so heavily used because they are
optimal in their own right and have served superbly well
in providing efficient insight into the underlying
structure of industrial processes.
Reasons
Classical
Designs May
Not Work
Cases do arise, however, for which the tabulated
classical designs do not cover a particular practical
situation. That is, user constraints preclude the use of
tabulated classical designs because such classical designs
do not accommodate user constraints. Such constraints
include:
1. Limited maximum number of runs:
User constraints in budget and time may dictate a
maximum allowable number of runs that is too
small or too "irregular" (e.g., "13") to be
accommodated by classical designs--even
fractional factorial designs.
2. Impossible factor combinations:
The user may have some factor combinations that
are impossible to run. Such combinations may at
times be specified (to maintain balance and
orthogonality) as part of a recommeded classical
design. If the user simply omits this impossible run
from the design, the net effect may be a reduction
in the quality and optimaltiy of the classical
design.
3. Too many levels:
The number of factors and/or the number of levels
of some factors intended for use may not be
included in tabulations of classical designs.
4. Complicated underlying model:
The user may be assuming an underlying model
that is too complicated (or too non-linear), so that
classical designs would be inappropriate.
What to Do If
Classical
Designs Do Not
Exist?
If user constraints are such that classical designs do not
exist to accommodate such constraints, then what is the
user to do?
The previous section's list of design criteria (capability
for the primary model, capability for the alternate model,
minimum variation of estimated coefficients, etc.) is a
good passive target to aim for in terms of desirable
design properties, but provides little help in terms of an
active formal construction methodology for generating a
design.
Common
Optimality
Criteria
To satisfy this need, an "optimal design" methodology
has been developed to generate a design when user
constraints preclude the use of tabulated classical
designs. Optimal designs may be optimal in many
different ways, and what may be an optimal design
according to one criterion may be suboptimal for other
criteria. Competing criteria have led to a literal alphabet-
soup collection of optimal design methodologies. The
four most popular ingredients in that "soup" are:
D-optimal
designs:
minimize the generalized variance of the
parameter estimators.
A-optimal
designs:
minimize the average variance of the
parameter estimators.
G-optimal
designs:
minimize the maximum variance of the
predicted values.
V-optimal
designs:
minimize the average variance of the
predicted values.
Need 1: a Model The motivation for optimal designs is the practical
constraints that the user has. The advantage of optimal
designs is that they do provide a reasonable design-
generating methodology when no other mechanism
exists. The disadvantage of optimal designs is that they
require a model from the user. The user may not have
this model.
All optimal designs are model-dependent, and so the
quality of the final engineering conclusions that result
from the ensuing design, data, and analysis is dependent
on the correctness of the analyst's assumed model. For
example, if the responses from a particular process are
actually being drawn from a cubic model and the analyst
assumes a linear model and uses the corresponding
optimal design to generate data and perform the data
analysis, then the final engineering conclusions will be
flawed and invalid. Hence one price for obtaining an in-
hand generated design is the designation of a model. All
optimal designs need a model; without a model, the
optimal design-generation methodology cannot be used,
and general design principles must be reverted to.
Need 2: a
Candidate Set of
Points
The other price for using optimal design methodology is
a user-specified set of candidate points. Optimal designs
will not generate the best design points from some
continuous region--that is too much to ask of the
mathematics. Optimal designs will generate the best
subset of points from a larger superset of candidate
points. The user must specify this candidate set of
points. Most commonly, the superset of candidate points
is the full factorial design over a fine-enough grid of the
factor space with which the analyst is comfortable. If the
grid is too fine, and the resulting superset overly large,
then the optimal design methodology may prove
computationally challenging.
Optimal
Designs are
Computationally
Intensive
The optimal design-generation methodology is
computationally intensive. Some of the designs (e.g., D-
optimal) are better than other designs (such as A-optimal
and G-optimal) in regard to efficiency of the underlying
search algorithm. Like most mathematical optimization
techniques, there is no iron-clad guarantee that the result
from the optimal design methodology is in fact the true
optimum. However, the results are usually satisfactory
from a practical point of view, and are far superior than
any ad hoc designs.
For further details about optimal designs, the analyst is
referred to Montgomery (2001).
4.3.5. How can I tell if a particular experimental design is good for my application?

4. Process Modeling
4.3.5. How can I tell if a particular experimental
design is good for my application?
Assess
Relative to
the Six
Design
Principles
If you have a design, generated by whatever method, in hand,
how can you assess its after-the-fact goodness? Such checks
can potentially parallel the list of the six general design
principles. The design can be assessed relative to each of
these six principles. For example, does it have capacity for
the primary model, does it have capacity for an alternative
model, etc.
Some of these checks are quantitative and complicated; other
checks are simpler and graphical. The graphical checks are
the most easily done and yet are among the most informative.
We include two such graphical checks and one quantitative
check.
Graphically
Check for
Univariate
Balance
If you have a design that claims to be globally good in k
factors, then generally that design should be locally good in
each of the individual k factors. Checking high-dimensional
global goodness is difficult, but checking low-dimensional
local goodness is easy. Generate k counts plots, with the
levels of factors plotted on the horizontal axis of each plot
and the number of design points for each level in factor on
the vertical axis. For most good designs, these counts should
be about the same (= balance) for all levels of a factor.
Exceptions exist, but such balance is a low-level
characteristic of most good designs.
Graphically
Check for
Bivariate
Balance
If you have a design that is purported to be globally good in k
factors, then generally that design should be locally good in
all pairs of the individual k factors. Graphically check for
such 2-way balance by generating plots for all pairs of
factors, where the horizontal axis of a given plot is and the
vertical axis is . The response variable does NOT come
into play in these plots. We are only interested in
characteristics of the design, and so only the variables are
involved. The 2-way plots of most good designs have a
certain symmetric and balanced look about them--all
combination points should be covered and each combination
point should have about the same number of points.
Check for For optimal designs, metrics exist (D-efficiency, A-
4.3.5. How can I tell if a particular experimental design is good for my application?
Minimal
Variation
efficiency, etc.) that can be computed and that reflect the
quality of the design. Further, relative ratios of standard
deviations of the coefficient estimators and relative ratios of
predicted values can be computed and compared for such
designs. Such calculations are commonly performed in
computer packages which specialize in the generation of
optimal designs.
4.4. Data Analysis for Process Modeling

4. Process Modeling
Building a
Good
Model
This section contains detailed discussions of the necessary
steps for developing a good process model after data have
been collected. A general model-building framework,
applicable to multiple statistical methods, is described with
method-specific points included when necessary.
Contents:
Section 4
1. What are the basic steps for developing an effective
process model?
2. How do I select a function to describe my process?
1. Incorporating Scientific Knowledge into Function
Selection
2. Using the Data to Select an Appropriate Function
3. Using Methods that Do Not Require Function
Specification
3. How are estimates of the unknown parameters
obtained?
1. Least Squares
2. Weighted Least Squares
4. How can I tell if a model fits my data?
1. How can I assess the sufficiency of the functional
part of the model?
2. How can I detect non-constant variation across
the data?
3. How can I tell if there was drift in the
measurement process?
4. How can I assess whether the random errors are
independent from one to the next?
5. How can I test whether or not the random errors
are normally distributed?
6. How can I test whether any significant terms are
missing or misspecified in the functional part of
the model?
7. How can I test whether all of the terms in the
functional part of the model are necessary?
5. If my current model does not fit the data well, how can I
improve it?
1. Updating the Function Based on Residual Plots
2. Accounting for Non-Constant Variation Across
the Data
3. Accounting for Errors with a Non-Normal
Distribution
4.4.1. What are the basic steps for developing an effective process model?

4. Process Modeling
4.4.1. What are the basic steps for developing an effective
process model?
Basic Steps
Provide
Universal
Framework
The basic steps used for model-building are the same across all modeling
methods. The details vary somewhat from method to method, but an
understanding of the common steps, combined with the typical underlying
assumptions needed for the analysis, provides a framework in which the results
from almost any method can be interpreted and understood.
Basic Steps
of Model
Building
The basic steps of the model-building process are:
1. model selection
2. model fitting, and
3. model validation.
These three basic steps are used iteratively until an appropriate model for the
data has been developed. In the model selection step, plots of the data, process
knowledge and assumptions about the process are used to determine the form
of the model to be fit to the data. Then, using the selected model and possibly
information about the data, an appropriate model-fitting method is used to
estimate the unknown parameters in the model. When the parameter estimates
have been made, the model is then carefully assessed to see if the underlying
assumptions of the analysis appear plausible. If the assumptions seem valid, the
model can be used to answer the scientific or engineering questions that
prompted the modeling effort. If the model validation identifies problems with
the current model, however, then the modeling process is repeated using
information from the model validation step to select and/or fit an improved
model.
A
Variation
on the
Basic Steps
The three basic steps of process modeling described in the paragraph above
assume that the data have already been collected and that the same data set can
be used to fit all of the candidate models. Although this is often the case in
model-building situations, one variation on the basic model-building sequence
comes up when additional data are needed to fit a newly hypothesized model
based on a model fit to the initial data. In this case two additional steps,
experimental design and data collection, can be added to the basic sequence
between model selection and model-fitting. The flow chart below shows the
basic model-fitting sequence with the integration of the related data collection
steps into the model-building process.
Model
Building
Sequence
Examples illustrating the model-building sequence in real applications can be
found in the case studies in Section 4.6. The specific tools and techniques used
in the basic model-building steps are described in the remainder of this section.
Design of
Initial
Experiment
Of course, considering the model selection and fitting before collecting the
initial data is also a good idea. Without data in hand, a hypothesis about what
the data will look like is needed in order to guess what the initial model should
be. Hypothesizing the outcome of an experiment is not always possible, of
course, but efforts made in the earliest stages of a project often maximize the
efficiency of the whole model-building process and result in the best possible
models for the process. More details about experimental design can be found in
Section 4.3 and in Chapter 5: Process Improvement.
4.4.2. How do I select a function to describe my process?

4. Process Modeling
4.4.2. How do I select a function to describe my
process?
Synthesis of
Process
Information
Necessary
Selecting a model of the right form to fit a set of data usually
requires the use of empirical evidence in the data, knowledge
of the process and some trial-and-error experimentation. As
mentioned on the previous page, model building is always an
iterative process. Much of the need to iterate stems from the
difficulty in initially selecting a function that describes the
data well. Details about the data are often not easily visible in
the data as originally observed. The fine structure in the data
can usually only be elicited by use of model-building tools
such as residual plots and repeated refinement of the model
form. As a result, it is important not to overlook any of the
sources of information that indicate what the form of the
model should be.
Answer Not
Provided
by
Statistics
Alone
Sometimes the different sources of information that need to
be integrated to find an effective model will be contradictory.
An open mind and a willingness to think about what the data
are saying is important. Maintaining balance and looking for
alternate sources for unusual effects found in the data are also
important. For example, in the load cell calibration case study
the statistical analysis pointed out that the model initially
thought to be appropriate did not account for all of the
structure in the data. A refined model was developed, but the
appearance of an unexpected result brings up the question of
whether the original understanding of the problem was
inaccurate, or whether the need for an alternate model was
due to experimental artifacts. In the load cell problem it was
easy to accept that the refined model was closer to the truth,
but in a more complicated case additional experiments might
have been needed to resolve the issue.
Knowing
Function
Types
Helps
Another helpful ingredient in model selection is a wide
knowledge of the shapes that different mathematical functions
can assume. Knowing something about the models that have
been found to work well in the past for different application
types also helps. A menu of different functions on the next
page, Section 4.4.2.1. (links provided below), provides one
way to learn about the function shapes and flexibility. Section
4.4.2.2. discusses how general function features and
qualitative scientific information can be combined to help
with model selection. Finally, Section 4.4.2.3. points to
4.4.2. How do I select a function to describe my process?
methods that don't require specification of a particular
function to be fit to the data, and how models of those types
can be refined.
1. Incorporating Scientific Knowledge into Function
Selection
2. Using the Data to Select an Appropriate Function
3. Using Methods that Do Not Require Function
Specification
4.4.3. How are estimates of the unknown parameters obtained?

4. Process Modeling
4.4.3. How are estimates of the unknown
parameters obtained?
Parameter
Estimation
in General
After selecting the basic form of the functional part of the
model, the next step in the model-building process is
estimation of the unknown parameters in the function. In
general, this is accomplished by solving an optimization
problem in which the objective function (the function being
minimized or maximized) relates the response variable and the
functional part of the model containing the unknown
parameters in a way that will produce parameter estimates that
will be close to the true, unknown parameter values. The
unknown parameters are, loosely speaking, treated as variables
to be solved for in the optimization, and the data serve as
known coefficients of the objective function in this stage of
the modeling process.
In theory, there are as many different ways of estimating
parameters as there are objective functions to be minimized or
maximized. However, a few principles have dominated
because they result in parameter estimators that have good
statistical properties. The two major methods of parameter
estimation for process models are maximum likelihood and
least squares. Both of these methods provide parameter
estimators that have many good properties. Both maximum
likelihood and least squares are sensitive to the presence of
outliers, however. There are also many newer methods of
parameter estimation, called robust methods, that try to
balance the efficiency and desirable properties of least squares
and maximum likelihood with a lower sensitivity to outliers.
Overview
of Section
4.3
Although robust techniques are valuable, they are not as well
developed as the more traditional methods and often require
specialized software that is not readily available. Maximum
likelihood also requires specialized algorithms in general,
although there are important special cases that do not have
such a requirement. For example, for data with normally
distributed random errors, the least squares and maximum
likelihood parameter estimators are identical. As a result of
these software and developmental issues, and the coincidence
of maximum likelihood and least squares in many
applications, this section currently focuses on parameter
estimation only by least squares methods. The remainder of
this section offers some intuition into how least squares works
4.4.3. How are estimates of the unknown parameters obtained?
and illustrates the effectiveness of this method.
Contents
of Section
4.3
1. Least Squares
2. Weighted Least Squares
4.4.4. How can I tell if a model fits my data?

4. Process Modeling
Is Not
Enough!
Model validation is possibly the most important step in the
model building sequence. It is also one of the most
overlooked. Often the validation of a model seems to consist
of nothing more than quoting the statistic from the fit
(which measures the fraction of the total variability in the
response that is accounted for by the model). Unfortunately, a
high value does not guarantee that the model fits the data
well. Use of a model that does not fit the data well cannot
provide good answers to the underlying engineering or
scientific questions under investigation.
Main
Tool:
Graphical
Residual
Analysis
There are many statistical tools for model validation, but the
primary tool for most process modeling applications is
graphical residual analysis. Different types of plots of the
residuals (see definition below) from a fitted model provide
information on the adequacy of different aspects of the model.
Numerical methods for model validation, such as the
statistic, are also useful, but usually to a lesser degree than
graphical methods. Graphical methods have an advantage over
numerical methods for model validation because they readily
illustrate a broad range of complex aspects of the relationship
between the model and the data. Numerical methods for model
validation tend to be narrowly focused on a particular aspect
of the relationship between the model and the data and often
try to compress that information into a single descriptive
number or test result.
Numerical
Methods'
Forte
Numerical methods do play an important role as confirmatory
methods for graphical techniques, however. For example, the
lack-of-fit test for assessing the correctness of the functional
part of the model can aid in interpreting a borderline residual
plot. There are also a few modeling situations in which
graphical methods cannot easily be used. In these cases,
numerical methods provide a fallback position for model
validation. One common situation when numerical validation
methods take precedence over graphical methods is when the
number of parameters being estimated is relatively close to the
size of the data set. In this situation residual plots are often
difficult to interpret due to constraints on the residuals
imposed by the estimation of the unknown parameters. One
area in which this typically happens is in optimization
applications using designed experiments. Logistic regression
with binary data is another area in which graphical residual
analysis can be difficult.
Residuals The residuals from a fitted model are the differences between
the responses observed at each combination values of the
explanatory variables and the corresponding prediction of the
response computed using the regression function.
Mathematically, the definition of the residual for the i
th
observation in the data set is written
,
with denoting the i
th
response in the data set and
represents the list of explanatory variables, each set at the
corresponding values found in the i
th
observation in the data
set.
Example The data listed below are from the Pressure/Temperature
example introduced in Section 4.1.1. The first column shows
the order in which the observations were made, the second
column indicates the day on which each observation was
made, and the third column gives the ambient temperature
recorded when each measurement was made. The fourth
column lists the temperature of the gas itself (the explanatory
variable) and the fifth column contains the observed pressure
of the gas (the response variable). Finally, the sixth column
gives the corresponding values from the fitted straight-line
regression function.
and the last column lists the residuals, the difference between
columns five and six.
Data,
Fitted
Values &
Residuals
Run Ambient
Fitted
Order Day Temperature Temperature Pressure
Value Residual
1 1 23.820 54.749 225.066
222.920 2.146
2 1 24.120 23.323 100.331
99.411 0.920
3 1 23.434 58.775 230.863
238.744 -7.881
4 1 23.993 25.854 106.160
109.359 -3.199
5 1 23.375 68.297 277.502
276.165 1.336
6 1 23.233 37.481 148.314
155.056 -6.741
7 1 24.162 49.542 197.562
202.456 -4.895
8 1 23.667 34.101 138.537
141.770 -3.232
9 1 24.056 33.901 137.969
140.983 -3.014
10 1 22.786 29.242 117.410
122.674 -5.263
11 2 23.785 39.506 164.442
163.013 1.429
12 2 22.987 43.004 181.044
176.759 4.285
13 2 23.799 53.226 222.179
216.933 5.246
14 2 23.661 54.467 227.010
221.813 5.198
15 2 23.852 57.549 232.496
233.925 -1.429
16 2 23.379 61.204 253.557
248.288 5.269
17 2 24.146 31.489 139.894
131.506 8.388
18 2 24.187 68.476 273.931
276.871 -2.940
19 2 24.159 51.144 207.969
208.753 -0.784
20 2 23.803 68.774 280.205
278.040 2.165
21 3 24.381 55.350 227.060
225.282 1.779
22 3 24.027 44.692 180.605
183.396 -2.791
23 3 24.342 50.995 206.229
208.167 -1.938
24 3 23.670 21.602 91.464
92.649 -1.186
25 3 24.246 54.673 223.869
222.622 1.247
26 3 25.082 41.449 172.910
170.651 2.259
27 3 24.575 35.451 152.073
147.075 4.998
28 3 23.803 42.989 169.427
176.703 -7.276
29 3 24.660 48.599 192.561
198.748 -6.188
30 3 24.097 21.448 94.448
92.042 2.406
31 4 22.816 56.982 222.794
231.697 -8.902
32 4 24.167 47.901 199.003
196.008 2.996
33 4 22.712 40.285 168.668
166.077 2.592
34 4 23.611 25.609 109.387
108.397 0.990
35 4 23.354 22.971 98.445
98.029 0.416
36 4 23.669 25.838 110.987
109.295 1.692
37 4 23.965 49.127 202.662
200.826 1.835
38 4 22.917 54.936 224.773
223.653 1.120
39 4 23.546 50.917 216.058
207.859 8.199
40 4 24.450 41.976 171.469
172.720 -1.251
Why Use
Residuals?
If the model fit to the data were correct, the residuals would
approximate the random errors that make the relationship
between the explanatory variables and the response variable a
statistical relationship. Therefore, if the residuals appear to
behave randomly, it suggests that the model fits the data well.
On the other hand, if non-random structure is evident in the
residuals, it is a clear sign that the model fits the data poorly.
The subsections listed below detail the types of plots to use to
test different aspects of a model and give guidance on the
correct interpretations of different results that could be
observed for each type of plot.
Model
Validation
1. How can I assess the sufficiency of the functional part
of the model?
Specifics 2. How can I detect non-constant variation across the data?
3. How can I tell if there was drift in the process?
4. How can I assess whether the random errors are
independent from one to the next?
5. How can I test whether or not the random errors are
distributed normally?
6. How can I test whether any significant terms are
missing or misspecified in the functional part of the
model?
7. How can I test whether all of the terms in the functional
part of the model are necessary?
4.4.5. If my current model does not fit the data well, how can I improve it?

4. Process Modeling
4.4.5. If my current model does not fit the data
well, how can I improve it?
What Next? Validating a model using residual plots, formal hypothesis
tests and descriptive statistics would be quite frustrating if
discovery of a problem meant restarting the modeling
process back at square one. Fortunately, however, there are
also techniques and tools to remedy many of the problems
uncovered using residual analysis. In some cases the model
validation methods themselves suggest appropriate changes
to a model at the same time problems are uncovered. This is
especially true of the graphical tools for model validation,
though tests on the parameters in the regression function
also offer insight into model refinement. Treatments for the
various model deficiencies that were diagnosed in Section
4.4.4. are demonstrated and discussed in the subsections
listed below.
Methods for
Model
Improvement
1. Updating the Function Based on Residual Plots
2. Accounting for Non-Constant Variation Across the
Data
3. Accounting for Errors with a Non-Normal
Distribution
4.5. Use and Interpretation of Process Models

4. Process Modeling
Overview
of Section
4.5
This section covers the interpretation and use of the models
developed from the collection and analysis of data using the
procedures discussed in Section 4.3 and Section 4.4. Three of
the main uses of such models, estimation, prediction and
calibration, are discussed in detail. Optimization, another
important use of this type of model, is primarily discussed in
Chapter 5: Process Improvement.
Contents
of Section
4.5
1. What types of predictions can I make using the model?
1. How do I estimate the average response for a
particular set of predictor variable values?
2. How can I predict the value and and estimate the
uncertainty of a single response?
2. How can I use my process model for calibration?
1. Single-Use Calibration Intervals
3. How can I optimize my process using the process
model?
4.5.1. What types of predictions can I make using the model?

4. Process Modeling
4.5.1. What types of predictions can I make
using the model?
Detailed
Information
on
Prediction
This section details some of the different types of predictions
that can be made using the various process models whose
development is discussed in Section 4.1 through Section 4.4.
Computational formulas or algorithms are given for each
different type of estimation or prediction, along with
simulation examples showing its probabilisitic interpretation.
An introduction to the different types of estimation and
prediction can be found in Section 4.1.3.1. A brief description
of estimation and prediction versus the other uses of process
models is given in Section 4.1.3.
Different
Types of
Predictions
1. How do I estimate the average response for a particular
set of predictor variable values?
2. How can I predict the value and and estimate the
uncertainty of a single response?
4.5.2. How can I use my process model for calibration?

4. Process Modeling
4.5.2. How can I use my process model for
calibration?
Detailed
Calibration
Information
This section details some of the different types of calibrations
that can be made using the various process models whose
development was discussed in previous sections.
Computational formulas or algorithms are given for each
different type of calibration, along with simulation examples
showing its probabilistic interpretation. An introduction to
calibration can be found in Section 4.1.3.2. A brief
comparison of calibration versus the other uses of process
models is given in Section 4.1.3. Additional information on
calibration is available in Section 3 of Chapter 2:
Measurement Process Characterization.
Calibration
Procedures
1. Single-Use Calibration Intervals
4.5.3. How can I optimize my process using the process model?

4. Process Modeling
4.5.3. How can I optimize my process using the
process model?
Detailed
Information
on Process
Optimization
Process optimization using models fit to data collected using
response surface designs is primarily covered in Section
5.5.3 of Chapter 5: Process Improvement. In that section
detailed information is given on how to determine the
correct process inputs to hit a target output value or to
maximize or minimize process output. Some background on
the use of process models for optimization can be found in
Section 4.1.3.3 of this chapter, however, and information on
the basic analysis of data from optimization experiments is
covered along with that of other types of models in Section
4.1 through Section 4.4 of this chapter.
Contents of
Chapter 5
Section 5.5.3.
1. Optimizing a Process
1. Single response case
1. Path of steepest ascent
2. Confidence region for search path
3. Choosing the step length
4. Optimization when there is adequate
quadratic fit
5. Effect of sampling error on optimal
solution
6. Optimization subject to experimental
region constraints
2. Multiple response case
2. Desirability function approach
3. Mathematical programming approach
4.6. Case Studies in Process Modeling

4. Process Modeling
Detailed,
Realistic
Examples
The general points of the first five sections are illustrated in
this section using data from physical science and engineering
applications. Each example is presented step-by-step in the
text and is often cross-linked with the relevant sections of the
chapter describing the analysis in general. Each analysis can
also be repeated using a worksheet linked to the appropriate
Dataplot macros. The worksheet is also linked to the step-by-
step analysis presented in the text for easy reference.
Contents:
Section 6
1. Load Cell Calibration
1. Background & Data
2. Selection of Initial Model
3. Model Fitting - Initial Model
4. Graphical Residual Analysis - Initial Model
5. Interpretation of Numerical Output - Initial Model
6. Model Refinement
7. Model Fitting - Model #2
8. Graphical Residual Analysis - Model #2
9. Interpretation of Numerical Output - Model #2
10. Use of the Model for Calibration
11. Work this Example Yourself
2. Alaska Pipeline Ultrasonic Calibration
2. Check for Batch Effect
3. Initial Linear Fit
4. Transformations to Improve Fit and Equalize
Variances
5. Weighting to Improve Fit
6. Compare the Fits
3. Ultrasonic Reference Block Study
2. Initial Non-Linear Fit
3. Transformations to Improve Fit
5. Compare the Fits
4. Thermal Expansion of Copper Case Study
2. Exact Rational Models
3. Initial Plot of Data
4. Fit Quadratic/Quadratic Model
5. Fit Cubic/Cubic Model
4.6.1. Load Cell Calibration

4. Process Modeling
4.6.1. Load Cell Calibration
Quadratic
Calibration
This example illustrates the construction of a linear regression
model for load cell data that relates a known load applied to a
load cell to the deflection of the cell. The model is then used
to calibrate future cell readings associated with loads of
unknown magnitude.
1. Background & Data
2. Selection of Initial Model
3. Model Fitting - Initial Model
4. Graphical Residual Analysis - Initial Model
5. Interpretation of Numerical Output - Initial Model
6. Model Refinement
7. Model Fitting - Model #2
8. Graphical Residual Analysis - Model #2
9. Interpretation of Numerical Output - Model #2
10. Use of the Model for Calibration
4.6.2. Alaska Pipeline

4. Process Modeling
4.6.2. Alaska Pipeline
Non-
Homogeneous
Variances
This example illustrates the construction of a linear
regression model for Alaska pipeline ultrasonic calibration
data. This case study demonstrates the use of
transformations and weighted fits to deal with the violation
of the assumption of constant standard deviations for the
random errors. This assumption is also called homogeneous
variances for the errors.
2. Check for a Batch Effect
3. Fit Initial Model
4. Transformations to Improve Fit and Equalize
Variances
6. Compare the Fits
4.6.3. Ultrasonic Reference Block Study

4. Process Modeling
4.6.3. Ultrasonic Reference Block Study
Non-Linear
Fit with Non-
Homogeneous
Variances
This example illustrates the construction of a non-linear
regression model for ultrasonic calibration data. This case
study demonstrates fitting a non-linear model and the use
of transformations and weighted fits to deal with the
violation of the assumption of constant standard deviations
for the errors. This assumption is also called homogeneous
variances for the errors.
2. Fit Initial Model
3. Transformations to Improve Fit
5. Compare the Fits
4.6.4. Thermal Expansion of Copper Case Study

4. Process Modeling
4.6.4. Thermal Expansion of Copper Case
Study
Rational
Function
Models
This case study illustrates the use of a class of nonlinear
models called rational function models. The data set used is
the thermal expansion of copper related to temperature.
This data set was provided by the NIST scientist Thomas
Hahn.
Contents 1. Background and Data
2. Rational Function Models
3. Initial Plot of Data
4. Fit Quadratic/Quadratic Model
5. Fit Cubic/Cubic Model
4.7. References For Chapter 4: Process Modeling

4. Process Modeling
4.7. References For Chapter 4: Process
Modeling
Handbook of Mathematical Functions with Formulas, Graphs and
Mathematical Tables (1964) Abramowitz M. and Stegun I. (eds.), U.S.
Government Printing Office, Washington, DC, 1046 p.
Berkson J. (1950) "Are There Two Regressions?," Journal of the
American Statistical Association, Vol. 45, pp. 164-180.
Carroll, R.J. and Ruppert D. (1988) Transformation and Weighting in
Regression, Chapman and Hall, New York.
Cleveland, W.S. (1979) "Robust Locally Weighted Regression and
Smoothing Scatterplots," Journal of the American Statistical Association,
Vol. 74, pp. 829-836.
Cleveland, W.S. and Devlin, S.J. (1988) "Locally Weighted Regression:
An Approach to Regression Analysis by Local Fitting," Journal of the
American Statistical Association, Vol. 83, pp. 596-610.
Fuller, W.A. (1987) Measurement Error Models, John Wiley and Sons,
New York.
Graybill, F.A. (1976) Theory and Application of the Linear Model,
Duxbury Press, North Sciutate, Massachusetts.
Graybill, F.A. and Iyer, H.K. (1994) Regression Analysis: Concepts and
Applications, Duxbury Press, Belmont, California.
Harter, H.L. (1983) "Least Squares," Encyclopedia of Statistical Sciences,
Kotz, S. and Johnson, N.L., eds., John Wiley & Sons, New York, pp.
593-598.
Montgomery, D.C. (2001) Design and Analysis of Experiments, 5th ed.,
Wiley, New York.
4.7. References For Chapter 4: Process Modeling
Neter, J., Wasserman, W., and Kutner, M. (1983) Applied Linear
Regression Models, Richard D. Irwin Inc., Homewood, IL.
Ryan, T.P. (1997) Modern Regression Methods, Wiley, New York
Seber, G.A.F and Wild, C.F. (1989) Nonlinear Regression, John Wiley
and Sons, New York.
Stigler, S.M. (1978) "Mathematical Statistics in the Early States," The
Annals of Statistics, Vol. 6, pp. 239-265.
Stigler, S.M. (1986) The History of Statistics: The Measurement of
Uncertainty Before 1900, The Belknap Press of Harvard University Press,
Cambridge, Massachusetts.
4.8. Some Useful Functions for Process Modeling

4. Process Modeling
4.8. Some Useful Functions for Process
Modeling
Overview
of Section
4.8
This section lists some functions commonly-used for process
modeling. Constructing an exhaustive list of useful functions
is impossible, of course, but the functions given here will often
provide good starting points when an empirical model must be
developed to describe a particular process.
Each function listed here is classified into a family of related
functions, if possible. Its statistical type, linear or nonlinear in
the parameters, is also given. Special features of each
function, such as asymptotes, are also listed along with the
function's domain (the set of allowable input values) and range
(the set of possible output values). Plots of some of the
different shapes that each function can assume are also
included.
Contents
of Section
4.8
1. Univariate Functions
1. Polynomials
2. Rational Functions
5. Process Improvement
http://www.itl.nist.gov/div898/handbook/pri/pri.htm[6/27/2012 2:26:20 PM]

1. Introduction
1. Definition of experimental
design
2. Uses
3. Steps
2. Assumptions
1. Measurement system capable
2. Process stable
3. Simple model
4. Residuals well-behaved
3. Choosing an Experimental
Design
1. Set objectives
2. Select process variables and
levels
3. Select experimental design
1. Completely randomized
designs
2. Randomized block
designs
3. Full factorial designs
4. Fractional factorial
designs
5. Plackett-Burman
designs
6. Response surface
designs
7. Adding center point
runs
8. Improving fractional
design resolution
9. Three-level full
factorial designs
10. Three-level, mixed-
level and fractional
factorial designs
4. Analysis of DOE Data
1. DOE analysis steps
2. Plotting DOE data
3. Modeling DOE data
4. Testing and revising DOE
models
5. Interpreting DOE results
6. Confirming DOE results
7. DOE examples
1. Full factorial example
2. Fractional factorial
example
3. Response surface
example
5. Advanced Topics
1. When classical designs don't
work
2. Computer-aided designs
1. D-Optimal designs
6. Case Studies
1. Eddy current probe sensitivity
study
2. Sonoluminescent light
intensity study
http://www.itl.nist.gov/div898/handbook/pri/pri.htm[6/27/2012 2:26:20 PM]
2. Repairing a design
3. Optimizing a process
4. Mixture designs
1. Mixture screening
designs
2. Simplex-lattice designs
3. Simplex-centroid
designs
4. Constrained mixture
designs
5. Treating mixture and
process variables
together
5. Nested variation
6. Taguchi designs
7. John's 3/4 fractional factorial
designs
8. Small composite designs
9. An EDA approach to
experiment design
7. A Glossary of DOE
Terminology
8. References
Click here for a detailed table of contents
http://www.itl.nist.gov/div898/handbook/pri/pri_d.htm[6/27/2012 2:23:22 PM]

5. Process Improvement - Detailed Table of Contents [5.]
1. Introduction [5.1.]
1. What is experimental design? [5.1.1.]
2. What are the uses of DOE? [5.1.2.]
3. What are the steps of DOE? [5.1.3.]
2. Assumptions [5.2.]
1. Is the measurement system capable? [5.2.1.]
2. Is the process stable? [5.2.2.]
3. Is there a simple model? [5.2.3.]
4. Are the model residuals well-behaved? [5.2.4.]
3. Choosing an experimental design [5.3.]
1. What are the objectives? [5.3.1.]
2. How do you select and scale the process variables? [5.3.2.]
3. How do you select an experimental design? [5.3.3.]
1. Completely randomized designs [5.3.3.1.]
2. Randomized block designs [5.3.3.2.]
1. Latin square and related designs [5.3.3.2.1.]
2. Graeco-Latin square designs [5.3.3.2.2.]
3. Hyper-Graeco-Latin square designs [5.3.3.2.3.]
3. Full factorial designs [5.3.3.3.]
1. Two-level full factorial designs [5.3.3.3.1.]
2. Full factorial example [5.3.3.3.2.]
3. Blocking of full factorial designs [5.3.3.3.3.]
4. Fractional factorial designs [5.3.3.4.]
1. A 2
3-1
design (half of a 2
3
) [5.3.3.4.1.]
2. Constructing the 2
3-1
half-fraction design [5.3.3.4.2.]
3. Confounding (also called aliasing) [5.3.3.4.3.]
4. Fractional factorial design specifications and design resolution [5.3.3.4.4.]
5. Use of fractional factorial designs [5.3.3.4.5.]
6. Screening designs [5.3.3.4.6.]
7. Summary tables of useful fractional factorial designs [5.3.3.4.7.]
5. Plackett-Burman designs [5.3.3.5.]
6. Response surface designs [5.3.3.6.]
1. Central Composite Designs (CCD) [5.3.3.6.1.]
2. Box-Behnken designs [5.3.3.6.2.]
3. Comparisons of response surface designs [5.3.3.6.3.]
4. Blocking a response surface design [5.3.3.6.4.]
7. Adding centerpoints [5.3.3.7.]
8. Improving fractional factorial design resolution [5.3.3.8.]
1. Mirror-Image foldover designs [5.3.3.8.1.]
2. Alternative foldover designs [5.3.3.8.2.]
9. Three-level full factorial designs [5.3.3.9.]
10. Three-level, mixed-level and fractional factorial designs [5.3.3.10.]
4. Analysis of DOE data [5.4.]
1. What are the steps in a DOE analysis? [5.4.1.]
2. How to "look" at DOE data [5.4.2.]
3. How to model DOE data [5.4.3.]
4. How to test and revise DOE models [5.4.4.]
5. How to interpret DOE results [5.4.5.]
6. How to confirm DOE results (confirmatory runs) [5.4.6.]
7. Examples of DOE's [5.4.7.]
1. Full factorial example [5.4.7.1.]
2. Fractional factorial example [5.4.7.2.]
3. Response surface model example [5.4.7.3.]
5. Advanced topics [5.5.]
1. What if classical designs don't work? [5.5.1.]
2. What is a computer-aided design? [5.5.2.]
1. D-Optimal designs [5.5.2.1.]
2. Repairing a design [5.5.2.2.]
3. How do you optimize a process? [5.5.3.]
1. Single response case [5.5.3.1.]
1. Single response: Path of steepest ascent [5.5.3.1.1.]
2. Single response: Confidence region for search path [5.5.3.1.2.]
3. Single response: Choosing the step length [5.5.3.1.3.]
4. Single response: Optimization when there is adequate quadratic fit [5.5.3.1.4.]
5. Single response: Effect of sampling error on optimal solution [5.5.3.1.5.]
6. Single response: Optimization subject to experimental region constraints [5.5.3.1.6.]
2. Multiple response case [5.5.3.2.]
1. Multiple responses: Path of steepest ascent [5.5.3.2.1.]
2. Multiple responses: The desirability approach [5.5.3.2.2.]
3. Multiple responses: The mathematical programming approach [5.5.3.2.3.]
4. What is a mixture design? [5.5.4.]
1. Mixture screening designs [5.5.4.1.]
2. Simplex-lattice designs [5.5.4.2.]
3. Simplex-centroid designs [5.5.4.3.]
4. Constrained mixture designs [5.5.4.4.]
5. Treating mixture and process variables together [5.5.4.5.]
5. How can I account for nested variation (restricted randomization)? [5.5.5.]
6. What are Taguchi designs? [5.5.6.]
7. What are John's 3/4 fractional factorial designs? [5.5.7.]
8. What are small composite designs? [5.5.8.]
9. An EDA approach to experimental design [5.5.9.]
1. Ordered data plot [5.5.9.1.]
2. DOE scatter plot [5.5.9.2.]
3. DOE mean plot [5.5.9.3.]
4. Interaction effects matrix plot [5.5.9.4.]
5. Block plot [5.5.9.5.]
6. DOE Youden plot [5.5.9.6.]
7. |Effects| plot [5.5.9.7.]
1. Statistical significance [5.5.9.7.1.]
2. Engineering significance [5.5.9.7.2.]
3. Numerical significance [5.5.9.7.3.]
4. Pattern significance [5.5.9.7.4.]
8. Half-normal probability plot [5.5.9.8.]
9. Cumulative residual standard deviation plot [5.5.9.9.]
1. Motivation: What is a Model? [5.5.9.9.1.]
2. Motivation: How do we Construct a Goodness-of-fit Metric for a Model? [5.5.9.9.2.]
3. Motivation: How do we Construct a Good Model? [5.5.9.9.3.]
4. Motivation: How do we Know When to Stop Adding Terms? [5.5.9.9.4.]
5. Motivation: What is the Form of the Model? [5.5.9.9.5.]
6. Motivation: Why is the 1/2 in the Model? [5.5.9.9.6.]
7. Motivation: What are the Advantages of the LinearCombinatoric Model? [5.5.9.9.7.]
8. Motivation: How do we use the Model to Generate Predicted Values? [5.5.9.9.8.]
9. Motivation: How do we Use the Model Beyond the Data Domain? [5.5.9.9.9.]
10. Motivation: What is the Best Confirmation Point for Interpolation? [5.5.9.9.10.]
11. Motivation: How do we Use the Model for Interpolation? [5.5.9.9.11.]
12. Motivation: How do we Use the Model for Extrapolation? [5.5.9.9.12.]
10. DOE contour plot [5.5.9.10.]
1. How to Interpret: Axes [5.5.9.10.1.]
2. How to Interpret: Contour Curves [5.5.9.10.2.]
3. How to Interpret: Optimal Response Value [5.5.9.10.3.]
4. How to Interpret: Best Corner [5.5.9.10.4.]
5. How to Interpret: Steepest Ascent/Descent [5.5.9.10.5.]
6. How to Interpret: Optimal Curve [5.5.9.10.6.]
7. How to Interpret: Optimal Setting [5.5.9.10.7.]
6. Case Studies [5.6.]
1. Eddy Current Probe Sensitivity Case Study [5.6.1.]
2. Initial Plots/Main Effects [5.6.1.2.]
3. Interaction Effects [5.6.1.3.]
4. Main and Interaction Effects: Block Plots [5.6.1.4.]
5. Estimate Main and Interaction Effects [5.6.1.5.]
6. Modeling and Prediction Equations [5.6.1.6.]
7. Intermediate Conclusions [5.6.1.7.]
8. Important Factors and Parsimonious Prediction [5.6.1.8.]
9. Validate the Fitted Model [5.6.1.9.]
10. Using the Fitted Model [5.6.1.10.]
11. Conclusions and Next Step [5.6.1.11.]
2. Sonoluminescent Light Intensity Case Study [5.6.2.]
2. Initial Plots/Main Effects [5.6.2.2.]
3. Interaction Effects [5.6.2.3.]
4. Main and Interaction Effects: Block Plots [5.6.2.4.]
5. Important Factors: Youden Plot [5.6.2.5.]
6. Important Factors: |Effects| Plot [5.6.2.6.]
7. Important Factors: Half-Normal Probability Plot [5.6.2.7.]
8. Cumulative Residual Standard Deviation Plot [5.6.2.8.]
9. Next Step: DOE Contour Plot [5.6.2.9.]
10. Summary of Conclusions [5.6.2.10.]
7. A Glossary of DOE Terminology [5.7.]
5.1. Introduction
http://www.itl.nist.gov/div898/handbook/pri/section1/pri1.htm[6/27/2012 2:23:28 PM]

5.1. Introduction
This section
describes
the basic
concepts of
the Design
of
Experiments
(DOE)
This section introduces the basic concepts, terminology, goals
and procedures underlying the proper statistical design of
experiments. Design of experiments is abbreviated as DOE
throughout this chapter.
Topics covered are:
What is experimental design or DOE?
What are the goals or uses of DOE?
What are the steps in DOE?
5.1.1. What is experimental design?

5.1. Introduction
Experimental
Design (or
DOE)
economically
maximizes
information
In an experiment, we deliberately change one or more process
variables (or factors) in order to observe the effect the changes have
on one or more response variables. The (statistical) design of
experiments (DOE) is an efficient procedure for planning experiments
so that the data obtained can be analyzed to yield valid and objective
conclusions.
DOE begins with determining the objectives of an experiment and
selecting the process factors for the study. An Experimental Design is
the laying out of a detailed experimental plan in advance of doing the
experiment. Well chosen experimental designs maximize the amount
of "information" that can be obtained for a given amount of
experimental effort.
The statistical theory underlying DOE generally begins with the
concept of process models.
Process Models for DOE
Black box
process
model
It is common to begin with a process model of the `black box' type,
with several discrete or continuous input factors that can be
controlled--that is, varied at will by the experimenter--and one or
more measured output responses. The output responses are assumed
continuous. Experimental data are used to derive an empirical
(approximation) model linking the outputs and inputs. These
empirical models generally contain first and second-order terms.
Often the experiment has to account for a number of uncontrolled
factors that may be discrete, such as different machines or operators,
and/or continuous such as ambient temperature or humidity. Figure
1.1 illustrates this situation.
Schematic
for a typical
process with
controlled
inputs,
outputs,
discrete
uncontrolled
factors and
continuous
uncontrolled
factors
FIGURE 1.1 A `Black Box' Process Model Schematic
Models for
DOE's
The most common empirical models fit to the experimental data take
either a linear form or quadratic form.
Linear model A linear model with two factors, X
1
and X
2
, can be written as
Here, Y is the response for given levels of the main effects X
1
and X
2
and the X
1
X
2
term is included to account for a possible interaction
effect between X
1
and X
2
. The constant
0
is the response of Y when
both main effects are 0.
For a more complicated example, a linear model with three factors
X
1
, X
2
, X
3
and one response, Y, would look like (if all possible terms
were included in the model)
The three terms with single "X's" are the main effects terms. There are
k(k-1)/2 = 3*2/2 = 3 two-way interaction terms and 1 three-way
interaction term (which is often omitted, for simplicity). When the
experimental data are analyzed, all the unknown " " parameters are
estimated and the coefficients of the "X" terms are tested to see which
ones are significantly different from 0.
Quadratic
model
A second-order (quadratic) model (typically used in response surface
DOE's with suspected curvature) does not include the three-way
interaction term but adds three more terms to the linear model,
namely
.
Note: Clearly, a full model could include many cross-product (or
interaction) terms involving squared X's. However, in general these
terms are not needed and most DOE software defaults to leaving them
out of the model.
5.1.2. What are the uses of DOE?

5.1. Introduction
DOE is a
multipurpose
tool that can
help in many
situations
Below are seven examples illustrating situations in which experimental design can
be used effectively:
Choosing Between Alternatives
Selecting the Key Factors Affecting a Response
Response Surface Modeling to:
Hit a Target
Reduce Variability
Maximize or Minimize a Response
Make a Process Robust (i.e., the process gets the "right" results even
though there are uncontrollable "noise" factors)
Seek Multiple Goals
Regression Modeling
Choosing Between Alternatives (Comparative Experiment)
A common
use is
planning an
experiment
to gather
data to make
a decision
between two
or more
alternatives
Supplier A vs. supplier B? Which new additive is the most effective? Is catalyst `x'
an improvement over the existing catalyst? These and countless other choices
between alternatives can be presented to us in a never-ending parade. Often we have
the choice made for us by outside factors over which we have no control. But in
many cases we are also asked to make the choice. It helps if one has valid data to
back up one's decision.
The preferred solution is to agree on a measurement by which competing choices
can be compared, generate a sample of data from each alternative, and compare
average results. The 'best' average outcome will be our preference. We have
performed a comparative experiment!
Types of
comparitive
studies
Sometimes this comparison is performed under one common set of conditions. This
is a comparative study with a narrow scope - which is suitable for some initial
comparisons of possible alternatives. Other comparison studies, intended to validate
that one alternative is perferred over a wide range of conditions, will purposely and
systematically vary the background conditions under which the primary comparison
is made in order to reach a conclusion that will be proven valid over a broad scope.
We discuss experimental designs for each of these types of comparisons in Sections
5.3.3.1 and 5.3.3.2.
Selecting the Key Factors Affecting a Response (Screening Experiments)
Selecting the
few that
Often there are many possible factors, some of which may be critical and others
which may have little or no effect on a response. It may be desirable, as a goal by
matter from
the many
possible
factors
itself, to reduce the number of factors to a relatively small set (2-5) so that attention
can be focussed on controlling those factors with appropriate specifications, control
charts, etc.
Screening experiments are an efficient way, with a minimal number of runs, of
determining the important factors. They may also be used as a first step when the
ultimate goal is to model a response with a response surface. We will discuss
experimental designs for screening a large number of factors in Sections 5.3.3.3,
5.3.3.4 and 5.3.3.5.
Response Surface Modeling a Process
Some
reasons to
model a
process
Once one knows the primary variables (factors) that affect the responses of interest,
a number of additional objectives may be pursued. These include:
Hitting a Target
Maximizing or Minimizing a Response
Reducing Variation
Making a Process Robust
Seeking Multiple Goals
What each of these purposes have in common is that experimentation is used to fit a
model that may permit a rough, local approximation to the actual surface. Given that
the particular objective can be met with such an approximate model, the
experimental effort is kept to a minimum while still achieving the immediate goal.
These response surface modeling objectives will now be briefly expanded upon.
Hitting a Target
Often we
want to "fine
tune" a
process to
consistently
hit a target
This is a frequently encountered goal for an experiment.
One might try out different settings until the desired target is `hit' consistently. For
example, a machine tool that has been recently overhauled may require some setup
`tweaking' before it runs on target. Such action is a small and common form of
experimentation. However, rather than experimenting in an ad hoc manner until we
happen to find a setup that hits the target, one can fit a model estimated from a
small experiment and use this model to determine the necessary adjustments to hit
the target.
More complex forms of experimentation, such as the determination of the correct
chemical mix of a coating that will yield a desired refractive index for the dried coat
(and simultaneously achieve specifications for other attributes), may involve many
ingredients and be very sensitive to small changes in the percentages in the mix.
Fitting suitable models, based on sequentially planned experiments, may be the only
way to efficiently achieve this goal of hitting targets for multiple responses
simultaneously.
Maximizing or Minimizing a Response
Optimizing a
process
output is a
common
Many processes are being run at sub-optimal settings, some of them for years, even
though each factor has been optimized individually over time. Finding settings that
increase yield or decrease the amount of scrap and rework represent opportunities
for substantial financial gain. Often, however, one must experiment with multiple
goal inputs to achieve a better output. Section 5.3.3.6 on second-order designs plus
material in Section 5.5.3 will be useful for these applications.
FIGURE 1.1 Pathway up the process response surface to an òptimum'
Reducing Variation
Processes
that are on
target, on
the average,
may still
have too
much
variability
A process may be performing with unacceptable consistency, meaning its internal
variation is too high.
Excessive variation can result from many causes. Sometimes it is due to the lack of
having or following standard operating procedures. At other times, excessive
variation is due to certain hard-to-control inputs that affect the critical output
characteristics of the process. When this latter situation is the case, one may
experiment with these hard-to-control factors, looking for a region where the surface
is flatter and the process is easier to manage. To take advantage of such flatness in
the surface, one must use designs - such as the second-order designs of Section
5.3.3.6 - that permit identification of these features. Contour or surface plots are
useful for elucidating the key features of these fitted models. See also 5.5.3.1.4.
Graph of
data before
variation
reduced
It might be possible to reduce the variation by altering the setpoints (recipe) of the
process, so that it runs in a more `stable' region.
Graph of
data after
process
variation
reduced
Finding this new recipe could be the subject of an experiment, especially if there are
many input factors that could conceivably affect the output.
Making a Process Robust
The less a
process or
product is
affected by
external
conditions,
the better it
is - this is
called
"Robustness"
An item designed and made under controlled conditions will be later `field tested' in
the hands of the customer and may prove susceptible to failure modes not seen in
the lab or thought of by design. An example would be the starter motor of an
automobile that is required to operate under extremes of external temperature. A
starter that performs under such a wide range is termed `robust' to temperature.
Designing an item so that it is robust calls for a special experimental effort. It is
possible to stress the item in the design lab and so determine the critical components
affecting its performance. A different gauge of armature wire might be a solution to
the starter motor, but so might be many other alternatives. The correct combination
of factors can be found only by experimentation.
Seeking Multiple Goals
Sometimes
we have
multiple
outputs and
we have to
compromise
to achieve
desirable
outcomes -
DOE can
help here
A product or process seldom has just one desirable output characteristic. There are
usually several, and they are often interrelated so that improving one will cause a
deterioration of another. For example: rate vs. consistency; strength vs. expense; etc.
Any product is a trade-off between these various desirable final characteristics.
Understanding the boundaries of the trade-off allows one to make the correct
choices. This is done by either constructing some weighted objective function
(`desirability function') and optimizing it, or examining contour plots of responses
generated by a computer program, as given below.
Sample
contour plot
of deposition
rate and
capability
FIGURE 1.4 Overlaid contour plot of Deposition Rate and Capability (Cp)
Regression Modeling
Regression
models
(Chapter 4)
are used to
fit more
precise
models
Sometimes we require more than a rough approximating model over a local region.
In such cases, the standard designs presented in this chapter for estimating first- or
second-order polynomial models may not suffice. Chapter 4 covers the topic of
experimental design and analysis for fitting general models for a single explanatory
factor. If one has multiple factors, and either a nonlinear model or some other
special model, the computer-aided designs of Section 5.5.2 may be useful.
5.1.3. What are the steps of DOE?

5.1. Introduction
Key steps for
DOE
Obtaining good results from a DOE involves these seven
steps:
1. Set objectives
2. Select process variables
3. Select an experimental design
4. Execute the design
5. Check that the data are consistent with the
experimental assumptions
6. Analyze and interpret the results
7. Use/present the results (may lead to further runs or
DOE's).
A checklist of
practical
considerations
Important practical considerations in planning and running
experiments are
Check performance of gauges/measurement devices
first.
Keep the experiment as simple as possible.
Check that all planned runs are feasible.
Watch out for process drifts and shifts during the
run.
Avoid unplanned changes (e.g., swap operators at
halfway point).
Allow some time (and back-up material) for
unexpected events.
Obtain buy-in from all parties involved.
Maintain effective ownership of each step in the
experimental plan.
Preserve all the raw data--do not keep only summary
averages!
Record everything that happens.
Reset equipment to its original state after the
experiment.
The Sequential or I terative Approach to DOE
Planning to
do a sequence
of small
experiments is
It is often a mistake to believe that òne big experiment
will give the answer.'
A more useful approach to experimental design is to
often better
than relying
on one big
experiment to
give you all
the answers
recognize that while one experiment might provide a
useful result, it is more common to perform two or three,
or maybe more, experiments before a complete answer is
attained. In other words, an iterative approach is best and,
in the end, most economical. Putting all one's eggs in one
basket is not advisable.
Each stage
provides
insight for
next stage
The reason an iterative approach frequently works best is
because it is logical to move through stages of
experimentation, each stage providing insight as to how the
next experiment should be run.
5.2. Assumptions

5.2. Assumptions
We should
check the
engineering
and model-
building
assumptions
that are
made in
most DOE's
In all model building we make assumptions, and we also
require certain conditions to be approximately met for
purposes of estimation. This section looks at some of the
engineering and mathematical assumptions we typically
make. These are:
Are the measurement systems capable for all of your
responses?
Is your process stable?
Are your responses likely to be approximated well by
simple polynomial models?
Are the residuals (the difference between the model
predictions and the actual observations) well behaved?
5.2.1. Is the measurement system capable?

5.2. Assumptions
5.2.1. Is the measurement system capable?
Metrology
capabilities
are a key
factor in
most
experiments
It is unhelpful to find, after you have finished all the
experimental runs, that the measurement devices you have at
your disposal cannot measure the changes you were hoping
to see. Plan to check this out before embarking on the
experiment itself. Measurement process characterization is
covered in Chapter 2.
SPC check
of
measurement
devices
In addition, it is advisable, especially if the experimental
material is planned to arrive for measurement over a
protracted period, that an SPC (i.e., quality control) check is
kept on all measurement devices from the start to the
conclusion of the whole experimental project. Strange
experimental outcomes can often be traced to `hiccups' in
the metrology system.
5.2.2. Is the process stable?

5.2. Assumptions
5.2.2. Is the process stable?
Plan to
examine
process
stability as
part of
your
experiment
Experimental runs should have control runs that are made at
the `standard' process setpoints, or at least at some standard
operating recipe. The experiment should start and end with
such runs. A plot of the outcomes of these control runs will
indicate if the underlying process itself has drifted or shifted
during the experiment.
It is desirable to experiment on a stable process. However, if
this cannot be achieved, then the process instability must be
accounted for in the analysis of the experiment. For example,
if the mean is shifting with time (or experimental trial run),
then it will be necessary to include a trend term in the
experimental model (i.e., include a time variable or a run
number variable).
5.2.3. Is there a simple model?

5.2. Assumptions
5.2.3. Is there a simple model?
Polynomial
approximation
models only
work for
smoothly
varying
outputs
In this chapter we restrict ourselves to the case for which the
response variable(s) are continuous outputs denoted as Y.
Over the experimental range, the outputs must not only be
continuous, but also reasonably smooth. A sharp falloff in Y
values is likely to be missed by the approximating
polynomials that we use because these polynomials assume a
smoothly curving underlying response surface.
Piecewise
smoothness
requires
separate
experiments
If the surface under investigation is known to be only
piecewise smooth, then the experiments will have to be
broken up into separate experiments, each investigating the
shape of the separate sections. A surface that is known to be
very jagged (i.e., non-smooth) will not be successfully
approximated by a smooth polynomial.
Examples of
piecewise
smooth and
jagged
responses
Piecewise Smooth Jagged
FIGURE 2.1 Examples of Piecewise
Smooth and Jagged Responses
5.2.4. Are the model residuals well-behaved?

5.2. Assumptions
Residuals are
the
differences
between the
observed and
predicted
responses
Residuals are estimates of experimental error obtained by subtracting the observed
responses from the predicted responses.
The predicted response is calculated from the chosen model, after all the unknown
model parameters have been estimated from the experimental data.
Examining residuals is a key part of all statistical modeling, including DOE's.
Carefully looking at residuals can tell us whether our assumptions are reasonable
and our choice of model is appropriate.
Residuals are
elements of
variation
unexplained
by fitted
model
Residuals can be thought of as elements of variation unexplained by the fitted
model. Since this is a form of error, the same general assumptions apply to the
group of residuals that we typically use for errors in general: one expects them to be
(roughly) normal and (approximately) independently distributed with a mean of 0
and some constant variance.
Assumptions
for residuals
These are the assumptions behind ANOVA and classical regression analysis. This
means that an analyst should expect a regression model to err in predicting a
response in a random fashion; the model should predict values higher than actual
and lower than actual with equal probability. In addition, the level of the error
should be independent of when the observation occurred in the study, or the size of
the observation being predicted, or even the factor settings involved in making the
prediction. The overall pattern of the residuals should be similar to the bell-shaped
pattern observed when plotting a histogram of normally distributed data.
We emphasize the use of graphical methods to examine residuals.
Departures
indicate
inadequate
model
Departures from these assumptions usually mean that the residuals contain structure
that is not accounted for in the model. Identifying that structure and adding term(s)
representing it to the original model leads to a better model.
Tests for Residual Normality
Plots for
examining
residuals
Any graph suitable for displaying the distribution of a set of data is suitable for
judging the normality of the distribution of a group of residuals. The three most
common types are:
1. histograms,
2. normal probability plots, and
3. dot plots.
Histogram
The histogram is a frequency plot obtained by placing the data in regularly spaced
cells and plotting each cell frequency versus the center of the cell. Figure 2.2
illustrates an approximately normal distribution of residuals produced by a model for
a calibration process. We have superimposed a normal density function on the
histogram.
Small sample
sizes
Sample sizes of residuals are generally small (<50) because experiments have
limited treatment combinations, so a histogram is not be the best choice for judging
the distribution of residuals. A more sensitive graph is the normal probability plot.
Normal
probability
plot
The steps in forming a normal probability plot are:
Sort the residuals into ascending order.
Calculate the cumulative probability of each residual using the formula:
P(i-th residual) = i/(N+1)
with P denoting the cumulative probability of a point, i is the order of the
value in the list and N is the number of entries in the list.
Plot the calculated p-values versus the residual value on normal probability
paper.
The normal probability plot should produce an approximately straight line if the
points come from a normal distribution.
Sample Figure 2.3 below illustrates the normal probability graph created from the same
normal
probability
plot with
overlaid dot
plot
group of residuals used for Figure 2.2.
This graph includes the addition of a dot plot. The dot plot is the collection of points
along the left y-axis. These are the values of the residuals. The purpose of the dot
plot is to provide an indication the distribution of the residuals.
"S" shaped
curves
indicate
bimodal
distribution
Small departures from the straight line in the normal probability plot are common,
but a clearly "S" shaped curve on this graph suggests a bimodal distribution of
residuals. Breaks near the middle of this graph are also indications of abnormalities
in the residual distribution.
NOTE: Studentized residuals are residuals converted to a scale approximately
representing the standard deviation of an individual residual from the center of the
residual distribution. The technique used to convert residuals to this form produces a
Student's t distribution of values.
Independence of Residuals Over Time
Run sequence
plot
If the order of the observations in a data table represents the order of execution of
each treatment combination, then a plot of the residuals of those observations versus
the case order or time order of the observations will test for any time dependency.
These are referred to as run sequence plots.
Sample run
sequence plot
that exhibits
a time trend
Sample run
sequence plot
that does not
exhibit a time
trend
Interpretation
of the sample
run sequence
plots
The residuals in Figure 2.4 suggest a time trend, while those in Figure 2.5 do not.
Figure 2.4 suggests that the system was drifting slowly to lower values as the
investigation continued. In extreme cases a drift of the equipment will produce
models with very poor ability to account for the variability in the data (low R
2
).
If the investigation includes centerpoints, then plotting them in time order may
produce a more clear indication of a time trend if one exists. Plotting the raw
responses in time sequence can also sometimes detect trend changes in a process
that residual plots might not detect.
Plot of Residuals Versus Corresponding Predicted Values
Check for
increasing
residuals as
size of fitted
value
increases
Plotting residuals versus the value of a fitted response should produce a distribution
of points scattered randomly about 0, regardless of the size of the fitted value. Quite
commonly, however, residual values may increase as the size of the fitted value
increases. When this happens, the residual cloud becomes "funnel shaped" with the
larger end toward larger fitted values; that is, the residuals have larger and larger
scatter as the value of the response increases. Plotting the absolute values of the
residuals instead of the signed values will produce a "wedge-shaped" distribution; a
smoothing function is added to each graph which helps to show the trend.
Sample
residuals
versus fitted
values plot
showing
increasing
residuals
Sample
residuals
versus fitted
values plot
that does not
show
increasing
residuals
Interpretation
of the
residuals
versus fitted
values plots
A residual distribution such as that in Figure 2.6 showing a trend to higher absolute
residuals as the value of the response increases suggests that one should transform
the response, perhaps by modeling its logarithm or square root, etc., (contractive
transformations). Transforming a response in this fashion often simplifies its
relationship with a predictor variable and leads to simpler models. Later sections
discuss transformation in more detail. Figure 2.7 plots the residuals after a
transformation on the response variable was used to reduce the scatter. Notice the
difference in scales on the vertical axes.
Independence of Residuals from Factor Settings
Sample
residuals
versus factor
setting plot
Sample
residuals
versus factor
setting plot
after adding
a quadratic
term
Interpreation
of residuals
versus factor
setting plots
Figure 2.8 shows that the size of the residuals changed as a function of a predictor's
settings. A graph like this suggests that the model needs a higher-order term in that
predictor or that one should transform the predictor using a logarithm or square root,
for example. Figure 2.9 shows the residuals for the same response after adding a
quadratic term. Notice the single point widely separated from the other residuals in
Figure 2.9. This point is an "outlier." That is, its position is well within the range of
values used for this predictor in the investigation, but its result was somewhat lower
than the model predicted. A signal that curvature is present is a trace resembling a
"frown" or a "smile" in these graphs.
Sample
residuals
versus factor
setting plot
lacking one
or more
higher-order
terms
Interpretation
of plot
The example given in Figures 2.8 and 2.9 obviously involves five levels of the
predictor. The experiment utilized a response surface design. For the simple factorial
design that includes center points, if the response model being considered lacked one
or more higher-order terms, the plot of residuals versus factor settings might appear
as in Figure 2.10.
Graph
indicates
prescence of
curvature
While the graph gives a definite signal that curvature is present, identifying the
source of that curvature is not possible due to the structure of the design. Graphs
generated using the other predictors in that situation would have very similar
appearances.
Additional
discussion of
residual
analysis
Note: Residuals are an important subject discussed repeatedly in this Handbook. For
example, graphical residual plots are discussed in Chapter 1 and the general
examination of residuals as a part of model building is discussed in Chapter 4.
5.3. Choosing an experimental design

Contents of
Section 3
This section describes in detail the process of choosing an
experimental design to obtain the results you need. The basic
designs an engineer needs to know about are described in
detail.
Note that
this section
describes
the basic
designs
used for
most
engineering
and
scientific
applications
1. Set objectives
2. Select process variables and levels
3. Select experimental design
1. Completely randomized designs
2. Randomized block designs
1. Latin squares
2. Graeco-Latin squares
3. Hyper-Graeco-Latin squares
3. Full factorial designs
1. Two-level full factorial designs
2. Full factorial example
3. Blocking of full factorial designs
4. Fractional factorial designs
1. A 2
3-1
half-fraction design
2. How to construct a 2
3-1
design
3. Confounding
4. Design resolution
5. Use of fractional factorial designs
6. Screening designs
7. Fractional factorial designs summary
tables
5. Plackett-Burman designs
6. Response surface (second-order) designs
1. Central composite designs
2. Box-Behnken designs
3. Response surface design comparisons
4. Blocking a response surface design
7. Adding center points
8. Improving fractional design resolution
1. Mirror-image foldover designs
2. Alternative foldover designs
9. Three-level full factorial designs
10. Three-level, mixed level and fractional factorial
designs
5.3.1. What are the objectives?

Planning
an
experiment
begins with
carefully
considering
what the
objectives
(or goals)
are
The objectives for an experiment are best determined by a
team discussion. All of the objectives should be written down,
even the "unspoken" ones.
The group should discuss which objectives are the key ones,
and which ones are "nice but not really necessary".
Prioritization of the objectives helps you decide which
direction to go with regard to the selection of the factors,
responses and the particular design. Sometimes prioritization
will force you to start over from scratch when you realize that
the experiment you decided to run does not meet one or more
critical objectives.
Types of
designs
Examples of goals were given earlier in Section 5.1.2, in
which we described four broad categories of experimental
designs, with various objectives for each. These were:
Comparative designs to:
choose between alternatives, with narrow scope,
suitable for an initial comparison (see Section
5.3.3.1)
choose between alternatives, with broad scope,
suitable for a confirmatory comparison (see
Section 5.3.3.2)
Screening designs to identify which factors/effects are
important
when you have 2 - 4 factors and can perform a
full factorial (Section 5.3.3.3)
when you have more than 3 factors and want to
begin with as small a design as possible (Section
5.3.3.4 and 5.3.3.5)
when you have some qualitative factors, or you
have some quantitative factors that are known to
have a non-monotonic effect (Section 3.3.3.10)
Note that some authors prefer to restrict the term
screening design to the case where you are trying to
extract the most important factors from a large (say >
5) list of initial factors (usually a fractional factorial
design). We include the case with a smaller number of
factors, usually a full factorial design, since the basic
purpose and analysis is similar.
Response Surface modeling to achieve one or more of
the following objectives:
hit a target
maximize or minimize a response
reduce variation by locating a region where the
process is easier to manage
make a process robust (note: this objective may
often be accomplished with screening designs
rather than with response surface designs - see
Section 5.5.6)
Regression modeling
to estimate a precise model, quantifying the
dependence of response variable(s) on process
inputs.
Based on
objective,
where to
go next
After identifying the objective listed above that corresponds
most closely to your specific goal, you can
proceed to the next section in which we discuss
selecting experimental factors
and then
select the appropriate design named in section 5.3.3 that
suits your objective (and follow the related links).
5.3.2. How do you select and scale the process variables?

5.3.2. How do you select and scale the process
variables?
Guidelines
to assist the
engineering
judgment
process of
selecting
process
variables
for a DOE
Process variables include both inputs and outputs - i.e., factors and
responses. The selection of these variables is best done as a team effort.
The team should
Include all important factors (based on engineering judgment).
Be bold, but not foolish, in choosing the low and high factor
levels.
Check the factor settings for impractical or impossible
combinations - i.e., very low pressure and very high gas flows.
Include all relevant responses.
Avoid using only responses that combine two or more
measurements of the process. For example, if interested in
selectivity (the ratio of two etch rates), measure both rates, not
just the ratio.
Be careful
when
choosing
the
allowable
range for
each factor
We have to choose the range of the settings for input factors, and it is
wise to give this some thought beforehand rather than just try extreme
values. In some cases, extreme values will give runs that are not
feasible; in other cases, extreme ranges might move one out of a
smooth area of the response surface into some jagged region, or close to
an asymptote.
Two-level
designs
have just a
"high" and
a "low"
setting for
each factor
The most popular experimental designs are two-level designs. Why
only two levels? There are a number of good reasons why two is the
most common choice amongst engineers: one reason is that it is ideal
for screening designs, simple and economical; it also gives most of the
information required to go to a multilevel response surface experiment
if one is needed.
Consider
adding
some center
points to
your two-
level design
The term "two-level design" is something of a misnomer, however, as
it is recommended to include some center points during the experiment
(center points are located in the middle of the design `box').
Notation for 2-Level Designs
Matrix
notation for
describing
an
experiment
The standard layout for a 2-level design uses +1 and -1 notation to
denote the "high level" and the "low level" respectively, for each factor.
For example, the matrix below
Factor 1 (X1) Factor 2 (X2)
Trial 1 -1 -1
Trial 2 +1 -1
Trial 3 -1 +1
Trial 4 +1 +1
describes an experiment in which 4 trials (or runs) were conducted with
each factor set to high or low during a run according to whether the
matrix had a +1 or -1 set for the factor during that trial. If the
experiment had more than 2 factors, there would be an additional
column in the matrix for each additional factor.
Note: Some authors shorten the matrix notation for a two-level design
by just recording the plus and minus signs, leaving out the "1's".
Coding the
data
The use of +1 and -1 for the factor settings is called coding the data.
This aids in the interpretation of the coefficients fit to any experimental
model. After factor settings are coded, center points have the value "0".
Coding is described in more detail in the DOE glossary.
The Model or Analysis Matrix
Design
matrices
If we add an "I" column and an "X1*X2" column to the matrix of 4
trials for a two-factor experiment described earlier, we obtain what is
known as the model or analysis matrix for this simple experiment,
which is shown below. The model matrix for a three-factor experiment
is shown later in this section.
I X1 X2 X1*X2
+1 -1 -1 +1
+1 +1 -1 -1
+1 -1 +1 -1
+1 +1 +1 +1
Model for
the
experiment
The model for this experiment is
and the "I" column of the design matrix has all 1's to provide for the
0
term. The X1*X2 column is formed by multiplying the "X1" and "X2"
columns together, row element by row element. This column gives
interaction term for each trial.
Model in
matrix
notation
In matrix notation, we can summarize this experiment by
Y = X + experimental error
for which Xis the 4 by 4 design matrix of 1's and -1's shown above, is
the vector of unknown model coefficients and Y is a
vector consisting of the four trial response observations.
Orthogonal Property of Scaling in a 2-Factor Experiment
Coding
produces
orthogonal
columns
Coding is sometime called "orthogonal coding" since all the columns
of a coded 2-factor design matrix (except the "I" column) are typically
orthogonal. That is, the dot product for any pair of columns is zero. For
example, for X1 and X2: (-1)(-1) + (+1)(-1) + (-1)(+1) + (+1)(+1) = 0.
5.3.3. How do you select an experimental design?

5.3.3. How do you select an experimental
design?
A design is
selected
based on the
experimental
objective
and the
number of
factors
The choice of an experimental design depends on the
objectives of the experiment and the number of factors to be
investigated.
Experimental Design Objectives
Types of
designs are
listed here
according to
the
experimental
objective
they meet
Types of designs are listed here according to the
experimental objective they meet.
Comparative objective: If you have one or several
factors under investigation, but the primary goal of
your experiment is to make a conclusion about one a-
priori important factor, (in the presence of, and/or in
spite of the existence of the other factors), and the
question of interest is whether or not that factor is
"significant", (i.e., whether or not there is a significant
change in the response for different levels of that
factor), then you have a comparative problem and you
need a comparative design solution.
Screening objective: The primary purpose of the
experiment is to select or screen out the few important
main effects from the many less important ones. These
screening designs are also termed main effects
designs.
Response Surface (method) objective: The
experiment is designed to allow us to estimate
interaction and even quadratic effects, and therefore
give us an idea of the (local) shape of the response
surface we are investigating. For this reason, they are
termed response surface method (RSM) designs. RSM
designs are used to:
Find improved or optimal process settings
Troubleshoot process problems and weak points
Make a product or process more robust against
external and non-controllable influences.
"Robust" means relatively insensitive to these
influences.
Optimizing responses when factors are proportions
of a mixture objective: If you have factors that are
proportions of a mixture and you want to know what
the "best" proportions of the factors are so as to
maximize (or minimize) a response, then you need a
mixture design.
Optimal fitting of a regression model objective: If
you want to model a response as a mathematical
function (either known or empirical) of a few
continuous factors and you desire "good" model
parameter estimates (i.e., unbiased and minimum
variance), then you need a regression design.
Mixture and
regression
designs
Mixture designs are discussed briefly in section 5 (Advanced
Topics) and regression designs for a single factor are
discussed in chapter 4. Selection of designs for the
remaining 3 objectives is summarized in the following table.
Summary
table for
choosing an
experimental
design for
comparative,
screening,
and
response
surface
designs
TABLE 3.1 Design Selection Guideline
Number
of
Factors
Comparative
Objective
Screening
Objective
Response
Surface
Objective
1
1-factor
completely
randomized
design
_ _
2 - 4
Randomized
block design
Full or
fractional
factorial
Central
composite or
Box-Behnken
5 or
more
Randomized
block design
Fractional
factorial or
Plackett-
Burman
Screen first to
reduce number
of factors
Resources
and degree
of control
over wrong
decisions
Choice of a design from within these various types depends
on the amount of resources available and the degree of
control over making wrong decisions (Type I and Type II
errors for testing hypotheses) that the experimenter desires.
Save some
runs for
center points
and "redos"
that might
It is a good idea to choose a design that requires somewhat
fewer runs than the budget permits, so that center point runs
can be added to check for curvature in a 2-level screening
design and backup resources are available to redo runs that
have processing mishaps.
be needed
5.3.3.1. Completely randomized designs

These designs
are for
studying the
effects of one
primary factor
without the
need to take
other nuisance
factors into
account
Here we consider completely randomized designs that
have one primary factor. The experiment compares the
values of a response variable based on the different levels
of that primary factor.
For completely randomized designs, the levels of the
primary factor are randomly assigned to the experimental
units. By randomization, we mean that the run sequence of
the experimental units is determined randomly. For
example, if there are 3 levels of the primary factor with
each level to be run 2 times, then there are 6 factorial
possible run sequences (or 6! ways to order the
experimental trials). Because of the replication, the
number of unique orderings is 90 (since 90 =
6!/(2!*2!*2!)). An example of an unrandomized design
would be to always run 2 replications for the first level,
then 2 for the second level, and finally 2 for the third
level. To randomize the runs, one way would be to put 6
slips of paper in a box with 2 having level 1, 2 having
level 2, and 2 having level 3. Before each run, one of the
slips would be drawn blindly from the box and the level
selected would be used for the next run of the experiment.
Randomization
typically
performed by
computer
software
In practice, the randomization is typically performed by a
computer program. However, the randomization can also
be generated from random number tables or by some
physical mechanism (e.g., drawing the slips of paper).
Three key
numbers
All completely randomized designs with one primary
factor are defined by 3 numbers:
k = number of factors (= 1 for these designs)
L = number of levels
n = number of replications
and the total sample size (number of runs) is N = k x L x
n.
Balance Balance dictates that the number of replications be the
same at each level of the factor (this will maximize the
sensitivity of subsequent statistical t (or F) tests).
Typical
example of a
completely
randomized
design
A typical example of a completely randomized design is
the following:
k = 1 factor (X1)
L = 4 levels of that single factor (called "1", "2",
"3", and "4")
n = 3 replications per level
N = 4 levels * 3 replications per level = 12 runs
A sample
randomized
sequence of
trials
The randomized sequence of trials might look like:
X1
3
1
4
2
2
1
3
4
1
2
4
3
Note that in this example there are 12!/(3!*3!*3!*3!) =
369,600 ways to run the experiment, all equally likely to
be picked by a randomization procedure.
Model for a
completely
randomized
design
The model for the response is
Y
i,j
= + T
i
+ random error
with
Y
i,j
being any observation for which X1 = i
(i and j denote the level of the factor and the
replication within the level of the factor,
respectively)
(or mu) is the general location parameter
T
i
is the effect of having treatment level i
Estimates and Statistical Tests
Estimating
and testing
model factor
Estimate for : = the average of all the data
Estimate for T
i
: -
levels
with = average of all Y for which X1 = i.
Statistical tests for levels of X1 are shown in the section on
one-way ANOVA in Chapter 7.
5.3.3.2. Randomized block designs

Blocking to
"remove" the
effect of
nuisance
factors
For randomized block designs, there is one factor or
variable that is of primary interest. However, there are also
several other nuisance factors.
Nuisance factors are those that may affect the measured
result, but are not of primary interest. For example, in
applying a treatment, nuisance factors might be the specific
operator who prepared the treatment, the time of day the
experiment was run, and the room temperature. All
experiments have nuisance factors. The experimenter will
typically need to spend some time deciding which nuisance
factors are important enough to keep track of or control, if
possible, during the experiment.
Blocking used
for nuisance
factors that
can be
controlled
When we can control nuisance factors, an important
technique known as blocking can be used to reduce or
eliminate the contribution to experimental error contributed
by nuisance factors. The basic concept is to create
homogeneous blocks in which the nuisance factors are held
constant and the factor of interest is allowed to vary.
Within blocks, it is possible to assess the effect of different
levels of the factor of interest without having to worry
about variations due to changes of the block factors, which
are accounted for in the analysis.
Definition of
blocking
factors
A nuisance factor is used as a blocking factor if every level
of the primary factor occurs the same number of times with
each level of the nuisance factor. The analysis of the
experiment will focus on the effect of varying levels of the
primary factor within each block of the experiment.
Block for a
few of the
most
important
nuisance
factors
The general rule is:
"Block what you can, randomize what you cannot."
Blocking is used to remove the effects of a few of the most
important nuisance variables. Randomization is then used
to reduce the contaminating effects of the remaining
nuisance variables.
Table of
randomized
One useful way to look at a randomized block experiment
is to consider it as a collection of completely randomized
block designs experiments, each run within one of the blocks of the total
experiment.
Randomized Block Designs (RBD)
Name of
Design
Number of
Factors
k
Number of
Runs
n
2-factor RBD 2 L
1
* L
2
3-factor RBD 3 L
1
* L
2
* L
3
4-factor RBD 4 L
1
* L
2
* L
3
* L
4
. . .
k-factor RBD k L
1
* L
2
* ... * L
k
with
L
1
= number of levels (settings) of factor 1
L
2
L
3
L
4
.
.
.

L
k
= number of levels (settings) of factor k
Example of a Randomized Block Design
Example of a
randomized
block design
Suppose engineers at a semiconductor manufacturing
facility want to test whether different wafer implant
material dosages have a significant effect on resistivity
measurements after a diffusion process taking place in a
furnace. They have four different dosages they want to try
and enough experimental wafers from the same lot to run
three wafers at each of the dosages.
Furnace run
is a nuisance
factor
The nuisance factor they are concerned with is "furnace
run" since it is known that each furnace run differs from
the last and impacts many process parameters.
Ideal would
be to
eliminate
nuisance
furnace factor
An ideal way to run this experiment would be to run all the
4x3=12 wafers in the same furnace run. That would
eliminate the nuisance furnace factor completely. However,
regular production wafers have furnace priority, and only a
few experimental wafers are allowed into any furnace run
at the same time.
Non-Blocked
method
A non-blocked way to run this experiment would be to run
each of the twelve experimental wafers, in random order,
one per furnace run. That would increase the experimental
error of each resistivity measurement by the run-to-run
furnace variability and make it more difficult to study the
effects of the different dosages. The blocked way to run
this experiment, assuming you can convince manufacturing
to let you put four experimental wafers in a furnace run,
would be to put four wafers with different dosages in each
of three furnace runs. The only randomization would be
choosing which of the three wafers with dosage 1 would go
into furnace run 1, and similarly for the wafers with
dosages 2, 3 and 4.
Description of
the
experiment
Let X1 be dosage "level" and X2 be the blocking factor
furnace run. Then the experiment can be described as
follows:
k = 2 factors (1 primary factor X1 and 1 blocking
factor X2)
L
1
= 4 levels of factor X1
L
2
= 3 levels of factor X2
n = 1 replication per cell
N =L
1
* L
2
= 4 * 3 = 12 runs
Design trial
before
randomization
Before randomization, the design trials look like:
X1 X2
1 1
1 2
1 3
2 1
2 2
2 3
3 1
3 2
3 3
4 1
4 2
4 3
Matrix
representation
An alternate way of summarizing the design trials would
be to use a 4x3 matrix whose 4 rows are the levels of the
treatment X1 and whose columns are the 3 levels of the
blocking variable X2. The cells in the matrix have indices
that match the X1, X2 combinations above.
By extension, note that the trials for any K-factor
randomized block design are simply the cell indices of a K
dimensional matrix.
Model for a Randomized Block Design
Model for a
randomized
block design
The model for a randomized block design with one
nuisance variable is
Y
i,j
= + T
i
+ B
j
+ random error
where
Y
i,j
is any observation for which X1 = i and X2 = j
X1 is the primary factor
X2 is the blocking factor
is the general location parameter (i.e., the mean)
T
i
is the effect for being in treatment i (of factor X1)
B
j
is the effect for being in block j (of factor X2)
Estimates for a Randomized Block Design
Estimating
factor effects
for a
randomized
block design
Estimate for T
i
: -
with = average of all Y for which X1 = i.
Estimate for B
j
: -
with = average of all Y for which X2 = j.
5.3.3.2.1. Latin square and related designs

Latin square
(and related)
designs are
efficient
designs to
block from 2
to 4 nuisance
factors
Latin square designs, and the related Graeco-Latin square
and Hyper-Graeco-Latin square designs, are a special type of
comparative design.
There is a single factor of primary interest, typically called
the treatment factor, and several nuisance factors. For Latin
square designs there are 2 nuisance factors, for Graeco-Latin
square designs there are 3 nuisance factors, and for Hyper-
Graeco-Latin square designs there are 4 nuisance factors.
Nuisance
factors used
as blocking
variables
The nuisance factors are used as blocking variables.
1. For Latin square designs, the 2 nuisance factors are
divided into a tabular grid with the property that each
row and each column receive each treatment exactly
once.
2. As with the Latin square design, a Graeco-Latin square
design is a kxk tabular grid in which k is the number of
levels of the treatment factor. However, it uses 3
blocking variables instead of the 2 used by the standard
Latin square design.
3. A Hyper-Graeco-Latin square design is also a kxk
tabular grid with k denoting the number of levels of the
treatment factor. However, it uses 4 blocking variables
instead of the 2 used by the standard Latin square
design.
Advantages
and
disadvantages
of Latin
square
designs
The advantages of Latin square designs are:
1. They handle the case when we have several nuisance
factors and we either cannot combine them into a
single factor or we wish to keep them separate.
2. They allow experiments with a relatively small number
of runs.
The disadvantages are:
1. The number of levels of each blocking variable must
equal the number of levels of the treatment factor.
2. The Latin square model assumes that there are no
interactions between the blocking variables or between
the treatment variable and the blocking variable.
Note that Latin square designs are equivalent to specific
fractional factorial designs (e.g., the 4x4 Latin square design
is equivalent to a 4
3-1
fractional factorial design).
Summary of
designs
Several useful designs are described in the table below.
Some Useful Latin Square, Graeco-Latin Square and
Hyper-Graeco-Latin Square Designs
Name of
Design
Number of
Factors
k
Number of
Runs
N
3-by-3 Latin Square 3 9

3-by-3 Graeco-Latin Square 4 9

4-by-4 Hyper-Graeco-Latin Square 5 16
5-by-5 Hyper-Graeco-Latin Square 5 25
Model for Latin Square and Related Designs
Latin square
design model
and estimates
for effect
levels
The model for a response for a latin square design is
with
Y
ijk
denoting any observation for which
X1 = i, X2 = j, X3 = k
X1 and X2 are blocking factors
X3 is the primary factor
denoting the general location parameter
R
i
denoting the effect for block i
C
j
denoting the effect for block j
T
k
denoting the effect for treatment k
Models for Graeco-Latin and Hyper-Graeco-Latin squares
are the obvious extensions of the Latin square model, with
additional blocking variables added.
Estimates for Latin Square Designs
Estimates
Estimate for R
i
: -
= average of all Y for which X1 = i
Estimate for C
j
: -
= average of all Y for which X2 = j
Estimate for T
k
: -
= average of all Y for which X3 = k
Randomize as
much as
design allows
Designs for Latin squares with 3-, 4-, and 5-level factors are
given next. These designs show what the treatment
combinations should be for each run. When using any of
these designs, be sure to randomize the treatment units and
trial order, as much as the design allows.
For example, one recommendation is that a Latin square
design be randomly selected from those available, then
randomize the run order.
Latin Square Designs for 3-, 4-, and 5-Level Factors
Designs for
3-level
factors (and 2
nuisance or
blocking
factors)
3-Level Factors
X1 X2 X3
row
blocking
factor
column
blocking
factor
treatment
factor
1 1 1
1 2 2
1 3 3
2 1 3
2 2 1
2 3 2
3 1 2
3 2 3
3 3 1
with
k = 3 factors (2 blocking factors and 1 primary factor)
L
1
= 3 levels of factor X1 (block)
L
2
L
3
= 3 levels of factor X3 (primary)
N = L1 * L2 = 9 runs
This can alternatively be represented as
A B C
C A B
B C A
Designs for
4-level
factors (and 2
nuisance or
blocking
factors)
4-Level Factors
X1 X2 X3
row
blocking
factor
column
blocking
factor
treatment
factor
1 1 1
1 2 2
1 3 4
1 4 3
2 1 4
2 2 3
2 3 1
2 4 2
3 1 2
3 2 4
3 3 3
3 4 1
4 1 3
4 2 1
4 3 2
4 4 4
with
L
1
L
2
L
3
N = L1 * L2 = 16 runs
A B D C
D C A B
B D C A
C A B D
Designs for 5-Level Factors
5-level
factors (and 2
nuisance or
blocking
factors)
X1 X2 X3
row
blocking
factor
column
blocking
factor
treatment
factor
1 1 1
1 2 2
1 3 3
1 4 4
1 5 5
2 1 3
2 2 4
2 3 5
2 4 1
2 5 2
3 1 5
3 2 1
3 3 2
3 4 3
3 5 4
4 1 2
4 2 3
4 3 4
4 4 5
4 5 1
5 1 4
5 2 5
5 3 1
5 4 2
5 5 3
with
L
1
L
2
L
3
N = L1 * L2 = 25 runs
A B C D E
C D E A B
E A B C D
B C D E A
D E A B C
Further
information
More details on Latin square designs can be found in Box,
Hunter, and Hunter (1978).
5.3.3.2.2. Graeco-Latin square designs

These
designs
handle 3
nuisance
factors
Graeco-Latin squares, as described on the previous page, are
efficient designs to study the effect of one treatment factor in
the presence of 3 nuisance factors. They are restricted,
however, to the case in which all the factors have the same
number of levels.
Randomize
as much as
design
allows
Designs for 3-, 4-, and 5-level factors are given on this page.
These designs show what the treatment combinations would
be for each run. When using any of these designs, be sure to
randomize the treatment units and trial order, as much as
the design allows.
For example, one recommendation is that a Graeco-Latin
square design be randomly selected from those available, then
Graeco-Latin Square Designs for 3-, 4-, and 5-Level
Factors
Designs for
3-level
factors
3-Level Factors
X1 X2 X3 X4
row
blocking
factor
column
blocking
factor
blocking
factor
treatment
factor
1 1 1 1
1 2 2 2
1 3 3 3
2 1 2 3
2 2 3 1
2 3 1 2
3 1 3 2
3 2 1 3
3 3 2 1
with
L
1
L
2
L
3
L
4
N = L1 * L2 = 9 runs
This can alternatively be represented as (A, B, and C
represent the treatment factor and 1, 2, and 3 represent the
blocking factor):
A1 B2 C3
C2 A3 B1
B3 C1 A2
Designs for
4-level
factors
4-Level Factors
X1 X2 X3 X4
row
blocking
factor
column
blocking
factor
blocking
factor
treatment
factor
1 1 1 1
1 2 2 2
1 3 3 3
1 4 4 4
2 1 2 4
2 2 1 3
2 3 4 2
2 4 3 1
3 1 3 2
3 2 4 1
3 3 1 4
3 4 2 3
4 1 4 3
4 2 3 4
4 3 2 1
4 4 1 2
with
L
1
L
2
L
3
L
4
N = L1 * L2 = 16 runs
This can alternatively be represented as (A, B, C, and D
represent the treatment factor and 1, 2, 3, and 4 represent the
blocking factor):
A1 B2 C3 D4
D2 C1 B4 A3
B3 A4 D1 C2
C4 D3 A2 B1
Designs for
5-level
factors
5-Level Factors
X1 X2 X3 X4
row
blocking
factor
column
blocking
factor
blocking
factor
treatment
factor
1 1 1 1
1 2 2 2
1 3 3 3
1 4 4 4
1 5 5 5
2 1 2 3
2 2 3 4
2 3 4 5
2 4 5 1
2 5 1 2
3 1 3 5
3 2 4 1
3 3 5 2
3 4 1 3
3 5 2 4
4 1 4 2
4 2 5 3
4 3 1 4
4 4 2 5
4 5 3 1
5 1 5 4
5 2 1 5
5 3 2 1
5 4 3 2
5 5 4 3
with
L
1
L
2
L
3
L
4
N = L1 * L2 = 25 runs
This can alternatively be represented as (A, B, C, D, and E
represent the treatment factor and 1, 2, 3, 4, and 5 represent
the blocking factor):
A1 B2 C3 D4 E5
C2 D3 E4 A5 B1
E3 A4 B5 C1 D2
B4 C5 D1 E2 A3
D5 E1 A2 B3 C4
Further
information
More designs are given in Box, Hunter, and Hunter (1978).
5.3.3.2.3. Hyper-Graeco-Latin square designs

These
designs
handle 4
nuisance
factors
Hyper-Graeco-Latin squares, as described earlier, are efficient
designs to study the effect of one treatment factor in the
presence of 4 nuisance factors. They are restricted, however,
to the case in which all the factors have the same number of
levels.
Randomize
as much as
design
allows
Designs for 4- and 5-level factors are given on this page.
These designs show what the treatment combinations should
be for each run. When using any of these designs, be sure to
randomize the treatment units and trial order, as much as
the design allows.
For example, one recommendation is that a hyper-Graeco-
Latin square design be randomly selected from those
available, then randomize the run order.
Hyper-Graeco-Latin Square Designs for 4- and 5-Level
Factors
Designs for
4-level
factors
(there are
no 3-level
factor
Hyper-
Graeco
Latin
square
designs)
4-Level Factors
X1 X2 X3 X4 X5
row
blocking
factor
column
blocking
factor
blocking
factor
blocking
factor
treatment
factor
1 1 1 1 1
1 2 2 2 2
1 3 3 3 3
1 4 4 4 4
2 1 4 2 3
2 2 3 1 4
2 3 2 4 1
2 4 1 3 2
3 1 2 3 4
3 2 1 4 3
3 3 4 1 2
3 4 3 2 1
4 1 3 4 2
4 2 4 3 1
4 3 1 2 4
4 4 2 1 3
with
L
1
L
2
L
3
L
4
L
5
N = L1 * L2 = 16 runs
This can alternatively be represented as (A, B, C, and D
represent the treatment factor and 1, 2, 3, and 4 represent the
blocking factors):
A11 B22 C33 D44
C42 D31 A24 B13
D23 C14 B41 A32
B34 A43 D12 C21
Designs for
5-level
factors
5-Level Factors
X1 X2 X3 X4 X5
row
blocking
factor
column
blocking
factor
blocking
factor
blocking
factor
treatment
factor
1 1 1 1 1
1 2 2 2 2
1 3 3 3 3
1 4 4 4 4
1 5 5 5 5
2 1 2 3 4
2 2 3 4 5
2 3 4 5 1
2 4 5 1 2
2 5 1 2 3
3 1 3 5 2
3 2 4 1 3
3 3 5 2 4
3 4 1 3 5
3 5 2 4 1
4 1 4 2 5
4 2 5 3 1
4 3 1 4 2
4 4 2 5 3
4 5 3 1 4
5 1 5 4 3
5 2 1 5 4
5 3 2 1 5
5 4 3 2 1
5 5 4 3 2
with
L
1
L
2
L
3
L
4
L
5
N = L1 * L2 = 25 runs
This can alternatively be represented as (A, B, C, D, and E
represent the treatment factor and 1, 2, 3, 4, and 5 represent
the blocking factors):
A11 B22 C33 D44 E55
D23 E34 A45 B51 C12
B35 C41 D52 E31 A24
E42 A53 B14 C25 D31
C54 D15 E21 A32 B43
Further
information
More designs are given in Box, Hunter, and Hunter (1978).
5.3.3.3. Full factorial designs

Full factorial designs in two levels
A design in
which every
setting of
every factor
appears with
every setting
of every other
factor is a
full factorial
design
A common experimental design is one with all input factors
set at two levels each. These levels are called `high' and
`low' or `+1' and `-1', respectively. A design with all
possible high/low combinations of all the input factors is
called a full factorial design in two levels.
If there are k factors, each at 2 levels, a full factorial
design has 2
k
runs.
TABLE 3.2 Number of Runs for a 2
k
Full Factorial
Number of Factors Number of Runs
2 4
3 8
4 16
5 32
6 64
7 128
Full factorial
designs not
recommended
for 5 or more
factors
As shown by the above table, when the number of factors is
5 or greater, a full factorial design requires a large number
of runs and is not very efficient. As recommended in the
Design Guideline Table, a fractional factorial design or a
Plackett-Burman design is a better choice for 5 or more
factors.
5.3.3.3.1. Two-level full factorial designs

Description
Graphical
representation
of a two-level
design with 3
factors
Consider the two-level, full factorial design for three factors,
namely the 2
3
design. This implies eight runs (not counting
replications or center point runs). Graphically, we can
represent the 2
3
design by the cube shown in Figure 3.1. The
arrows show the direction of increase of the factors. The
numbers `1' through `8' at the corners of the design box
reference the `Standard Order' of runs (see Figure 3.1).
FIGURE 3.1 A 2
3
two-level, full factorial design; factors
X1, X2, X3
The design
matrix
In tabular form, this design is given by:
TABLE 3.3 A 2
3
two-level, full factorial
design table showing runs in `Standard
Order'
run X1 X2 X3
1 -1 -1 -1
2 1 -1 -1
3 -1 1 -1
4 1 1 -1
5 -1 -1 1
6 1 -1 1
7 -1 1 1
8 1 1 1
The left-most column of Table 3.3, numbers 1 through 8,
specifies a (non-randomized) run order called the `Standard
Order.' These numbers are also shown in Figure 3.1. For
example, run 1 is made at the `low' setting of all three
factors.
Standard Order for a 2
k
Level Factorial Design
Rule for
writing a 2
k
full factorial
in "standard
order"
We can readily generalize the 2
3
standard order matrix to a
2-level full factorial with k factors. The first (X1) column
starts with -1 and alternates in sign for all 2
k
runs. The
second (X2) column starts with -1 repeated twice, then
alternates with 2 in a row of the opposite sign until all 2
k
places are filled. The third (X3) column starts with -1
repeated 4 times, then 4 repeats of +1's and so on. In general,
the i-th column (X
i
) starts with 2
i-1
repeats of -1 folowed by
2
i-1
repeats of +1.
Example of a 2
3
Experiment
Analysis
matrix for the
3-factor
complete
factorial
An engineering experiment called for running three factors;
namely, Pressure (factor X1), Table speed (factor X2) and
Down force (factor X3), each at a `high' and `low' setting, on
a production tool to determine which had the greatest effect
on product uniformity. Two replications were run at each
setting. A (full factorial) 2
3
design with 2 replications calls
for 8*2=16 runs.
TABLE 3.4 Model or Analysis Matrix for a 2
3
Experiment
Model Matrix Response
Variables
I X1 X2 X1*X2 X3 X1*X3 X2*X3 X1*X2*X3
Rep
1
Rep
2
+1 -1 -1 +1 -1 +1 +1 -1 -3 -1
+1 +1 -1 -1 -1 -1 +1 +1 0 -1
+1 -1 +1 -1 -1 +1 -1 +1 -1 0
+1 +1 +1 +1 -1 -1 -1 -1 +2 +3
+1 -1 -1 +1 +1 -1 -1 +1 -1 0
+1 +1 -1 -1 +1 +1 -1 -1 +2 +1
+1 -1 +1 -1 +1 -1 +1 -1 +1 +1
+1 +1 +1 +1 +1 +1 +1 +1 +6 +5
The block with the 1's and -1's is called the Model Matrix or
the Analysis Matrix. The table formed by the columns X1,
X2 and X3 is called the Design Table or Design Matrix.
Orthogonality Properties of Analysis Matrices for 2-
Factor Experiments
Eliminate
correlation
between
estimates of
main effects
and
interactions
When all factors have been coded so that the high value is
"1" and the low value is "-1", the design matrix for any full
(or suitably chosen fractional) factorial experiment has
columns that are all pairwise orthogonal and all the columns
(except the "I" column) sum to 0.
The orthogonality property is important because it eliminates
correlation between the estimates of the main effects and
interactions.
5.3.3.3.2. Full factorial example

A Full Factorial Design Example
An example of
a full factorial
design with 3
factors
The following is an example of a full factorial design with
3 factors that also illustrates replication, randomization, and
added center points.
Suppose that we wish to improve the yield of a polishing
operation. The three inputs (factors) that are considered
important to the operation are Speed (X1), Feed (X2), and
Depth (X3). We want to ascertain the relative importance of
each of these factors on Yield (Y).
Speed, Feed and Depth can all be varied continuously along
their respective scales, from a low to a high setting. Yield is
observed to vary smoothly when progressive changes are
made to the inputs. This leads us to believe that the ultimate
response surface for Y will be smooth.
Table of factor
level settings
TABLE 3.5 High (+1), Low (-1), and Standard (0)
Settings for a Polishing Operation
Low (-1) Standard (0) High (+1) Units
Speed 16 20 24 rpm
Feed 0.001 0.003 0.005 cm/sec
Depth 0.01 0.015 0.02 cm/sec
Factor Combinations
Graphical
representation
of the factor
level settings
We want to try various combinations of these settings so as
to establish the best way to run the polisher. There are eight
different ways of combining high and low settings of
Speed, Feed, and Depth. These eight are shown at the
corners of the following diagram.
FIGURE 3.2 A 2
3
Two-level, Full Factorial Design;
Factors X1, X2, X3. (The arrows show the direction of
increase of the factors.)
2
3
implies 8
runs
Note that if we have k factors, each run at two levels, there
will be 2
k
different combinations of the levels. In the
present case, k = 3 and 2
3
= 8.
Full Model Running the full complement of all possible factor
combinations means that we can estimate all the main and
interaction effects. There are three main effects, three two-
factor interactions, and a three-factor interaction, all of
which appear in the full model as follows:
A full factorial design allows us to estimate all eight `beta'
coefficients .
Standard order
Coded
variables in
standard order
The numbering of the corners of the box in the last figure
refers to a standard way of writing down the settings of an
experiment called `standard order'. We see standard order
displayed in the following tabular representation of the
eight-cornered box. Note that the factor settings have been
coded, replacing the low setting by -1 and the high setting
by 1.
Factor settings
in tabular
form
TABLE 3.6 A 2
3
Two-level, Full Factorial
Design Table Showing Runs in `Standard
Order'
X1 X2 X3
1 -1 -1 -1
2 +1 -1 -1
3 -1 +1 -1
4 +1 +1 -1
5 -1 -1 +1
6 +1 -1 +1
7 -1 +1 +1
8 +1 +1 +1
Replication
Replication
provides
information on
variability
Running the entire design more than once makes for easier
data analysis because, for each run (i.e., `corner of the
design box') we obtain an average value of the response as
well as some idea about the dispersion (variability,
consistency) of the response at that setting.
Homogeneity
of variance
One of the usual analysis assumptions is that the response
dispersion is uniform across the experimental space. The
technical term is `homogeneity of variance'. Replication
allows us to check this assumption and possibly find the
setting combinations that give inconsistent yields, allowing
us to avoid that area of the factor space.
Factor settings
in standard
order with
replication
We now have constructed a design table for a two-level full
factorial in three factors, replicated twice.
TABLE 3.7 The 2
3
Full Factorial
Replicated Twice and Presented in
Standard Order
Speed, X1 Feed, X2 Depth, X3
1 16, -1 .001, -1 .01, -1
2 24, +1 .001, -1 .01, -1
3 16, -1 .005, +1 .01, -1
4 24, +1 .005, +1 .01, -1
5 16, -1 .001, -1 .02, +1
6 24, +1 .001, -1 .02, +1
7 16, -1 .005, +1 .02, +1
8 24, +1 .005, +1 .02, +1
9 16, -1 .001, -1 .01, -1
10 24, +1 .001, -1 .01, -1
11 16, -1 .005, +1 .01, -1
12 24, +1 .005, +1 .01, -1
13 16, -1 .001, -1 .02, +1
14 24, +1 .001, -1 .02, +1
15 16, -1 .005, +1 .02, +1
16 24, +1 .005, +1 .02, +1
Randomization
No
randomization
and no center
points
If we now ran the design as is, in the order shown, we
would have two deficiencies, namely:
1. no randomization, and
2. no center points.
Randomization
provides
protection
against
extraneous
factors
affecting the
results
The more freely one can randomize experimental runs, the
more insurance one has against extraneous factors possibly
affecting the results, and hence perhaps wasting our
experimental time and effort. For example, consider the
`Depth' column: the settings of Depth, in standard order,
follow a `four low, four high, four low, four high' pattern.
Suppose now that four settings are run in the day and four
at night, and that (unknown to the experimenter) ambient
temperature in the polishing shop affects Yield. We would
run the experiment over two days and two nights and
conclude that Depth influenced Yield, when in fact ambient
temperature was the significant influence. So the moral is:
Randomize experimental runs as much as possible.
Table of factor
settings in
randomized
order
Here's the design matrix again with the rows randomized.
The old standard order column is also shown for
comparison and for re-sorting, if desired, after the runs are
in.
TABLE 3.8 The 2
3
Full Factorial
Replicated Twice with Random Run Order
Indicated
Random
Order
Standard
Order X1 X2 X3
1 5 -1 -1 +1
2 15 -1 +1 +1
3 9 -1 -1 -1
4 7 -1 +1 +1
5 3 -1 +1 -1
6 12 +1 +1 -1
7 6 +1 -1 +1
8 4 +1 +1 -1
9 2 +1 -1 -1
10 13 -1 -1 +1
11 8 +1 +1 +1
12 16 +1 +1 +1
13 1 -1 -1 -1
14 14 +1 -1 +1
15 11 -1 +1 -1
16 10 +1 -1 -1
Table showing
design matrix
with
randomization
and center
points
This design would be improved by adding at least 3
centerpoint runs placed at the beginning, middle and end of
the experiment. The final design matrix is shown below:
TABLE 3.9 The 2
3
Full Factorial
Replicated Twice with Random Run Order
Indicated and Center Point Runs Added
Random
Order
Standard
Order X1 X2 X3
1 0 0 0
2 5 -1 -1 +1
3 15 -1 +1 +1
4 9 -1 -1 -1
5 7 -1 +1 +1
6 3 -1 +1 -1
7 12 +1 +1 -1
8 6 +1 -1 +1
9 0 0 0
10 4 +1 +1 -1
11 2 +1 -1 -1
12 13 -1 -1 +1
13 8 +1 +1 +1
14 16 +1 +1 +1
15 1 -1 -1 -1
16 14 +1 -1 +1
17 11 -1 +1 -1
18 10 +1 -1 -1
19 0 0 0
5.3.3.3.3. Blocking of full factorial designs

Eliminate the
influence of
extraneous
factors by
"blocking"
We often need to eliminate the influence of extraneous
factors when running an experiment. We do this by
"blocking".
Previously, blocking was introduced when randomized
block designs were discussed. There we were concerned
with one factor in the presence of one of more nuisance
factors. In this section we look at a general approach that
enables us to divide 2-level factorial experiments into
blocks.
For example, assume we anticipate predictable shifts will
occur while an experiment is being run. This might happen
when one has to change to a new batch of raw materials
halfway through the experiment. The effect of the change in
raw materials is well known, and we want to eliminate its
influence on the subsequent data analysis.
Blocking in a
2
3
factorial
design
In this case, we need to divide our experiment into two
halves (2 blocks), one with the first raw material batch and
the other with the new batch. The division has to balance
out the effect of the materials change in such a way as to
eliminate its influence on the analysis, and we do this by
blocking.
Example
Example: An eight-run 2
3
full factorial has to be blocked
into two groups of four runs each. Consider the design `box'
for the 2
3
full factorial. Blocking can be achieved by
assigning the first block to the dark-shaded corners and the
second block to the open circle corners.
Graphical
representation
of blocking
scheme
FIGURE 3.3 Blocking Scheme for a 2
3
Using Alternate
Corners
Three-factor
interaction
confounded
with the block
effect
This works because we are in fact assigning the èstimation'
of the (unwanted) blocking effect to the three-factor
interaction, and because of the special property of two-level
designs called orthogonality. That is, the three-factor
interaction is "confounded" with the block effect as will be
seen shortly.
Orthogonality Orthogonality guarantees that we can always estimate the
effect of one factor or interaction clear of any influence due
to any other factor or interaction. Orthogonality is a very
desirable property in DOE and this is a major reason why
two-level factorials are so popular and successful.
Table
showing
blocking
scheme
Formally, consider the 2
3
design table with the three-factor
interaction column added.
TABLE 3.10 Two Blocks for a 2
3
Design
SPEED
X1
FEED
X2
DEPTH
X3 X1*X2*X3
BLOCK
-1 -1 -1 -1 I
+1 -1 -1 +1 II
-1 +1 -1 +1 II
+1 +1 -1 -1 I
-1 -1 +1 +1 II
+1 -1 +1 -1 I
-1 +1 +1 -1 I
+1 +1 +1 +1 II
Block by
assigning the
"Block effect"
Rows that have a `-1' in the three-factor interaction column
are assigned to `Block I' (rows 1, 4, 6, 7), while the other
rows are assigned to `Block II' (rows 2, 3, 5, 8). Note that
to a high-
order
interaction
the Block I rows are the open circle corners of the design
`box' above; Block II are dark-shaded corners.
Most DOE
software will
do blocking
for you
The general rule for blocking is: use one or a combination
of high-order interaction columns to construct blocks. This
gives us a formal way of blocking complex designs. Apart
from simple cases in which you can design your own
blocks, your statistical/DOE software will do the blocking if
asked, but you do need to understand the principle behind
it.
Block effects
are
confounded
with higher-
order
interactions
The price you pay for blocking by using high-order
interaction columns is that you can no longer distinguish the
high-order interaction(s) from the blocking effect - they
have been `confounded,' or àliased.' In fact, the blocking
effect is now the sum of the blocking effect and the high-
order interaction effect. This is fine as long as our
assumption about negligible high-order interactions holds
true, which it usually does.
Center points
within a block
Within a block, center point runs are assigned as if the
block were a separate experiment - which in a sense it is.
Randomization takes place within a block as it would for
any non-blocked DOE.
5.3.3.4. Fractional factorial designs

Full
factorial
experiments
can require
many runs
The ASQC (1983) Glossary & Tables for Statistical Quality
Control defines fractional factorial design in the following
way: "A factorial experiment in which only an adequately
chosen fraction of the treatment combinations required for
the complete factorial experiment is selected to be run."
A carefully
chosen
fraction of
the runs
may be all
that is
necessary
Even if the number of factors, k, in a design is small, the 2
k
runs specified for a full factorial can quickly become very
large. For example, 2
6
= 64 runs is for a two-level, full
factorial design with six factors. To this design we need to
add a good number of centerpoint runs and we can thus
quickly run up a very large resource requirement for runs
with only a modest number of factors.
Later
sections
will show
how to
choose the
"right"
fraction for
2-level
designs -
these are
both
balanced
and
orthogonal
The solution to this problem is to use only a fraction of the
runs specified by the full factorial design. Which runs to
make and which to leave out is the subject of interest here. In
general, we pick a fraction such as , , etc. of the runs
called for by the full factorial. We use various strategies that
ensure an appropriate choice of runs. The following sections
will show you how to choose an appropriate fraction of a full
factorial design to suit your purpose at hand. Properly chosen
fractional factorial designs for 2-level experiments have the
desirable properties of being both balanced and orthogonal.
2-Level
fractional
factorial
designs
emphasized
Note: We will be emphasizing fractions of two-level designs
only. This is because two-level fractional designs are, in
engineering at least, by far the most popular fractional
designs. Fractional factorials where some factors have three
levels will be covered briefly in Section 5.3.3.10.
5.3.3.4.1. A 23-1 design (half of a 23)

5.3.3.4.1.
A 2
3-1
design (half of a 2
3
)
We can run a
fraction of a
full factorial
experiment
and still be
able to
estimate main
effects
Consider the two-level, full factorial design for three
factors, namely the 2
3
design. This implies eight runs (not
counting replications or center points). Graphically, as
shown earlier, we can represent the 2
3
design by the
following cube:
FIGURE 3.4 A 2
3
Full Factorial Design;
Factors X
1
, X
2
, X
3
. (The arrows show the direction of
increase of the factors. Numbers `1' through `8' at the
corners of the design cube reference the `Standard
Order' of runs)
Tabular
representation
of the design
In tabular form, this design (also showing eight
observations `y
j
'
(j = 1,...,8) is given by
TABLE 3.11 A 2
3
Two-level, Full Factorial Design
Table Showing Runs in `Standard Order,' Plus
Observations (y
j
)
X1 X2 X3 Y
1 -1 -1 -1 y
1
= 33
2 +1 -1 -1 y
2
= 63
3 -1 +1 -1 y
3
= 41
4 +1 +1 -1 Y
4
= 57
5 -1 -1 +1 y
5
= 57
6 +1 -1 +1 y
6
= 51
7 -1 +1 +1 y
7
= 59
8 +1 +1 +1 y
8
= 53
Responses in
standard
order
The right-most column of the table lists `y
1
' through `y
8
' to
indicate the responses measured for the experimental runs
when listed in standard order. For example, `y
1
' is the
response (i.e., output) observed when the three factors were
all run at their `low' setting. The numbers entered in the "y"
column will be used to illustrate calculations of effects.
Computing X1
main effect
From the entries in the table we are able to compute all
èffects' such as main effects, first-order ìnteraction'
effects, etc. For example, to compute the main effect
estimate `c
1
' of factor X
1
, we compute the average response
at all runs with X
1
at the `high' setting, namely (1/4)(y
2
+
y
4
+ y
6
+ y
8
), minus the average response of all runs with
X
1
set at `low,' namely (1/4)(y
1
+ y
3
+ y
5
+ y
7
). That is,
c
1
= (1/4) (y
2
+ y
4
+ y
6
+ y
8
) - (1/4)(y
1
+ y
3
+ y
5
+
y
7
) or
c
1
= (1/4)(63+57+51+53 ) - (1/4)(33+41+57+59) =
8.5
Can we
estimate X1
main effect
with four
runs?
Suppose, however, that we only have enough resources to
do four runs. Is it still possible to estimate the main effect
for X
1
? Or any other main effect? The answer is yes, and
there are even different choices of the four runs that will
accomplish this.
Example of
computing the
main effects
using only
four runs
For example, suppose we select only the four light
(unshaded) corners of the design cube. Using these four
runs (1, 4, 6 and 7), we can still compute c
1
as follows:
c
1
= (1/2) (y
4
+ y
6
) - (1/2) (y
1
+ y
7
) or
c
1
= (1/2) (57+51) - (1/2) (33+59) = 8.
Simarly, we would compute c
2
, the effect due to X
2
, as
c
2
= (1/2) (y
4
+ y
7
) - (1/2) (y
1
+ y
6
) or
c
2
= (1/2) (57+59) - (1/2) (33+51) = 16.
Finally, the computation of c
3
for the effect due to X
3
would be
c
3
= (1/2) (y
6
+ y
7
) - (1/2) (y
1
+ y
4
) or
c
3
= (1/2) (51+59) - (1/2) (33+57) = 10.
Alternative
runs for
computing
main effects
We could also have used the four dark (shaded) corners of
the design cube for our runs and obtained similiar, but
slightly different, estimates for the main effects. In either
case, we would have used half the number of runs that the
full factorial requires. The half fraction we used is a new
design written as 2
3-1
. Note that 2
3-1
= 2
3
/2 = 2
2
= 4,
which is the number of runs in this half-fraction design. In
the next section, a general method for choosing fractions
that "work" will be discussed.
Example of
how
fractional
factorial
experiments
often arise in
industry
Example: An engineering experiment calls for running
three factors, namely Pressure, Table speed, and Down
force, each at a `high' and a `low' setting, on a production
tool to determine which has the greatest effect on product
uniformity. Interaction effects are considered negligible, but
uniformity measurement error requires that at least two
separate runs (replications) be made at each process setting.
In addition, several `standard setting' runs (centerpoint runs)
need to be made at regular intervals during the experiment
to monitor for process drift. As experimental time and
material are limited, no more than 15 runs can be planned.
A full factorial 2
3
design, replicated twice, calls for 8x2 =
16 runs, even without centerpoint runs, so this is not an
option. However a 2
3-1
design replicated twice requires
only 4x2 = 8 runs, and then we would have 15-8 = 7 spare
runs: 3 to 5 of these spare runs can be used for centerpoint
runs and the rest saved for backup in case something goes
wrong with any run. As long as we are confident that the
interactions are negligbly small (compared to the main
effects), and as long as complete replication is required,
then the above replicated 2
3-1
fractional factorial design
(with center points) is a very reasonable choice.
On the other hand, if interactions are potentially large (and
if the replication required could be set aside), then the usual
2
3
full factorial design (with center points) would serve as a
good design.
5.3.3.4.2. Constructing the 23-1 half-fraction design

5.3.3.4.2.
Constructing the 2
3-1
half-fraction
design
Construction
of a 2
3-1
half fraction
design by
staring with
a 2
2
full
factorial
design
First note that, mathematically, 2
3-1
= 2
2
. This gives us the
first step, which is to start with a regular 2
2
full factorial
design. That is, we start with the following design table.
TABLE 3.12 A Standard
Order 2
2
Full Factorial
Design Table
X1 X2
1 -1 -1
2 +1 -1
3 -1 +1
4 +1 +1
Assign the
third factor
to the
interaction
column of a
2
2
design
This design has four runs, the right number for a half-
fraction of a 2
3
, but there is no column for factor X3. We
need to add a third column to take care of this, and we do it
by adding the X1*X2 interaction column. This column is, as
you will recall from full factorial designs, constructed by
multiplying the row entry for X1 with that of X2 to obtain the
row entry for X1*X2.
TABLE 3.13 A 2
2
Design
Table Augmented with the
X1*X2 Interaction Column
`X1*X2'
X1 X2 X1*X2
1 -1 -1 +1
2 +1 -1 -1
3 -1 +1 -1
4 +1 +1 +1
Design table
with X3 set
to X1*X2
We may now substitute `X3' in place of `X1*X2' in this
table.
TABLE 3.15 A 2
3-1
Design
Table with Column X3 set
to X1*X2
X1 X2 X3
1 -1 -1 +1
2 +1 -1 -1
3 -1 +1 -1
4 +1 +1 +1
Design table
with X3 set
to -X1*X2
Note that the rows of Table 3.14 give the dark-shaded
corners of the design in Figure 3.4. If we had set X3 = -
X1*X2 as the rule for generating the third column of our 2
3-
1
design, we would have obtained:
TABLE 3.15 A 2
3-1
Design
Table with Column X3 set
to - X1*X2
X1 X2 X3
1 -1 -1 -1
2 +1 -1 +1
3 -1 +1 +1
4 +1 +1 -1
Main effect
estimates
from
fractional
factorial not
as good as
full factorial
This design gives the light-shaded corners of the box of
Figure 3.4. Both 2
3-1
designs that we have generated are
equally good, and both save half the number of runs over the
original 2
3
full factorial design. If c
1
, c
2
, and c
3
are our
estimates of the main effects for the factors X1, X2, X3 (i.e.,
the difference in the response due to going from "low" to
"high" for an effect), then the precision of the estimates c
1
,
c
2
, and c
3
are not quite as good as for the full 8-run factorial
because we only have four observations to construct the
averages instead of eight; this is one price we have to pay
for using fewer runs.
Example Example: For the `Pressure (P), Table speed (T), and Down
force (D)' design situation of the previous example, here's a
replicated 2
3-1
in randomized run order, with five
centerpoint runs (`000') interspersed among the runs. This
design table was constructed using the technique discussed
above, with D = P*T.
Design table
for the
example
TABLE 3.16 A 2
3-1
Design Replicated
Twice, with Five Centerpoint Runs Added
Pattern P T D
Center
Point
1 000 0 0 0 1
2 +-- +1 -1 -1 0
3 -+- -1 +1 -1 0
4 000 0 0 0 1
5 +++ +1 +1 +1 0
6 --+ -1 -1 +1 0
7 000 0 0 0 1
8 +-- +1 -1 -1 0
9 --+ -1 -1 +1 0
10 000 0 0 0 1
11 +++ +1 +1 +1 0
12 -+- -1 +1 -1 0
13 000 0 0 0 1
5.3.3.4.3. Confounding (also called aliasing)

Confounding
means we
have lost the
ability to
estimate
some effects
and/or
interactions
One price we pay for using the design table column X1*X2 to
obtain column X3 in Table 3.14 is, clearly, our inability to obtain
an estimate of the interaction effect for X1*X2 (i.e., c
12
) that is
separate from an estimate of the main effect for X3. In other
words, we have confounded the main effect estimate for factor
X3 (i.e., c
3
) with the estimate of the interaction effect for X1 and
X2 (i.e., with c
12
). The whole issue of confounding is
fundamental to the construction of fractional factorial designs,
and we will spend time discussing it below.
Sparsity of
effects
assumption
In using the 2
3-1
design, we also assume that c
12
is small
compared to c
3
; this is called a `sparsity of effects' assumption.
Our computation of c
3
is in fact a computation of c
3
+ c
12
. If the
desired effects are only confounded with non-significant
interactions, then we are OK.
A Notation and Method for Generating Confounding or
Aliasing
A short way
of writing
factor
column
multiplication
A short way of writing `X3 = X1*X2' (understanding that we are
talking about multiplying columns of the design table together)
is: `3 = 12' (similarly 3 = -12 refers to X3 = -X1*X2). Note that
`12' refers to column multiplication of the kind we are using to
construct the fractional design and any column multiplied by
itself gives the identity column of all 1's.
Next we multiply both sides of 3=12 by 3 and obtain 33=123, or
I=123 since 33=I (or a column of all 1's). Playing around with
this "algebra", we see that 2I=2123, or 2=2123, or 2=1223, or
2=13 (since 2I=2, 22=I, and 1I3=13). Similarly, 1=23.
Definition of
"design
generator" or
"generating
relation" and
"defining
relation"
I=123 is called a design generator or a generating relation for
this 2
3-1
design (the dark-shaded corners of Figure 3.4). Since
there is only one design generator for this design, it is also the
defining relation for the design. Equally, I=-123 is the design
generator (and defining relation) for the light-shaded corners of
Figure 3.4. We call I=123 the defining relation for the 2
3-1
design because with it we can generate (by "multiplication") the
complete confounding pattern for the design. That is, given
I=123, we can generate the set of {1=23, 2=13, 3=12, I=123},
which is the complete set of aliases, as they are called, for this
2
3-1
fractional factorial design. With I=123, we can easily
generate all the columns of the half-fraction design 2
3-1
.
Principal
fraction
Note: We can replace any design generator by its negative
counterpart and have an equivalent, but different fractional
design. The fraction generated by positive design generators is
sometimes called the principal fraction.
All main
effects of 2
3-1
design
confounded
with two-
factor
interactions
The confounding pattern described by 1=23, 2=13, and 3=12
tells us that all the main effects of the 2
3-1
design are confounded
with two-factor interactions. That is the price we pay for using
this fractional design. Other fractional designs have different
confounding patterns; for example, in the typical quarter-fraction
of a 2
6
design, i.e., in a 2
6-2
design, main effects are confounded
with three-factor interactions (e.g., 5=123) and so on. In the case
of 5=123, we can also readily see that 15=23 (etc.), which alerts
us to the fact that certain two-factor interactions of a 2
6-2
are
confounded with other two-factor interactions.
A useful
summary
diagram for a
fractional
factorial
design
Summary: A convenient summary diagram of the discussion so
far about the 2
3-1
design is as follows:
FIGURE 3.5 Essential Elements of a 2
3-1
Design
The next section will add one more item to the above box, and
then we will be able to select the right two-level fractional
factorial design for a wide range of experimental tasks.
5.3.3.4.4. Fractional factorial design specifications and design resolution

5.3.3.4.4. Fractional factorial design
specifications and design resolution
Generating
relation and
diagram for
the 2
8-3
fractional
factorial
design
We considered the 2
3-1
design in the previous section and
saw that its generator written in "I = ... " form is {I = +123}.
Next we look at a one-eighth fraction of a 2
8
design, namely
the 2
8-3
fractional factorial design. Using a diagram similar
to Figure 3.5, we have the following:
FIGURE 3.6 Specifications for a 2
8-3
Design
2
8-3
design
has 32 runs
Figure 3.6 tells us that a 2
8-3
design has 32 runs, not
including centerpoint runs, and eight factors. There are three
generators since this is a 1/8 = 2
-3
fraction (in general, a 2
k-p
fractional factorial needs p generators which define the
settings for p additional factor columns to be added to the
2
k-p
full factorial design columns - see the following
detailed description for the 2
8-3
design).
How to Construct a Fractional Factorial Design From
the Specification
Rule for
constructing
a fractional
factorial
design
In order to construct the design, we do the following:
1. Write down a full factorial design in standard order
for k-p factors (8-3 = 5 factors for the example
5
above). In the specification above we start with a 2
full factorial design. Such a design has 2
5
= 32 rows.
2. Add a sixth column to the design table for factor 6,
using 6 = 345 (or 6 = -345) to manufacture it (i.e.,
create the new column by multiplying the indicated
old columns together).
3. Do likewise for factor 7 and for factor 8, using the
appropriate design generators given in Figure 3.6.
4. The resultant design matrix gives the 32 trial runs for
an 8-factor fractional factorial design. (When actually
running the experiment, we would of course
Design
generators
We note further that the design generators, written in Ì = ...'
form, for the principal 2
8-3
fractional factorial design are:
{ I = + 3456; I = + 12457; I = +12358 }.
These design generators result from multiplying the "6 =
345" generator by "6" to obtain "I = 3456" and so on for the
other two generqators.
"Defining
relation" for
a fractional
factorial
design
The total collection of design generators for a factorial
design, including all new generators that can be formed as
products of these generators, is called a defining relation.
There are seven "words", or strings of numbers, in the
defining relation for the 2
8-3
design, starting with the
original three generators and adding all the new "words" that
can be formed by multiplying together any two or three of
these original three words. These seven turn out to be I =
3456 = 12457 = 12358 = 12367 = 12468 = 3478 = 5678. In
general, there will be (2
p
-1) words in the defining relation
for a 2
k-p
fractional factorial.
Definition of
"Resolution"
The length of the shortest word in the defining relation is
called the resolution of the design. Resolution describes the
degree to which estimated main effects are aliased (or
confounded) with estimated 2-level interactions, 3-level
interactions, etc.
Notation for
resolution
(Roman
numerals)
The length of the shortest word in the defining relation for
the 2
8-3
design is four. This is written in Roman numeral
script, and subscripted as . Note that the 2
3-1
design has
only one word, "I = 123" (or "I = -123"), in its defining
relation since there is only one design generator, and so this
fractional factorial design has resolution three; that is, we
may write .
Diagram for
a 2
8-3
design
showing
resolution
Now Figure 3.6 may be completed by writing it as:
FIGURE 3.7 Specifications for a 2
8-3
, Showing
Resolution IV
Resolution
and
confounding
The design resolution tells us how badly the design is
confounded. Previously, in the 2
3-1
design, we saw that the
main effects were confounded with two-factor interactions.
However, main effects were not confounded with other main
effects. So, at worst, we have 3=12, or 2=13, etc., but we do
not have 1=2, etc. In fact, a resolution II design would be
pretty useless for any purpose whatsoever!
Similarly, in a resolution IV design, main effects are
confounded with at worst three-factor interactions. We can
see, in Figure 3.7, that 6=345. We also see that 36=45,
34=56, etc. (i.e., some two-factor interactions are
confounded with certain other two-factor interactions) etc.;
but we never see anything like 2=13, or 5=34, (i.e., main
effects confounded with two-factor interactions).
The
complete
first-order
interaction
confounding
for the given
2
8-3
design
The complete confounding pattern, for confounding of up to
two-factor interactions, arising from the design given in
Figure 3.7 is
34 = 56 = 78
35 = 46
36 = 45
37 = 48
38 = 47
57 = 68
58 = 67
All of these relations can be easily verified by multiplying
the indicated two-factor interactions by the generators. For
example, to verify that 38= 47, multiply both sides of
8=1235 by 3 to get 38=125. Then, multiply 7=1245 by 4 to
get 47=125. From that it follows that 38=47.
One or two
factors
suspected of
possibly
having
significant
first-order
interactions
can be
assigned in
such a way
as to avoid
having them
aliased
For this fractional factorial design, 15 two-factor
interactions are aliased (confounded) in pairs or in a group
of three. The remaining 28 - 15 = 13 two-factor interactions
are only aliased with higher-order interactions (which are
generally assumed to be negligible). This is verified by
noting that factors "1" and "2" never appear in a length-4
word in the defining relation. So, all 13 interactions
involving "1" and "2" are clear of aliasing with any other
two factor interaction.
If one or two factors are suspected of possibly having
significant first-order interactions, they can be assigned in
such a way as to avoid having them aliased.
Higher
resoulution
designs have
less severe
confounding,
but require
more runs
A resolution IV design is "better" than a resolution III design
because we have less-severe confounding pattern in the ÌV'
than in the ÌII' situation; higher-order interactions are less
likely to be significant than low-order interactions.
A higher-resolution design for the same number of factors
will, however, require more runs and so it is `worse' than a
lower order design in that sense.
Resolution V
designs for 8
factors
Similarly, with a resolution V design, main effects would be
confounded with four-factor (and possibly higher-order)
interactions, and two-factor interactions would be
confounded with certain three-factor interactions. To obtain
a resolution V design for 8 factors requires more runs than
the 2
8-3
design. One option, if estimating all main effects
and two-factor interactions is a requirement, is a
design. However, a 48-run alternative (John's 3/4 fractional
factorial) is also available.
There are
many
choices of
fractional
factorial
designs -
some may
have the
same
number of
runs and
resolution,
but different
aliasing
patterns.
Note: There are other fractional designs that can be
derived starting with different choices of design generators
for the "6", "7" and "8" factor columns. However, they are
either equivalent (in terms of the number of words of length
of length of four) to the fraction with generators 6 = 345, 7 =
1245, 8 = 1235 (obtained by relabeling the factors), or they
are inferior to the fraction given because their defining
relation contains more words of length four (and therefore
more confounded two-factor interactions). For example, the
design with generators 6 = 12345, 7 = 135, and 8 = 245
has five length-four words in the defining relation (the
defining relation is I = 123456 = 1357 = 2458 = 2467 =
1368 = 123478 = 5678). As a result, this design would
confound more two factor-interactions (23 out of 28 possible
two-factor interactions are confounded, leaving only "12",
"14", "23", "27" and "34" as estimable two-factor
interactions).
Diagram of As an example of an equivalent "best" fractional
an
alternative
way for
generating
the 2
8-3
design
factorial design, obtained by "relabeling", consider the
design specified in Figure 3.8.
FIGURE 3.8 Another Way of Generating the 2
8-3
Design
This design is equivalent to the design specified in Figure
3.7 after relabeling the factors as follows: 1 becomes 5, 2
becomes 8, 3 becomes 1, 4 becomes 2, 5 becomes 3, 6
remains 6, 7 becomes 4 and 8 becomes 7.
Minimum
aberration
A table given later in this chapter gives a collection of
useful fractional factorial designs that, for a given k and p,
maximize the possible resolution and minimize the number
of short words in the defining relation (which minimizes
two-factor aliasing). The term for this is "minimum
aberration".
Design Resolution Summary
Commonly
used design
Resolutions
The meaning of the most prevalent resolution levels is as
follows:
Resolution III Designs
Main effects are confounded (aliased) with two-factor
interactions.
Resolution IV Designs
No main effects are aliased with two-factor interactions, but
two-factor interactions are aliased with each other.
Resolution V Designs
No main effect or two-factor interaction is aliased with any
other main effect or two-factor interaction, but two-factor
interactions are aliased with three-factor interactions.
5.3.3.4.5. Use of fractional factorial designs

5.3.3.4.5. Use of fractional factorial designs
Use low-
resolution
designs for
screening
among main
effects and use
higher-
resolution
designs when
interaction
effects and
response
surfaces need
to be
investigated
The basic purpose of a fractional factorial design is to
economically investigate cause-and-effect relationships of
significance in a given experimental setting. This does not
differ in essence from the purpose of any experimental
design. However, because we are able to choose fractions
of a full design, and hence be more economical, we also
have to be aware that different factorial designs serve
different purposes.
Broadly speaking, with designs of resolution three, and
sometimes four, we seek to screen out the few important
main effects from the many less important others. For this
reason, these designs are often termed main effects
designs, or screening designs.
On the other hand, designs of resolution five, and higher,
are used for focusing on more than just main effects in an
experimental situation. These designs allow us to estimate
interaction effects and such designs are easily augmented
to complete a second-order design - a design that permits
estimation of a full second-order (quadratic) model.
Different
purposes for
screening/RSM
designs
Within the screening/RSM strategy of design, there are a
number of functional purposes for which designs are used.
For example, an experiment might be designed to
determine how to make a product better or a process more
robust against the influence of external and non-
controllable influences such as the weather. Experiments
might be designed to troubleshoot a process, to determine
bottlenecks, or to specify which component(s) of a product
are most in need of improvement. Experiments might also
be designed to optimize yield, or to minimize defect
levels, or to move a process away from an unstable
operating zone. All these aims and purposes can be
achieved using fractional factorial designs and their
appropriate design enhancements.
5.3.3.4.6. Screening designs

5.3.3.4.6. Screening designs
Screening
designs are
an efficient
way to
identify
significant
main effects
The term `Screening Design' refers to an experimental plan
that is intended to find the few significant factors from a list
of many potential ones. Alternatively, we refer to a design as
a screening design if its primary purpose is to identify
significant main effects, rather than interaction effects, the
latter being assumed an order of magnitude less important.
Use
screening
designs
when you
have many
factors to
consider
Even when the experimental goal is to eventually fit a
response surface model (an RSM analysis), the first
experiment should be a screening design when there are
many factors to consider.
Screening
designs are
usually
resolution
III or IV
Screening designs are typically of resolution III. The reason is
that resolution III designs permit one to explore the effects of
many factors with an efficient number of runs.
Sometimes designs of resolution IV are also used for
screening designs. In these designs, main effects are
confounded with, at worst, three-factor interactions. This is
better from the confounding viewpoint, but the designs
require more runs than a resolution III design.
Plackett-
Burman
designs
Another common family of screening designs is the Plackett-
Burman set of designs, so named after its inventors. These
designs are of resolution III and will be described later.
Economical
plans for
determing
significant
main effects
In short, screening designs are economical experimental plans
that focus on determining the relative significance of many
main effects.
5.3.3.4.7. Summary tables of useful fractional factorial designs

5.3.3.4.7. Summary tables of useful fractional
factorial designs
Useful
fractional
factorial
designs for
up to 10
factors are
summarized
here
There are very useful summaries of two-level fractional
factorial designs for up to 11 factors, originally published in
the book Statistics for Experimenters by G.E.P. Box, W.G.
Hunter, and J.S. Hunter (New York, John Wiley & Sons,
1978). and also given in the book Design and Analysis of
Experiments, 5th edition by Douglas C. Montgomery (New
York, John Wiley & Sons, 2000).
Generator
column
notation can
use either
numbers or
letters for
the factor
columns
They differ in the notation for the design generators. Box,
Hunter, and Hunter use numbers (as we did in our earlier
discussion) and Montgomery uses capital letters according to
the following scheme:
Notice the absence of the letter I. This is usually reserved for
the intercept column that is identically 1. As an example of
the letter notation, note that the design generator "6 = 12345"
is equivalent to "F = ABCDE".
Details of
the design
generators,
the defining
relation, the
confounding
structure,
and the
design
matrix
TABLE 3.17 catalogs these useful fractional factorial designs
using the notation previously described in FIGURE 3.7.
Clicking on the specification for a given design
provides details (courtesy of Dataplot files) of the design
generators, the defining relation, the confounding structure
(as far as main effects and two-level interactions are
concerned), and the design matrix. The notation used follows
our previous labeling of factors with numbers, not letters.
Click on the
design
specification
in the table
below and a
TABLE 3.17 Summary of Useful Fractional Factorial
Designs
Number of
Factors, k
Design
Specification
Number of Runs
N
text file with
details
about the
design can
be viewed
or saved

3 2
III
3-1
4
4 2
IV
4-1
8
5 2
V
5-1
16
5 2
III
5-2
8
6 2
VI
6-1
32
6 2
IV
6-2
16
6 2
III
6-3
8
7 2
VII
7-1
64
7 2
IV
7-2
32
7 2
IV
7-3
16
7 2
III
7-4
8
8 2
VIII
8-1
128
8 2
V
8-2
64
8 2
IV
8-3
32
8 2
IV
8-4
16
9 2
VI
9-2
128
9 2
IV
9-3
64
9 2
IV
9-4
32
9 2
III
9-5
16
10 2
V
10-3
128
10 2
IV
10-4
64
10 2
IV
10-5
32
10 2
III
10-6
16
11 2
V
11-4
128
11 2
IV
11-5
64
11 2
IV
11-6
32
11 2
III
11-7
16
15 2
III
15-11
16
31 2
III
31-26
32

5.3.3.5. Plackett-Burman designs

Plackett-
Burman
designs
In 1946, R.L. Plackett and J.P. Burman published their now famous paper "The Design
of Optimal Multifactorial Experiments" in Biometrika (vol. 33). This paper described the
construction of very economical designs with the run number a multiple of four (rather
than a power of 2). Plackett-Burman designs are very efficient screening designs when
only main effects are of interest.
These
designs
have run
numbers
that are a
multiple of
4
Plackett-Burman (PB) designs are used for screening experiments because, in a PB
design, main effects are, in general, heavily confounded with two-factor interactions. The
PB design in 12 runs, for example, may be used for an experiment containing up to 11
factors.
12-Run
Plackett-
Burnam
design
TABLE 3.18 Plackett-Burman Design in 12 Runs for up to 11
Factors
Pattern X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
1 +++++++++++ +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1
2 -+-+++---+- -1 +1 -1 +1 +1 +1 -1 -1 -1 +1 -1
3 --+-+++---+ -1 -1 +1 -1 +1 +1 +1 -1 -1 -1 +1
4 +--+-+++--- +1 -1 -1 +1 -1 +1 +1 +1 -1 -1 -1
5 -+--+-+++-- -1 +1 -1 -1 +1 -1 +1 +1 +1 -1 -1
6 --+--+-+++- -1 -1 +1 -1 -1 +1 -1 +1 +1 +1 -1
7 ---+--+-+++ -1 -1 -1 +1 -1 -1 +1 -1 +1 +1 +1
8 +---+--+-++ +1 -1 -1 -1 +1 -1 -1 +1 -1 +1 +1
9 ++---+--+-+ +1 +1 -1 -1 -1 +1 -1 -1 +1 -1 +1
10 +++---+--+- +1 +1 +1 -1 -1 -1 +1 -1 -1 +1 -1
11 -+++---+--+ -1 +1 +1 +1 -1 -1 -1 +1 -1 -1 +1
12 +-+++---+-- +1 -1 +1 +1 +1 -1 -1 -1 +1 -1 -1
Saturated
Main Effect
designs
PB designs also exist for 20-run, 24-run, and 28-run (and higher) designs. With a 20-run
design you can run a screening experiment for up to 19 factors, up to 23 factors in a 24-
run design, and up to 27 factors in a 28-run design. These Resolution III designs are
known as Saturated Main Effect designs because all degrees of freedom are utilized to
estimate main effects. The designs for 20 and 24 runs are shown below.
20-Run TABLE 3.19 A 20-Run Plackett-Burman Design
Plackett-
Burnam
design
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19
1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1
2 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1
3 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1
4 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1
5 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1
6 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1
7 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1
8 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1
9 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1
10 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1
11 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1
12 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1
13 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1
14 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1
15 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1
16 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1
17 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1
18 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1
19 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1
20 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1
24-Run
Plackett-
Burnam
design
TABLE 3.20 A 24-Run Plackett-Burman Design
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1
3 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1
4 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1
5 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1
6 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1
7 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1
8 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1
9 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1
10 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1
11 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1
12 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1
13 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1
14 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1
15 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1
16 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1
17 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1
18 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1
19 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1
20 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1
21 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1
22 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1
23 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1
24 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1
No defining
relation
These designs do not have a defining relation since interactions are not identically equal
to main effects. With the designs, a main effect column X
i
is either orthogonal to
X
i
X
j
or identical to plus or minus X
i
X
j
. For Plackett-Burman designs, the two-factor
interaction column X
i
X
j
is correlated with every X
k
(for k not equal to i or j).
Economical
for
detecting
large main
effects
However, these designs are very useful for economically detecting large main effects,
assuming all interactions are negligible when compared with the few important main
effects.
5.3.3.6. Response surface designs

Response
surface
models may
involve just
main effects
and
interactions
or they may
also have
quadratic
and possibly
cubic terms
to account
for curvature
Earlier, we described the response surface method (RSM) objective.
Under some circumstances, a model involving only main effects
and interactions may be appropriate to describe a response surface
when
1. Analysis of the results revealed no evidence of "pure
quadratic" curvature in the response of interest (i.e., the
response at the center approximately equals the average of
the responses at the factorial runs).
2. The design matrix originally used included the limits of the
factor settings available to run the process.
Equations for
quadratic
and cubic
models
In other circumstances, a complete description of the process
behavior might require a quadratic or cubic model:
Quadratic
Cubic
These are the full models, with all possible terms, rarely would all
of the terms be needed in an application.
Quadratic
models
almost
always
sufficient for
industrial
applications
If the experimenter has defined factor limits appropriately and/or
taken advantage of all the tools available in multiple regression
analysis (transformations of responses and factors, for example),
then finding an industrial process that requires a third-order model
is highly unusual. Therefore, we will only focus on designs that are
useful for fitting quadratic models. As we will see, these designs
often provide lack of fit detection that will help determine when a
higher-order model is needed.
General
quadratic
surface types
Figures 3.9 to 3.12 identify the general quadratic surface types that
an investigator might encounter

FIGURE 3.9 A Response
Surface "Peak"
Surface "Hillside"
Surface "Rising Ridge"
Surface "Saddle"
Factor Levels for Higher-Order Designs
Possible
behaviors of
responses as
functions of
factor
settings
Figures 3.13 through 3.15 illustrate possible behaviors of responses
as functions of factor settings. In each case, assume the value of the
response increases from the bottom of the figure to the top and that
the factor settings increase from left to right.
FIGURE 3.13
Linear Function
FIGURE 3.14
Quadratic Function
FIGURE 3.15
Cubic Function
A two-level
experiment
with center
points can
detect, but
not fit,
quadratic
If a response behaves as in Figure 3.13, the design matrix to
quantify that behavior need only contain factors with two levels --
low and high. This model is a basic assumption of simple two-level
factorial and fractional factorial designs. If a response behaves as in
Figure 3.14, the minimum number of levels required for a factor to
quantify that behavior is three. One might logically assume that
adding center points to a two-level design would satisfy that
effects requirement, but the arrangement of the treatments in such a matrix
confounds all quadratic effects with each other. While a two-level
design with center points cannot estimate individual pure quadratic
effects, it can detect them effectively.
Three-level
factorial
design
A solution to creating a design matrix that permits the estimation of
simple curvature as shown in Figure 3.14 would be to use a three-
level factorial design. Table 3.21 explores that possibility.
Four-level
factorial
design
Finally, in more complex cases such as illustrated in Figure 3.15,
the design matrix must contain at least four levels of each factor to
characterize the behavior of the response adequately.
3-level
factorial
designs can
fit quadratic
models but
they require
many runs
when there
are more
than 4
factors
TABLE 3.21 Three-level Factorial Designs
Number
of
Factors
Treatment
Combinations
3
k
Factorial
Number of Coefficients
Quadratic Empirical
Model
2 9 6
3 27 10
4 81 15
5 243 21
6 729 28
Fractional
factorial
designs
created to
avoid such a
large number
of runs
Two-level factorial designs quickly become too large for practical
application as the number of factors investigated increases. This
problem was the motivation for creating `fractional factorial'
designs. Table 3.21 shows that the number of runs required for a 3
k
factorial becomes unacceptable even more quickly than for 2
k
designs. The last column in Table 3.21 shows the number of terms
present in a quadratic model for each case.
Number of
runs large
even for
modest
number of
factors
With only a modest number of factors, the number of runs is very
large, even an order of magnitude greater than the number of
parameters to be estimated when k isn't small. For example, the
absolute minimum number of runs required to estimate all the
terms present in a four-factor quadratic model is 15: the intercept
term, 4 main effects, 6 two-factor interactions, and 4 quadratic
terms.
The corresponding 3
k
design for k = 4 requires 81 runs.
Complex
alias
structure and
lack of
rotatability
for 3-level
fractional
factorial
Considering a fractional factorial at three levels is a logical step,
given the success of fractional designs when applied to two-level
designs. Unfortunately, the alias structure for the three-level
fractional factorial designs is considerably more complex and
harder to define than in the two-level case.
Additionally, the three-level factorial designs suffer a major flaw in
their lack of `rotatability.'
designs
Rotatability of Designs
"Rotatability"
is a desirable
property not
present in 3-
level
factorial
designs
In a rotatable design, the variance of the predicted values of y is a
function of the distance of a point from the center of the design and
is not a function of the direction the point lies from the center.
Before a study begins, little or no knowledge may exist about the
region that contains the optimum response. Therefore, the
experimental design matrix should not bias an investigation in any
direction.
Contours of
variance of
predicted
values are
concentric
circles
In a rotatable design, the contours associated with the variance of
the predicted values are concentric circles. Figures 3.16 and 3.17
(adapted from Box and Draper, Èmpirical Model Building and
Response Surfaces,' page 485) illustrate a three-dimensional plot
and contour plot, respectively, of the ìnformation function'
associated with a 3
2
design.
Information
function
The information function is:
with V denoting the variance (of the predicted value ).
Each figure clearly shows that the information content of the design
is not only a function of the distance from the center of the design
space, but also a function of direction.
Graphs of
the
information
function for a
rotatable
quadratic
design
Figures 3.18 and 3.19 are the corresponding graphs of the
information function for a rotatable quadratic design. In each of
these figures, the value of the information function depends only on
the distance of a point from the center of the space.
FIGURE 3.16 Three-
Dimensional Illustration for the
Information Function of a 3
2
Design
FIGURE 3.17
Contour Map of the
Information Function for a
3
2
Design
FIGURE 3.18 Three-
Dimensional Illustration of the
Information Function for a
Rotatable Quadratic Design for
Two Factors
FIGURE 3.19 Contour Map
of the Information Function
for a Rotatable Quadratic
Design for Two Factors
Classical Quadratic Designs
Central
composite
and Box-
Behnken
designs
Introduced during the 1950's, classical quadratic designs fall into
two broad categories: Box-Wilson central composite designs and
Box-Behnken designs. The next sections describe these design
classes and their properties.
5.3.3.6.1. Central Composite Designs (CCD)

Box-Wilson Central Composite Designs
CCD designs
start with a
factorial or
fractional
factorial
design (with
center points)
and add
"star" points
to estimate
curvature
A Box-Wilson Central Composite Design, commonly
called à central composite design,' contains an imbedded
factorial or fractional factorial design with center points
that is augmented with a group of `star points' that allow
estimation of curvature. If the distance from the center of
the design space to a factorial point is 1 unit for each
factor, the distance from the center of the design space to a
star point is with | | > 1. The precise value of
depends on certain properties desired for the design and on
the number of factors involved.
Similarly, the number of centerpoint runs the design is to
contain also depends on certain properties required for the
design.
Diagram of
central
composite
design
generation for
two factors
FIGURE 3.20 Generation of a Central Composite
Design for Two Factors
A CCD design
with k factors
has 2k star
points
A central composite design always contains twice as many
star points as there are factors in the design. The star points
represent new extreme values (low and high) for each
factor in the design. Table 3.22 summarizes the properties
of the three varieties of central composite designs. Figure
3.21 illustrates the relationships among these varieties.
Description of TABLE 3.22 Central Composite Designs
3 types of
CCD designs,
which depend
on where the
star points
are placed
Central
Composite
Design
Type
Terminology Comments
Circumscribed CCC
CCC designs are the
original form of the central
composite design. The star
points are at some distance
from the center based on
the properties desired for
the design and the number
of factors in the design.
The star points establish
new extremes for the low
and high settings for all
factors. Figure 5 illustrates
a CCC design. These
designs have circular,
spherical, or hyperspherical
symmetry and require 5
levels for each factor.
Augmenting an existing
factorial or resolution V
fractional factorial design
with star points can
produce this design.
Inscribed CCI
For those situations in
which the limits specified
for factor settings are truly
limits, the CCI design uses
the factor settings as the
star points and creates a
factorial or fractional
factorial design within
those limits (in other
words, a CCI design is a
scaled down CCC design
with each factor level of
the CCC design divided by
to generate the CCI
design). This design also
requires 5 levels of each
factor.
Face
Centered
CCF
In this design the star
points are at the center of
each face of the factorial
space, so = 1. This
variety requires 3 levels of
each factor. Augmenting an
existing factorial or
resolution V design with
appropriate star points can
also produce this design.
Pictorial
representation
of where the
star points
are placed for
the 3 types of
CCD designs

FIGURE 3.21 Comparison of the Three Types of
Central Composite Designs
Comparison
of the 3
central
composite
designs
The diagrams in Figure 3.21 illustrate the three types of
central composite designs for two factors. Note that the
CCC explores the largest process space and the CCI
explores the smallest process space. Both the CCC and
CCI are rotatable designs, but the CCF is not. In the CCC
design, the design points describe a circle circumscribed
about the factorial square. For three factors, the CCC
design points describe a sphere around the factorial cube.
Determining in Central Composite Designs
The value of
is chosen to
maintain
rotatability
To maintain rotatability, the value of depends on the
number of experimental runs in the factorial portion of the
central composite design:
If the factorial is a full factorial, then
However, the factorial portion can also be a fractional
factorial design of resolution V.
Table 3.23 illustrates some typical values of as a
function of the number of factors.
Values of
depending on
the number of
factors in the
factorial part
of the design
TABLE 3.23 Determining for
Rotatability
Number of
Factors
Factorial
Portion
Scaled Value for
Relative to 1
2
2
2
2
2/4
= 1.414
3
2
3
2
3/4
= 1.682
4
2
4
2
4/4
= 2.000
5
2
5-1
2
4/4
= 2.000
5
2
5
2
5/4
= 2.378
6
2
6-1
2
5/4
= 2.378
6
2
6
2
6/4
= 2.828
Orthogonal
blocking
The value of also depends on whether or not the design
is orthogonally blocked. That is, the question is whether or
not the design is divided into blocks such that the block
effects do not affect the estimates of the coefficients in the
2nd order model.
Example of
both
rotatability
and
orthogonal
blocking for
two factors
Under some circumstances, the value of allows
simultaneous rotatability and orthogonality. One such
example for k = 2 is shown below:
BLOCK X1 X2

1 -1 -1
1 1 -1
1 -1 1
1 1 1
1 0 0
1 0 0
2 -1.414 0
2 1.414 0
2 0 -1.414
2 0 1.414
2 0 0
2 0 0
Additional
central
composite
designs
Examples of other central composite designs will be given
after Box-Behnken designs are described.
5.3.3.6.2. Box-Behnken designs

An
alternate
choice for
fitting
quadratic
models
that
requires 3
levels of
each
factor and
is
rotatable
(or
"nearly"
rotatable)
The Box-Behnken design is an independent quadratic design
in that it does not contain an embedded factorial or fractional
factorial design. In this design the treatment combinations are
at the midpoints of edges of the process space and at the
center. These designs are rotatable (or near rotatable) and
require 3 levels of each factor. The designs have limited
capability for orthogonal blocking compared to the central
composite designs.
Figure 3.22 illustrates a Box-Behnken design for three factors.
Box-
Behnken
design for
3 factors
FIGURE 3.22 A Box-Behnken Design for Three Factors
Geometry
of the
design
The geometry of this design suggests a sphere within the
process space such that the surface of the sphere protrudes
through each face with the surface of the sphere tangential to
the midpoint of each edge of the space.
Examples of Box-Behnken designs are given on the next page.
5.3.3.6.3. Comparisons of response surface designs

Choosing a Response Surface Design
Various
CCD
designs and
Box-
Behnken
designs are
compared
and their
properties
discussed
Table 3.24 contrasts the structures of four common quadratic designs
one might use when investigating three factors. The table combines
CCC and CCI designs because they are structurally identical.
For three factors, the Box-Behnken design offers some advantage in
requiring a fewer number of runs. For 4 or more factors, this advantage
disappears.
Structural
comparisons
of CCC
(CCI), CCF,
and Box-
Behnken
designs for
three factors
TABLE 3.24 Structural Comparisons of CCC (CCI), CCF,
and Box-Behnken Designs for Three Factors
CCC (CCI) CCF Box-Behnken
Rep X1 X2 X3 Rep X1 X2 X3 Rep X1 X2 X3
1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 0
1 +1 -1 -1 1 +1 -1 -1 1 +1 -1 0
1 -1 +1 -1 1 -1 +1 -1 1 -1 +1 0
1 +1 +1 -1 1 +1 +1 -1 1 +1 +1 0
1 -1 -1 +1 1 -1 -1 +1 1 -1 0 -1
1 +1 -1 +1 1 +1 -1 +1 1 +1 0 -1
1 -1 +1 +1 1 -1 +1 +1 1 -1 0 +1
1 +1 +1 +1 1 +1 +1 +1 1 +1 0 +1
1 -1.682 0 0 1 -1 0 0 1 0 -1 -1
1 1.682 0 0 1 +1 0 0 1 0 +1 -1
1 0 -1.682 0 1 0 -1 0 1 0 -1 +1
1 0 1.682 0 1 0 +1 0 1 0 +1 +1
1 0 0 -1.682 1 0 0 -1 3 0 0 0
1 0 0 1.682 1 0 0 +1
6 0 0 0 6 0 0 0
Total Runs = 20 Total Runs = 20 Total Runs = 15
Factor
settings for
CCC and
CCI three
factor
designs
Table 3.25 illustrates the factor settings required for a central
composite circumscribed (CCC) design and for a central composite
inscribed (CCI) design (standard order), assuming three factors, each
with low and high settings of 10 and 20, respectively. Because the
CCC design generates new extremes for all factors, the investigator
must inspect any worksheet generated for such a design to make
certain that the factor settings called for are reasonable.
In Table 3.25, treatments 1 to 8 in each case are the factorial points in
the design; treatments 9 to 14 are the star points; and 15 to 20 are the
system-recommended center points. Notice in the CCC design how the
low and high values of each factor have been extended to create the
star points. In the CCI design, the specified low and high values
become the star points, and the system computes appropriate settings
for the factorial part of the design inside those boundaries.
TABLE 3.25 Factor Settings for CCC and CCI Designs for
Three Factors
Central Composite
Circumscribed CCC
Central Composite
Inscribed CCI
Sequence
Number X1 X2 X3
Sequence
Number X1 X2 X3
1 10 10 10 1 12 12 12
2 20 10 10 2 18 12 12
3 10 20 10 3 12 18 12
4 20 20 10 4 18 18 12
5 10 10 20 5 12 12 18
6 20 10 20 6 18 12 18
7 10 20 20 7 12 12 18
8 20 20 20 8 18 18 18
9 6.6 15 15 * 9 10 15 15
10 23.4 15 15 * 10 20 15 15
11 15 6.6 15 * 11 15 10 15
12 15 23.4 15 * 12 15 20 15
13 15 15 6.6 * 13 15 15 10
14 15 15 23.4 * 14 15 15 20
15 15 15 15 15 15 15 15
16 15 15 15 16 15 15 15
17 15 15 15 17 15 15 15
18 15 15 15 18 15 15 15
19 15 15 15 19 15 15 15
20 15 15 15 20 15 15 15
* are star points
Factor
settings for
CCF and
Box-
Behnken
three factor
designs
Table 3.26 illustrates the factor settings for the corresponding central
composite face-centered (CCF) and Box-Behnken designs. Note that
each of these designs provides three levels for each factor and that the
Box-Behnken design requires fewer runs in the three-factor case.
TABLE 3.26 Factor Settings for CCF and Box-Behnken
Designs for Three Factors
Central Composite
Face-Centered CCC
Box-Behnken
Sequence
Number X1 X2 X3
Sequence
Number X1 X2 X3
1 10 10 10 1 10 10 15
2 20 10 10 2 20 10 15
3 10 20 10 3 10 20 15
4 20 20 10 4 20 20 15
5 10 10 20 5 10 15 10
6 20 10 20 6 20 15 10
7 10 20 20 7 10 15 20
8 20 20 20 8 20 15 20
9 10 15 15 * 9 15 10 10
10 20 15 15 * 10 15 20 10
11 15 10 15 * 11 15 10 20
12 15 20 15 * 12 15 20 20
13 15 15 10 * 13 15 15 15
14 15 15 20 * 14 15 15 15
15 15 15 15 15 15 15 15
16 15 15 15
17 15 15 15
18 15 15 15
19 15 15 15
20 15 15 15
* are star points for the CCC
Properties
of classical
response
surface
designs
Table 3.27 summarizes properties of the classical quadratic designs.
Use this table for broad guidelines when attempting to choose from
among available designs.
TABLE 3.27 Summary of Properties of Classical Response
Surface Designs
Design
Type
Comment
CCC designs provide high quality predictions over the
entire design space, but require factor settings outside the
range of the factors in the factorial part. Note: When the
CCC
possibility of running a CCC design is recognized before
starting a factorial experiment, factor spacings can be
reduced to ensure that for each coded factor
corresponds to feasible (reasonable) levels.
Requires 5 levels for each factor.
CCI
CCI designs use only points within the factor ranges
originally specified, but do not provide the same high
quality prediction over the entire space compared to the
CCC.
Requires 5 levels of each factor.
CCF
CCF designs provide relatively high quality predictions over
the entire design space and do not require using points
outside the original factor range. However, they give poor
precision for estimating pure quadratic coefficients.
Box-
Behnken
These designs require fewer treatment combinations than a
central composite design in cases involving 3 or 4 factors.
The Box-Behnken design is rotatable (or nearly so) but it
contains regions of poor prediction quality like the CCI. Its
"missing corners" may be useful when the experimenter
should avoid combined factor extremes. This property
prevents a potential loss of data in those cases.
Number of
runs
required by
central
composite
and Box-
Behnken
designs
Table 3.28 compares the number of runs required for a given number
of factors for various Central Composite and Box-Behnken designs.
TABLE 3.28 Number of Runs Required by Central Composite and
Box-Behnken Designs
Number of
Factors Central Composite
Box-
Behnken
2 13 (5 center points) -
3 20 (6 centerpoint runs) 15
4 30 (6 centerpoint runs) 27
5 33 (fractional factorial) or 52 (full
factorial)
46
6 54 (fractional factorial) or 91 (full
factorial)
54
Desirable Features for Response Surface Designs
A summary
of desirable
properties
for response
G. E. P. Box and N. R. Draper in "Empirical Model Building and
Response Surfaces," John Wiley and Sons, New York, 1987, page 477,
identify desirable properties for a response surface design:
Satisfactory distribution of information across the experimental
surface
designs
region.
- rotatability
Fitted values are as close as possible to observed values.
- minimize residuals or error of prediction
Good lack of fit detection.
Internal estimate of error.
Constant variance check.
Transformations can be estimated.
Suitability for blocking.
Sequential construction of higher order designs from simpler
designs
Minimum number of treatment combinations.
Good graphical analysis through simple data patterns.
Good behavior when errors in settings of input variables occur.
5.3.3.6.4. Blocking a response surface design

How can we block a response surface design?
When
augmenting
a resolution
V design to
a CCC
design by
adding star
points, it
may be
desirable to
block the
design
If an investigator has run either a 2
k
full factorial or a 2
k-p
fractional
factorial design of at least resolution V, augmentation of that design
to a central composite design (either CCC of CCF) is easily
accomplished by adding an additional set (block) of star and
centerpoint runs. If the factorial experiment indicated (via the t test)
curvature, this composite augmentation is the best follow-up option
(follow-up options for other situations will be discussed later).
An
orthogonal
blocked
response
surface
design has
advantages
An important point to take into account when choosing a response
surface design is the possibility of running the design in blocks.
Blocked designs are better designs if the design allows the estimation
of individual and interaction factor effects independently of the block
effects. This condition is called orthogonal blocking. Blocks are
assumed to have no impact on the nature and shape of the response
surface.
CCF
designs
cannot be
orthogonally
blocked
The CCF design does not allow orthogonal blocking and the Box-
Behnken designs offer blocking only in limited circumstances,
whereas the CCC does permit orthogonal blocking.
Axial and
factorial
blocks
In general, when two blocks are required there should be an axial
block and a factorial block. For three blocks, the factorial block is
divided into two blocks and the axial block is not split. The blocking
of the factorial design points should result in orthogonality between
blocks and individual factors and between blocks and the two factor
interactions.
The following Central Composite design in two factors is broken into
two blocks.
Table of TABLE 3.29 CCD: 2 Factors, 2 Blocks
CCD design
with 2
factors and
2 blocks
Pattern Block X1 X2 Comment
-- 1 -1 -1 Full Factorial
-+ 1 -1 +1 Full Factorial
+- 1 +1 -1 Full Factorial
++ 1 +1 +1 Full Factorial
00 1 0 0 Center-Full Factorial
-0 2 -1.414214 0 Axial
+0 2 +1.414214 0 Axial
0- 2 0 -1.414214 Axial
0+ 2 0 +1.414214 Axial
00 2 0 0 Center-Axial
Note that the first block includes the full factorial points and three
centerpoint replicates. The second block includes the axial points and
another three centerpoint replicates. Naturally these two blocks
should be run as two separate random sequences.
Table of
CCD design
with 3
factors and
3 blocks
The following three examples show blocking structure for various
designs.
TABLE 3.30 CCD: 3 Factors 3 Blocks, Sorted by Block
Pattern Block X1 X2 X3 Comment
--- 1 -1 -1 -1 Full Factorial
-++ 1 -1 +1 +1 Full Factorial
+-+ 1 +1 -1 +1 Full Factorial
++- 1 +1 +1 -1 Full Factorial
000 1 0 0 0 Center-Full Factorial
--+ 2 -1 -1 +1 Full Factorial
-+- 2 -1 +1 -1 Full Factorial
+-- 2 +1 -1 -1 Full Factorial
+++ 2 +1 +1 +1 Full Factorial
-00 3 -1.63299 0 0 Axial
+00 3 +1.63299 0 0 Axial
0-0 3 0 -1.63299 0 Axial
0+0 3 0 +1.63299 0 Axial
00- 3 0 0 -1.63299 Axial
00+ 3 0 0 +1.63299 Axial
000 3 0 0 0 Axial
000 3 0 0 0 Axial
Table of
CCD design
with 4
factors and
3 blocks
TABLE 3.31 CCD: 4 Factors, 3 Blocks
Pattern Block X1 X2 X3 X4 Comment
---+ 1 -1 -1 -1 +1 Full Factorial
--+- 1 -1 -1 +1 -1 Full Factorial
-+-- 1 -1 +1 -1 -1 Full Factorial
-+++ 1 -1 +1 +1 +1 Full Factorial
+--- 1 +1 -1 -1 -1 Full Factorial
+-++ 1 +1 -1 +1 +1 Full Factorial
++-+ 1 +1 +1 -1 +1 Full Factorial
+++- 1 +1 +1 +1 -1 Full Factorial
0000 1 0 0 0 0 Center-Full Factorial
---- 2 -1 -1 -1 -1 Full Factorial
--++ 2 -1 -1 +1 +1 Full Factorial
-+-+ 2 -1 +1 -1 +1 Full Factorial
-++- 2 -1 +1 +1 -1 Full Factorial
+--+ 2 +1 -1 -1 +1 Full Factorial
+-+- 2 +1 -1 +1 -1 Full Factorial
++-- 2 +1 +1 -1 -1 Full Factorial
++++ 2 +1 +1 +1 +1 Full Factorial
-000 3 -2 0 0 0 Axial
+000 3 +2 0 0 0 Axial
+000 3 +2 0 0 0 Axial
0-00 3 0 -2 0 0 Axial
0+00 3 0 +2 0 0 Axial
00-0 3 0 0 -2 0 Axial
00+0 3 0 0 +2 0 Axial
000- 3 0 0 0 -2 Axial
000+ 3 0 0 0 +2 Axial
0000 3 0 0 0 0 Center-Axial
Table
of
CCD
design
with 5
factors
and 2
blocks
TABLE 3.32 CCD: 5 Factors, 2 Blocks
Pattern Block X1 X2 X3 X4 X5 Comment
----+ 1 -1 -1 -1 -1 +1 Fractional
Factorial
---+- 1 -1 -1 -1 +1 -1 Fractional
Factorial
--+-- 1 -1 -1 +1 -1 -1 Fractional
Factorial
--+++ 1 -1 -1 +1 +1 +1 Fractional
Factorial
-+--- 1 -1 +1 -1 -1 -1 Fractional
Factorial
-+-++ 1 -1 +1 -1 +1 +1 Fractional
Factorial
-++-+ 1 -1 +1 +1 -1 +1 Fractional
Factorial
-+++- 1 -1 +1 +1 +1 -1 Fractional
Factorial
+---- 1 +1 -1 -1 -1 -1 Fractional
Factorial
+--++ 1 +1 -1 -1 +1 +1 Fractional
Factorial
+-+-+ 1 +1 -1 +1 -1 +1 Fractional
Factorial
+-++- 1 +1 -1 +1 +1 -1 Fractional
Factorial
++--+ 1 +1 +1 -1 -1 +1 Fractional
Factorial
++-+- 1 +1 +1 -1 +1 -1 Fractional
Factorial
+++-- 1 +1 +1 +1 -1 -1 Fractional
Factorial
+++++ 1 +1 +1 +1 +1 +1 Fractional
Factorial
00000 1 0 0 0 0 0 Center-
Fractional
Factorial
00000 1 0 0 0 0 0 Center-
Fractional
Factorial
00000 1 0 0 0 0 0 Center-
Fractional
Factorial
00000 1 0 0 0 0 0 Center-
Fractional
Factorial
00000 1 0 0 0 0 0 Center-
Fractional
Factorial
00000 1 0 0 0 0 0 Center-
Fractional
Factorial
-0000 2 -2 0 0 0 0 Axial
+0000 2 +2 0 0 0 0 Axial
0-000 2 0 -2 0 0 0 Axial
0+000 2 0 +2 0 0 0 Axial
00-00 2 0 0 -2 0 0 Axial
00+00 2 0 0 +2 0 0 Axial
000-0 2 0 0 0 -2 0 Axial
000+0 2 0 0 0 +2 0 Axial
0000- 2 0 0 0 0 -2 Axial
0000+ 2 0 0 0 0 +2 Axial
00000 2 0 0 0 0 0 Center-
Axial
5.3.3.7. Adding centerpoints

Center point, or `Control' Runs
Centerpoint
runs
provide a
check for
both
process
stability and
possible
curvature
As mentioned earlier in this section, we add centerpoint runs
interspersed among the experimental setting runs for two
purposes:
1. To provide a measure of process stability
and inherent variability
2. To check for curvature.
Centerpoint
runs are not
randomized
Centerpoint runs should begin and end the experiment, and
should be dispersed as evenly as possible throughout the
design matrix. The centerpoint runs are not randomized!
There would be no reason to randomize them as they are
there as guardians against process instability and the best way
to find instability is to sample the process on a regular basis.
Rough rule
of thumb is
to add 3 to
5 center
point runs
to your
design
With this in mind, we have to decide on how many
centerpoint runs to do. This is a tradeoff between the
resources we have, the need for enough runs to see if there is
process instability, and the desire to get the experiment over
with as quickly as possible. As a rough guide, you should
generally add approximately 3 to 5 centerpoint runs to a full
or fractional factorial design.
Table of
randomized,
replicated
2
3
full
factorial
design with
centerpoints
In the following Table we have added three centerpoint runs
to the otherwise randomized design matrix, making a total of
nineteen runs.
TABLE 3.32 Randomized, Replicated 2
3
Full Factorial
Design Matrix with Centerpoint Control Runs Added

Random
Order
Standard
Order
SPEED FEED DEPTH
1 not applicable not applicable 0 0 0
2 1 5 -1 -1 1
3 2 15 -1 1 1
4 3 9 -1 -1 -1
5 4 7 -1 1 1
6 5 3 -1 1 -1
7 6 12 1 1 -1
8 7 6 1 -1 1
9 8 4 1 1 -1
11 9 2 1 -1 -1
12 10 13 -1 -1 1
13 11 8 1 1 1
14 12 16 1 1 1
15 13 1 -1 -1 -1
16 14 14 1 -1 1
17 15 11 -1 1 -1
18 16 10 1 -1 -1
Preparing a
worksheet
for operator
of
experiment
To prepare a worksheet for an operator to use when running
the experiment, delete the columns `RandOrd' and `Standard
Order.' Add an additional column for the output (Yield) on
the right, and change all `-1', `0', and `1' to original factor
levels as follows.
Operator
worksheet
TABLE 3.33 DOE Worksheet Ready to Run
Sequence
Number Speed Feed Depth Yield
1 20 0.003 0.015
2 16 0.001 0.02
3 16 0.005 0.02
4 16 0.001 0.01
5 16 0.005 0.02
6 16 0.005 0.01
7 24 0.005 0.01
8 24 0.001 0.02
9 24 0.005 0.01
10 20 0.003 0.015
11 24 0.001 0.01
12 16 0.001 0.02
13 24 0.005 0.02
14 24 0.005 0.02
15 16 0.001 0.01
16 24 0.001 0.02
17 16 0.005 0.01
18 24 0.001 0.01
19 20 0.003 0.015
Note that the control (centerpoint) runs appear at rows 1, 10,
and 19.
This worksheet can be given to the person who is going to do
the runs/measurements and asked to proceed through it from
first row to last in that order, filling in the Yield values as
they are obtained.
Pseudo Center points
Center
points for
discrete
factors
One often runs experiments in which some factors are
nominal. For example, Catalyst "A" might be the (-1) setting,
catalyst "B" might be coded (+1). The choice of which is
"high" and which is "low" is arbitrary, but one must have
some way of deciding which catalyst setting is the "standard"
one.
These standard settings for the discrete input factors together
with center points for the continuous input factors, will be
regarded as the "center points" for purposes of design.
Center Points in Response Surface Designs
Uniform
precision
In an unblocked response surface design, the number of
center points controls other properties of the design matrix.
The number of center points can make the design orthogonal
or have "uniform precision." We will only focus on uniform
precision here as classical quadratic designs were set up to
have this property.
Variance of
prediction
Uniform precision ensures that the variance of prediction is
the same at the center of the experimental space as it is at a
unit distance away from the center.
Protection
against bias
In a response surface context, to contrast the virtue of
uniform precision designs over replicated center-point
orthogonal designs one should also consider the following
guidance from Montgomery ("Design and Analysis of
Experiments," Wiley, 1991, page 547), "A uniform precision
design offers more protection against bias in the regression
coefficients than does an orthogonal design because of the
presence of third-order and higher terms in the true surface.
Controlling
and the
number of
center
points
Myers, Vining, et al, ["Variance Dispersion of Response
Surface Designs," Journal of Quality Technology, 24, pp. 1-
11 (1992)] have explored the options regarding the number of
center points and the value of somewhat further: An
investigator may control two parameters, and the number
of center points (n
c
), given k factors. Either set = 2
(k/4)
(for rotatability) or -- an axial point on perimeter of
design region. Designs are similar in performance with
preferable as k increases. Findings indicate that the best
overall design performance occurs with and 2 n
c

5.
5.3.3.8. Improving fractional factorial design resolution

5.3.3.8. Improving fractional factorial design
resolution
Foldover
designs
increase
resolution
Earlier we saw how fractional factorial designs resulted in an
alias structure that confounded main effects with certain
interactions. Often it is useful to know how to run a few
additional treatment combinations to remove alias structures
that might be masking significant effects or interactions.
Partial
foldover
designs
break up
specific
alias
patterns
Two methods will be described for selecting these additional
treatment combinations:
Mirror-image foldover designs (to build a
resolution IV design from a resolution III
design)
Alternative foldover designs (to break up
specific alias patterns).
5.3.3.8.1. Mirror-Image foldover designs

A foldover
design is
obtained
from a
fractional
factorial
design by
reversing
the signs of
all the
columns
A mirror-image fold-over (or foldover, without the hyphen)
design is used to augment fractional factorial designs to
increase the resolution of and Plackett-Burman designs.
It is obtained by reversing the signs of all the columns of the
original design matrix. The original design runs are combined
with the mirror-image fold-over design runs, and this
combination can then be used to estimate all main effects
clear of any two-factor interaction. This is referred to as:
breaking the alias link between main effects and two-factor
interactions.
Before we illustrate this concept with an example, we briefly
review the basic concepts involved.
Review of Fractional 2
k-p
Designs
A
resolution
III design,
combined
with its
mirror-
image
foldover,
becomes
resolution
IV
In general, a design type that uses a specified fraction of the
runs from a full factorial and is balanced and orthogonal is
called a fractional factorial.
A 2-level fractional factorial is constructed as follows: Let the
number of runs be 2
k-p
. Start by constructing the full factorial
for the k-p variables. Next associate the extra factors with
higher-order interaction columns. The Table shown
previously details how to do this to achieve a minimal amount
of confounding.
For example, consider the 2
5-2
design (a resolution III
design). The full factorial for k = 5 requires 2
5
= 32 runs. The
fractional factorial can be achieved in 2
5-2
= 8 runs, called a
quarter (1/4) fractional design, by setting X4 = X1*X2 and X5
= X1*X3.
Design
matrix for
a 2
5-2
fractional
factorial
The design matrix for a 2
5-2
fractional factorial looks like:
TABLE 3.34 Design Matrix for a 2
5-2
Fractional
Factorial
run X1 X2 X3 X4 = X1X2 X5 = X1X3
1 -1 -1 -1 +1 +1
2 +1 -1 -1 -1 -1
3 -1 +1 -1 -1 +1
4 +1 +1 -1 +1 -1
5 -1 -1 +1 +1 -1
6 +1 -1 +1 -1 +1
7 -1 +1 +1 -1 -1
8 +1 +1 +1 +1 +1
Design Generators, Defining Relation and the Mirror-
Image Foldover
Increase to
resolution
IV design
by
augmenting
design
matrix
In this design the X1X2 column was used to generate the X4
main effect and the X1X3 column was used to generate the X5
main effect. The design generators are: 4 = 12 and 5 = 13 and
the defining relation is I = 124 = 135 = 2345. Every main
effect is confounded (aliased) with at least one first-order
interaction (see the confounding structure for this design).
We can increase the resolution of this design to IV if we
augment the 8 original runs, adding on the 8 runs from the
mirror-image fold-over design. These runs make up another
1/4 fraction design with design generators 4 = -12 and 5 = -13
and defining relation I = -124 = -135 = 2345. The augmented
runs are:
Augmented
runs for the
design
matrix
run X1 X2 X3 X4 = -X1X2 X5 = -X1X3
9 +1 +1 +1 -1 -1
10 -1 +1 +1 +1 +1
11 +1 -1 +1 +1 -1
12 -1 -1 +1 -1 +1
13 +1 +1 -1 -1 +1
14 -1 +1 -1 +1 -1
15 +1 -1 -1 +1 +1
16 -1 -1 -1 -1 -1
Mirror-
image
foldover
design
reverses all
signs in
original
design
matrix
A mirror-image foldover design is the original design with all
signs reversed. It breaks the alias chains between every main
factor and two-factor interactionof a resolution III design.
That is, we can estimate all the main effects clear of any two-
factor interaction.
A 1/16 Design Generator Example
2
7-3
example
Now we consider a more complex example.
We would like to study the effects of 7 variables. A full 2-
level factorial, 2
7
, would require 128 runs.
Assume economic reasons restrict us to 8 runs. We will build
a 2
7-4
= 2
3
full factorial and assign certain products of
columns to the X4, X5, X6 and X7 variables. This will
generate a resolution III design in which all of the main
effects are aliased with first-order and higher interaction
terms. The design matrix (see the previous Table for a
complete description of this fractional factorial design) is:
Design
matrix for
2
7-3
fractional
factorial
Design Matrix for a 2
7-3
Fractional Factorial
run X1 X2 X3
X4 =
X1X2
X5 =
X1X3
X6 =
X2X3
X7 =
X1X2X3
1 -1 -1 -1 +1 +1 +1 -1
2 +1 -1 -1 -1 -1 +1 +1
3 -1 +1 -1 -1 +1 -1 +1
4 +1 +1 -1 +1 -1 -1 -1
5 -1 -1 +1 +1 -1 -1 +1
6 +1 -1 +1 -1 +1 -1 -1
7 -1 +1 +1 -1 -1 +1 -1
8 +1 +1 +1 +1 +1 +1 +1
Design
generators
and
defining
relation for
this
example
The design generators for this 1/16 fractional factorial design
are:
4 = 12, 5 = 13, 6 = 23 and 7 = 123
From these we obtain, by multiplication, the defining relation:
I = 124 = 135 = 236 = 347 = 257 = 167 = 456 = 1237 =
2345 = 1346 = 1256 = 1457 = 2467 = 3567 = 1234567.
Computing
alias
structure
for
complete
design
Using this defining relation, we can easily compute the alias
structure for the complete design, as shown previously in the
link to the fractional design Table given earlier. For example,
to figure out which effects are aliased (confounded) with
factor X1 we multiply the defining relation by 1 to obtain:
1 = 24 = 35 = 1236 = 1347 = 1257 = 67 = 1456 = 237
= 12345 = 346 = 256 = 457 = 12467 = 13567 = 234567
In order to simplify matters, let us ignore all interactions with
3 or more factors; we then have the following 2-factor alias
pattern for X1: 1 = 24 = 35 = 67 or, using the full notation, X1
= X2*X4 = X3*X5 = X6*X7.
The same procedure can be used to obtain all the other aliases
for each of the main effects, generating the following list:
1 = 24 = 35 = 67
2 = 14 = 36 = 57
3 = 15 = 26 = 47
4 = 12 = 37 = 56
5 = 13 = 27 = 46
6 = 17 = 23 = 45
7 = 16 = 25 = 34
Signs in
every
column of
original
design
matrix
reversed
for mirror-
image
foldover
design
The chosen design used a set of generators with all positive
signs. The mirror-image foldover design uses generators with
negative signs for terms with an even number of factors or, 4
= -12, 5 = -13, 6 = -23 and 7 = 123. This generates a design
matrix that is equal to the original design matrix with every
sign in every column reversed.
If we augment the initial 8 runs with the 8 mirror-image
foldover design runs (with all column signs reversed), we can
de-alias all the main effect estimates from the 2-way
interactions. The additional runs are:
Design
matrix for
mirror-
image
foldover
runs
Design Matrix for the Mirror-Image Foldover
Runs of the 2
7-3
Fractional Factorial
run X1 X2 X3
X4 =
X1X2
X5 =
X1X3
X6 =
X2X3
X7 =
X1X2X3
1 +1 +1 +1 -1 -1 -1 +1
2 -1 +1 +1 +1 +1 -1 -1
3 +1 -1 +1 +1 -1 +1 -1
4 -1 -1 +1 -1 +1 +1 +1
5 +1 +1 -1 -1 +1 +1 -1
6 -1 +1 -1 +1 -1 +1 +1
7 +1 -1 -1 +1 +1 -1 +1
8 -1 -1 -1 -1 -1 -1 -1
Alias
structure
for
augmented
runs
Following the same steps as before and making the same
assumptions about the omission of higher-order interactions
in the alias structure, we arrive at:
1 = -24 = -35 = -67
2 = -14 = -36 =- 57
3 = -15 = -26 = -47
4 = -12 = -37 = -56
5 = -13 = -27 = -46
6 = -17 = -23 = -45
7 = -16 = -25 = -34
With both sets of runs, we can now estimate all the main
effects free from two factor interactions.
Build a
resolution
IV design
from a
resolution
III design
Note: In general, a mirror-image foldover design is a method
to build a resolution IV design from a resolution III design. It
is never used to follow-up a resolution IV design.
5.3.3.8.2. Alternative foldover designs

Alternative
foldover
designs can
be an
economical
way to break
up a selected
alias pattern
The mirror-image foldover (in which signs in all columns are
reversed) is only one of the possible follow-up fractions that
can be run to augment a fractional factorial design. It is the
most common choice when the original fraction is resolution
III. However, alternative foldover designs with fewer runs
can often be utilized to break up selected alias patterns. We
illustrate this by looking at what happens when the signs of a
single factor column are reversed.
Example of
de-aliasing a
single factor
Previously, we described how we de-alias all the factors of a
2
7-4
experiment. Suppose that we only want to de-alias the
X4 factor. This can be accomplished by only changing the
sign of X4 = X1X2 to X4 = -X1X2. The resulting design is:
Table
showing
design
matrix of a
reverse X4
foldover
design
TABLE 3.36 A "Reverse X4" Foldover Design
run X1 X2 X3
X4 =
-X1X2
X5 =
-X1X3
X6 =
X2X3
X7 =
X1X2X3
1 -1 -1 -1 -1 +1 +1 -1
2 +1 -1 -1 +1 -1 +1 +1
3 -1 +1 -1 +1 +1 -1 +1
4 +1 +1 -1 -1 -1 -1 -1
5 -1 -1 +1 -1 -1 -1 +1
6 +1 -1 +1 +1 +1 -1 -1
7 -1 +1 +1 +1 -1 +1 -1
8 +1 +1 +1 -1 +1 +1 +1
Alias
patterns and
effects that
can be
estimated in
the example
design
The two-factor alias patterns for X4 are: Original experiment:
X4 = X1X2 = X3X7 = X5X6; "Reverse X4" foldover
experiment: X4 = -X1X2 = -X3X7 = -X5X6.
The following effects can be estimated by combining the
original with the "Reverse X4" foldover fraction:
X1 + X3X5 + X6X7
X2 + X3X6 + X5X7
X3 + X1X5 + X2X6
X4
X5 + X1X3 + X2X7
X6 + X2X3 + X1X7
X7 + X2X5 + X1X6
X1X4
X2X4
X3X4
X4X5
X4X6
X4X7
X1X2 + X3X7 + X5X6
Note: The 16 runs allow estimating the above 14 effects,
with one degree of freedom left over for a possible block
effect.
Advantage
and
disadvantage
of this
example
design
The advantage of this follow-up design is that it permits
estimation of the X4 effect and each of the six two-factor
interaction terms involving X4.
The disadvantage is that the combined fractions still yield a
resolution III design, with all main effects other than X4
aliased with two-factor interactions.
Case when
purpose is
simply to
estimate all
two-factor
interactions
of a single
factor
Reversing a single factor column to obtain de-aliased two-
factor interactions for that one factor works for any
resolution III or IV design. When used to follow-up a
resolution IV design, there are relatively few new effects to
be estimated (as compared to designs). When the
original resolution IV fraction provides sufficient precision,
and the purpose of the follow-up runs is simply to estimate
all two-factor interactions for one factor, the semifolding
option should be considered.
Semifolding
Number of
runs can be
reduced for
resolution IV
designs
For resolution IV fractions, it is possible to economize on the
number of runs that are needed to break the alias chains for
all two-factor interactions of a single factor. In the above
case we needed 8 additional runs, which is the same number
of runs that were used in the original experiment. This can be
improved upon.
Additional
information
on John's 3/4
designs
We can repeat only the points that were set at the high levels
of the factor of choice and then run them at their low settings
in the next experiment. For the given example, this means an
additional 4 runs instead 8. We mention this technique only
in passing, more details may be found in the references (or
see John's 3/4 designs).
5.3.3.9. Three-level full factorial designs

Three-level
designs are
useful for
investigating
quadratic
effects
The three-level design is written as a 3
k
factorial design. It means
that k factors are considered, each at 3 levels. These are (usually)
referred to as low, intermediate and high levels. These levels are
numerically expressed as 0, 1, and 2. One could have considered the
digits -1, 0, and +1, but this may be confusing with respect to the 2-
level designs since 0 is reserved for center points. Therefore, we will
use the 0, 1, 2 scheme. The reason that the three-level designs were
proposed is to model possible curvature in the response function and
to handle the case of nominal factors at 3 levels. A third level for a
continuous factor facilitates investigation of a quadratic relationship
between the response and each of the factors.
Three-level
design may
require
prohibitive
number of
runs
Unfortunately, the three-level design is prohibitive in terms of the
number of runs, and thus in terms of cost and effort. For example a
two-level design with center points is much less expensive while it
still is a very good (and simple) way to establish the presence or
absence of curvature.
The 3
2
design
The simplest
3-level design
- with only 2
factors
This is the simplest three-level design. It has two factors, each at
three levels. The 9 treatment combinations for this type of design can
be shown pictorially as follows:
FIGURE 3.23 A 3
2
Design Schematic
A notation such as "20" means that factor A is at its high level (2)
and factor B is at its low level (0).
The 3
3
design
The model
and treatment
runs for a 3
factor, 3-level
design
This is a design that consists of three factors, each at three levels. It
can be expressed as a 3 x 3 x 3 = 3
3
design. The model for such an
experiment is
where each factor is included as a nominal factor rather than as a
continuous variable. In such cases, main effects have 2 degrees of
freedom, two-factor interactions have 2
2
= 4 degrees of freedom and
k-factor interactions have 2
k
degrees of freedom. The model contains
2 + 2 + 2 + 4 + 4 + 4 + 8 = 26 degrees of freedom. Note that if there
is no replication, the fit is exact and there is no error term (the
epsilon term) in the model. In this no replication case, if one assumes
that there are no three-factor interactions, then one can use these 8
degrees of freedom for error estimation.
In this model we see that i = 1, 2, 3, and similarly for j and k, making
27 treatments.
Table of
treatments for
the 3
3
design
These treatments may be displayed as follows:
TABLE 3.37 The 3
3
Design
Factor A
Factor B Factor C 0 1 2
0 0 000 100 200
0 1 001 101 201
0 2 002 102 202
1 0 010 110 210
1 1 011 111 211
1 2 012 112 212
2 0 020 120 220
2 1 021 121 221
2 2 022 122 222
Pictorial
representation
of the 3
3
design
The design can be represented pictorially by
FIGURE 3.24 A 3
3
Design Schematic
Two types of
3
k
designs
Two types of fractions of 3
k
designs are employed:
Box-Behnken designs whose purpose is to estimate a second-
order model for quantitative factors (discussed earlier in
section 5.3.3.6.2)
3
k-p
orthogonal arrays.
5.3.3.10. Three-level, mixed-level and fractional factorial designs
http://www.itl.nist.gov/div898/handbook/pri/section3/pri33a.htm[6/27/2012 2:24:20 PM]

5.3.3.10. Three-level, mixed-level and fractional
factorial designs
Mixed level
designs have
some factors
with, say, 2
levels, and
some with 3
levels or 4
levels
The 2
k
and 3
k
experiments are special cases of factorial
designs. In a factorial design, one obtains data at every
combination of the levels. The importance of factorial designs,
especially 2-level factorial designs, was stated by Montgomery
(1991): It is our belief that the two-level factorial and
fractional factorial designs should be the cornerstone of
industrial experimentation for product and process
development and improvement. He went on to say: There are,
however, some situations in which it is necessary to include a
factor (or a few factors) that have more than two levels.
This section will look at how to add three-level factors
starting with two-level designs, obtaining what is called a
mixed-level design. We will also look at how to add a four-
level factor to a two-level design. The section will conclude
with a listing of some useful orthogonal three-level and
mixed-level designs (a few of the so-called Taguchi "L"
orthogonal array designs), and a brief discussion of their
benefits and disadvantages.
Generating a Mixed Three-Level and Two-Level Design
Montgomery
scheme for
generating a
mixed
design
Montgomery (1991) suggests how to derive a variable at three
levels from a 2
3
design, using a rather ingenious scheme. The
objective is to generate a design for one variable, A, at 2 levels
and another, X, at three levels. This will be formed by
combining the -1 and 1 patterns for the B and C factors to
form the levels of the three-level factor X:
TABLE 3.38 Generating a Mixed Design
Two-Level Three-Level
B C X
-1 -1
x
1
+1 -1
x
2
-1 +1
x
2
+1 +1
x
3
Similar to the 3
k
case, we observe that X has 2 degrees of
freedom, which can be broken out into a linear and a
quadratic component. To illustrate how the 2
3
design leads to
the design with one factor at two levels and one factor at three
levels, consider the following table, with particular attention
focused on the column labels.
Table
illustrating
the
generation
of a design
with one
factor at 2
levels and
another at 3
levels from a
2
3
design
A X
L
X
L
AX
L
AX
L
X
Q
AX
Q
TRT MNT
Run A B C AB AC BC ABC A X
1 -1 -1 -1 +1 +1 +1 -1 Low Low
2 +1 -1 -1 -1 -1 +1 +1 High Low
3 -1 +1 -1 -1 +1 -1 +1 Low Medium
4 +1 +1 -1 +1 -1 -1 -1 High Medium
5 -1 -1 +1 +1 -1 -1 +1 Low Medium
6 +1 -1 +1 -1 +1 -1 -1 High Medium
7 -1 +1 +1 -1 -1 +1 -1 Low High
8 +1 +1 +1 +1 +1 +1 +1 High High
If quadratic
effect
negligble,
we may
include a
second two-
level factor
If we believe that the quadratic effect is negligible, we may
include a second two-level factor, D, with D = ABC. In fact,
we can convert the design to exclusively a main effect
(resolution III) situation consisting of four two-level factors
and one three-level factor. This is accomplished by equating
the second two-level factor to AB, the third to AC and the
fourth to ABC. Column BC cannot be used in this manner
because it contains the quadratic effect of the three-level
factor X.
More than one three-level factor
3-Level
factors from
2
4
and 2
5
designs
We have seen that in order to create one three-level factor, the
starting design can be a 2
3
factorial. Without proof we state
that a 2
4
can split off 1, 2 or 3 three-level factors; a 2
5
is able
to generate 3 three-level factors and still maintain a full
factorial structure. For more on this, see Montgomery (1991).
Generating a Two- and Four-Level Mixed Design
Constructing
a design
with one 4-
level factor
and two 2-
level factors
We may use the same principles as for the three-level factor
example in creating a four-level factor. We will assume that
the goal is to construct a design with one four-level and two
two-level factors.
Initially we wish to estimate all main effects and interactions.
It has been shown (see Montgomery, 1991) that this can be
accomplished via a 2
4
(16 runs) design, with columns A and B
used to create the four level factor X.
Table
showing
TABLE 3.39 A Single Four-level Factor and Two Two-
level Factors in 16 runs
design with
4-level, two
2-level
factors in 16
runs
Run (A B) = X C D
1 -1 -1 x
1
-1 -1
2 +1 -1 x
2
-1 -1
3 -1 +1 x
3
-1 -1
4 +1 +1 x
4
-1 -1
5 -1 -1 x
1
+1 -1
6 +1 -1 x
2
+1 -1
7 -1 +1 x
3
+1 -1
8 +1 +1 x
4
+1 -1
9 -1 -1 x
1
-1 +1
10 +1 -1 x
2
-1 +1
11 -1 +1 x
3
-1 +1
12 +1 +1 x
4
-1 +1
13 -1 -1 x
1
+1 +1
14 +1 -1 x
2
+1 +1
15 -1 +1 x
3
+1 +1
16 +1 +1 x
4
+1 +1
Some Useful (Taguchi) Orthogonal "L" Array Designs
L
9
design
L
9
- A 3
4-2
Fractional Factorial Design 4
Factors at Three Levels (9 runs)
Run X1 X2 X3 X4
1 1 1 1 1
2 1 2 2 2
3 1 3 3 3
4 2 1 2 3
5 2 2 3 1
6 2 3 1 2
7 3 1 3 2
8 3 2 1 3
9 3 3 2 1
L
18
design
L
18
- A 2 x 3
7-5
Fractional Factorial (Mixed-Level) Design
1 Factor at Two Levels and Seven Factors at 3 Levels (18
Runs)
Run X1 X2 X3 X4 X5 X6 X7 X8
1 1 1 1 1 1 1 1 1
2 1 1 2 2 2 2 2 2
3 1 1 3 3 3 3 3 3
4 1 2 1 1 2 2 3 3
5 1 2 2 2 3 3 1 1
6 1 2 3 3 1 1 2 2
7 1 3 1 2 1 3 2 3
8 1 3 2 3 2 1 3 1
9 1 3 3 1 3 2 1 2
10 2 1 1 3 3 2 2 1
11 2 1 2 1 1 3 3 2
12 2 1 3 2 2 1 1 3
13 2 2 1 2 3 1 3 2
14 2 2 2 3 1 2 1 3
15 2 2 3 1 2 3 2 1
16 2 3 1 3 2 3 1 2
17 2 3 2 1 3 1 2 3
18 2 3 3 2 1 2 3 1
L
27
design
L
27
- A 3
13-10
Fractional Factorial Design
Thirteen Factors at Three Levels (27 Runs)
Run X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13
1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 2 2 2 2 2 2 2 2 2
3 1 1 1 1 3 3 3 3 3 3 3 3 3
4 1 2 2 2 1 1 1 2 2 2 3 3 3
5 1 2 2 2 2 2 2 3 3 3 1 1 1
6 1 2 2 2 3 3 3 1 1 1 2 2 2
7 1 3 3 3 1 1 1 3 3 3 2 2 2
8 1 3 3 3 2 2 2 1 1 1 3 3 3
9 1 3 3 3 3 3 3 2 2 2 1 1 1
10 2 1 2 3 1 2 3 1 2 3 1 2 3
11 2 1 2 3 2 3 1 2 3 1 2 3 1
12 2 1 2 3 3 1 2 3 1 2 3 1 2
13 2 2 3 1 1 2 3 2 3 1 3 1 2
14 2 2 3 1 2 3 1 3 1 2 1 2 3
15 2 2 3 1 3 1 2 1 2 3 2 3 1
16 2 3 1 2 1 2 3 3 1 2 2 3 1
17 2 3 1 2 2 3 1 1 2 3 3 1 2
18 2 3 1 2 3 1 2 2 3 1 1 2 3
19 3 1 3 2 1 3 2 1 3 2 1 3 2
20 3 1 3 2 2 1 3 2 1 3 2 1 3
21 3 1 3 2 3 2 1 3 2 1 3 2 1
22 3 2 1 3 1 3 2 2 1 3 3 2 1
23 3 2 1 3 2 1 3 3 2 1 1 3 2
24 3 2 1 3 3 2 1 1 3 2 2 1 3
25 3 3 2 1 1 3 2 3 2 1 2 1 3
26 3 3 2 1 2 1 3 1 3 2 3 2 1
27 3 3 2 1 3 2 1 2 1 3 1 3 2
L
36
design
L36 - A Fractional Factorial (Mixed-Level) Design Eleven Factors at Two Levels and Twelve Factors at 3
Levels (36 Runs)
Run X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2
3 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3
4 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3
5 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 1 1 1 1
6 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2
7 1 1 2 2 2 1 1 1 2 2 2 1 1 2 3 1 2 3 3 1 2 2 3
8 1 1 2 2 2 1 1 1 2 2 2 2 2 3 1 2 3 1 1 2 3 3 1
9 1 1 2 2 2 1 1 1 2 2 2 3 3 1 2 3 1 2 2 3 1 1 2
10 1 2 1 2 2 1 2 2 1 1 2 1 1 3 2 1 3 2 3 2 1 3 2
11 1 2 1 2 2 1 2 2 1 1 2 2 2 1 3 2 1 3 1 3 2 1 3
12 1 2 1 2 2 1 2 2 1 1 2 3 3 2 1 3 2 1 2 1 3 2 1
13 1 2 2 1 2 2 1 2 1 2 1 1 2 3 1 3 2 1 3 3 2 1 2
14 1 2 2 1 2 2 1 2 1 2 1 2 3 1 2 1 3 2 1 1 3 2 3
15 1 2 2 1 2 2 1 2 1 2 1 3 1 2 3 2 1 3 2 2 1 3 1
16 1 2 2 2 1 2 2 1 2 1 1 1 2 3 2 1 1 3 2 3 3 2 1
17 1 2 2 2 1 2 2 1 2 1 1 2 3 1 3 2 2 1 3 1 1 3 2
18 1 2 2 2 1 2 2 1 2 1 1 3 1 2 1 3 3 2 1 2 2 1 3
19 2 1 2 2 1 1 2 2 1 2 1 1 2 1 3 3 3 1 2 2 1 2 3
20 2 1 2 2 1 1 2 2 1 2 1 2 3 2 1 1 1 2 3 3 2 3 1
21 2 1 2 2 1 1 2 2 1 2 1 3 1 3 2 2 2 3 1 1 3 1 2
22 2 1 2 1 2 2 2 1 1 1 2 1 2 2 3 3 1 2 1 1 3 3 2
23 2 1 2 1 2 2 2 1 1 1 2 2 3 3 1 1 2 3 2 2 1 1 3
24 2 1 2 1 2 2 2 1 1 1 2 3 1 1 2 2 3 1 3 3 2 2 1
25 2 1 1 2 2 2 1 2 2 1 1 1 3 2 1 2 3 3 1 3 1 2 2
26 2 1 1 2 2 2 1 2 2 1 1 2 1 3 2 3 1 1 2 1 2 3 3
27 2 1 1 2 2 2 1 2 2 1 1 3 2 1 3 1 2 2 3 2 3 1 1
28 2 2 2 1 1 1 1 2 2 1 2 1 3 2 2 2 1 1 3 2 3 1 3
29 2 2 2 1 1 1 1 2 2 1 2 2 1 3 3 3 2 2 1 3 1 2 1
30 2 2 2 1 1 1 1 2 2 1 2 3 2 1 1 1 3 3 2 1 2 3 2
31 2 2 1 2 1 2 1 1 1 2 2 1 3 3 3 2 3 2 2 1 2 1 1
32 2 2 1 2 1 2 1 1 1 2 2 2 1 1 1 3 1 3 3 2 3 2 2
33 2 2 1 2 1 2 1 1 1 2 2 3 2 2 2 1 2 1 1 3 1 3 3
34 2 2 1 1 2 1 2 1 2 2 1 1 3 1 2 3 2 3 1 2 2 3 1
35 2 2 1 1 2 1 2 1 2 2 1 2 1 2 3 1 3 1 2 3 3 1 2
36 2 2 1 1 2 1 2 1 2 2 1 3 2 3 1 2 1 2 3 1 1 2 3
Advantages and Disadvantages of Three-Level and
Mixed-Level "L" Designs
Advantages The good features of these designs are:
and
disadvantages
of three-level
mixed-level
designs
They are orthogonal arrays. Some analysts believe
this simplifies the analysis and interpretation of
results while other analysts believe it does not.
They obtain a lot of information about the main
effects in a relatively few number of runs.
You can test whether non-linear terms are needed in
the model, at least as far as the three-level factors are
concerned.
On the other hand, there are several undesirable features of
these designs to consider:
They provide limited information about interactions.
They require more runs than a comparable
2
k-p
design, and a two-level design will often suffice
when the factors are continuous and monotonic
(many three-level designs are used when two-level
designs would have been adequate).
5.4. Analysis of DOE data

Contents of
this section
Assuming you have a starting model that you want to fit to
your experimental data and the experiment was designed
correctly for your objective, most DOE software packages
will analyze your DOE data. This section will illustrate how
to analyze DOE's by first going over the generic basic steps
and then showing software examples. The contents of the
section are:
DOE analysis steps
Plotting DOE data
Modeling DOE data
Testing and revising DOE models
Interpreting DOE results
Confirming DOE results
DOE examples
Full factorial example
Fractional factorial example
Response surface example
Prerequisite
statistical
tools and
concepts
needed for
DOE
analyses
The examples in this section assume the reader is familiar
with the concepts of
ANOVA tables (see Chapter 3 or Chapter 7)
p-values
Residual analysis
Model Lack of Fit tests
Data transformations for normality and linearity
5.4.1. What are the steps in a DOE analysis?

General
flowchart
for
analyzing
DOE data
Flowchart of DOE Analysis Steps
DOE Analysis Steps
Analysis
steps:
graphics,
theoretical
model,
actual
model,
validate
model, use
model
The following are the basic steps in a DOE analysis.
1. Look at the data. Examine it for outliers, typos and obvious problems.
Construct as many graphs as you can to get the big picture.
Response distributions (histograms, box plots, etc.)
Responses versus time order scatter plot (a check for possible time
effects)
Responses versus factor levels (first look at magnitude of factor effects)
Typical DOE plots (which assume standard models for effects and
errors)
Main effects mean plots
Block plots
Normal or half-normal plots of the effects
Interaction plots
Sometimes the right graphs and plots of the data lead to obvious
answers for your experimental objective questions and you can skip to
step 5. In most cases, however, you will want to continue by fitting and
validating a model that can be used to answer your questions.
2. Create the theoretical model (the experiment should have been designed with
this model in mind!).
3. Create a model from the data. Simplify the model, if possible, using stepwise
regression methods and/or parameter p-value significance information.
4. Test the model assumptions using residual graphs.
If none of the model assumptions were violated, examine the ANOVA.
Simplify the model further, if appropriate. If reduction is
appropriate, then return to step 3 with a new model.
If model assumptions were violated, try to find a cause.
Are necessary terms missing from the model?
Will a transformation of the response help? If a transformation is
used, return to step 3 with a new model.
5. Use the results to answer the questions in your experimental objectives --
finding important factors, finding optimum settings, etc.
Flowchart
is a
guideline,
not a
hard-and
-fast rule
Note: The above flowchart and sequence of steps should not be regarded as a "hard-
and-fast rule" for analyzing all DOE's. Different analysts may prefer a different
sequence of steps and not all types of experiments can be analyzed with one set
procedure. There still remains some art in both the design and the analysis of
experiments, which can only be learned from experience. In addition, the role of
engineering judgment should not be underestimated.
5.4.2. How to "look" at DOE data

The
importance
of looking
at the data
with a wide
array of
plots or
visual
displays
cannot be
over-
stressed
The right graphs, plots or visual displays of a dataset can
uncover anomalies or provide insights that go beyond what
most quantitative techniques are capable of discovering.
Indeed, in many cases quantitative techniques and models are
tools used to confirm and extend the conclusions an analyst
has already formulated after carefully "looking" at the data.
Most software packages have a selection of different kinds of
plots for displaying DOE data. Some of these useful ways of
looking at data are mentioned below, with links to detailed
explanations in Chapter 1 (Exploratory Data Analysis or
EDA) or to other places where they are illustrated and
explained. In addition, examples and detailed explanations of
visual (EDA) DOE techniques can be found in section 5.5.9.
Plots for
viewing the
response
data
First "Look" at the Data
Histogram of responses
Run-sequence plot (pay special attention to results at
center points)
Scatter plot (for pairs of response variables)
Lag plot
Normal probability plot
Autocorrelation plot
Plots for
viewing
main effects
and 2-factor
interactions,
explanation
of normal
or half-
normal
plots to
detect
possible
important
effects
Subsequent Plots: Main Effects, Comparisons and 2-Way
Interactions
Quantile-quantile (q-q) plot
Block plot
Box plot
Bi-histogram
DOE scatter plot
DOE mean plot
DOE interaction plots
Normal or half-normal probability plots for effects.
Note: these links show how to generate plots to test for
normal (or half-normal) data with points lining up
along a straight line, approximately, if the plotted
points were from the assumed normal (or half-normal)
distribution. For two-level full factorial and fractional
factorial experiments, the points plotted are the
estimates of all the model effects, including possible
interactions. Those effects that are really negligible
should have estimates that resemble normally
distributed noise, with mean zero and a constant
variance. Significant effects can be picked out as the
ones that do not line up along the straight line. Normal
effect plots use the effect estimates directly, while
half-normal plots use the absolute values of the effect
estimates.
Youden plots
Plots for
testing and
validating
models
Model testing and Validation
Response vs predictions
Residuals vs predictions
Residuals vs independent variables
Residuals lag plot
Residuals histogram
Normal probability plot of residuals
Plots for
model
prediction
Model Predictions
Contour plots
5.4.3. How to model DOE data

DOE models
should be
consistent
with the
goal of the
experiment
In general, the trial model that will be fit to DOE data should
be consistent with the goal of the experiment and has been
predetermined by the goal of the experiment and the
experimental design and data collection methodology.
Comparative
designs
Models were given earlier for comparative designs
(completely randomized designs, randomized block designs
and Latin square designs).
Full
factorial
designs
For full factorial designs with k factors (2
k
runs, not
counting any center points or replication runs), the full
model contains all the main effects and all orders of
interaction terms. Usually, higher-order (three or more
factors) interaction terms are included initially to construct
the normal (or half-normal) plot of effects, but later dropped
when a simpler, adequate model is fit. Depending on the
software available or the analyst's preferences, various
techniques such as normal or half-normal plots, Youden
plots, p-value comparisons and stepwise regression routines
are used to reduce the model to the minimum number of
needed terms. An example of model selection is shown later
in this section and an example of Yates algorithm is given
as a case study.
Fractional
factorial
designs
For fractional factorial screening designs, it is necessary to
know the alias structure in order to write an appropriate
starting model containing only the interaction terms the
experiment was designed to estimate (assuming all terms
confounded with these selected terms are insignificant). This
is illustrated by the fractional factorial example later in this
section. The starting model is then possibly reduced by the
same techniques described above for full factorial models.
Response
surface
designs
Response surface initial models include quadratic terms and
may occasionally also include cubic terms. These models
were described in section 3.
Model
validation
Of course, as in all cases of model fitting, residual analysis
and other tests of model fit are used to confirm or adjust
models, as needed.
5.4.4. How to test and revise DOE models

5.4.4. How to test and revise DOE models
Tools for
testing,
revising,
and
selecting
models
All the tools and procedures for testing, revising and selecting
final DOE models are covered in various sections of the
Handbook. The outline below gives many of the most
common and useful techniques and has links to detailed
explanations.
Outline of Model Testing and Revising: Tools and
Procedures
An outline
(with
links)
covers
most of the
useful
tools and
procedures
for testing
and
revising
DOE
models
Graphical Indicators for testing models (using residuals)
Response vs predictions
Residuals vs predictions
Residuals vs independent variables
Residuals lag plot
Residuals histogram
Normal probability plot of residuals
Overall numerical indicators for testing models and
model terms
R Squared and R Squared adjusted
p-values
Model selection tools or procedures
p-values
Residual analysis
Data transformations for normality and linearity
Stepwise regression procedures
Normal or half-normal plots of effects (primarily
for two-level full and fractional factorial
experiments)
Youden plots
Other methods
5.4.5. How to interpret DOE results

5.4.5. How to interpret DOE results
Final model
used to
make
conclusions
and
decisions
Assume that you have a final model that has passed all the
relevant tests (visual and quantitative) and you are ready to
make conclusions and decisions. These should be responses
to the questions or outputs dictated by the original
experimental goals.
Checklist relating DOE conclusions or outputs to
experimental goals or experimental purpose:
A checklist
of how to
compare
DOE results
to the
experimental
goals
Do the responses differ significantly over the factor
levels? (comparative experiment goal)
Which are the significant effects or terms in the final
model? (screening experiment goal)
What is the model for estimating responses?
Full factorial case (main effects plus significant
interactions)
Fractional factorial case (main effects plus
significant interactions that are not confounded
with other possibly real effects)
RSM case (allowing for quadratic or possibly
cubic models, if needed)
What responses are predicted and how can responses
be optimized? (RSM goal)
Contour plots
Settings for confirmation runs and prediction
intervals for results
5.4.6. How to confirm DOE results (confirmatory runs)

5.4.6. How to confirm DOE results
(confirmatory runs)
Definition of
confirmation
runs
When the analysis of the experiment is complete, one must
verify that the predictions are good. These are called
confirmation runs.
The interpretation and conclusions from an experiment may
include a "best" setting to use to meet the goals of the
experiment. Even if this "best" setting were included in the
design, you should run it again as part of the confirmation
runs to make sure nothing has changed and that the response
values are close to their predicted values. would get.
At least 3
confirmation
runs should
be planned
In an industrial setting, it is very desirable to have a stable
process. Therefore, one should run more than one test at the
"best" settings. A minimum of 3 runs should be conducted
(allowing an estimate of variability at that setting).
If the time between actually running the experiment and
conducting the confirmation runs is more than a few hours,
the experimenter must be careful to ensure that nothing else
has changed since the original data collection.
Carefully
duplicate the
original
environment
The confirmation runs should be conducted in an
environment as similar as possible to the original
experiment. For example, if the experiment were conducted
in the afternoon and the equipment has a warm-up effect,
the confirmation runs should be conducted in the afternoon
after the equipment has warmed up. Other extraneous factors
that may change or affect the results of the confirmation runs
are: person/operator on the equipment, temperature,
humidity, machine parameters, raw materials, etc.
Checks for
when
confirmation
runs give
surprises
What do you do if you don't obtain the results you expected?
If the confirmation runs don't produce the results you
expected:
1. check to see that nothing has changed since the
original data collection
2. verify that you have the correct settings for the
confirmation runs
3. revisit the model to verify the "best" settings from the
analysis
5.4.6. How to confirm DOE results (confirmatory runs)
4. verify that you had the correct predicted value for the
confirmation runs.
If you don't find the answer after checking the above 4
items, the model may not predict very well in the region you
decided was "best". You still learned from the experiment
and you should use the information gained from this
experiment to design another follow-up experiment.
Even when
the
experimental
goals are
not met,
something
was learned
that can be
used in a
follow-up
experiment
Every well-designed experiment is a success in that you
learn something from it. However, every experiment will not
necessarily meet the goals established before
experimentation. That is why it makes sense to plan to
experiment sequentially in order to meet the goals.
5.4.7. Examples of DOE's

5.4.7. Examples of Designed Experiments
Three
detailed
examples
Perhaps one of the best ways to illustrate how to analyze data
from a designed experiment is to work through a detailed
example, explaining each step in the analysis.
Detailed analyses are presented for three basic types of
designed experiments:
1. A full factorial experiment
2. A fractional factorial experiment
3. A response surface experiment
Software Most analyses of designed experiments are performed by
statistical software packages. Good statistical software enables
the analyst to view graphical displays, build models, and test
assumptions. Occasionally, the goals of the experiment can be
achieved by simply examining appropriate graphical displays
of the experimental responses. In other cases, a satisfactory
model has to be fit in order to determine the most significant
factors or the optimal contours of the response surface. In any
case, the software will perform the appropriate calculations as
long as the analyst knows what to request and how to interpret
the program outputs.
5.4.7.1. Full factorial example

Data Source
This example
uses data from
a NIST high
performance
ceramics
experiment
This data set was taken from an experiment that was performed a few years ago at NIST by
Said Jahanmir of the Ceramics Division in the Material Science and Engineering Laboratory.
The original analysis was performed primarily by Lisa Gill of the Statistical Engineering
Division. The example shown here is an independent analysis of a modified portion of the
original data set.
The original data set was part of a high performance ceramics experiment with the goal of
characterizing the effect of grinding parameters on sintered reaction-bonded silicon nitride,
reaction bonded silicone nitride, and sintered silicon nitride.
Only modified data from the first of the three ceramic types (sintered reaction-bonded silicon
nitride) will be discussed in this illustrative example of a full factorial data analysis.
The reader can download the data as a text file.
Description of Experiment: Response and Factors
Response and
factor
variables
Purpose: To determine the effect of machining factors on ceramic strength
Response variable = mean (over 15 repetitions) of the ceramic strength
Number of observations = 32 (a complete 2
5
factorial design)
Response Variable Y = Mean (over 15 reps) of Ceramic Strength
Factor 1 = Table Speed (2 levels: slow (.025 m/s) and fast (.125 m/s))
Factor 2 = Down Feed Rate (2 levels: slow (.05 mm) and fast (.125 mm))
Factor 3 = Wheel Grit (2 levels: 140/170 and 80/100)
Factor 4 = Direction (2 levels: longitudinal and transverse)
Factor 5 = Batch (2 levels: 1 and 2)
Since two factors were qualitative (direction and batch) and it was reasonable to expect
monotone effects from the quantitative factors, no centerpoint runs were included.
The data The design matrix, with measured ceramic strength responses, appears below. The actual
randomized run order is given in the last column. (The interested reader may download the
data as a text file.)
speed rate grit direction batch strength order
1 -1 -1 -1 -1 -1 680.45 17
2 1 -1 -1 -1 -1 722.48 30
3 -1 1 -1 -1 -1 702.14 14
4 1 1 -1 -1 -1 666.93 8
5 -1 -1 1 -1 -1 703.67 32
6 1 -1 1 -1 -1 642.14 20
7 -1 1 1 -1 -1 692.98 26
8 1 1 1 -1 -1 669.26 24
9 -1 -1 -1 1 -1 491.58 10
10 1 -1 -1 1 -1 475.52 16
11 -1 1 -1 1 -1 478.76 27
12 1 1 -1 1 -1 568.23 18
13 -1 -1 1 1 -1 444.72 3
14 1 -1 1 1 -1 410.37 19
15 -1 1 1 1 -1 428.51 31
16 1 1 1 1 -1 491.47 15
17 -1 -1 -1 -1 1 607.34 12
18 1 -1 -1 -1 1 620.80 1
19 -1 1 -1 -1 1 610.55 4
20 1 1 -1 -1 1 638.04 23
21 -1 -1 1 -1 1 585.19 2
22 1 -1 1 -1 1 586.17 28
23 -1 1 1 -1 1 601.67 11
24 1 1 1 -1 1 608.31 9
25 -1 -1 -1 1 1 442.90 25
26 1 -1 -1 1 1 434.41 21
27 -1 1 -1 1 1 417.66 6
28 1 1 -1 1 1 510.84 7
29 -1 -1 1 1 1 392.11 5
30 1 -1 1 1 1 343.22 13
31 -1 1 1 1 1 385.52 22
32 1 1 1 1 1 446.73 29
Analysis of the Experiment
Five basic
steps
The experimental data will be analyzed following the previously described five basic steps.
The analyses shown in this page can be generated using R code.
Step 1: Look at the data
Plot the
response
variable
We start by plotting the response data several ways to see if any trends or anomalies appear
that would not be accounted for by the standard linear response models.
First, we look at the distribution of the response variable regardless of factor levels by
generating the following four plots.
1. The first plot is a normal probability plot of the response variable. The red line is the
theoretical normal distribution.
2. The second plot is a box plot of the response variable.
3. The third plot is a histogram of the response variable.
4. The fourth plot is the response versus the run order.
Clearly there is "structure" that we hope to account for when we fit a response model. For
example, the response variable is separated into two roughly equal-sized clumps in the
histogram. The first clump is centered approximately around the value 450 while the second
clump is centered approximately around the value 650. As hoped for, the run-order plot does
not indicate a significant time effect.
Box plots of
response by
factor
variables
Next, we look at box plots of the response for each factor.
Several factors, most notably "Direction" followed by "Batch" and possibly "Wheel Grit",
appear to change the average response level.
Step 2: Create the theoretical model
Theoretical
model: assume
all four-factor
and higher
interaction
terms are not
significant
For a 2
5
full factorial experiment we can fit a model containing a mean term, five main effect
terms, ten two-factor interaction terms, ten three-factor interaction terms, five four-factor
interaction terms, and a five-factor interaction term (32 parameters). However, we start by
assuming all four-factor and higher interaction terms are non-existent. It's very rare for such
high-order interactions to be significant, and they are very difficult to interpret from an
engineering viewpoint. The assumption allows us to accumulate the sums of squares for these
terms and use them to estimate an error term. We start with a theoretical model with 26
unknown constants, hoping the data will clarify which of these are the significant main effects
and interactions we need for a final model.
Step 3: Fit model to the data
Results from The ANOVA table for the 26-parameter model (intercept not shown) follows.
fitting up to
and including
third-order
interaction
terms
Summary of Fit
RSquare 0.995127
RSquare Adj 0.974821
Root Mean Square Error 17.81632
Mean of Response 546.8959
Observations 32
Sum
Source DF of Squares F Ratio Prob>F
X1: Table Speed 1 894.33 2.8175 0.1442
X2: Feed Rate 1 3497.20 11.0175 0.0160
X1: Table Speed* 1 4872.57 15.3505 0.0078
X2: Feed Rate
X3: Wheel Grit 1 12663.96 39.8964 0.0007
X1: Table Speed* 1 1838.76 5.7928 0.0528
X3: Wheel Grit
X2: Feed Rate* 1 307.46 0.9686 0.3630
X3: Wheel Grit
X1:Table Speed* 1 357.05 1.1248 0.3297
X2: Feed Rate*
X3: Wheel Grit
X4: Direction 1 315132.65 992.7901 <.0001
X1: Table Speed* 1 1637.21 5.1578 0.0636
X4: Direction
X2: Feed Rate* 1 1972.71 6.2148 0.0470
X4: Direction
X1: Table Speed 1 5895.62 18.5735 0.0050
X2: Feed Rate*
X4: Direction
X3: Wheel Grit* 1 3158.34 9.9500 0.0197
X4: Direction
X1: Table Speed* 1 2.12 0.0067 0.9376
X3: Wheel Grit*
X4: Direction
X2: Feed Rate* 1 44.49 0.1401 0.7210
X3: Wheel Grit*
X4: Direction
X5: Batch 1 33653.91 106.0229 <.0001
X1: Table Speed* 1 465.05 1.4651 0.2716
X5: Batch
X2: Feed Rate* 1 199.15 0.6274 0.4585
X5: Batch
X1: Table Speed* 1 144.71 0.4559 0.5247
X2: Feed Rate*
X5: Batch
X3: Wheel Grit* 1 29.36 0.0925 0.7713
X5: Batch
X1: Table Speed* 1 30.36 0.0957 0.7676
X3: Wheel Grit*
X5: Batch
X2: Feed Rate* 1 25.58 0.0806 0.7860
X3: Wheel Grit*
X5: Batch
X4: Direction * 1 1328.83 4.1863 0.0867
X5: Batch
X1: Table Speed* 1 544.58 1.7156 0.2382
X4: Directio*
X5: Batch
X2: Feed Rate* 1 167.31 0.5271 0.4952
X4: Direction*
X5: Batch
X3: Wheel Grit* 1 32.46 0.1023 0.7600
X4: Direction*
X5: Batch
This fit has a large R
2
and adjusted R
2
, but the high number of large (>0.10) p-values (in the
"Prob>F" column) makes it clear that the model has many unnecessary terms.
Stepwise
regression
Starting with the 26 terms, we use stepwise regression to eliminate unnecessary terms. By a
combination of stepwise regression and the removal of remaining terms with a p-value larger
than 0.05, we quickly arrive at a model with an intercept and 12 significant effect terms.
Results from
fitting the 12-
Summary of Fit
RSquare 0.989114
term model
Observations (or Sum Wgts) 32
Sum
Source DF of Squares F Ratio Prob>F
X1: Table Speed 1 894.33 3.9942 0.0602
X2: Feed Rate 1 3497.20 15.6191 0.0009
X1: Table Speed* 1 4872.57 21.7618 0.0002
X2: Feed Rate
X3: Wheel Grit 1 12663.96 56.5595 <.0001
X1: Table Speed* 1 1838.76 8.2122 0.0099
X3: Wheel Grit
X4: Direction 1 315132.65 1407.4390 <.0001
X1: Table Speed* 1 1637.21 7.3121 0.0141
X4: Direction
X2: Feed Rate* 1 1972.71 8.8105 0.0079
X4: Direction
X1: Table Speed* 1 5895.62 26.3309 <.0001
X2: Feed Rate*
X4:Direction
X3: Wheel Grit* 1 3158.34 14.1057 0.0013
X4: Direction
X5: Batch 1 33653.91 150.3044 <.0001
X4: Direction* 1 1328.83 5.9348 0.0249
X5: Batch
Normal plot of
the effects
Non-significant effects should effectively follow an approximately normal distribution with
the same location and scale. Significant effects will vary from this normal distribution.
Therefore, another method of determining significant effects is to generate a normal
probability plot of all 31 effects. The effects that deviate substantially from the straight line fit
to the data are considered significant. Although this is a somewhat subjective criteria, it tends
to work well in practice. It is helpful to use both the numerical output from the fit and
graphical techniques such as the normal probability plot in deciding which terms to keep in
the model.
A normal probability plot of the effects is shown below. (To reduce the scale of the y-axis, the
largest two effects, X4: Direction and X5: Batch, are not shown on the plot. In addition, these
two effects were not used to compute the normal reference line.) The effects we consider to
be significant are labeled. In this case, we have arrived at the exact same 12 terms by looking
at the normal probability plot as we did from the stepwise regression.
Most of the effects cluster close to the center (zero) line and follow the fitted normal model
straight line. The effects that appear to be above or below the line by more than a small
amount are the same effects identified using the stepwise routine, with the exception of X1.
Some analysts prefer to include a main effect term when it has several significant interactions
even if the main effect term itself does not appear to be significant.
Model appears
to account for
most of the
variability
At this stage, the model appears to account for most of the variability in the response,
achieving an adjusted R
2
of 0.982. All the main effects are significant, as are six 2-factor
interactions and one 3-factor interaction. The only interaction that makes little physical sense
is the " X4: Direction*X5: Batch" interaction - why would the response using one batch of
material react differently when the batch is cut in a different direction as compared to another
batch of the same formulation?
However, before accepting any model, residuals need to be examined.
Step 4: Test the model assumptions using residual graphs (adjust and simplify as
needed)
Plot of
residuals
versus
predicted
responses
First we look at the residuals plotted versus the predicted responses.
The residuals appear to spread out more with larger values of predicted strength, which should
not happen when there is a common variance.
Next we examine the distribution of the residuals with a normal quantile plot, a box plot, a
histogram, and a run-order plot.
None of these plots appear to show typical normal residuals and the boxplot indicates that
there may be outliers.
Step 4 continued: Transform the data and fit the model again
Box-Cox
Transformation
We next look at whether we can model a transformation of the response variable and obtain
residuals with the assumed properties. We calculate an optimum Box-Cox transformation by
finding the value of that maximizes the negative log likelihood.
The optimum is found at = 0.2. A new Y: Strength variable is calculated using:
Fit model to
transformed
data
When the 12-effect model is fit to the transformed data, the "X4: Direction * X5: Batch"
interaction term is no longer significant. The 11-effect model fit is shown below, with
parameter estimates and p-values.
The fitted
model after
applying Box-
Cox
transformation
The 11-Effect Model Fit to Tranformed Response Data
Response: Y:NewStrength
Summary of Fit
RSquare 0.99041
Parameter
Effect Estimate p-value
Intercept 1917.115 <.0001
X1: Table Speed 5.777 0.0282
X2: Feed Rate 11.691 0.0001
X1: Table Speed* -14.467 <.0001
X2: Feed Rate
X3: Wheel Grit -21.649 <.0001
X1: Table Speed* 7.339 0.007
X3: Wheel Grit
X4: Direction -99.272 <.0001
X1: Table Speed* -7.188 0.0080
X4: Direction
X2: Feed Rate* -9.160 0.0013
X4: Direction
X1: Table Speed* 15.325 <.0001
X2: Feed Rate*
X4:Direction
X3: Wheel Grit* 12.965 <.0001
X4: Direction
X5: Batch -31.871 <.0001
Model has high
R
2
This model has a very large R
2
and adjusted R
2
. The residual plots (shown below) are quite a
bit better behaved than before
Residual plots
from model
with
transformed
response
The plot of the residuals versus the predicted values indicates that the transformation has
resolved the problem of increasing variace with increasing strength.
The normal probability plot, box plot, and the histogram of the residuals do not indicate any
serious violations of the model assumptions. The run sequence plot of the residuals does not
indicate any time dependent patterns.
Step 5. Answer the questions in your experimental objectives
Important main
effects and
interaction
effects
The magnitudes of the effect estimates show that "Direction" is by far the most important
factor. "Batch" plays the next most critical role, followed by "Wheel Grit". Then, there are
several important interactions followed by "Feed Rate". "Table Speed" plays a role in almost
every significant interaction term, but is the least important main effect on its own. Note that
large interactions can obscure main effects.
Plots of the
main effects
and significant
2-way
interactions
Plots of the main effects and the significant 2-way interactions are shown below.
Next, we plot 2-way interaction plot showing means for all combinations of levels for the two
factors.
The labels located in the diagonal spaces of the plot grid have two purposes. First, the label
indicates the factor associated with the x-axis for all plots in the same row. Second, the label
indicates the factor defining the two lines for plots in the same column.
For example, the plot labeled r*s contains averages for low and high levels of the rate variable
(x-axis) for both levels of speed. The blue line represents the low level of speed and the pink
line represents the high level of speed. The two lines in the r*s plot cross, indicating that there
is interaction between rate and speed. Parallel lines indicate that there is no interaction
between the two factors.
Optimal
Settings
Based on the analyses, we can select factor settings that maximize ceramic strength.
Translating from "-1" and "+1" back to the actual factor settings, we have: Table speed at "1"
or .125m/s; Down Feed Rate at "1" or .125 mm; Wheel Grit at "-1" or 140/170; and Direction
at "-1" or longitudinal.
Unfortunately, "Batch" is also a very significant factor, with the first batch giving higher
strengths than the second. Unless it is possible to learn what worked well with this batch, and
how to repeat it, not much can be done about this factor.
Comments
Analyses with
value of
Direction fixed
1. One might ask what an analysis of just the 2
4
factorial with "Direction" kept at -1 (i.e.,
longitudinal) would yield. This analysis turns out to have a very simple model; only
"Wheel Grit" and "Batch" are significant main effects and no interactions are
indicates
complex model
is needed only
for transverse
cut
significant.
If, on the other hand, we do an analysis of the 2
4
factorial with "Direction" kept at +1
(i.e., transverse), then we obtain a 7-parameter model with all the main effects and
interactions we saw in the 2
5
analysis, except, of course, any terms involving
"Direction".
So it appears that the complex model of the full analysis came from the physical
properties of a transverse cut, and these complexities are not present for longitudinal
cuts.
Half fraction
design
2. If we had assumed that three-factor and higher interactions were negligible before
experimenting, a half fraction design might have been chosen. In hindsight, we
would have obtained valid estimates for all main effects and two-factor interactions
except for X3 and X5, which would have been aliased with X1*X2*X4 in that half
fraction.
Natural log
transformation
3. Finally, we note that many analysts might prefer to adopt a natural logarithm
transformation (i.e., use ln Y) as the response instead of using a Box-Cox
transformation with an exponent of 0.2. The natural logarithm transformation
corresponds to an exponent of = 0 in the Box-Cox graph.
5.4.7.2. Fractional factorial example

A "Catapult" Fractional Factorial Experiment
A step-by-step
analysis of a
fractional
factorial
"catapult"
experiment
This experiment was conducted by a team of students on a catapult a table-top wooden
device used to teach design of experiments and statistical process control. The catapult has
several controllable factors and a response easily measured in a classroom setting. It has been
used for over 10 years in hundreds of classes.

Catapult
Description of Experiment: Response and Factors
The experiment
has five factors
that might
affect the
distance the
golf ball
travels
Purpose: To determine the significant factors that affect the distance the ball is thrown by the
catapult, and to determine the settings required to reach three different distances (30, 60 and
90 inches).
Response Variable: The distance in inches from the front of the catapult to the spot where the
ball lands. The ball is a plastic golf ball.
Number of observations: 20 (a 2
5-1
resolution V design with 4 center points).
Variables:
1. Response Variable Y = distance
2. Factor 1 = band height (height of the pivot point for the rubber bands levels were 2.25
and 4.75 inches with a centerpoint level of 3.5)
3. Factor 2 = start angle (location of the arm when the operator releases starts the forward
motion of the arm levels were 0 and 20 degrees with a centerpoint level of 10
degrees)
4. Factor 3 = rubber bands (number of rubber bands used on the catapult levels were 1
and 2 bands)
5. Factor 4 = arm length (distance the arm is extended levels were 0 and 4 inches with a
centerpoint level of 2 inches)
6. Factor 5 = stop angle (location of the arm where the forward motion of the arm is
stopped and the ball starts flying levels were 45 and 80 degrees with a centerpoint
level of 62 degrees)
Design matrix
and responses
(in run order)
The design matrix appears below in (randomized) run order.
distance height start bands length stop order
28.00 3.25 0 1 0 80 1
99.00 4 10 2 2 62 2
126.50 4.75 20 2 4 80 3
126.50 4.75 0 2 4 45 4
45.00 3.25 20 2 4 45 5
35.00 4.75 0 1 0 45 6
45.00 4 10 1 2 62 7
28.25 4.75 20 1 0 80 8
85.00 4.75 0 1 4 80 9
8.00 3.25 20 1 0 45 10
36.50 4.75 20 1 4 45 11
33.00 3.25 0 1 4 45 12
84.50 4 10 2 2 62 13
28.50 4.75 20 2 0 45 14
33.50 3.25 0 2 0 45 15
36.00 3.25 20 2 0 80 16
84.00 4.75 0 2 0 80 17
45.00 3.25 20 1 4 80 18
37.50 4 10 1 2 62 19
106.00 3.25 0 2 4 80 20
One discrete
factor
Note that four of the factors are continuous, and one number of rubber bands is discrete.
Due to the presence of this discrete factor, we actually have two different centerpoints, each
with two runs. Runs 7 and 19 are with one rubber band, and the center of the other factors,
while runs 2 and 13 are with two rubber bands and the center of the other factors.
Five
confirmatory
runs
After analyzing the 20 runs and determining factor settings needed to achieve predicted
distances of 30, 60 and 90 inches, the team was asked to conduct five confirmatory runs at
each of the derived settings.
Analysis of the Experiment
Step 1: Look at the data
Histogram,
box plot,
normal
probability
plot, and run
order plot of
the response
We start by plotting the data several ways to see if any trends or anomalies appear that would
not be accounted for by the models.
We can see the large spread of the data and a pattern to the data that should be explained by
the analysis. The run order plot does not indicate an obvious time sequence. The four
highlighted points in the run order plot are the center points in the design. Recall that runs 2
and 13 had two rubber bands and runs 7 and 19 had one rubber band. There may be a slight
aging of the rubber bands in that the second center point resulted in a distance that was a little
shorter than the first for each pair.
Plots of
responses
versus factor
columns
Next look at the plots of responses sorted by factor columns.
Several factors appear to change the average response level and most have large spread at
each of the levels.
Step 2: Create the theoretical model
The resolution
V design can
estimate main
effects and all
two-factor
interactions
With a resolution V design we are able to estimate all the main effects and all two-factor
interactions without worrying about confounding. Therefore, the initial model will have 16
terms the intercept term, the 5 main effects, and the 10 two-factor interactions.
Step 3: Create the actual model from the data
Variable
coding
Note we have used the orthogonally coded columns for the analysis, and have abbreviated the
factor names as follows:
Height (h) = band height
Start (s) = start angle
Bands (b) = number of rubber bands
Stop (e) = stop angle
Length (l) = arm length.
Trial model
with all main
factors and
two-factor
interactions
The results of fitting the trial model that includes all main factors and two-factor interactions
follow.
Source Estimate Std. Error t value Pr(>|t|)
--------- -------- ---------- ------- --------
Intercept 57.5375 2.9691 19.378 4.18e-05 ***
h 13.4844 3.3196 4.062 0.01532 *
s -11.0781 3.3196 -3.337 0.02891 *
b 19.4125 2.9691 6.538 0.00283 **
l 20.1406 3.3196 6.067 0.00373 **
e 12.0469 3.3196 3.629 0.02218 *
h*s -2.7656 3.3196 -0.833 0.45163
h*b 4.6406 3.3196 1.398 0.23467
h*l 4.7031 3.3196 1.417 0.22950
h*e 0.1094 3.3196 0.033 0.97529
s*b -3.1719 3.3196 -0.955 0.39343
s*l -1.1094 3.3196 -0.334 0.75502
s*e 2.6719 3.3196 0.805 0.46601
b*l 7.6094 3.3196 2.292 0.08365 .
b*e 2.8281 3.3196 0.852 0.44225
l*e 3.1406 3.3196 0.946 0.39768
Significance codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
Residual standard error: 13.28, based on 4 degrees of freedom
Multiple R-squared: 0.9709
Adjusted R-squared: 0.8619
F-statistic: 8.905, based on 15 and 4 degrees of freedom
p-value: 0.02375
Use p-values
and a normal
probability
plot to help
select
significant
effects
The model has a good R
2
value, but the fact that R
2
adjusted is considerably smaller indicates
that we undoubtedly have some terms in our model that are not significant. Scanning the
column of p-values (labeled Pr(>|t|)) for small values shows five significant effects at the 0.05
level and another one at the 0.10 level.
A normal probability plot of effects is a useful graphical tool to determine significant effects.
The graph below shows that there are nine terms in the model that can be assumed to be
noise. That would leave six terms to be included in the model. Whereas the output above
shows a p-value of 0.0836 for the interaction of Bands (b) and Length (l), the normal plot
suggests we treat this interaction as significant.
Refit using just
the effects that
appear to
matter
Remove the non-significant terms from the model and refit to produce the following analysis
of variance table.
Source Df Sum of Sq Mean Sq F value Pr(>F)
----------- -- --------- ------- ------- ------
Model 6 22148.55 3691.6
Total error 13 2106.99 162.1 22.77 3.5e-06
Lack-of-fit 11 1973.74 179.4
Pure error 2 133.25 66.6 2.69 0.3018
Residual standard error: 12.73 based on 13 degrees of freedom
p-value:
R
2
is OK and
there is no
significant
model "lack of
fit"
The R
2
and R
2
adjusted values are acceptable. The ANOVA table shows us that the model is
significant, and the lack-of-fit test is not significant. Parameter estimates are below.
--------- -------- ---------- ------- --------
Intercept 57.537 2.847 20.212 3.33e-11 ***
h 13.484 3.183 4.237 0.00097 ***
s -11.078 3.183 -3.481 0.00406 **
b 19.412 2.847 6.819 1.23e-05 ***
l 20.141 3.183 6.328 2.62e-05 ***
e 12.047 3.183 3.785 0.00227 **
b*l 7.609 3.183 2.391 0.03264 *
Significance codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
Step 4: Test the model assumptions using residual graphs (adjust and simplify as
needed)
Diagnostic
residual plots
To examine the assumption that the residuals are approximately normally distributed, are
independent, and have equal variances, we generate four plots of the residuals: a normal
probability plot, box plot, histogram, and a run-order plot of the residuals. In the run-order
plot, the highlighted points are the centerpoint values. Recall that run numbers 2 and 13 had
two rubber bands while run numbers 7 and 19 had only one rubber band.
The residuals do appear to have, at least approximately, a normal distributed.
Plot of
residuals
versus
predicted
values
Next we plot the residuals versus the predicted values.
There does not appear to be a pattern to the residuals. One observation about the graph, from
a single point, is that the model performs poorly in predicting a short distance. In fact, run
number 10 had a measured distance of 8 inches, but the model predicts -11 inches, giving a
residual of 19 inches. The fact that the model predicts an impossible negative distance is an
obvious shortcoming of the model. We may not be successful at predicting the catapult
settings required to hit a distance less than 25 inches. This is not surprising since there is only
one data value less than 28 inches. Recall that the objective is to achieve distances of 30, 60,
and 90 inches.
Plots of
residuals
versus the
factor
variables
Next we look at the residual values versus each of the factors.
The residual
graphs are not
ideal, although
the model
passes the
lack-of-fit test
Most of the residual graphs versus the factors appear to have a slight "frown" on the graph
(higher residuals in the center). This may indicate a lack of fit, or sign of curvature at the
centerpoint values. The lack- of-fit test, however, indicates that the lack of fit is not
significant.
Consider a
transformation
of the response
variable to see
if we can
obtain a better
model
At this point, since there are several unsatisfactory features of the model we have fit and the
resultant residuals, we should consider whether a simple transformation of the response
variable (Y = "Distance") might improve the situation.
There are at least two good reasons to suspect that using the logarithm of distance as the
response might lead to a better model.
1. A linear model fit to ln(Y) will always predict a positive distance when converted back
to the original scale for any possible combination of X factor values.
2. Physical considerations suggest that a realistic model for distance might require
quadratic terms since gravity plays a key role - taking logarithms often reduces the
impact of non-linear terms.
To see whether using ln(Y) as the response leads to a more satisfactory model, we return to
step 3.
Step 3a: Fit the full model using ln(Y) as the response
First a main
effects and
two-factor
interaction
model is fit to
the log
distance
responses
Proceeding as before, using the coded values of the factor levels and the natural logarithm of
distance as the response, we obtain the following parameter estimates.
--------- -------- ---------- ------- --------
(Intercept) 3.85702 0.06865 56.186 6.01e-07 ***
h 0.25735 0.07675 3.353 0.02849 *
s -0.24174 0.07675 -3.150 0.03452 *
b 0.34880 0.06865 5.081 0.00708 **
l 0.39437 0.07675 5.138 0.00680 **
e 0.26273 0.07675 3.423 0.02670 *
h*s -0.02582 0.07675 -0.336 0.75348
h*b -0.02035 0.07675 -0.265 0.80403
h*l -0.01396 0.07675 -0.182 0.86457
h*e -0.04873 0.07675 -0.635 0.55999
s*b 0.00853 0.07675 0.111 0.91686
s*l 0.06775 0.07675 0.883 0.42724
s*e 0.07955 0.07675 1.036 0.35855
b*l 0.01499 0.07675 0.195 0.85472
b*e -0.01152 0.07675 -0.150 0.88794
l*e -0.01120 0.07675 -0.146 0.89108
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
F-statistic: 5.845 based on 15 and 4 degrees of freedom
p-value: 0.0502
A simpler
model with just
main effects
has a
satisfactory fit
Examining the p-values of the 16 model coefficients, only the intercept and the 5 main effect
terms appear significant. Refitting the model with just these terms yields the following results.
Source Df Sum of Sq Mean Sq F value Pr(>F)
----------- -- --------- ------- ------- ------
Model 5 8.02079 1.60416 36.285 1.6e-07
Total error 14 0.61896 0.04421
Lack-of-fit 12 0.58980 0.04915
Pure error 2 0.02916 0.01458 3.371 0.2514
--------- -------- ---------- ------- --------
Intercept 3.85702 0.04702 82.035 < 2e-16 ***
h 0.25735 0.05257 4.896 0.000236 ***
s -0.24174 0.05257 -4.599 0.000413 ***
b 0.34880 0.04702 7.419 3.26e-06 ***
l 0.39437 0.05257 7.502 2.87e-06 ***
e 0.26273 0.05257 4.998 0.000195 ***
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
This is a simpler model than previously obtained in Step 3 (no interaction term). All the terms
are highly significant and there is no indication of a significant lack of fit.
We next look at the residuals for this new model fit.
Step 4a: Test the (new) model assumptions using residual graphs (adjust and simplify as
needed)
Normal
probability
plot, box plot,
histogram, and
run-order plot
of the residuals
The following normal plot, box plot, histogram and run-order plot of the residuals shows no
problems.
Residuals plotted versus run order again show a possible slight decreasing trend (rubber band
fatigue?).
Plot of
residuals
versus
predicted ln(Y)
values
A plot of the residuals versus the predicted ln(Y) values looks reasonable, although there might
be a tendency for the model to overestimate slightly for high predicted values.
Plot of
residuals
versus the
factor
variables
Next we look at the residual values versus each of the factors.
The residuals
for the main
effects model
(fit to natural
log of
distance) are
reasonably
well behaved
These plots still appear to have a slight "frown" on the graph (higher residuals in the center).
However, the model is generally an improvement over the previous model and will be
accepted as possibly the best that can be done without conducting a new experiment designed
to fit a quadratic model.
Step 5: Use the results to answer the questions in your experimental objectives
Final step:
Predict the
settings that
should be used
to obtain
desired
distances
Based on the analyses and plots, we can select factor settings that maximize the log-
transaformed distance. Translating from "-1", "0", and "+1" back to the actual factor settings,
we have: band height at "0" or 3.5 inches; start angle at "0" or 10 degrees; number of rubber
bands at "1" or 2 bands; arm length at "1" or 4 inches , and stop angle at "0" or 80 degrees.
"Confirmation"
runs were
successful
In the confirmatory runs that followed the experiment, the team was successful at hitting all
three targets, but did not hit them all five times. The model discovery and fitting process, as
illustrated in this analysis, is often an iterative process.
5.4.7.3. Response surface model example

Data Source
A CCD with
two
responses
This example uses experimental data published in Czitrom and Spagon,
(1997), Statistical Case Studies for Industrial Process Improvement. The
material is copyrighted by the American Statistical Association and the
Society for Industrial and Applied Mathematics, and is used with their
permission. Specifically, Chapter 15, titled "Elimination of TiN Peeling
During Exposure to CVD Tungsten Deposition Process Using Designed
Experiments", describes a semiconductor wafer processing experiment
(labeled Experiment 2).
Goal,
response
variables,
and factor
variables
The goal of this experiment was to fit response surface models to the two
responses, deposition layer Uniformity and deposition layer Stress, as a
function of two particular controllable factors of the chemical vapor
deposition (CVD) reactor process. These factors were Pressure (measured in
torr) and the ratio of the gaseous reactants H
2
and WF
6
(called H
2
/WF
6
).
The experiment also included an important third (categorical) response - the
presence or absence of titanium nitride (TiN) peeling. The third response has
been omitted in this example in order to focus on the response surface
aspects of the experiment.
To summarize, the goal is to obtain a response surface model for two
responses, Uniformity and Stress. The factors are: Pressure and H
2
/WF
6
.
Experiment Description
The design is
a 13-run
CCI design
with 3
centerpoints
The minimum and maximum values chosen for Pressure were 4 torr and 80
torr (0.5333 kPa and 10.6658 kPa). Although the international system of
units indicates that the standard unit for pressure is Pascal, or 1 N/m
2
, we
use torr to be consistent with the analysis appearing in the paper by Czitrom
and Spagon.
The minimum and maximum H
2
/WF
6
ratios were chosen to be 2 and 10.
Since response curvature, especially for Uniformity, was a distinct
possibility, an experimental design that allowed estimating a second order
(quadratic) model was needed. The experimenters decided to use a central
composite inscribed (CCI) design. For two factors, this design is typically
recommended to have 13 runs with 5 centerpoint runs. However, the
experimenters, perhaps to conserve a limited supply of wafer resources,
chose to include only 3 centerpoint runs. The design is still rotatable, but the
uniform precision property has been sacrificed.
Table
containing
the CCI
design points
and
experimental
responses
The table below shows the CCI design and experimental responses, in the
order in which they were run (presumably randomized). The last two
columns show coded values of the factors.
Run Pressure
H
2
/WF
6 Uniformity Stress
Coded
Pressure
Coded
H
2/
WF
6
1 80 6 4.6 8.04 1 0
2 42 6 6.2 7.78 0 0
3 68.87 3.17 3.4 7.58 0.71 -0.71
4 15.13 8.83 6.9 7.27 -0.71 0.71
5 4 6 7.3 6.49 -1 0
6 42 6 6.4 7.69 0 0
7 15.13 3.17 8.6 6.66 -0.71 -0.71
8 42 2 6.3 7.16 0 -1
9 68.87 8.83 5.1 8.33 0.71 0.71
10 42 10 5.4 8.19 0 1
11 42 6 5.0 7.90 0 0
Low values
of both
responses
are better
than high
Uniformity is calculated from four-point probe sheet resistance
measurements made at 49 different locations across a wafer. The value in the
table is the standard deviation of the 49 measurements divided by their
mean, expressed as a percentage. So a smaller value of Uniformity indicates
a more uniform layer - hence, lower values are desirable. The Stress
calculation is based on an optical measurement of wafer bow, and again
lower values are more desirable.
Analysis of DOE Data
Steps for
fitting a
response
surface
model
The steps for fitting a response surface (second-order or quadratic) model are
as follows:
1. Fit the full model to the first response.
2. Use stepwise regression, forward selection, or backward elimination to
identify important variables.
3. When selecting variables for inclusion in the model, follow the
hierarchy principle and keep all main effects that are part of significant
higher-order terms or interactions, even if the main effect p-value is
larger than you would like (note that not all analysts agree with this
principle).
4. Generate diagnostic residual plots (histograms, box plots, normal
plots, etc.) for the model selected.
5. Examine the fitted model plot, interaction plots, and ANOVA statistics
(R
2
, adjusted R
2
, lack-of-fit test, etc.). Use all these plots and statistics
to determine whether the model fit is satisfactory.
6. Use contour plots of the response surface to explore the effect of
changing factor levels on the response.
7. Repeat all the above steps for the second response variable.
8. After satisfactory models have been fit to both responses, you can
overlay the surface contours for both responses.
9. Find optimal factor settings.
Fitting a Model to the Uniformity Response, Simplifying the Model and
Checking Residuals
Fit full
quadratic
model to
Uniformity
response
------ -------- ---------- ------- --------
Intercept 5.86613 0.41773 14.043 3.29e-05
Pressure -1.90967 0.36103 -5.289 0.00322
H2/WF6 -0.22408 0.36103 -0.621 0.56201
Pressure*H2/WF6 1.68617 0.71766 2.350 0.06560
Pressure^2 0.13373 0.60733 0.220 0.83442
H2/WF6^2 0.03373 0.60733 0.056 0.95786
p-value: 0.0278
Stepwise
regression
for
Uniformity
Start: AIC=-3.79
Model: Uniformity ~ Pressure + H2/WF6 + Pressure*H2/WF6 +
Pressure^2 + H2/WF6^2
Step 1: Remove H2/WF6^2, AIC=-5.79
Model: Uniformity ~ Pressure + H2/WF6 + Pressure*H2/WF6 +
Pressure^2
Step 2: Remove Pressure^2, AIC=-7.69
Model: Uniformity ~ Pressure + H2/WF6 + Pressure*H2/WF6
Step 3: Remove H2/WF6, AIC=-8.88
Model: Uniformity ~ Pressure + Pressure*H2/WF6
The stepwise routine selects a model containing the intercept, Pressure, and
the interaction term. However, many statisticians do not think an interaction
term should be included in a model unless both main effects are also
included. Thus, we will use the model from Step 2 that included Pressure,
H2/WF6, and the interaction term. Interaction plots confirm the need for an
interaction term in the model.
Analysis of
model
selected by
stepwise
regression
for
Uniformity
Source DF Sum of Sq Mean Sq F value Pr(>F)
------ -- --------- ------- ------- ------
Model 3 17.739 5.9130 15.66 0.0017
Total error 7 2.643 0.3776
Lack-of-fit 5 1.4963 0.2993 0.52 0.7588
Pure error 2 1.1467 0.5734
------ -------- ---------- ------- --------
Intercept 5.9273 0.1853 31.993 7.54e-09
Pressure -1.9097 0.3066 -6.228 0.000433
H2/WF6 -0.2241 0.3066 -0.731 0.488607
Pressure*H2/WF6 1.6862 0.6095 2.767 0.027829
A contour plot and perspective plot of Uniformity provide a visual display of
the response surface.

Residual
plots
We perform a residuals analysis to validate the model assumptions. We
generate a normal plot, a box plot, a histogram and a run-order plot of the
residuals.
The residual plots do not indicate problems with the underlying assumptions.
Conclusions
from the
analysis
From the above output, we make the following conclusions.
The R
2
is reasonable for fitting Uniformity (well known to be a
difficult response to model).
The lack-of-fit test is not significant (very small "Prob > F " would
indicate a lack of fit).
The residual plots do not reveal any major violations of the underlying
assumptions.
The interaction plot shows why an interaction term is needed (parallel
lines would suggest no interaction).
Fitting a Model to the Stress Response, Simplifying the Model and
Checking Residuals
Fit full
quadratic
model to
Stress
------ -------- --------- ------- --------
Intercept 8.056791 0.179561 44.869 1.04e-07
Pressure 0.735933 0.038524 19.103 7.25e-06
H2/WF6 0.852099 0.198192 4.299 0.00772
Pressure*H2/WF6 0.069431 0.076578 0.907 0.40616
response
Pressure^2 -0.528848 0.064839 -8.156 0.00045
H2/WF6^2 -0.007414 0.004057 -1.827 0.12722
p-value: 3.358e-05
Stepwise
regression
for Stress
Start: AIC=-53.02
Model: Stress ~ Pressure + H2/WF6 + Pressure*H2/WF6 +
Pressure^2 + H2/WF6^2
Step 1: AIC=-53.35
Model: Stress ~ Pressure + H2/WF6 + Pressure^2 + H2/WF6^2
The stepwise routine identifies a model containing the intercept, the main
effects, and both squared terms. However, the fit of the full quadratic model
indicates that neither the H2/WF6 squared term nor the interaction term are
significant. A comparison of the full model and the model containing just the
main effects and squared pressure terms indicates that there is no significant
difference between the two models.
Model 1: Stress ~ Pressure + H2/WF6 + Pressure^2
Model 2: Stress ~ Pressure + H2/WF6 + Pressure^2 +
Pressure*H2/WF6 + H2/WF6^2
------ -- --------- ------- ------- -------
Model 1 2 0.024802 0.01240 2.08 0.22
Model 2 5 0.029804 0.00596
In addition, interaction plots do not indicate any significant interaction.
Thus, we will proceed with the model containing main effects and the
squared pressure term.
The fact that the stepwise procedure selected a model for Stress containing a
term that was not significant indicates that all output generated by statistical
software should be carefully examined. In this case, the stepwise procedure
identified the model with the lowest AIC (Akaike information criterion), but
did not take into account contributions by individual terms. Other software
using a different criteria may identify a different model, so it is important to
understand the algorithms being used.
Analysis of
reduced
model for
Stress
------ -- --------- ------- ------- -------
Model 3 3.5454 1.1818 151.5 9.9e-07
Total error 7 0.0546 0.0078
Lack-of-fit 5 0.032405 0.00065 0.58 0.73
Pure error 2 0.022200 0.01110
------ -------- ---------- ------- --------
Intercept 7.73410 0.03715 208.185 1.56e-14
Pressure 0.73593 0.04407 16.699 6.75e-07
H2/WF6 0.49686 0.04407 11.274 9.65e-06
Pressure^2 -0.49426 0.07094 -6.967 0.000218
A contour plot and perspective plot of Stress provide a visual representation
of the response surface.

Residual
plots
We perform a residuals analysis to validate the model by generating a run-
order plot, box plot, histogram, and normal probability plot of the residuals.
The residual plots do not indicate any major violations of the underlying
assumptions.
Conclusions From the above output, we make the following conclusions.
The R
2
is very good for fitting Stress.
The lack-of-fit test is not significant (very small "Prob > F " would
indicate a lack of fit).
The residual plots do not reveal any major violations of the underlying
assumptions.
The nearly parallel lines in the interaction plots show why an
interaction term is not needed.
Response Surface Contours for Both Responses
Overlay
contour plots
We overlay the contour plots for the two responses to visually compare the
surfaces over the region of interest.
Summary
Final
response
surface
models
The response surface models fit to (coded) Uniformity and Stress were:
Uniformity = 5.93 - 1.91*Pressure - 0.22*H
2
/WF
6
+
1.70*Pressure*H
2
/WF
6
Stress = 7.73 + 0.74*Pressure + 0.50*H
2
/WF
6
- 0.49*Pressure
2
Trade-offs
are often
needed for
multiple
responses
The models and the corresponding contour plots show that trade-offs have to
be made when trying to achieve low values for both Uniformity and Stress
since a high value of Pressure is good for Uniformity while a low value of
Pressure is good for Stress. While low values of H
2
/WF
6
are good for both
responses, the situation is further complicated by the fact that the Peeling
response (not considered in this analysis) was unacceptable for values of
H
2
/WF
6
below approximately 5.
Uniformity
was chosen
as more
important
In this case, the experimenters chose to focus on optimizing Uniformity
while keeping H
2
/WF
6
at 5. That meant setting Pressure at 80 torr.
Confirmation
runs
validated the
model
A set of 16 verification runs at the chosen conditions confirmed that all
goals, except those for the Stress response, were met by this set of process
settings.
projections
5.5. Advanced topics

Contents
of
"Advanced
Topics"
section
This section builds on the basics of DOE described in the
preceding sections by adding brief or survey descriptions of a
selection of useful techniques. Subjects covered are:
1. When classical designs don't work
2. Computer-aided designs
1. D-Optimal designs
2. Repairing a design
3. Optimizing a Process
2. Confidence region for search path
3. Choosing the step length
4. Optimization when there is adequate
quadratic fit
5. Effect of sampling error on optimal solution
6. Optimization subject to experimental region
constraints
2. Desirability function approach
3. Mathematical programming approach
4. Mixture designs
1. Mixture screening designs
2. Simplex-lattice designs
3. Simplex-Centroid designs
4. Constrained mixture designs
5. Treating mixture and process variables together
5. Nested variation
6. Taguchi designs
7. John's 3/4 fractional factorial designs
8. Small composite designs
9. An EDA approach to experimental design
2. DOE scatter plot
3. DOE mean plot
4. Interaction effects matrix plot
5. Block plot
6. DOE Youden plot
7. |Effects| plot
8. Half-normal probability plot
10. DOE contour plot
5.5.1. What if classical designs don't work?

5.5.1. What if classical designs don't work?
Reasons
designs
don't work
Most experimental situations call for standard designs that can
be constructed with many statistical software packages.
Standard designs have assured degrees of precision,
orthogonality, and other optimal properties that are important
for the exploratory nature of most experiments. In some
situations, however, standard designs are not appropriate or
are impractical. These may include situations where
1. The required blocking structure or blocking size of the
experimental situation does not fit into a standard
blocked design
2. Not all combinations of the factor settings are feasible,
or for some other reason the region of experimentation is
constrained or irregularly shaped.
3. A classical design needs to be 'repaired'. This can
happen due to improper planning with the original
design treatment combinations containing forbidden or
unreachable combinations that were not considered
before the design was generated.
4. A nonlinear model is appropriate.
5. A quadratic or response surface design is required in the
presence of qualitative factors.
6. The factors in the experiment include both components
of a mixture and other process variables.
7. There are multiple sources of variation leading to nested
or hierarchical data structures and restrictions on what
can be randomized.
8. A standard fractional factorial design requires too many
treatment combinations for the given amount of time
and/or resources.
Computer-
aided
designs
When situations such as the above exist, computer-aided
designs are a useful option. In some situations, computer-
aided designs are the only option an experimenter has.
5.5.2. What is a computer-aided design?

Computer-
aided
designs are
generated
by a
computer
algorithm
and
constructed
to be
optimal for
certain
models
according
to one of
many types
of
optimality
criteria
Designs generated from a computer algorithm are referred to
as computer-aided designs. Computer-aided designs are
experimental designs that are generated based on a particular
optimality criterion and are generally 'optimal' only for a
specified model. As a result, they are sometimes referred to as
optimal designs and generally do not satisfy the desirable
properties such as independence among the estimators that
standard classical designs do. The design treatment runs that
are generated by the algorithms are chosen from an overall
candidate set of possible treatment combinations. The
candidate set consists of all the possible treatment
combinations that one wishes to consider in an experiment.
Optimality
critieria
There are various forms of optimality criteria that are used to
select the points for a design.
D-
Optimality
One popular criterion is D-optimality, which seeks to
maximize |X'X|, the determinant of the information matrix
X'X of the design. This criterion results in minimizing the
generalized variance of the parameter estimates based on a
pre-specified model.
A-
Optimality
Another criterion is A-optimality, which seeks to minimize the
trace of the inverse of the information matrix. This criterion
results in minimizing the average variance of the parameter
estimates based on a pre-specified model.
G-
Optimality
A third criterion is G-optimality, which seeks to minimize the
maximum prediction variance, i.e., minimize max. [d=x'(X'X)
-
1
x], over a specified set of design points.
V-
Optimality
A fourth criterion is V-optimality, which seeks to minimize
the average prediction variance over a specified set of design
points.
Optimality
of a given
design is
model
dependent
Since the optimality criterion of most computer-aided designs
is based on some function of the information matrix, the
'optimality' of a given design is model dependent. That is, the
experimenter must specify a model for the design and the
final number of design points desired before the 'optimal'
design' can be generated. The design generated by the
computer algorithm is 'optimal' only for that model.
5.5.2.1. D-Optimal designs

D-optimal
designs
are often
used when
classical
designs do
not apply
D-optimal designs are one form of design provided by a
computer algorithm. These types of computer-aided designs
are particularly useful when classical designs do not apply.
Unlike standard classical designs such as factorials and
fractional factorials, D-optimal design matrices are usually not
orthogonal and effect estimates are correlated.
These types of designs are always an option regardless of the
type of model the experimenter wishes to fit (for example,
first order, first order plus some interactions, full quadratic,
cubic, etc.) or the objective specified for the experiment (for
example, screening, response surface, etc.). D-optimal designs
are straight optimizations based on a chosen optimality
criterion and the model that will be fit. The optimality
criterion used in generating D-optimal designs is one of
maximizing |X'X|, the determinant of the information matrix
X'X.
This optimality criterion results in minimizing the generalized
variance of the parameter estimates for a pre-specified model.
As a result, the 'optimality' of a given D-optimal design is
model dependent. That is, the experimenter must specify a
model for the design before a computer can generate the
specific treatment combinations. Given the total number of
treatment runs for an experiment and a specified model, the
computer algorithm chooses the optimal set of design runs
from a candidate set of possible design treatment runs. This
candidate set of treatment runs usually consists of all possible
combinations of various factor levels that one wishes to use in
the experiment.
In other words, the candidate set is a collection of treatment
combinations from which the D-optimal algorithm chooses
the treatment combinations to include in the design. The
computer algorithm generally uses a stepping and exchanging
process to select the set of treatment runs.
Note: There is no guarantee that the design the computer
generates is actually D-optimal.
D-optimal The reasons for using D-optimal designs instead of standard
designs
are useful
when
resources
are limited
or there
are
constraints
on factor
settings
classical designs generally fall into two categories:
1. standard factorial or fractional factorial designs require
too many runs for the amount of resources or time
allowed for the experiment
2. the design space is constrained (the process space
contains factor settings that are not feasible or are
impossible to run).
Example Suppose an industrial process has three design variables (k =
3), and engineering judgment specifies the following model as
an appropriate representation of the process.
The levels being considered by the researcher are (coded)
X1: 5 levels (-1, -0.5, 0, 0.5, 1)
X2: 2 levels (-1, 1)
X3: 2 levels (-1, 1)
Due to resource limitations, only n = 12 data points can be
collected.
Create the
candidate
set
Given the experimental specifications, the first step in
generating the design is to create a candidate set of points.
The candidate set is a data table with a row for each point
(run) to be considered for the design, often a full factorial. For
our problem, the candidate set is a full factorial in all factors
containing 5*2*2 = 20 possible design runs.
Table
containing
the
candidate
set
TABLE 5.1 Candidate Set for Variables X1, X2,
X3
X1 X2 X3
-1 -1 -1
-1 -1 +1
-1 +1 -1
-1 +1 +1
-0.5 -1 -1
-0.5 -1 +1
-0.5 +1 -1
-0.5 +1 +1
0 -1 -1
0 -1 +1
0 +1 -1
0 +1 +1
0.5 -1 -1
0.5 -1 +1
0.5 +1 -1
0.5 +1 +1
+1 -1 -1
+1 -1 +1
+1 +1 -1
+1 +1 +1
Generating
a D-
optimal
design
D-optimal designs maximize the D-efficiency, which is a
volume criterion on the generalized variance of the parameter
estimates. The D-efficiency of the standard fractional factorial
is 100 %, but it is not possible to achieve 100 % D-efficiency
when pure quadratic terms such as (X1)
2
are included in the
model.
The D-efficiency values are a function of the number of points
in the design, the number of independent variables in the
model, and the maximum standard error for prediction over
the design points. The best design is the one with the highest
D-efficiency. Other reported efficiencies (e.g. A, G, I) help
choose an optimal design when various models produce
similar D-efficiencies.
D-optimal
design
The D-optimal design (D=0.6825575, A=2.2, G=1, I=4.6625)
using 12 runs is shown in Table 5.2 in standard order. The
standard error of prediction is also shown. The design runs
should be randomized before the treatment combinations are
executed.
TABLE 5.2 Final D-optimal Design
X1 X2 X3 OptStdPred
-1 -1 -1 0.645497
-1 -1 +1 0.645497
-1 +1 -1 0.645497
-1 +1 +1 0.645497
0 -1 -1 0.645497
0 -1 +1 0.645497
0 +1 -1 0.645497
0 +1 +1 0.645497
+1 -1 -1 0.645497
+1 -1 +1 0.645497
+1 +1 -1 0.645497
+1 +1 +1 0.645497
Software
note
Software packages may have different procedures for
generating D-optimal designs, so the final design may be
different depending on the software packaged being used.
5.5.2.2. Repairing a design

5.5.2.2. Repairing a design
Repair or
augment
classical
designs
Computer-aided designs are helpful in either repairing or
augmenting a current experimental design. They can be used
to repair a 'broken' standard classical design.
Original
design
matrix
may
contain
runs that
were lost
or
impossible
to acieve
There may be situations in which, due to improper planning or
other issues, the original design matrix contains forbidden or
unreachable combinations of the factor settings. A computer-
aided design (for example a D-optimal design) can be used to
'replace' those runs from the original design that were
unattainable. The runs from the original design that are
attainable are labeled as 'inclusion' runs and will be included
in the final computer-aided design.
Computer-
aided
design can
generate
additional
attainable
runs
Given a pre-specified model, the computer-aided design can
generate the additional attainable runs that are necessary in
order to estimate the model of interest. As a result, the
computer-aided design is just replacing those runs in the
original design that were unattainable with a new set of runs
that are attainable, and which still allows the experimenter to
obtain information regarding the factors from the experiment.
Properties
of this
final
design
may not
compare
with those
of the
original
design
The properties of this final design will probably not compare
with those of the original design and there may exist some
correlation among the estimates. However, instead of not
being able to use any of the data for analysis, generating the
replacement runs from a computer-aided design, a D-optimal
design for example, allows one to analyze the data.
Furthermore, computer-aided designs can be used to augment
a classical design with treatment combinations that will break
alias chains among the terms in the model or permit the
estimation of curvilinear effects.
5.5.3. How do you optimize a process?

How do you determine the optimal region to run a
process?
Often the
primary
DOE goal is
to find the
operating
conditions
that
maximize (or
minimize)
the system
responses
The optimal region to run a process is usually determined
after a sequence of experiments has been conducted and a
series of empirical models obtained. In many engineering
and science applications, experiments are conducted and
empirical models are developed with the objective of
improving the responses of interest. From a mathematical
point of view, the objective is to find the operating
conditions (or factor levels) X
1
, X
2
, ..., X
k
that maximize or
minimize the r system response variables Y
1
, Y
2
, ..., Y
r
. In
experimental optimization, different optimization techniques
are applied to the fitted response equations .
Provided that the fitted equations approximate adequately
the true (unknown) system responses, the optimal operating
conditions of the model will be "close" to the optimal
operating conditions of the true system.
The DOE
approach to
optimization
The experimental optimization of response surface models
differs from classical optimization techniques in at least
three ways:
Find
approximate
(good)
models and
iteratively
search for
(near)
optimal
operating
conditions
1. Experimental optimization is an iterative process; that
is, experiments conducted in one set of experiments
result in fitted models that indicate where to search for
improved operating conditions in the next set of
experiments. Thus, the coefficients in the fitted
equations (or the form of the fitted equations) may
change during the optimization process. This is in
contrast to classical optimization in which the
functions to optimize are supposed to be fixed and
given.
Randomness
(sampling
variability)
affects the
final
2. The response models are fit from experimental data
that usually contain random variability due to
uncontrollable or unknown causes. This implies that
an experiment, if repeated, will result in a different
fitted response surface model that might lead to
answers and
should be
taken into
account
different optimal operating conditions. Therefore,
sampling variability should be considered in
experimental optimization.
In contrast, in classical optimization techniques the
functions are deterministic and given.
Optimization
process
requires
input of the
experimenter
3. The fitted responses are local approximations,
implying that the optimization process requires the
input of the experimenter (a person familiar with the
process). This is in contrast with classical optimization
which is always automated in the form of some
computer algorithm.
5.5.3.1. Single response case

Optimizing
of a single
response
usually
starts with
line searches
in the
direction of
maximum
improvement
The experimental optimization of a single response is usually conducted in two
phases or steps, following the advice of Box and Wilson. The first phase consists of
a sequence of line searches in the direction of maximum improvement. Each search
in the sequence is continued until there is evidence that the direction chosen does not
result in further improvements. The sequence of line searches is performed as long
as there is no evidence of lack of fit for a simple first-order model of the form
If there is
lack of fit
for linear
models,
quadratic
models are
tried next
The second phase is performed when there is lack of linear fit in Phase I, and
instead, a second-order or quadratic polynomial regression model of the general
form
is fit. Not all responses will require quadratic fit, and in such cases Phase I is
stopped when the response of interest cannot be improved any further. Each phase is
explained and illustrated in the next few sections.
"Flowchart"
for two
phases of
experimental
optimization
The following is a flow chart showing the two phases of experimental optimization.
FIGURE 5.1: The Two Phases of Experimental Optimization
5.5.3.1.1. Single response: Path of steepest ascent

Starting at
the current
operating
conditions, fit
a linear
model
If experimentation is initially performed in a new, poorly understood production
process, chances are that the initial operating conditions X
1
, X
2
, ...,X
k
are located
far from the region where the factors achieve a maximum or minimum for the
response of interest, Y. A first-order model will serve as a good local
approximation in a small region close to the initial operating conditions and far
from where the process exhibits curvature. Therefore, it makes sense to fit a
simple first-order (or linear polynomial) model of the form:

Experimental strategies for fitting this type of model were discussed earlier.
Usually, a 2
k-p
fractional factorial experiment is conducted with repeated runs at
the current operating conditions (which serve as the origin of coordinates in
orthogonally coded factors).
Determine the
directions of
steepest
ascent and
continue
experimenting
until no
further
improvement
occurs - then
iterate the
process
The idea behind "Phase I" is to keep experimenting along the direction of steepest
ascent (or descent, as required) until there is no further improvement in the
response. At that point, a new fractional factorial experiment with center runs is
conducted to determine a new search direction. This process is repeated until at
some point significant curvature in is detected. This implies that the operating
conditions X
1
, X
2
, ...,X
k
are close to where the maximum (or minimum, as
required) of Y occurs. When significant curvature, or lack of fit, is detected, the
experimenter should proceed with "Phase II". Figure 5.2 illustrates a sequence of
line searches when seeking a region where curvature exists in a problem with 2
factors (i.e., k=2).

FIGURE 5.2: A Sequence of Line Searches for a 2-Factor Optimization
Problem
Two main
decisions:
search
direction and
length of step
There are two main decisions an engineer must make in Phase I:
1. determine the search direction;
2. determine the length of the step to move from the current operating
conditions.
Figure 5.3 shows a flow diagram of the different iterative tasks required in Phase
I. This diagram is intended as a guideline and should not be automated in such a
way that the experimenter has no input in the optimization process.
Flow chart of
iterative
search
process

FIGURE 5.3: Flow Chart for the First Phase of the Experimental
Optimization Procedure
Procedure for Finding the Direction of Maximum Improvement
The direction
of steepest
ascent is
Suppose a first-order model (like above) has been fit and provides a useful
approximation. As long as lack of fit (due to pure quadratic curvature and
interactions) is very small compared to the main effects, steepest ascent can be
determined by
the gradient
of the fitted
model
attempted. To determine the direction of maximum improvement we use
1. the estimated direction of steepest ascent, given by the gradient of , if the
objective is to maximize Y;
2. the estimated direction of steepest descent, given by the negative of the
gradient of , if the objective is to minimize Y.
The direction
of steepest
ascent
depends on
the scaling
convention -
equal
variance
scaling is
recommended
The direction of the gradient, g, is given by the values of the parameter estimates,
that is, g' = (b
1
, b
2
, ..., b
k
). Since the parameter estimates b
1
, b
2
, ..., b
k
depend on
the scaling convention for the factors, the steepest ascent (descent) direction is also
scale dependent. That is, two experimenters using different scaling conventions
will follow different paths for process improvement. This does not diminish the
general validity of the method since the region of the search, as given by the signs
of the parameter estimates, does not change with scale. An equal variance scaling
convention, however, is recommended. The coded factors x
i
, in terms of the
factors in the original units of measurement, X
i
, are obtained from the relation
This coding convention is recommended since it provides parameter estimates that
are scale independent, generally leading to a more reliable search direction. The
coordinates of the factor settings in the direction of steepest ascent, positioned a
distance from the origin, are given by:

Solution is a
simple
equation
This problem can be solved with the aid of an optimization solver (e.g., like the
solver option of a spreadsheet). However, in this case this is not really needed, as
the solution is a simple equation that yields the coordinates

Equation can
be computed
for increasing
values of
An engineer can compute this equation for different increasing values of and
obtain different factor settings, all on the steepest ascent direction.
To see the details that explain this equation, see Technical Appendix 5A.
Example: Optimization of a Chemical Process
Optimization
by search
example
It has been concluded (perhaps after a factor screening experiment) that the yield
(Y, in %) of a chemical process is mainly affected by the temperature (X
1
, in C)
and by the reaction time (X
2
, in minutes). Due to safety reasons, the region of
operation is limited to
Factor levels The process is currently run at a temperature of 200 C and a reaction time of 200
minutes. A process engineer decides to run a 2
2
full factorial experiment with
factor levels at
factor low center high
X
1
170 200 230
X
2
150 200 250
Orthogonally
coded factors
Five repeated runs at the center levels are conducted to assess lack of fit. The
orthogonally coded factors are
Experimental
results
The experimental results were:
x
1
x
2
X
1
X
2
Y (= yield)
-1 -1 170 150 32.79
+1 -1 230 150 24.07
-1 +1 170 250 48.94
+1 +1 230 250 52.49
0 0 200 200 38.89
0 0 200 200 48.29
0 0 200 200 29.68
0 0 200 200 46.50
0 0 200 200 44.15
ANOVA table The corresponding ANOVA table for a first-order polynomial model is
SUM OF MEAN F
SOURCE SQUARES DF SQUARE VALUE PROB>F
MODEL 503.3035 2 251.6517 4.7972 0.0687
CURVATURE 8.2733 1 8.2733 0.1577 0.7077
RESIDUAL 262.2893 5 52.4579
LACK OF FIT 37.6382 1 37.6382 0.6702 0.4590
PURE ERROR 224.6511 4 56.1628
COR TOTAL 773.8660 8
Resulting
model
It can be seen from the ANOVA table that there is no significant lack of linear fit
due to an interaction term and there is no evidence of curvature. Furthermore,
there is evidence that the first-order model is significant. The resulting model (in
the coded variables) is
Diagnostic
checks
The usual diagnostic checks show conformance to the regression assumptions,
although the R
2
value is not very high: R
2
= 0.6504.
Determine
level of
factors for
next run
using
direction of
steepest
ascent
To maximize , we use the direction of steepest ascent. The engineer selects = 1
since a point on the steepest ascent direction one unit (in the coded units) from the
origin is desired. Then from the equation above for the predicted Y response, the
coordinates of the factor levels for the next run are given by:
and
This means that to improve the process, for every (-0.1152)(30) = -3.456 C that
temperature is varied (decreased), the reaction time should be varied by
(0.9933)(50) = 49.66 minutes.
===========================================================
Technical Appendix 5A: finding the factor settings on the steepest ascent
direction a specified distance from the origin
Details of
how to
determine the
path of
steepest
ascent
The problem of finding the factor settings on the steepest ascent/descent direction
that are located a distance from the origin is given by the optimization problem,

Solve using a
Lagrange
multiplier
approach
To solve it, use a Lagrange multiplier approach. First, add a penalty for
solutions not satisfying the constraint (since we want a direction of steepest
ascent, we maximize, and therefore the penalty is negative). For steepest descent
we minimize and the penalty term is added instead.
Compute the partials and equate them to zero
Solve two
equations in
two unknowns
These two equations have two unknowns (the vector x and the scalar ) and thus
can be solved yielding the desired solution:
or, in non-vector notation:
Multiples of
the direction
of the
gradient
From this equation we can see that any multiple of the direction of the gradient
(given by ) will lead to points on the steepest ascent direction. For steepest
descent, use instead -b
i
in the numerator of the equation above.
5.5.3.1.2. Single response: Confidence region for search path

"Randomness"
means that the
steepest
ascent
direction is
just an
estimate and it
is possible to
construct a
confidence
"cone" around
this direction
estimate
The direction given by the gradient g' = (b
0
, b
2
, ... , b
k
) constitutes only a single
(point) estimate based on a sample of N runs. If a different set of N runs were
conducted, these would provide different parameter estimates, which in turn would
give a different gradient. To account for this sampling variability, Box and Draper
gave a formula for constructing a "cone" around the direction of steepest ascent that
with certain probability contains the true (unknown) system gradient given by
. The width of the confidence cone is useful to assess how reliable
an estimated search direction is.
Figure 5.4 shows such a cone for the steepest ascent direction in an experiment with
two factors. If the cone is so wide that almost every possible direction is inside the
cone, an experimenter should be very careful in moving too far from the current
operating conditions along the path of steepest ascent or descent. Usually this will
happen when the linear fit is quite poor (i.e., when the R
2
value is low). Thus,
plotting the confidence cone is not so important as computing its width.
If you are interested in the details on how to compute such a cone (and its width),
see Technical Appendix 5B.
Graph of a
confidence
cone for the
steepest
ascent
direction
FIGURE 5.4: A Confidence Cone for the Steepest Ascent Direction in an
Experiment with 2 Factors
=============================================================
Technical Appendix 5B: Computing a Confidence Cone on the Direction of
Steepest Ascent
Details of how
to construct a
confidence
cone for the
direction of
steepest
ascent
Suppose the response of interest is adequately described by a first-order polynomial
model. Consider the inequality

with
C
jj
is the j-th diagonal element of the matrix (X'X)
-1
(for j = 1, ..., k these values are
all equal if the experimental design is a 2
k-p
factorial of at least Resolution III), and
X is the model matrix of the experiment (including columns for the intercept and
second-order terms, if any). Any operating condition with coordinates x' = (x
1
, x
2
,
..., x
k
) that satisfies this inequality generates a direction that lies within the 100(1-
) % confidence cone of steepest ascent if
or inside the 100(1- ) % confidence cone of steepest descent if
Inequality
defines a cone
The inequality defines a cone with the apex at the origin and center line located
along the gradient of .
A measure of
goodnes of fit:
A measure of "goodness" of a search direction is given by the fraction of directions
excluded by the 100(1- ) % confidence cone around the steepest ascent/descent
direction (see Box and Draper, 1987) which is given by:
with T
k-1
() denoting the complement of the Student's t distribution function with k-1
degrees of freedom (that is, T
k-1
(x) = P(t
k-1
x)) and F
, k-1, n-p
denotes an
percentage point of the F distribution with k-1 and n-p degrees of freedom, with n-p
denoting the error degrees of freedom. The value of represents the fraction of
directions included by the confidence cone. The smaller is, the wider the cone is,
with . Note that the inequality equation and the "goodness measure"
equation are valid when operating conditions are given in coded units.
Example: Computing
Compute
from ANOVA
table and C
jj
From the ANOVA table in the chemical experiment discussed earlier
since C
jj
= 1/4 (j=2,3) for a 2
2
factorial. The fraction of directions excluded by a 95
% confidence cone in the direction of steepest ascent is:
Compute
Conclusions
for this
example
since F
0.05,1,6
= 5.99. Thus 71 % of the possible directions from the current
operating point are excluded with 95 % confidence. This is useful information that
can be used to select a step length. The smaller is, the shorter the step should be,
as the steepest ascent direction is less reliable. In this example, with high confidence,
the true steepest ascent direction is within this cone of 29 % of possible directions.
For k=2, 29 % of 360
o
= 104.4
o
, so we are 95 % confident that our estimated
steepest ascent path is within plus or minus 52.2
o
of the true steepest path. In this
case, we should not use a large step along the estimated steepest ascent path.
5.5.3.1.3. Single response: Choosing the step length

5.5.3.1.3. Single response: Choosing the step
length
A procedure
for choosing
how far
along the
direction of
steepest
ascent to go
for the next
trial run
Once the search direction is determined, the second decision
needed in Phase I relates to how far in that direction the
process should be "moved". The most common procedure for
selecting a step length is based on choosing a step size in
one factor and then computing step lengths in the other
factors proportional to their parameter estimates. This
provides a point on the direction of maximum improvement.
The procedure is given below. A similar approach is
obtained by choosing increasing values of in
.
However, the procedure below considers the original units of
measurement which are easier to deal with than the coded
"distance" .
Procedure: selection of step length
Procedure
for selecting
the step
length
The following is the procedure for selecting the step length.
1. Choose a step length X
j
(in natural units of
measurement) for some factor j. Usually, factor j is
chosen to be the one engineers feel more comfortable
varying, or the one with the largest |b
j
|. The value of
X
j
can be based on the width of the confidence cone
around the steepest ascent/descent direction. Very
wide cones indicate that the estimated steepest
ascent/descent direction is not reliable, and thus X
j
should be small. This usually occurs when the R
2
value is low. In such a case, additional experiments
can be conducted in the current experimental region to
obtain a better model fit and a better search direction.
2. Transform to coded units:
with s
j
denoting the scale factor used for factor j (e.g.,
s
j
= range
j
/2).
3. Set for all other factors i.
4. Transform all the x
i
's to natural units: X
i
= (
x
i
)(s
i
).
Example: Step Length Selection.
An example
of step
length
selection
The following is an example of the step length selection
procedure.
For the chemical process experiment described
previously, the process engineer selected X
2
= 50
minutes. This was based on process engineering
considerations. It was also felt that X
2
= 50 does not
move the process too far away from the current region
of experimentation. This was desired since the R
2
value of 0.6580 for the fitted model is quite low,
providing a not very reliable steepest ascent direction
(and a wide confidence cone, see Technical Appendix
5B).
.
.
X
2
= (-0.1160)(30) = -3.48
o
C.
Thus the step size is X' = (-3.48
o
C, 50 minutes).
Procedure: Conducting Experiments Along the Direction
of Maximum Improvement
Procedure
for
conducting
experiments
along the
direction of
maximum
improvement
The following is the procedure for conducting experiments
along the direction of maximum improvement.
1. Given current operating conditions = (X
1
, X
2
, ...,
X
k
) and a step size X' = ( X
1
, X
2
, ..., X
k
),
perform experiments at factor levels X
0
+ X, X
0
+ 2
X, X
0
+ 3 X, ... as long as improvement in the
response Y (decrease or increase, as desired) is
observed.
2. Once a point has been reached where there is no
further improvement, a new first-order experiment
(e.g., a 2
k-p
fractional factorial) should be performed
with repeated center runs to assess lack of fit. If there
is no significant evidence of lack of fit, the new first-
order model will provide a new search direction, and
another iteration is performed as indicated in Figure
5.3. Otherwise (there is evidence of lack of fit), the
experimental design is augmented and a second-order
model should be fitted. That is, the experimenter
should proceed to "Phase II".
Example: Experimenting Along the Direction of
Maximum Improvement
Step 1:
increase
factor levels
by
Step 1:
Given X
0
= (200
o
C, 200 minutes) and X = (-3.48
o
C, 50
minutes), the next experiments were performed as follows
(the step size in temperature was rounded to -3.5
o
C for
practical reasons):
X
1
X
2
x
1
x
2
Y (= yield)
X
0
200 200 0 0
X
0
+ X 196.5 250 -0.1160 1 56.2
X
0
+ 2 X 193.0 300 -0.2320 2 71.49
X
0
+ 3 X 189.5 350 -0.3480 3 75.63
X
0
+ 4 X 186.0 400 -0.4640 4 72.31
X
0
+ 5 X 182.5 450 -0.5800 5 72.10
Since the goal is to maximize Y, the point of maximum
observed response is X
1
= 189.5
o
C, X
2
= 350 minutes.
Notice that the search was stopped after 2 consecutive drops
in response, to assure that we have passed by the "peak" of
the "hill".
Step 2: new
factorial
experiment
Step 2:
A new 2
2
factorial experiment is performed with X' =
(189.5, 350) as the origin. Using the same scaling factors as
before, the new scaled controllable factors are:
Five center runs (at X
1
= 189.5, X
2
= 350) were repeated to
assess lack of fit. The experimental results were:
x
1
x
2
X
1
X
2
Y (= yield)
-1 -1 159.5 300 64.33
+1 -1 219.5 300 51.78
-1 +1 159.5 400 77.30
+1 +1 219.5 400 45.37
0 0 189.5 350 62.08
0 0 189.5 350 79.36
0 0 189.5 350 75.29
0 0 189.5 350 73.81
0 0 189.5 350 69.45
The corresponding ANOVA table for a linear model is
SUM OF MEAN F
SOURCE SQUARES DF SQUARE VALUE
PROB > F
MODEL 505.376 2 252.688 4.731
0.0703
CURVATURE 336.364 1 336.364 6.297
0.0539
RESIDUAL 267.075 5 53.415
LACK OF FIT 93.896 1 93.896 2.168
0.2149
PURE ERROR 173.179 4 43.295
COR TOTAL 1108.815 8
From the table, the linear effects (model) are significant and
there is no evidence of lack of fit. However, there is a
significant curvature effect (at the 5.4 % significance level),
which implies that the optimization should proceed with
Phase II; that is, the fit and optimization of a second-order
model.
5.5.3.1.4. Single response: Optimization when there is adequate quadratic fit

5.5.3.1.4. Single response: Optimization when there is adequate
quadratic fit
Regions
where
quadratic
models or
even cubic
models are
needed occur
in many
instances in
industry
After a few steepest ascent (or descent) searches, a first-order model will eventually lead to
no further improvement or it will exhibit lack of fit. The latter case typically occurs when
operating conditions have been changed to a region where there are quadratic (second-
order) effects present in the response. A second-order polynomial can be used as a local
approximation of the response in a small region where, hopefully, optimal operating
conditions exist. However, while a quadratic fit is appropriate in most of the cases in
industry, there will be a few times when a quadratic fit will not be sufficiently flexible to
explain a given response. In such cases, the analyst generally does one of the following:
1. Uses a transformation of Y or the X
i
s to improve the fit.
2. Limits use of the model to a smaller region in which the model does fit.
3. Adds other terms to the model.
Procedure: obtaining the estimated optimal operating conditions
Second-
order
polynomial
model
Once a linear model exhibits lack of fit or when significant curvature is detected, the
experimental design used in Phase I (recall that a 2
k-p
factorial experiment might be used)
should be augmented with axial runs on each factor to form what is called a central
composite design. This experimental design allows estimation of a second-order polynomial
of the form
Steps to find
optimal
operating
conditions
If the corresponding analysis of variance table indicates no lack of fit for this model, the
engineer can proceed to determine the estimated optimal operating conditions.
1. Using some graphics software, obtain a contour plot of the fitted response. If the
number of factors (k) is greater than 2, then plot contours in all planes corresponding
to all the possible pairs of factors. For k greater than, say, 5, this could be too
cumbersome (unless the graphic software plots all pairs automatically). In such a
case, a "canonical analysis" of the surface is recommended (see Technical Appendix
5D).
2. Use an optimization solver to maximize or minimize (as desired) the estimated
response .
3. Perform a confirmation experiment at the estimated optimal operating conditions
given by the solver in step 2.
Chemical
experiment
example
We illustrate these steps using the chemical experiment discussed previously. For a
technical description of a formula that provides the coordinates of the stationary point of
the surface, see Technical Appendix 5C.
Example: Second Phase Optimization of Chemical Process
Experimental
results for
axial runs
Recall that in the chemical experiment, the ANOVA table, obtained from using an
experiment run around the coordinates X
1
= 189.5, X
2
= 350, indicated significant
curvature effects. Augmenting the 2
2
factorial experiment with axial runs at to
achieve a rotatable central composite experimental design, the following experimental
results were obtained:
x
1
x
2
X
1
X
2
Y (= yield)
-1.414 0 147.08 350 72.58
+1.414 0 231.92 350 37.42
0 -1.414 189.5 279.3 54.63
0 +1.414 189.5 420.7 54.18
ANOVA table The ANOVA table corresponding to a cubic model with an interaction term (contained in
the quadratic sum-of-squares partition) is
SUM OF MEAN F
SOURCE SQUARES DF SQUARE VALUE PROB > F
MEAN 51418.2 1 51418.2
Linear 1113.7 2 556.8 5.56 0.024
Quadratic 768.1 3 256.0 7.69 0.013
Cubic 9.9 2 5.0 0.11 0.897
RESIDUAL 223.1 5 44.6
TOTAL 53533.0 13
Lack-of-fit
tests and
auxillary
diagnostic
statistics
From the ANOVA table, the linear and quadratic effects are significant. The lack-of-fit
tests and auxiliary diagnostic statistics for linear, quadratic, and cubic models are:
SUM OF MEAN F
MODEL SQUARES DF SQUARE VALUE PROB > F
Linear 827.9 6 138.0 3.19 0.141
Quadratic 59.9 3 20.0 0.46 0.725
Cubic 49.9 1 49.9 1.15 0.343
PURE ERROR 173.2 4 43.3
ROOT ADJ PRED
MODEL MSE R-SQR R-SQR R-SQR PRESS
Linear 10.01 0.5266 0.4319 0.2425 1602.02
Quadratic 5.77 0.8898 0.8111 0.6708 696.25
Cubic 6.68 0.8945 0.7468 -0.6393 3466.71
The quadratic model has a larger p-value for the lack of fit test, higher adjusted R
2
, and a
lower PRESS statistic; thus it should provide a reliable model. The fitted quadratic
equation, in coded units, is
Step 1:
Contour plot
of the fitted
response
function
A contour plot of this function (Figure 5.5) shows that it appears to have a single optimum
point in the region of the experiment (this optimum is calculated below to be (-0.9285,
0.3472), in coded x
1
, x
2
units, with a predicted response value of 77.59).

FIGURE 5.5: Contour Plot of the Fitted Response in the Example
3D plot of the
fitted
response
function
Since there are only two factors in this example, we can also obtain a 3D plot of the fitted
response against the two factors (Figure 5.6).

FIGURE 5.6: 3D Plot of the Fitted Response in the Example
Step 2:
Optimization
point
An optimization routine was used to maximize . The results are = 161.64
o
C, =
367.32 minutes. The estimated yield at the optimal point is (X
*
) = 77.59 %.
Step 3:
Confirmation
experiment
A confirmation experiment was conducted by the process engineer at settings X
1
= 161.64,
X
2
= 367.32. The observed response was (X
*
) = 76.5 %, which is satisfactorily close to
the estimated optimum.
==================================================================
Technical Appendix 5C: Finding the Factor Settings for the Stationary Point of a
Quadratic Response
How to find
the maximum
or minimum
point for a
quadratic
response
1. Rewrite the fitted equation using matrix notation as
where b' = (b
1
, b
2
, ..., b
k
) denotes a vector of first-order parameter estimates,
is a matrix of second-order parameter estimates and x' = (x
1
, x
2
, ..., x
k
) is the vector
of controllable factors. Notice that the off-diagonal elements of B are equal to half
the two-factor interaction coefficients.
2. Equating the partial derivatives of with respect to x to zeroes and solving the
resulting system of equations, the coordinates of the stationary point of the response
are given by
Nature of the
stationary
point is
determined by
B
The nature of the stationary point (whether it is a point of maximum response, minimum
response, or a saddle point) is determined by the matrix B. The two-factor interactions do
not, in general, let us "see" what type of point x
*
is. One thing that can be said is that if the
diagonal elements of B (b
ii
) have mixed signs, x
*
is a saddle point. Otherwise, it is
necessary to look at the characteristic roots or eigenvalues of B to see whether B is
"positive definite" (so x
*
is a point of minimum response) or "negative definite" (the case
in which x
*
is a point of maximum response). This task is easier if the two-factor
interactions are "eliminated" from the fitted equation as is described in Technical Appendix
5D.
Example: computing the stationary point, Chemical Process experiment
Example of
computing the
stationary
point
The fitted quadratic equation in the chemical experiment discussed in Section 5.5.3.1.1 is,
in coded units,
from which we obtain b' = (-11.78, 0.74),
and
Transforming back to the original units of measurement, the coordinates of the stationary
point are
.
The predicted response at the stationary point is (X
*
) = 77.59 %.
Technical Appendix 5D: "Canonical Analysis" of Quadratic Responses
Case for a
single
controllable
response
Whether the stationary point X
*
represents a point of maximum or minimum response, or is
just a saddle point, is determined by the matrix of second-order coefficients, B. In the
simpler case of just a single controllable factor (k=1), B is a scalar proportional to the
second derivative of (x) with respect to x. If d
2
/dx
2
is positive, recall from calculus that
the function (x) is convex ("bowl shaped") and x
*
is a point of minimum response.
Case for
multiple
controllable
responses not
so easy
Unfortunately, the multiple factor case (k>1) is not so easy since the two-factor interactions
(the off-diagonal elements of B) obscure the picture of what is going on. A recommended
procedure for analyzing whether B is "positive definite" (we have a minimum) or "negative
definite" (we have a maximum) is to rotate the axes x
1
, x
2
, ..., x
k
so that the two-factor
interactions disappear. It is also customary (Box and Draper, 1987; Khuri and Cornell,
1987; Myers and Montgomery, 1995) to translate the origin of coordinates to the stationary
point so that the intercept term is eliminated from the equation of (x). This procedure is
called the canonical analysis of (x).
Procedure: Canonical Analysis
Steps for
performing
the canonical
analysis
1. Define a new axis z = x - x
*
(translation step). The fitted equation becomes
.
2. Define a new axis w = E'z, with E'BE = D and D a diagonal matrix to be defined
(rotation step). The fitted equation becomes
.
This is the so-called canonical form of the model. The elements on the diagonal of D,
i
(i = 1, 2, ..., k) are the eigenvalues of B. The columns of E', e
i
, are the
orthonormal eigenvectors of B, which means that the e
i
satisfy (B -
i
)e
i
= 0, =
0 for i j, and = 1.0.
3. If all the
i
are negative, x
*
is a point of maximum response. If all the
i
are positive,
x
*
is a point of minimum response. Finally, if the
i
are of mixed signs, the response
is a saddle function and x
*
is the saddle point.
Eigenvalues
that are
approximately
zero
If some
i
0, the fitted ellipsoid
is elongated (i.e., it is flat) along the direction of the w
i
axis. Points along the w
i
axis will
have an estimated response close to optimal; thus the process engineer has flexibility in
choosing "good" operating conditions. If two eigenvalues (say
i
and
j
) are close to zero, a
plane in the (w
i
, w
j
) coordinates will have close to optimal operating conditions, etc.
Canonical
analysis
typically
performed by
software
Software is available to compute the eigenvalues
i
and the orthonormal eigenvectors e
i
;
thus there is no need to do a canonical analysis by hand.
Example: Canonical Analysis of Yield Response in Chemical Experiment
B matrix for
this example
Let us return to the chemical experiment example to illustrate the method. Keep in mind
that when the number of factors is small (e.g., k=2 as in this example) canonical analysis is
not recommended in practice since simple contour plotting will provide sufficient
information. The fitted equation of the model yields
Compute the
eigenvalues
and find the
orthonormal
eigenvectors
To compute the eigenvalues
i
, we have to find all roots of the expression that results from
equating the determinant of B -
i
I to zero. Since B is symmetric and has real coefficients,
there will be k real roots
i
, i = 1, 2, ..., k. To find the orthonormal eigenvectors, solve the
simultaneous equations (B -
i
I )e
i
= 0 and = 1.
Canonical
analysis
results
The results of the canonical analysis are as follows:
Eigenvectors
Eigenvalues X1 X2
-4.973187 0.728460 -0.685089
-9.827317 0.685089 0.728460
Notice that the eigenvalues are the two roots of
As mentioned previously, the stationary point is (x
*
)' = (-0.9278, 0.3468), which
corresponds to X
*
' = (161.64, 367.36). Since both eigenvalues are negative, x
*
is a point of
maximum response. To obtain the directions of the axis of the fitted ellipsoid, compute
w
1
= 0.7285(x
1
+ 0.9278) - 0.6851(x
2
- 0.3468) = 0.9143 + 0.7285x
1
- 0.6851x
2
and
w
2
= 0.6851(x
1
+ 0.9278) - 0.7285(x
2
- 0.3468) = 0.8830 + 0.6851x
1
+ 0.7285x
2
Since |
1
| < |
2
|, there is somewhat more elongation in the w
i
direction. However, since
both eigenvalues are quite far from zero, there is not much flexibility in choosing operating
conditions. It can be seen from Figure 5.5 that the fitted ellipses do not have a great
elongation in the w
1
direction, the direction of the major axis. It is important to emphasize
that confirmation experiments at x
*
should be performed to check the validity of the
estimated optimal solution.
5.5.3.1.5. Single response: Effect of sampling error on optimal solution

5.5.3.1.5. Single response: Effect of sampling
error on optimal solution
Experimental
error means
all derived
optimal
operating
conditions
are just
estimates -
confidence
regions that
are likely to
contain the
optimal
points can
be derived
Process engineers should be aware that the estimated
optimal operating conditions x
*
represent a single estimate
of the true (unknown) system optimal point. That is, due to
sampling (experimental) error, if the experiment is repeated,
a different quadratic function will be fitted which will yield
a different stationary point x
*
. Some authors (Box and
Hunter, 1954; Myers and Montgomery, 1995) provide a
procedure that allows one to compute a region in the factor
space that, with a specified probability, contains the system
stationary point. This region is useful information for a
process engineer in that it provides a measure of how
"good" the point estimate x
*
is. In general, the larger this
region is, the less reliable the point estimate x
*
is. When the
number of factors, k, is greater than 3, these confidence
regions are difficult to visualize.
Confirmation
runs are very
important
Awareness of experimental error should make a process
engineer realize the importance of performing confirmation
runs at x
*
, the estimated optimal operating conditions.
5.5.3.1.6. Single response: Optimization subject to experimental region constraints

5.5.3.1.6. Single response: Optimization subject
to experimental region constraints
Optimal
operating
conditions
may fall
outside
region where
experiment
conducted
Sometimes the optimal operating conditions x
*
simply fall
outside the region where the experiment was conducted. In
these cases, constrained optimization techniques can be
used to find the solution x
*
that optimizes without
leaving the region in the factor space where the experiment
took place.
Ridge
analysis is a
method for
finding
optimal
factor
settings that
satisfy
certain
constraints
"Ridge Analysis", as developed by Hoerl (1959), Hoerl
(1964) and Draper (1963), is an optimization technique that
finds factor settings x
*
such that they
optimize (x) = b
0
+ b'x + x'Bx
subject to: x'x =
2
The solution x
*
to this problem provides operating
conditions that yield an estimated absolute maximum or
minimum response on a sphere of radius . Different
solutions can be obtained by trying different values of .
Solve with
non-linear
programming
software
The original formulation of Ridge Analysis was based on
the eigenvalues of a stationarity system. With the wide
availability of non-linear programming codes, Ridge
Analysis problems can be solved without recourse to
eigenvalue analysis.
5.5.3.2. Multiple response case

When there
are multiple
responses, it is
often
impossible to
simultaneously
optimize each
one - trade-
offs must be
made
In the multiple response case, finding process operating
conditions that simultaneously maximize (or minimize, as
desired) all the responses is quite difficult, and often
impossible. Almost inevitably, the process engineer must
make some trade-offs in order to find process operating
conditions that are satisfactory for most (and hopefully all)
the responses. In this subsection, we examine some
effective ways to make these trade-offs.
Path of steepest ascent
The desirability function approach
The mathematical programming approach
Dual response systems
More than 2 responses
5.5.3.2.1. Multiple responses: Path of steepest ascent

Objective:
consider and
balance the
individual
paths of
maximum
improvement
When the responses exhibit adequate linear fit (i.e., the response models
are all linear), the objective is to find a direction or path that
simultaneously considers the individual paths of maximum improvement
and balances them in some way. This case is addressed next.
When there is a mix of linear and higher-order responses, or when all
empirical response models are of higher-order, see sections 5.5.3.2.2 and
5.5.3.2.3. The desirability method (section 5.5.3.2.2) can also be used
when all response models are linear.
Procedure: Path of Steepest Ascent, Multiple Responses.
A weighted
priority
strategy is
described
using the
path of
steepest
ascent for
each
response
The following is a weighted priority strategy using the path of steepest
ascent for each response.
1. Compute the gradients g
i
(i = 1, 2, . . ., k) of all responses as
explained in section 5.5.3.1.1. If one of the responses is clearly of
primary interest compared to the others, use only the gradient of
this response and follow the procedure of section 5.5.3.1.1.
Otherwise, continue with step 2.
2. Determine relative priorities for each of the k responses. Then,
the weighted gradient for the search direction is given by
and the weighted direction is
Weighting
factors
based on R
2
The confidence cone for the direction of maximum improvement
explained in section 5.5.3.1.2 can be used to weight down "poor"
response models that provide very wide cones and unreliable directions.
Since the width of the cone is proportional to (1 - R
2
), we can use
Single
response
steepest
ascent
procedure
Given a weighted direction of maximum improvement, we can follow
the single response steepest ascent procedure as in section 5.5.3.1.1 by
selecting points with coordinates x
*
= d
i
, i = 1, 2, ..., k. These and
related issues are explained more fully in Del Castillo (1996).
Example: Path of Steepest Ascent, Multiple Response Case
An example
using the
weighted
priority
method
Suppose the response model:
with = 0.8968 represents the average yield of a production process
obtained from a replicated factorial experiment in the two controllable
factors (in coded units). From the same experiment, a second response
model for the process standard deviation of the yield is obtained and
given by
with = 0.5977. We wish to maximize the mean yield while
minimizing the standard deviation of the yield.
Step 1: compute the gradients:
Compute the
gradients
We compute the gradients as follows.
(recall we wish to minimize y
2
).
Step 2: find relative priorities:
Find relative
priorities
Since there are no clear priorities, we use the quality of fit as the
priority:
Then, the weighted gradient is
g' = (0.6(0.3124) + 0.4(-0.7088), 0.6(0.95) + 0.4(-0.7054)) = (-
0.096, 0.2878)
which, after scaling it (by dividing each coordinate by
), gives the weighted direction d' = (-.03164,
0.9486).
Therefore, if we want to move = 1 coded units along the path of
maximum improvement, we will set x
1
= (1)(-0.3164) = -0.3164, x
2
=
(1)(0.9486) = 0.9486 in the next run or experiment.
5.5.3.2.2. Multiple responses: The desirability approach

The
desirability
approach is
a popular
method that
assigns a
"score" to a
set of
responses
and chooses
factor
settings that
maximize
that score
The desirability function approach is one of the most widely used methods in
industry for the optimization of multiple response processes. It is based on the idea
that the "quality" of a product or process that has multiple quality characteristics,
with one of them outside of some "desired" limits, is completely unacceptable. The
method finds operating conditions x that provide the "most desirable" response
values.
For each response Y
i
(x), a desirability function d
i
(Y
i
) assigns numbers between 0
and 1 to the possible values of Y
i
, with d
i
(Y
i
) = 0 representing a completely
undesirable value of Y
i
and d
i
(Y
i
) = 1 representing a completely desirable or ideal
response value. The individual desirabilities are then combined using the geometric
mean, which gives the overall desirability D:
with k denoting the number of responses. Notice that if any response Y
i
is
completely undesirable (d
i
(Y
i
) = 0), then the overall desirability is zero. In practice,
fitted response values
i
are used in place of the Y
i
.
Desirability
functions of
Derringer
and Suich
Depending on whether a particular response Y
i
is to be maximized, minimized, or
assigned a target value, different desirability functions d
i
(Y
i
) can be used. A useful
class of desirability functions was proposed by Derringer and Suich (1980). Let L
i
,
U
i
and T
i
be the lower, upper, and target values, respectively, that are desired for
response Y
i
, with L
i
T
i
U
i
.
Desirability
function for
"target is
best"
If a response is of the "target is best" kind, then its individual desirability function is
with the exponents s and t determining how important it is to hit the target value.
For s = t = 1, the desirability function increases linearly towards T
i
; for s < 1, t < 1,
the function is convex, and for s > 1, t > 1, the function is concave (see the example
below for an illustration).
Desirability
function for
maximizing a
response
If a response is to be maximized instead, the individual desirability is defined as
with T
i
in this case interpreted as a large enough value for the response.
Desirability
function for
minimizing a
response
Finally, if we want to minimize a response, we could use
with T
i
denoting a small enough value for the response.
Desirability
approach
steps
The desirability approach consists of the following steps:
1. Conduct experiments and fit response models for all k responses;
2. Define individual desirability functions for each response;
3. Maximize the overall desirability D with respect to the controllable factors.
Example:
An example
using the
desirability
approach
Derringer and Suich (1980) present the following multiple response experiment
arising in the development of a tire tread compound. The controllable factors are: x
1
,
hydrated silica level, x
2
, silane coupling agent level, and x
3
, sulfur level. The four
responses to be optimized and their desired ranges are:
Factor and
response
variables
Source Desired range
PICO Abrasion index, Y
1
120 < Y
1
200% modulus, Y
2
1000 < Y
2
Elongation at break, Y
3
400 < Y
3
< 600
Hardness, Y
4
60 < Y
4
< 75
The first two responses are to be maximized, and the value s=1 was chosen for their
desirability functions. The last two responses are "target is best" with T
3
= 500 and
T
4
= 67.5. The values s=t=1 were chosen in both cases.
Experimental The following experiments were conducted using a central composite design.
runs from a
central
composite
design
Run
Number
x
1
x
2
x
3
Y
1
Y
2
Y
3
Y
4
1 -1.00 -1.00 -1.00 102 900 470 67.5
2 +1.00 -1.00 -1.00 120 860 410 65.0
3 -1.00 +1.00 -1.00 117 800 570 77.5
4 +1.00 +1.00 -1.00 198 2294 240 74.5
5 -1.00 -1.00 +1.00 103 490 640 62.5
6 +1.00 -1.00 +1.00 132 1289 270 67.0
7 -1.00 +1.00 +1.00 132 1270 410 78.0
8 +1.00 +1.00 +1.00 139 1090 380 70.0
9 -1.63 0.00 0.00 102 770 590 76.0
10 +1.63 0.00 0.00 154 1690 260 70.0
11 0.00 -1.63 0.00 96 700 520 63.0
12 0.00 +1.63 0.00 163 1540 380 75.0
13 0.00 0.00 -1.63 116 2184 520 65.0
14 0.00 0.00 +1.63 153 1784 290 71.0
15 0.00 0.00 0.00 133 1300 380 70.0
16 0.00 0.00 0.00 133 1300 380 68.5
17 0.00 0.00 0.00 140 1145 430 68.0
18 0.00 0.00 0.00 142 1090 430 68.0
19 0.00 0.00 0.00 145 1260 390 69.0
20 0.00 0.00 0.00 142 1344 390 70.0
Fitted
response
Using ordinary least squares and standard diagnostics, the fitted responses are:
(R
2
= 0.8369 and adjusted R
2
= 0.6903);
(R
2
2
= 0.4562);
(R
2
2
= 0.6224);
(R
2
2
= 0.7466).
Note that no interactions were significant for response 3 and that the fit for response
2 is quite poor.
Best Solution
The best solution is (x
*
)' = (-0.10, 0.15, -1.0) and results in:
d
1
(
1
) = 0.34 (
1
(x
*
) = 136.4)
d
2
(
2
) = 1.0 (
2
(x
*
) = 1571.05)
d
3
(
3
) = 0.49 (
3
(x
*
) = 450.56)
d
4
(
4
) = 0.76 (
4
(x
*
) = 69.26)
The overall desirability for this solution is 0.596. All responses are predicted to be
within the desired limits.
3D plot of
the overall
desirability
function
Figure 5.8 shows a 3D plot of the overall desirability function D(x) for the (x
2
, x
3
)
plane when x
1
is fixed at -0.10. The function D(x) is quite "flat" in the vicinity of
the optimal solution, indicating that small variations around x
*
are predicted to not
change the overall desirability drastically. However, the importance of performing
confirmatory runs at the estimated optimal operating conditions should be
emphasized. This is particularly true in this example given the poor fit of the
response models (e.g.,
2
).
FIGURE 5.8 Overall Desirability Function for Example Problem
5.5.3.2.3. Multiple responses: The mathematical programming approach

5.5.3.2.3. Multiple responses: The mathematical
programming approach
The
mathematical
programming
approach
maximizes or
minimizes a
primary
response,
subject to
appropriate
constraints
on all other
responses
The analysis of multiple response systems usually involves
some type of optimization problem. When one response can
be chosen as the "primary", or most important response, and
bounds or targets can be defined on all other responses, a
mathematical programming approach can be taken. If this is
not possible, the desirability approach should be used instead.
In the mathematical programming approach, the primary
response is maximized or minimized, as desired, subject to
appropriate constraints on all other responses. The case of two
responses ("dual" responses) has been studied in detail by
some authors and is presented first. Then, the case of more
than 2 responses is illustrated.
More than 2 responses
Optimization
of dual
response
systems
The optimization of dual response systems (DRS) consists of
finding operating conditions x that
with T denoting the target value for the secondary response, p
the number of primary responses (i.e., responses to be
optimized), s the number of secondary responses (i.e.,
responses to be constrained), and is the radius of a
spherical constraint that limits the region in the controllable
factor space where the search should be undertaken. The
value of should be chosen with the purpose of avoiding
solutions that extrapolate too far outside the region where the
experimental data were obtained. For example, if the
experimental design is a central composite design, choosing
(axial distance) is a logical choice. Bounds of the form
L x
i
U can be used instead if a cubical experimental
region were used (e.g., when using a factorial experiment).
Note that a Ridge Analysis problem is related to a DRS
problem when the secondary constraint is absent. Thus, any
algorithm or solver for DRS's will also work for the Ridge
Analysis of single response systems.
Nonlinear
programming
software
required for
DRS
In a DRS, the response models and can be linear,
quadratic or even cubic polynomials. A nonlinear
programming algorithm has to be used for the optimization of
a DRS. For the particular case of quadratic responses, an
equality constraint for the secondary response, and a spherical
region of experimentation, specialized optimization
algorithms exist that guarantee global optimal solutions. In
such a case, the algorithm DRSALG can be used (download
from http://www.stat.cmu.edu/jqt/29-3), but a Fortran
compiler is necessary.
More general
case
In the more general case of inequality constraints or a cubical
region of experimentation, a general purpose nonlinear solver
must be used and several starting points should be tried to
avoid local optima. This is illustrated in the next section.
Example for more than 2 responses
Example:
problem
setup
The values of three components (x
1
, x
2
, x
3
) of a propellant
need to be selected to maximize a primary response, burning
rate (Y
1
), subject to satisfactory levels of two secondary
reponses; namely, the variance of the burning rate (Y
2
) and
the cost (Y
3
). The three components must add to 100% of the
mixture. The fitted models are:
The
optimization
problem
The optimization problem is therefore:
maximize
1
(x)
subject to:
2
(x) 4.5

3
(x) 20

x
1
+ x
2
+ x
3
= 1.0

0 x
1
1

0 x
2
1

0 x
3
1
Solution
The solution is (x
*
)' = (0.212, 0.343, 0.443) which provides
1
= 106.62,
2
= 4.17, and
3
= 18.23. Therefore, both
secondary responses are below the specified upper bounds.
The optimization should be implemented using a variety of
starting points to avoid local optima. Once again,
confirmatory experiments should be conducted at the
estimated optimal operating conditions.
The solution to the optimization problem can be obtained
using R code.
5.5.4. What is a mixture design?

When the
factors are
proportions
of a blend,
you need to
use a
mixture
design
In a mixture experiment, the independent factors are
proportions of different components of a blend. For example,
if you want to optimize the tensile strength of stainless steel,
the factors of interest might be the proportions of iron,
copper, nickel, and chromium in the alloy. The fact that the
proportions of the different factors must sum to 100%
complicates the design as well as the analysis of mixture
experiments.
Standard
mixture
designs and
constrained
mixture
designs
When the mixture components are subject to the constraint
that they must sum to one, there are standard mixture designs
for fitting standard models, such as Simplex-Lattice designs
and Simplex-Centroid designs. When mixture components
are subject to additional constraints, such as a maximum
and/or minimum value for each component, designs other
than the standard mixture designs, referred to as constrained
mixture designs or Extreme-Vertices designs, are appropriate.
Measured
response
assumed to
depend only
on relative
proportions
In mixture experiments, the measured response is assumed to
depend only on the relative proportions of the ingredients or
components in the mixture and not on the amount of the
mixture. The amount of the mixture could also be studied as
an additional factor in the experiment; however, this would
be an example of mixture and process variables being treated
together.
Proportions
of each
variable
must sum to
1
The main distinction between mixture experiments and
independent variable experiments is that with the former, the
input variables or components are non-negative proportionate
amounts of the mixture, and if expressed as fractions of the
mixture, they must sum to one. If for some reason, the sum
of the component proportions is less than one, the variable
proportions can be rewritten as scaled fractions so that the
scaled fractions sum to one.
Purpose of
a mixture
design
In mixture problems, the purpose of the experiment is to
model the blending surface with some form of mathematical
equation so that:
1. Predictions of the response for any mixture or
combination of the ingredients can be made
empirically, or
2. Some measure of the influence on the response of each
component singly and in combination with other
components can be obtained.
Assumptions
for mixture
experiments
The usual assumptions made for factorial experiments are
also made for mixture experiments. In particular, it is
assumed that the errors are independent and identically
distributed with zero mean and common variance. Another
assumption that is made, as with factorial designs, is that the
true underlying response surface is continuous over the
region being studied.
Steps in
planning a
mixture
experiment
Planning a mixture experiment typically involves the
following steps (Cornell and Piepel, 1994):
1. Define the objectives of the experiment.
2. Select the mixture components and any other factors to
be studied. Other factors may include process variables
or the total amount of the mixture.
3. Identify any constraints on the mixture components or
other factors in order to specify the experimental
region.
4. Identify the response variable(s) to be measured.
5. Propose an appropriate model for modeling the
response data as functions of the mixture components
and other factors selected for the experiment.
6. Select an experimental design that is sufficient not
only to fit the proposed model, but which allows a test
of model adequacy as well.
5.5.4.1. Mixture screening designs

5.5.4.1. Mixture screening designs
Screening
experiments
can be used
to identify
the
important
mixture
factors
In some areas of mixture experiments, for example, certain
chemical industries, there is often a large number, q, of
potentially important components that can be considered
candidates in an experiment. The objective of these types of
experiments is to screen the components to identify the ones
that are most important. In this type of situation, the
experimenter should consider a screening experiment to
reduce the number of possible components.
A first
order
mixture
model
The construction of screening designs and their
corresponding models often begins with the first-order or
first-degree mixture model
for which the beta coefficients are non-negative and sum to
one.
Choices of
types of
screening
designs
depend on
constraints
If the experimental region is a simplex, it is generally a good
idea to make the ranges of the components as similar as
possible. Then the relative effects of the components can be
assessed by ranking the ratios of the parameter estimates (i.e.,
the estimates of the
i
), relative to their standard errors.
Simplex screening designs are recommended when it is
possible to experiment over the total simplex region.
Constrained mixture designs are suggested when the
proportions of some or all of the components are restricted by
upper and lower bounds. If these designs are not feasible in
this situation, then D-optimal designs for a linear model are
always an option.
5.5.4.2. Simplex-lattice designs

Definition of
simplex-
lattice points
A {q, m} simplex-lattice design for q components consists of
points defined by the following coordinate settings: the
proportions assumed by each component take the m+1 equally
spaced values from 0 to 1,
x
i
= 0, 1/m, 2/m, ... , 1 for i = 1, 2, ... , q
and all possible combinations (mixtures) of the proportions from
this equation are used.
Except for
the center, all
design points
are on the
simplex
boundaries
Note that the standard Simplex-Lattice and the Simplex-Centroid
designs (described later) are boundary-point designs; that is, with
the exception of the overall centroid, all the design points are on
the boundaries of the simplex. When one is interested in
prediction in the interior, it is highly desirable to augment the
simplex-type designs with interior design points.
Example of a
three-
component
simplex
lattice design
Consider a three-component mixture for which the number of
equally spaced levels for each component is four (i.e., x
i
= 0,
0.333, 0.667, 1). In this example q = 3 and m = 3. If one uses all
possible blends of the three components with these proportions,
the {3, 3} simplex-lattice then contains the 10 blending
coordinates listed in the table below. The experimental region and
the distribution of design runs over the simplex region are shown
in the figure below. There are 10 design runs for the {3, 3}
simplex-lattice design.
Design table TABLE 5.3 Simplex Lattice
Design
X1 X2 X3
0 0 1
0 0.667 0.333
0 1 0
0.333 0 0.667
0.333 0.333 0.333
0.333 0.6667 0
0.667 0 0.333
0.667 0.333 0
1 0 0
Diagram
showing
configuration
of design
runs

FIGURE 5.9 Configuration of Design Runs for a {3,3}
Simplex-Lattice Design
The number of design points in the simplex-lattice is (q+m-
1)!/(m!(q-1)!).
Definition of
canonical
polynomial
model used
in mixture
experiments
Now consider the form of the polynomial model that one might fit
to the data from a mixture experiment. Due to the restriction x
1
+
x
2
+ ... + x
q
= 1, the form of the regression function that is fit to
the data from a mixture experiment is somewhat different from
the traditional polynomial fit and is often referred to as the
canonical polynomial. Its form is derived using the general form
of the regression function that can be fit to data collected at the
points of a {q, m} simplex-lattice design and substituting into this
function the dependence relationship among the x
i
terms. The
number of terms in the {q, m} polynomial is (q+m-1)!/(m!(q-1)!),
as stated previously. This is equal to the number of points that
make up the associated {q, m} simplex-lattice design.
Example for
a {q, m=1}
simplex-
lattice design
For example, the equation that can be fit to the points from a {q,
m=1} simplex-lattice design is
Multiplying
0
by (x
1
+ x
2
+ ... + x
q
= 1), the resulting equation
is
with =
0
+
i
for all i = 1, ..., q.
First- This is called the canonical form of the first-order mixture model.
order
canonical
form
In general, the canonical forms of the mixture models (with the
asterisks removed from the parameters) are as follows:
Summary of
canonical
mixture
models
Linear
Quadratic
Cubic
Special
Cubic
Linear
blending
portion
The terms in the canonical mixture polynomials have simple
interpretations. Geometrically, the parameter
i
in the above
equations represents the expected response to the pure mixture
x
i
=1, x
j
=0, i j, and is the height of the mixture surface at the
vertex x
i
=1. The portion of each of the above polynomials given
by
is called the linear blending portion. When blending is strictly
additive, then the linear model form above is an appropriate
model.
Three-
component
mixture
example
The following example is from Cornell (1990) and consists of a
three-component mixture problem. The three components are
Polyethylene (X1), polystyrene (X2), and polypropylene (X3),
which are blended together to form fiber that will be spun into
yarn. The product developers are only interested in the pure and
binary blends of these three materials. The response variable of
interest is yarn elongation in kilograms of force applied. A {3,2}
simplex-lattice design is used to study the blending process. The
simplex region and the six design runs are shown in the figure
below. The design and the observed responses are listed in Table
5.4. There were two replicate observations run at each of the pure
blends. There were three replicate observations run at the binary
blends. There are 15 observations with six unique design runs.
Diagram
showing the
designs runs
for this
example

FIGURE 5.10 Design Runs for the {3,2} Simplex-Lattice Yarn
Elongation Problem
Table
showing the
simplex-
lattice design
and observed
responses
TABLE 5.4 Simplex-Lattice Design for
Yarn Elongation Problem
X1 X2 X3
Observed
Elongation Values
0.0 0.0 1.0 16.8, 16.0
0.0 0.5 0.5 10.0, 9.7, 11.8
0.0 1.0 0.0 8.8, 10.0
0.5 0.0 0.5 17.7, 16.4, 16.6
0.5 0.5 0.0 15.0, 14.8, 16.1
1.0 0.0 0.0 11.0, 12.4
Fit a
quadratic
mixture
model
The design runs listed in the above table are in standard order.
The actual order of the 15 treatment runs was completely
randomized. Since there are three levels of each of the three
mixture components, a quadratic mixture model can be fit to the
data. The results of the model fit are shown below. Note that there
was no intercept in the model.
Summary of Fit
RSquare 0.951356
Source DF Sum of Squares Mean Square F Ratio
Prob > F
Model 5 2878.27 479.7117 658.141
1.55e-13
Error 9 6.56 0.7289
C Total 14 2884.83
Parameter Estimates
Term Estimate Std Error t Ratio Prob>|t|
X1 11.7 0.603692 19.38 <.0001
X2 9.4 0.603692 15.57 <.0001
X3 16.4 0.603692 27.17 <.0001
X2*X1 19 2.608249 7.28 <.0001
X3*X1 11.4 2.608249 4.37 0.0018
X3*X2 -9.6 2.608249 -3.68 0.0051
Interpretation
of results
Under the parameter estimates section of the output are the
individual t-tests for each of the parameters in the model. The
three cross product terms are significant (X1*X2, X3*X1,
X3*X2), indicating a significant quadratic fit.
The fitted
quadratic
model
The fitted quadratic mixture model is
Conclusions
from the
fitted
quadratic
model
Since b
3
> b
1
> b
2
, one can conclude that component 3
(polypropylene) produces yarn with the highest elongation.
Additionally, since b
12
and b
13
are positive, blending components
1 and 2 or components 1 and 3 produces higher elongation values
than would be expected just by averaging the elongations of the
pure blends. This is an example of 'synergistic' blending effects.
Components 2 and 3 have antagonistic blending effects because
b
23
is negative.
Contour plot
of the
predicted
elongation
values
The figure below is the contour plot of the elongation values.
From the plot it can be seen that if maximum elongation is
desired, a blend of components 1 and 3 should be chosen
consisting of about 75% - 80% component 3 and 20% - 25%
component 1.

FIGURE 5.11 Contour Plot of Predicted Elongation Values
from {3,2} Simplex-Lattice Design
The analyses in this page can be obtained using R code.
5.5.4.3. Simplex-centroid designs

5.5.4.3. Simplex-centroid designs
Definition
of simplex-
centroid
designs
A second type of mixture design is the simplex-centroid design. In
the q-component simplex-centroid design, the number of distinct
points is 2
q
- 1. These points correspond to q permutations of (1, 0,
0, ..., 0) or q single component blends, the permutations of
(.5, .5, 0, ..., 0) or all binary mixtures, the permutations of
(1/3, 1/3, 1/3, 0, ..., 0), ..., and so on, with finally the overall centroid
point (1/q, 1/q, ..., 1/q) or q-nary mixture.
The design points in the Simplex-Centroid design will support the
polynomial
Model
supported
by simplex-
centroid
designs
which is the qth-order mixture polynomial. For q = 2, this is the
quadratic model. For q = 3, this is the special cubic model.
Example of
runs for
three and
four
components
For example, the fifteen runs for a four component (q = 4) simplex-
centroid design are:
(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (.5,.5,0,0), (.5,0,.5,0) ...,
(0,0,.5,.5), (1/3,1/3,1/3,0), ...,(0,1/3,1/3,1/3), (1/4,1/4,1/4,1/4).
The runs for a three component simplex-centroid design of degree 2
are
(1,0,0), (0,1,0), (0,0,1), (.5,.5,0), (.5,0,.5), (0,.5,.5), (1/3, 1/3,
1/3).
However, in order to fit a first-order model with q =4, only the five
runs with a "1" and all "1/4's" would be needed. To fit a second-
order model, add the six runs with a ".5" (this also fits a saturated
third-order model, with no degrees of freedom left for error).
5.5.4.4. Constrained mixture designs

Upper and/or
lower bound
constraints
may be present
In mixture designs when there are constraints on the
component proportions, these are often upper and/or
lower bound constraints of the form L
i
x
i
U
i
, i = 1,
2,..., q, where L
i
is the lower bound for the i-th
component and U
i
the upper bound for the i-th
component. The general form of the constrained mixture
problem is
Typical
additional
constraints
x
1
+ x
2
+ ... + x
q
= 1
L
i
x
i
U
i
, for i = 1, 2,..., q
with L
i
0 and U
i
1.
Example using
only lower
bounds
Consider the following case in which only the lower
bounds in the above equation are imposed, so that the
constrained mixture problem becomes
x
1
+ x
2
+ ... + x
q
= 1
L
i
x
i
1, for i = 1, 2,..., q
Assume we have a three-component mixture problem
with constraints
0.3 x
1
0.4 x
2
0.1 x
3
Feasible
mixture region
The feasible mixture space is shown in the figure below.
Note that the existence of lower bounds does not affect
the shape of the mixture region, it is still a simplex
region. In general, this will always be the case if only
lower bounds are imposed on any of the component
proportions.
Diagram
showing the
feasible
mixture space

FIGURE 5.12 The Feasible Mixture Space (Shaded
Region) for Three Components with Lower Bounds
A simple
transformation
helps in design
construction
and analysis
Since the new region of the experiment is still a simplex,
it is possible to define a new set of components that take
on the values from 0 to 1 over the feasible region. This
will make the design construction and the model fitting
easier over the constrained region of interest. These new
components ( ) are called pseudo components and are
defined using the following formula
Formula for
pseudo
components
with
denoting the sum of all the lower bounds.
Computation of
the pseudo
components for
the example
In the three component example above, the pseudo
components are
Constructing
the design in
the pseudo
components
Constructing a design in the pseudo components is
accomplished by specifying the design points in terms of
the and then converting them to the original
component settings using
x
i
= L
i
+ (1 - L)
Select
appropriate
design
In terms of the pseudo components, the experimenter has
the choice of selecting a Simplex-Lattice or a Simplex-
Centroid design, depending on the objectives of the
experiment.
Simplex-
centroid design
example (after
transformation)
Suppose, we decided to use a Simplex-centroid design for
the three-component experiment. The table below shows
the design points in the pseudo components, along with
the corresponding setting for the original components.
Table showing
the design
points in both
the pseudo
components
and the
original
components
TABLE 5.5 Pseudo Component Settings and
Original Component Settings, Three-
Component Simplex-Centroid Design
Pseudo Components Original Components
X
1
X
2
X
3

1 0 0 0.5 0.4 0.1
0 1 0 0.3 0.6 0.1
0 0 1 0.3 0.4 0.3
0.5 0.5 0 0.4 0.5 0.1
0.5 0 0.5 0.4 0.4 0.2
0 0.5 0.5 0.3 0.5 0.2
0.3333 0.3333 0.3333 0.3667 0.4667 0.1666
Use of pseudo
components
(after
transformation)
is
recommended
It is recommended that the pseudo components be used to
fit the mixture model. This is due to the fact that the
constrained design space will usually have relatively high
levels of multicollinearity among the predictors. Once the
final predictive model for the pseudo components has
been determined, the equation in terms of the original
components can be determined by substituting the
relationship between x
i
and .
D-optimal
designs can
also be used
Computer-aided designs (D-optimal, for example) can be
used to select points for a mixture design in a constrained
region. See Myers and Montgomery (1995) for more
details on using D-optimal designs in mixture
experiments.
Extreme vertice
designs anre
another option
Note: There are other mixture designs that cover only a
sub-portion or smaller space within the simplex. These
types of mixture designs (not covered here) are referred to
as extreme vertices designs. (See chapter 11 of Myers and
Montgomery (1995) or Cornell (1990).
5.5.4.5. Treating mixture and process variables together

5.5.4.5. Treating mixture and process variables
together
Options for
setting up
experiments
for
processes
that have
both
standard
process
variables
and mixture
variables
Consider a mixture experiment consisting of q mixture
components and k process variables. First consider the case in
which each of the process variables to be studied has only
two levels. Orthogonally scaled factor settings for the process
variables will be used (i.e., -1 is the low level, 1 is the high
level, and 0 is the center point). Also assume that each of the
components x
i
can range from 0 to 1. The region of interest
then for the process variables is a k-dimensional hypercube.
The region of interest for the mixture components is the (q-
1)-dimensional simplex. The combined region of interest for
both the process variables and the mixture components is of
dimensionality q - 1 + k.
Example of
three
mixture
components
and three
process
variables
For example, consider three mixture components (x
1
, x
2
, x
3
)
with three process variables (z
1
, z
2
, z
3
). The dimensionality
of the region is 5. The combined region of interest for the
three mixture components and three process variables is
shown in the two figures below. The complete space of the
design can be viewed in either of two ways. The first diagram
shows the idea of a full factorial at each vertex of the three-
component simplex region. The second diagram shows the
idea of a three-component simplex region at each point in the
full factorial. In either case, the same overall process space is
being investigated.
Diagram
showing
simplex
region of a
3-
component
mixture
with a 2^3
full
factorial at
each pure
mixture run

FIGURE 5.13 Simplex Region of a Three Component
Mixture with a 2
3
Full Factorial at Each Pure Mixture
Run
Diagram
showing
process
space of a
2
3
full
factorial
with the 3-
component
simplex
region at
each point
of the full
factorial

FIGURE 5.14 Process Space of a 2
3
Full Factorial with
the Three Component Simplex Region at Each Point of
the Full Factorial
Additional
options
available
As can be seen from the above diagrams, setting up the
design configurations in the process variables and mixture
components involves setting up either a mixture design at
each point of a configuration in the process variables, or
similarly, creating a factorial arrangement in the process
variables at each point of composition in the mixture
components. For the example depicted in the above two
diagrams, this is not the only design available for this number
of mixture components with the specified number of process
variables. Another option might be to run a fractional
factorial design at each vertex or point of the mixture design,
with the same fraction run at each mixture design point. Still
another option might be to run a fractional factorial design at
each vertex or point of the mixture design, with a different
fraction run at each mixture design point.
5.5.5. How can I account for nested variation (restricted randomization)?

5.5.5. How can I account for nested variation
(restricted randomization)?
Nested data
structures are
common and
lead to many
sources of
variability
Many processes have more than one source of variation in
them. In order to reduce variation in processes, these multiple
sources must be understood, and that often leads to the
concept of nested or hierarchical data structures. For
example, in the semiconductor industry, a batch process may
operate on several wafers at a time (wafers are said to be
nested within batch). Understanding the input variables that
control variation among those wafers, as well as
understanding the variation across each wafer in a run, is an
important part of the strategy for minimizing the total
variation in the system.
Example of
nested data
Figure 5.15 below represents a batch process that uses 7
monitor wafers in each run. The plan further calls for
measuring response on each wafer at each of 9 sites. The
organization of the sampling plan has a hierarchical or nested
structure: the batch run is the topmost level, the second level
is an individual wafer, and the third level is the site on the
wafer.
The total amount of data generated per batch run will be 7*9
= 63 data points. One approach to analyzing these data would
be to compute the mean of all these points as well as their
standard deviation and use those results as responses for each
run.
Diagram
illustrating
the example
FIGURE 5.15 Hierarchical Data Structure Example
Sites nested
within wafers
and wafers
are nested
within runs
Analyzing the data as suggested above is not absolutely
incorrect, but doing so loses information that one might
otherwise obtain. For example, site 1 on wafer 1 is physically
different from site 1 on wafer 2 or on any other wafer. The
same is true for any of the sites on any of the wafers.
Similarly, wafer 1 in run 1 is physically different from wafer
1 in run 2, and so on. To describe this situation one says that
sites are nested within wafers while wafers are nested within
runs.
Nesting places
restrictions on
the
randomization
As a consequence of this nesting, there are restrictions on the
randomization that can occur in the experiment. This kind of
restricted randomization always produces nested sources of
variation. Examples of nested variation or restricted
randomization discussed on this page are split-plot and strip-
plot designs.
Wafer-to-
wafer and
site-to-site
variations are
often "noise
factors" in an
experiment
The objective of an experiment with the type of sampling
plan described in Figure 5.15 is generally to reduce the
variability due to sites on the wafers and wafers within runs
(or batches) in the process. The sites on the wafers and the
wafers within a batch become sources of unwanted variation
and an investigator seeks to make the system robust to those
sources -- in other words, one could treat wafers and sites as
noise factors in such an experiment.
Treating
wafers and
sites as
random
effects allows
calculation of
variance
estimates
Because the wafers and the sites represent unwanted sources
of variation and because one of the objectives is to reduce the
process sensitivity to these sources of variation, treating
wafers and sites as random effects in the analysis of the data
is a reasonable approach. In other words, nested variation is
often another way of saying nested random effects or nested
sources of noise. If the factors "wafers" and "sites", are
treated as random effects, then it is possible to estimate a
variance component due to each source of variation through
analysis of variance techniques. Once estimates of the
variance components have been obtained, an investigator is
then able to determine the largest source of variation in the
process under experimentation, and also determine the
magnitudes of the other sources of variation in relation to the
largest source.
Nested
random
effects same
as nested
variation
If an experiment or process has nested variation, the
experiment or process has multiple sources of random error
that affect its output. Having nested random effects in a
model is the same thing as having nested variation in a
model.
Split-Plot Designs
Split-plot
designs often
arise when
Split-plot designs result when a particular type of restricted
randomization has occurred during the experiment. A simple
factorial experiment can result in a split-plot type of design
some factors
are "hard to
vary" or when
batch
processes are
run
because of the way the experiment was actually executed.
In many industrial experiments, three situations often occur:
1. some of the factors of interest may be 'hard to vary'
while the remaining factors are easy to vary. As a
result, the order in which the treatment combinations
for the experiment are run is determined by the
ordering of these 'hard-to-vary' factors
2. experimental units are processed together as a batch
for one or more of the factors in a particular treatment
combination
3. experimental units are processed individually, one
right after the other, for the same treatment
combination without resetting the factor settings for
that treatment combination.
A split-plot
experiment
example
An experiment run under one of the above three situations
usually results in a split-plot type of design. Consider an
experiment to examine electroplating of aluminum (non-
aqueous) on copper strips. The three factors of interest are:
current (A); solution temperature (T); and the solution
concentration of the plating agent (S). Plating rate is the
measured response. There are a total of 16 copper strips
available for the experiment. The treatment combinations to
be run (orthogonally scaled) are listed below in standard
order (i.e., they have not been randomized):
Table
showing the
design matrix
TABLE 5.6 Orthogonally Scaled
Treatment Combinations from a 2
3
Full
Factorial
Current Temperature Concentration
-1 -1 -1
-1 -1 +1
-1 +1 -1
-1 +1 +1
+1 -1 -1
+1 -1 +1
+1 +1 -1
+1 +1 +1
Concentration
is hard to
vary, so
minimize the
number of
times it is
changed
Consider running the experiment under the first condition
listed above, with the factor solution concentration of the
plating agent (S) being hard to vary. Since this factor is hard
to vary, the experimenter would like to randomize the
treatment combinations so that the solution concentration
factor has a minimal number of changes. In other words, the
randomization of the treatment runs is restricted somewhat by
the level of the solution concentration factor.
Randomize so
that all runs
for one level
of
concentration
are run first
As a result, the treatment combinations might be randomized
such that those treatment runs corresponding to one level of
the concentration (-1) are run first. Each copper strip is
individually plated, meaning only one strip at a time is
placed in the solution for a given treatment combination.
Once the four runs at the low level of solution concentration
have been completed, the solution is changed to the high
level of concentration (1), and the remaining four runs of the
experiment are performed (where again, each strip is
individually plated).
Performing
replications
Once one complete replicate of the experiment has been
completed, a second replicate is performed with a set of four
copper strips processed for a given level of solution
concentration before changing the concentration and
processing the remaining four strips. Note that the levels for
the remaining two factors can still be randomized. In
addition, the level of concentration that is run first in the
replication runs can also be randomized.
Whole plot
and subplot
factors
Running the experiment in this way results in a split-plot
design. Solution concentration is known as the whole plot
factor and the subplot factors are the current and the solution
temperature.
Definition of
experimental
units and
whole plot
and subplot
factors for
this
experiment
A split-plot design has more than one size experimental unit.
In this experiment, one size experimental unit is an individual
copper strip. The treatments or factors that were applied to
the individual strips are solution temperature and current
(these factors were changed each time a new strip was placed
in the solution). The other or larger size experimental unit is
a set of four copper strips. The treatment or factor that was
applied to a set of four strips is solution concentration (this
factor was changed after four strips were processed). The
smaller size experimental unit is referred to as the subplot
experimental unit, while the larger experimental unit is
referred to as the whole plot unit.
Each size of
experimental
unit leads to
an error term
in the model
for the
experiment
There are 16 subplot experimental units for this experiment.
Solution temperature and current are the subplot factors in
this experiment. There are four whole-plot experimental units
in this experiment. Solution concentration is the whole-plot
factor in this experiment. Since there are two sizes of
experimental units, there are two error terms in the model,
one that corresponds to the whole-plot error or whole-plot
experimental unit and one that corresponds to the subplot
error or subplot experimental unit.
Partial
ANOVA table
The ANOVA table for this experiment would look, in part, as
follows:
Source DF
Replication 1
Concentration 1
Error (Whole plot) = Rep*Conc 1
Temperature 1
Rep*Temp 1
Current 1
Rep*Current 1
Temp*Conc 1
Rep*Temp*Conc 1
Temp*Current 1
Rep*Temp*Current 1
Current*Conc 1
Rep*Current*Conc 1
Temp*Current*Conc 1
Error (Subplot) =Rep*Temp*Current*Conc 1
The first three sources are from the whole-plot level, while
the next 12 are from the subplot portion. A normal
probability plot of the 12 subplot term estimates could be
used to look for significant terms.
A batch
process leads
to a different
experiment -
also a strip-
plot
Consider running the experiment under the second condition
listed above (i.e., a batch process) for which four copper
strips are placed in the solution at one time. A specified level
of current can be applied to an individual strip within the
solution. The same 16 treatment combinations (a replicated
2
3
factorial) are run as were run under the first scenario.
However, the way in which the experiment is performed
would be different. There are four treatment combinations of
solution temperature and solution concentration: (-1, -1), (-1,
1), (1, -1), (1, 1). The experimenter randomly chooses one of
these four treatments to set up first. Four copper strips are
placed in the solution. Two of the four strips are randomly
assigned to the low current level. The remaining two strips
are assigned to the high current level. The plating is
performed and the response is measured. A second treatment
combination of temperature and concentration is chosen and
the same procedure is followed. This is done for all four
temperature / concentration combinations.
This also a
split-plot
design
Running the experiment in this way also results in a split-plot
design in which the whole-plot factors are now solution
concentration and solution temperature, and the subplot
factor is current.
Defining
experimental
units
In this experiment, one size experimental unit is again an
individual copper strip. The treatment or factor that was
applied to the individual strips is current (this factor was
changed each time for a different strip within the solution).
The other or larger size experimental unit is again a set of
four copper strips. The treatments or factors that were applied
to a set of four strips are solution concentration and solution
temperature (these factors were changed after four strips
were processed).
Subplot
experimental
unit
The smaller size experimental unit is again referred to as the
subplot experimental unit. There are 16 subplot experimental
units for this experiment. Current is the subplot factor in this
experiment.
Whole-plot
experimental
unit
The larger-size experimental unit is the whole-plot
experimental unit. There are four whole plot experimental
units in this experiment and solution concentration and
solution temperature are the whole plot factors in this
experiment.
Two error
terms in the
model
There are two sizes of experimental units and there are two
error terms in the model: one that corresponds to the whole-
plot error or whole-plot experimental unit, and one that
corresponds to the subplot error or subplot experimental unit.
Partial
ANOVA table
The ANOVA for this experiment looks, in part, as follows:
Source DF
Concentration 1
Temperature 1
Error (Whole plot) = Conc*Temp 1
Current 1
Conc*Current 1
Temp*Current 1
Conc*Temp*Current 1
Error (Subplot) 8
The first three sources come from the whole-plot level and
the next 5 come from the subplot level. Since there are 8
degrees of freedom for the subplot error term, this MSE can
be used to test each effect that involves current.
Running the
experiment
under the
third scenario
Consider running the experiment under the third scenario
listed above. There is only one copper strip in the solution at
one time. However, two strips, one at the low current and
one at the high current, are processed one right after the
other under the same temperature and concentration setting.
Once two strips have been processed, the concentration is
changed and the temperature is reset to another combination.
Two strips are again processed, one after the other, under this
temperature and concentration setting. This process is
continued until all 16 copper strips have been processed.
This also a
split-plot
design
Running the experiment in this way also results in a split-plot
design in which the whole-plot factors are again solution
concentration and solution temperature and the subplot factor
is current. In this experiment, one size experimental unit is an
individual copper strip. The treatment or factor that was
applied to the individual strips is current (this factor was
changed each time for a different strip within the solution).
The other or larger-size experimental unit is a set of two
copper strips. The treatments or factors that were applied to a
pair of two strips are solution concentration and solution
temperature (these factors were changed after two strips were
processed). The smaller size experimental unit is referred to
as the subplot experimental unit.
Current is the There are 16 subplot experimental units for this experiment.
subplot factor
and
temperature
and
concentration
are the whole
plot factors
Current is the subplot factor in the experiment. There are
eight whole-plot experimental units in this experiment.
Solution concentration and solution temperature are the
whole plot factors. There are two error terms in the model,
one that corresponds to the whole-plot error or whole-plot
experimental unit, and one that corresponds to the subplot
error or subplot experimental unit.
Partial
ANOVA table
The ANOVA for this (third) approach is, in part, as follows:
Source DF
Concentration 1
Temperature 1
Conc*Temp 1
Error (Whole plot) 4
Current 1
Conc*Current 1
Temp*Current 1
Conc*Temp*Current 1
Error (Subplot) 4
The first four terms come from the whole-plot analysis and
the next 5 terms come from the subplot analysis. Note that
we have separate error terms for both the whole plot and the
subplot effects, each based on 4 degrees of freedom.
Primary
distinction of
split-plot
designs is that
they have
more than one
experimental
unit size (and
therefore
more than one
error term)
As can be seen from these three scenarios, one of the major
differences in split-plot designs versus simple factorial
designs is the number of different sizes of experimental units
in the experiment. Split-plot designs have more than one size
experimental unit, i.e., more than one error term. Since these
designs involve different sizes of experimental units and
different variances, the standard errors of the various mean
comparisons involve one or more of the variances.
Specifying the appropriate model for a split-plot design
involves being able to identify each size of experimental unit.
The way an experimental unit is defined relative to the design
structure (for example, a completely randomized design
versus a randomized complete block design) and the
treatment structure (for example, a full 2
3
factorial, a
resolution V half fraction, a two-way treatment structure with
a control group, etc.). As a result of having greater than one
size experimental unit, the appropriate model used to analyze
split-plot designs is a mixed model.
Using wrong
model can
lead to invalid
conclusions
If the data from an experiment are analyzed with only one
error term used in the model, misleading and invalid
conclusions can be drawn from the results. For a more
detailed discussion of these designs and the appropriate
analysis procedures, see Milliken, Analysis of Messy Data,
Vol. 1.
Strip-Plot Designs
Strip-plot Similar to a split-plot design, a strip-plot design can result
desgins often
result from
experiments
that are
conducted
over two or
more process
steps
when some type of restricted randomization has occurred
during the experiment. A simple factorial design can result in
a strip-plot design depending on how the experiment was
conducted. Strip-plot designs often result from experiments
that are conducted over two or more process steps in which
each process step is a batch process, i.e., completing each
treatment combination of the experiment requires more than
one processing step with experimental units processed
together at each process step. As in the split-plot design,
strip-plot designs result when the randomization in the
experiment has been restricted in some way. As a result of
the restricted randomization that occurs in strip-plot designs,
there are multiple sizes of experimental units. Therefore,
there are different error terms or different error variances that
are used to test the factors of interest in the design. A
traditional strip-plot design has three sizes of experimental
units.
Example with
two steps and
three factor
variables
Consider the following example from the semiconductor
industry. An experiment requires an implant step and an
anneal step. At both the anneal and the implant steps there are
three factors to test. The implant process accommodates 12
wafers in a batch, and implanting a single wafer under a
specified set of conditions is not practical nor does doing so
represent economical use of the implanter. The anneal
furnace can handle up to 100 wafers.
Explanation
of the
diagram that
illustrates the
design
structure of
the example
The figure below shows the design structure for how the
experiment was run. The rectangles at the top of the diagram
represent the settings for a two-level factorial design for the
three factors in the implant step (A, B, C). Similarly, the
rectangles at the lower left of the diagram represent a two-
level factorial design for the three factors in the anneal step
(D, E, F).
The arrows connecting each set of rectangles to the grid in
the center of the diagram represent a randomization of trials
in the experiment. The horizontal elements in the grid
represent the experimental units for the anneal factors. The
vertical elements in the grid represent the experimental units
for the implant factors. The intersection of the vertical and
horizontal elements represents the experimental units for the
interaction effects between the implant factors and the anneal
factors. Therefore, this experiment contains three sizes of
experimental units, each of which has a unique error term for
estimating the significance of effects.
Diagram of
the split-plot
design
FIGURE 5.16 Diagram of a strip-plot design involving
two process steps with three factors in each step
Physical
meaning of
the
experimental
units
To put actual physical meaning to each of the experimental
units in the above example, consider each cell in the grid as
an individual wafer. A batch of eight wafers goes through the
implant step first. According to the figure, treatment
combination #3 in factors A, B, and C is the first implant
treatment run. This implant treatment is applied to all eight
wafers at once. Once the first implant treatment is finished,
another set of eight wafers is implanted with treatment
combination #5 of factors A, B, and C. This continues until
the last batch of eight wafers is implanted with treatment
combination #6 of factors A, B, and C. Once all of the eight
treatment combinations of the implant factors have been run,
the anneal step starts. The first anneal treatment combination
to be run is treatment combination #5 of factors D, E, and F.
This anneal treatment combination is applied to a set of eight
wafers, with each of these eight wafers coming from one of
the eight implant treatment combinations. After this first
batch of wafers has been annealed, the second anneal
treatment is applied to a second batch of eight wafers, with
these eight wafers coming from one each of the eight implant
treatment combinations. This is continued until the last batch
of eight wafers has been implanted with a particular
combination of factors D, E, and F.
Three sizes of Running the experiment in this way results in a strip-plot
experimental
units
design with three sizes of experimental units. A set of eight
wafers that are implanted together is the experimental unit for
the implant factors A, B, and C and for all of their
interactions. There are eight experimental units for the
implant factors. A different set of eight wafers are annealed
together. This different set of eight wafers is the second size
experimental unit and is the experimental unit for the anneal
factors D, E, and F and for all of their interactions. The third
size experimental unit is a single wafer. This is the
experimental unit for all of the interaction effects between the
implant factors and the anneal factors.
Replication Actually, the above figure of the strip-plot design represents
one block or one replicate of this experiment. If the
experiment contains no replication and the model for the
implant contains only the main effects and two-factor
interactions, the three-factor interaction term A*B*C (1
degree of freedom) provides the error term for the estimation
of effects within the implant experimental unit. Invoking a
similar model for the anneal experimental unit produces the
three-factor interaction term D*E*F for the error term (1
degree of freedom) for effects within the anneal experimental
unit.
Further
information
For more details about strip-plot designs, see Milliken and
Johnson (1987) or Miller (1997).
5.5.6. What are Taguchi designs?

Taguchi
designs are
related to
fractional
factorial
designs -
many of which
are large
screening
designs
Genichi Taguchi, a Japanese engineer, proposed several approaches to
experimental designs that are sometimes called "Taguchi Methods."
These methods utilize two-, three-, and mixed-level fractional factorial
designs. Large screening designs seem to be particularly favored by
Taguchi adherents.
Taguchi refers to experimental design as "off-line quality control"
because it is a method of ensuring good performance in the design stage
of products or processes. Some experimental designs, however, such as
when used in evolutionary operation, can be used on-line while the
process is running. He has also published a booklet of design
nomograms ("Orthogonal Arrays and Linear Graphs," 1987, American
Supplier Institute) which may be used as a design guide, similar to the
table of fractional factorial designs given previously in Section 5.3.
Some of the well-known Taguchi orthogonal arrays (L9, L18, L27 and
L36) were given earlier when three-level, mixed-level and fractional
factorial designs were discussed.
If these were the only aspects of "Taguchi Designs," there would be
little additional reason to consider them over and above our previous
discussion on factorials. "Taguchi" designs are similar to our familiar
fractional factorial designs. However, Taguchi has introduced several
noteworthy new ways of conceptualizing an experiment that are very
valuable, especially in product development and industrial engineering,
and we will look at two of his main ideas, namely Parameter Design and
Tolerance Design.
Parameter Design
Taguchi
advocated
using inner
and outer
array designs
to take into
account noise
factors (outer)
and design
factors (inner)
The aim here is to make a product or process less variable (more robust)
in the face of variation over which we have little or no control. A simple
fictitious example might be that of the starter motor of an automobile
that has to perform reliably in the face of variation in ambient
temperature and varying states of battery weakness. The engineer has
control over, say, number of armature turns, gauge of armature wire, and
ferric content of magnet alloy.
Conventionally, one can view this as an experiment in five factors.
Taguchi has pointed out the usefulness of viewing it as a set-up of three
inner array factors (turns, gauge, ferric %) over which we have design
control, plus an outer array of factors over which we have control only
in the laboratory (temperature, battery voltage).
Pictorial
representation
of Taguchi
designs
Pictorially, we can view this design as being a conventional design in
the inner array factors (compare Figure 3.1) with the addition of a
"small" outer array factorial design at each corner of the "inner array"
box.
Let I1 = "turns," I2 = "gauge," I3 = "ferric %," E1 = "temperature," and
E2 = "voltage." Then we construct a 2
3
design "box" for the I's, and at
each of the eight corners so constructed, we place a 2
2
design "box" for
the E's, as is shown in Figure 5.17.
FIGURE 5.17 Inner 2
3
and outer 2
2
arrays for robust design
with Ì' the inner array, È' the outer array.
An example of
an inner and
outer array
designed
experiment
We now have a total of 8x4 = 32 experimental settings, or runs. These
are set out in Table 5.7, in which the 2
3
design in the I's is given in
standard order on the left of the table and the 2
2
design in the E's is
written out sideways along the top. Note that the experiment would not
be run in the standard order but should, as always, have its runs
randomized. The output measured is the percent of (theoretical)
maximum torque.
Table showing
the Taguchi
design and the
responses
from the
experiment
TABLE 5.7 Design table, in standard order(s) for the
parameter design of Figure 5.9
Run
Number 1 2 3 4

I1 I2 I3
E1
E2
-1
-1
+1
-1
-1
+1
+1
+1
Output
MEAN
Output
STD. DEV

1 -1 -1 -1 75 86 67 98 81.5 13.5
2 +1 -1 -1 87 78 56 91 78.0 15.6
3 -1 +1 -1 77 89 78 8 63.0 37.1
4 +1 +1 -1 95 65 77 95 83.0 14.7
5 -1 -1 +1 78 78 59 94 77.3 14.3
6 +1 -1 +1 56 79 67 94 74.0 16.3
7 -1 +1 +1 79 80 66 85 77.5 8.1
8 +1 +1 +1 71 80 73 95 79.8 10.9
Interpretation
of the table
Note that there are four outputs measured on each row. These
correspond to the four òuter array' design points at each corner of the
òuter array' box. As there are eight corners of the outer array box, there
are eight rows in all.
Each row yields a mean and standard deviation % of maximum torque.
Ideally there would be one row that had both the highest average torque
and the lowest standard deviation (variability). Row 4 has the highest
torque and row 7 has the lowest variability, so we are forced to
compromise. We can't simply `pick the winner.'
Use contour
plots to see
inside the box
One might also observe that all the outcomes occur at the corners of the
design `box', which means that we cannot see ìnside' the box. An
optimum point might occur within the box, and we can search for such a
point using contour plots. Contour plots were illustrated in the example
of response surface design analysis given in Section 4.
Fractional
factorials
Note that we could have used fractional factorials for either the inner or
outer array designs, or for both.
Tolerance Design
Taguchi also
advocated
tolerance
studies to
determine,
based on a
loss or cost
function,
which
variables have
critical
tolerances
that need to
be tightened
This section deals with the problem of how, and when, to specify
tightened tolerances for a product or a process so that quality and
performance/productivity are enhanced. Every product or process has a
numberperhaps a large numberof components. We explain here
how to identify the critical components to target when tolerances have to
be tightened.
It is a natural impulse to believe that the quality and performance of any
item can easily be improved by merely tightening up on some or all of
its tolerance requirements. By this we mean that if the old version of the
item specified, say, machining to 1 micron, we naturally believe that
we can obtain better performance by specifying machining to
micron.
This can become expensive, however, and is often not a guarantee of
much better performance. One has merely to witness the high initial and
maintenance costs of such tight-tolerance-level items as space vehicles,
expensive automobiles, etc. to realize that tolerance designthe
selection of critical tolerances and the re-specification of those critical
tolerancesis not a task to be undertaken without careful thought. In
fact, it is recommended that only after extensive parameter design
studies have been completed should tolerance design be performed as a
last resort to improve quality and productivity.
Example
Example:
measurement
of electronic
component
made up of
two
components
Customers for an electronic component complained to their supplier that
the measurement reported by the supplier on the as-delivered items
appeared to be imprecise. The supplier undertook to investigate the
matter.
The supplier's engineers reported that the measurement in question was
made up of two components, which we label x and y, and the final
measurement M was reported according to the standard formula
M = K x/y
with `K' a known physical constant. Components x and y were measured
separately in the laboratory using two different techniques, and the
results combined by software to produce M. Buying new measurement
devices for both components would be prohibitively expensive, and it
was not even known by how much the x or y component tolerances
should be improved to produce the desired improvement in the precision
of M.
Taylor series
expansion
Assume that in a measurement of a standard item the `true' value of x is
x
o
and for y it is y
o
. Let f(x, y) = M; then the Taylor Series expansion
for f(x, y) is
with all the partial derivatives, `df/dx', etc., evaluated at (x
o
, y
o
).
Apply formula
to M
Applying this formula to M(x, y) = Kx/y, we obtain
It is assumed known from experience that the measurements of x show a
distribution with an average value x
o
, and with a standard deviation
x
=
0.003 x-units.
Assume
distribution of
x is normal
In addition, we assume that the distribution of x is normal. Since 99.74%
of a normal distribution's range is covered by 6 , we take 3
x
= 0.009 x-
units to be the existing tolerance T
x
for measurements on x. That is, T
x
=
0.009 x-units is the `play' around x
o
that we expect from the existing
measurement system.
Assume
distribution of
y is normal
It is also assumed known that the y measurements show a normal
distribution around y
o
, with standard deviation
y
= 0.004 y-units. Thus
T
y
= 3
y
= 0.012.
Worst case
values
Now T
x
and T
y
may be thought of as `worst case' values for (x-x
o
)
and (y-y
o
). Substituting T
x
for (x-x
o
) and T
y
for (y-y
o
) in the expanded
formula for M(x, y), we have
Drop some
terms
The and T
x
T
y
terms, and all terms of higher order, are going to be at
least an order of magnitude smaller than terms in T
x
and in T
y
, and for
this reason we drop them, so that
Worst case
Euclidean
distance
Thus, a `worst case' Euclidean distance of M(x, y) from its ideal value
Kx
o
/y
o
is (approximately)
This shows the relative contributions of the components to the variation
in the measurement.
Economic
decision
As y
o
is a known quantity and reduction in T
x
and in T
y
each carries its
own price tag, it becomes an economic decision whether one should
spend resources to reduce T
x
or T
y
, or both.
Simulation an
alternative to
Taylor series
approximation
In this example, we have used a Taylor series approximation to obtain a
simple expression that highlights the benefit of T
x
and T
y
. Alternatively,
one might simulate values of M = K*x/y, given a specified (T
x
,T
y
) and
(x
0
,y
0
), and then summarize the results with a model for the variability
of M as a function of (T
x
,T
y
).
Functional
form may not
be available
In other applications, no functional form is available and one must use
experimentation to empirically determine the optimal tolerance design.
See Bisgaard and Steinberg (1997).
5.5.7. What are John's 3/4 fractional factorial designs?

John's
designs
require only
3/4 of the
number of
runs a full
2
n
factorial
would
require
Three-quarter () designs are two-level factorial designs that require
only three-quarters of the number of runs of the òriginal' design. For
example, instead of making all of the sixteen runs required for a 2
4
fractional factorial design, we need only run 12 of them. Such designs
were invented by Professor Peter John of the University of Texas, and
are sometimes called`John's designs.'
Three-quarter fractional factorial designs can be used to save on
resources in two different contexts. In one scenario, we may wish to
perform additional runs after having completed a fractional factorial, so
as to de-alias certain specific interaction patterns. Second , we may wish
to use a design to begin with and thus save on 25% of the run
requirement of a regular design.
Semifolding Example
Four
experimental
factors
We have four experimental factors to investigate, namely X1, X2, X3,
and X4, and we have designed and run a 2
4-1
fractional factorial design.
Such a design has eight runs, or rows, if we don't count center point runs
(or replications).
Resolution
IV design
The 2
4-1
design is of resolution IV, which means that main effects are
confounded with, at worst, three-factor interactions, and two-factor
interactions are confounded with other two factor interactions.
Design
matrix
The design matrix, in standard order, is shown in Table 5.8 along with all
the two-factor interaction columns. Note that the column for X4 is
constructed by multiplying columns for X1, X2, and X3 together (i.e.,
4=123).
Table 5.8 The 2
4-1
design plus 2-factor interaction columns shown in
standard order. Note that 4=123.
Run Two-Factor Interaction Columns
Number X1 X2 X3 X4 X1*X2 X1*X3 X1*X4 X2*X3 X2*X4 X3*X4

1 -1 -1 -1 -1 +1 +1 +1 +1 +1 +1
2 +1 -1 -1 +1 -1 -1 +1 +1 -1 -1
3 -1 +1 -1 +1 -1 +1 -1 -1 +1 -1
4 +1 +1 -1 -1 +1 -1 -1 -1 -1 +1
5 -1 -1 +1 +1 +1 -1 -1 -1 -1 +1
6 +1 -1 +1 -1 -1 +1 -1 -1 +1 -1
7 -1 +1 +1 -1 -1 -1 +1 +1 -1 -1
8 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1
Confounding
of two-
factor
interactions
Note also that 12=34, 13=24, and 14=23. These follow from the
generating relationship 4=123 and tells us that we cannot estimate any
two-factor interaction that is free of some other two-factor alias.
Estimating
two-factor
interactions
free of
confounding
Suppose that we became interested in estimating some or all of the two-
factor interactions that involved factor X1; that is, we want to estimate
one or more of the interactions 12, 13, and 14 free of two-factor
confounding.
One way of doing this is to run the òther half' of the designan
additional eight rows formed from the relationship 4 = -123. Putting these
two `halves' togetherthe original one and the new one, we'd obtain a 2
4
design in sixteen runs. Eight of these runs would already have been run,
so all we'd need to do is run the remaining half.
Alternative
method
requiring
fewer runs
There is a way, however, to obtain what we want while adding only four
more runs. These runs are selected in the following manner: take the four
rows of Table 5.8 that have `-1' in the `X1' column and switch the `-' sign
under X1 to `+' to obtain the four-row table of Table 5.9. This is called a
foldover on X1, choosing the subset of runs with X1 = -1. Note that this
choice of 4 runs is not unique, and that if the initial design suggested that
X1 = -1 were a desirable level, we would have chosen to experiment at
the other four treatment combinations that were omitted from the initial
design.
Table of the
additional
design
points
TABLE 5.9 Foldover on `X1' of the
2
4-1
design of Table 5.5
Run
Number X1 X2 X3 X4
9 +1 -1 -1 -1
10 +1 +1 -1 +1
11 +1 -1 +1 +1
12 +1 +1 +1 -1
Table with
new design
points added
to the
original
design
points
Add this new block of rows to the bottom of Table 5.8 to obtain a design
in twelve rows. We show this in Table 5.10 and also add in the two-
factor interactions as well for illustration (not needed when we do the
runs).
TABLE 5.10 A twelve-run design based on the 2
4-1
also showing all
two-factor interaction columns
Run Two-Factor Interaction Columns
Number X1 X2 X3 X4 X1*X2 X1*X3 X1*X4 X2*X3 X2*X4 X3*X4

1 -1 -1 -1 -1 +1 +1 +1 +1 +1 +1
2 +1 -1 -1 +1 -1 -1 +1 +1 -1 -1
3 -1 +1 -1 +1 -1 +1 -1 -1 +1 -1
4 +1 +1 -1 -1 +1 -1 -1 -1 -1 +1
5 -1 -1 +1 +1 +1 -1 -1 -1 -1 +1
6 +1 -1 +1 -1 -1 +1 -1 -1 +1 -1
7 -1 +1 +1 -1 -1 -1 +1 +1 -1 -1
8 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1
1 +1 -1 -1 -1 -1 -1 -1 +1 +1 +1
10 +1 +1 -1 +1 +1 -1 +1 -1 +1 -1
11 +1 -1 +1 +1 -1 +1 +1 -1 -1 +1
12 +1 +1 +1 -1 +1 +1 -1 +1 -1 -1
Design is
resolution V
Examine the two-factor interaction columns and convince yourself that
no two are alike. This means that no two-factor interaction involving X1
is aliased with any other two-factor interaction. Thus, the design is
resolution V, which is not always the case when constructing these types
of foldover designs.
Estimating
X1 two-
factor
interactions
What we now have is a design with 12 runs, with which we can estimate
all the two-factor interactions involving X1 free of aliasing with any
other two-factor interaction. It is called a design because it has the
number of rows of the next regular factorial design (a 2
4
).
Standard
errors of
effect
estimates
If one fits a model with an intercept, a block effect, the four main effects
and the six two-factor interactions, then each coefficient has a standard
error of /8
1/2
- instead of /12
1/2
- because the design is not orthogonal
and each estimate is correlated with two other estimates. Note that no
degrees of freedom exists for estimating . Instead, one should plot the
10 effect estimates using a normal (or half-normal) effects plot to judge
which effects to declare significant.
Further
information
For more details on fractions obtained by adding a follow-up design
that is half the size of the original design, see Mee and Peralta (2000).
Next we consider an example in which a fraction arises when the ()
2
k-p
design is planned from the start because it is an efficient design that
allows estimation of a sufficient number of effects.
A 48-Run 3/4 Design Example
Estimate all
main effects
and two-
factor
interactions
Suppose we wish to run an experiment for k=8 factors, with which we
want to estimate all main effects and two-factor interactions. We could
use the design described in the summary table of fractional factorial
designs, but this would require a 64-run experiment to estimate the 1 + 8
+ 28 = 37 desired coefficients. In this context, and especially for larger
for 8 factors
resolution V designs, of the design points will generally suffice.
Construction
of the 48-
run design
The 48 run-design is constructed as follows: start by creating the full
design using the generators 7 = 1234 and 8 = 1256. The defining
relation is I = 12347 = 12568 = 345678 (see the summary table details for
this design).
Next, arrange these 64 treatment combinations into four blocks of size 16,
blocking on the interactions 135 and 246 (i.e., block 1 has 135 = 246 = -1
runs, block 2 has 135 = -1, 246 = +1, block 3 has 135 = +1, 246 = -1 and
block 4 has 135 = 246 = +1). If we exclude the first block in which 135 =
246 = -1, we have the desired design reproduced below (the reader can
verify that these are the runs described in the summary table, excluding
the runs numbered 1, 6, 11, 16, 18, 21, 28, 31, 35, 40, 41,46, 52, 55, 58
and 61).
Table
containing
the design
matrix
X1 X2 X3 X4 X5 X6 X7 X8
+1 -1 -1 -1 -1 -1 -1 -1
-1 +1 -1 -1 -1 -1 -1 -1
+1 +1 -1 -1 -1 -1 +1 +1
-1 -1 +1 -1 -1 -1 -1 +1
-1 +1 +1 -1 -1 -1 +1 -1
+1 +1 +1 -1 -1 -1 -1 +1
-1 -1 -1 +1 -1 -1 -1 +1
+1 -1 -1 +1 -1 -1 +1 -1
+1 +1 -1 +1 -1 -1 -1 +1
-1 -1 +1 +1 -1 -1 +1 +1
+1 -1 +1 +1 -1 -1 -1 -1
-1 +1 +1 +1 -1 -1 -1 -1
-1 -1 -1 -1 +1 -1 +1 -1
-1 +1 -1 -1 +1 -1 -1 +1
+1 +1 -1 -1 +1 -1 +1 -1
+1 -1 +1 -1 +1 -1 +1 +1
-1 +1 +1 -1 +1 -1 +1 +1
+1 +1 +1 -1 +1 -1 -1 -1
-1 -1 -1 +1 +1 -1 -1 -1
+1 -1 -1 +1 +1 -1 +1 +1
-1 +1 -1 +1 +1 -1 +1 +1
-1 -1 +1 +1 +1 -1 +1 -1
+1 -1 +1 +1 +1 -1 -1 +1
+1 +1 +1 +1 +1 -1 +1 -1
-1 -1 -1 -1 -1 +1 +1 -1
+1 -1 -1 -1 -1 +1 -1 +1
+1 +1 -1 -1 -1 +1 +1 -1
-1 -1 +1 -1 -1 +1 -1 -1
+1 -1 +1 -1 -1 +1 +1 +1
-1 +1 +1 -1 -1 +1 +1 +1
+1 -1 -1 +1 -1 +1 +1 +1
-1 +1 -1 +1 -1 +1 +1 +1
+1 +1 -1 +1 -1 +1 -1 -1
-1 -1 +1 +1 -1 +1 +1 -1
-1 +1 +1 +1 -1 +1 -1 +1
+1 +1 +1 +1 -1 +1 +1 -1
-1 -1 -1 -1 +1 +1 +1 +1
+1 -1 -1 -1 +1 +1 -1 -1
-1 +1 -1 -1 +1 +1 -1 -1
-1 -1 +1 -1 +1 +1 -1 +1
+1 -1 +1 -1 +1 +1 +1 -1
+1 +1 +1 -1 +1 +1 -1 +1
-1 -1 -1 +1 +1 +1 -1 +1
-1 +1 -1 +1 +1 +1 +1 -1
+1 +1 -1 +1 +1 +1 -1 +1
+1 -1 +1 +1 +1 +1 -1 -1
-1 +1 +1 +1 +1 +1 -1 -1
+1 +1 +1 +1 +1 +1 +1 +1
Good
precision for
coefficient
estimates
This design provides 11 degrees of freedom for error and also provides
good precision for coefficient estimates (some of the coefficients have a
standard error of and some have a standard error of
).
Further
information
More about John's designs can be found in John (1971) or Diamond
(1989).
5.5.8. What are small composite designs?

Small
composite
designs save
runs,
compared to
Resolution V
response
surface
designs, by
adding star
points to a
Resolution
III design
Response surface designs (RSD) were described earlier. A
typical RSD requires about 13 runs for 2 factors, 20 runs for
3 factors, 31 runs for 4 factors, and 32 runs for 5 factors. It
is obvious that, once you have four or more factors you wish
to include in a RSD, you will need more than one lot (i.e.,
batch) of experimental units for your basic design. This is
what most statistical software today will give you. However,
there is a way to cut down on the number of runs, as
suggested by H.O. Hartley in his paper 'Smallest Composite
Designs for Quadratic Response Surfaces', published in
Biometrics, December 1959.
This method addresses the theory that using a Resolution V
design as the smallest fractional design to create a RSD is
unnecessary. The method adds star points to designs of
Resolution III and uses the star points to clear the main
effects of aliasing with the two-factor interactions. The
resulting design allows estimation of the higher-order
interactions. It also provides poor interaction coefficient
estimates and should not be used unless the error variability
is negligible compared to the systematic effects of the
factors.
Useful for 4
or 5 factors
This could be particularly useful when you have a design
containing four or five factors and you wish to only use the
experimental units from one lot (i.e., batch).
Table
containing
design
matrix for
four factors
The following is a design for four factors. You would want
to randomize these runs before implementing them; -1 and
+1 represent the low and high settings, respectively, of each
factor.
TABLE 5.11 Four factors: Factorial design section
is based on a generator of I = X1*X2*X3,
Resolution III; - and + are the star points,
calculated beyond the factorial range; 0 represents
the midpoint of the factor range.
Row X1 X2 X3 X4
1 +1 -1 -1 -1
2 -1 +1 -1 -1
3 -1 -1 +1 -1
4 +1 +1 +1 -1
5 +1 -1 -1 +1
6 -1 +1 -1 +1
7 -1 -1 +1 +1
8 +1 +1 +1 +1
9 - 0 0 0
10 0 0 0
11 0 - 0 0
12 0 0 0
13 0 0 - 0
14 0 0 0
15 0 0 0 -
16 0 0 0
17 0 0 0 0
18 0 0 0 0
19 0 0 0 0
20 0 0 0 0
Determining in Small Composite Designs
based on
number of
treatment
combinations
in the
factorial
portion
To maintain rotatability for usual CCD's, the value of is
determined by the number of treatment combinations in the
factorial portion of the central composite design:
Small
composite
designs not
rotatable
However, small composite designs are not rotatable,
regardless of the choice of . For small composite designs,
should not be smaller than [number of factorial runs]
1/4
nor larger than k
1/2
.
5.5.9. An EDA approach to experimental design

Introduction This section presents an exploratory data analysis (EDA)
approach to analyzing the data from a designed experiment.
This material is meant to complement, not replace, the more
model-based approach for analyzing experiment designs
given in section 4 of this chapter.
Choosing an appropriate design is discussed in detail in
section 3 of this chapter.
Starting point
Problem
category
The problem category we will address is the screening
problem. Two characteristics of screening problems are:
1. There are many factors to consider.
2. Each of these factors may be either continuous or
discrete.
Desired
output
The desired output from the analysis of a screening problem
is:
A ranked list (by order of importance) of factors.
The best settings for each of the factors.
A good model.
Insight.
Problem
essentials
The essentials of the screening problem are:
There are k factors with n observations.
The generic model is:
Y = f(X
1
, X
2
, ..., X
k
) +
Design type
In particular, the EDA approach is applied to 2
k
full factorial
and 2
k-p
fractional factorial designs.
An EDA approach is particularly applicable to screening
designs because we are in the preliminary stages of
understanding our process.
EDA
philosophy
EDA is not a single technique. It is an approach to analyzing
data.
EDA is data-driven. That is, we do not assume an
initial model. Rather, we attempt to let the data speak
for themselves.
EDA is question-based. That is, we select a technique
to answer one or more questions.
EDA utilizes multiple techniques rather than depending
on a single technique. Different plots have a different
basis, focus, and sensitivities, and therefore may bring
out different aspects of the data. When multiple
techniques give us a redundancy of conclusions, this
increases our confidence that our conclusions are valid.
When they give conflicting conclusions, this may be
giving us a clue as to the nature of our data.
EDA tools are often graphical. The primary objective is
to provide insight into the data, which graphical
techniques often provide more readily than quantitative
techniques.
10-Step
process
The following is a 10-step EDA process for analyzing the
data from 2
k
full factorial and 2
k-p
fractional factorial
designs.
2. DOE scatter plot
3. DOE mean plot
4. Interaction effects matrix plot
5. Block plot
6. DOE Youden plot
7. |Effects| plot
8. Half-normal probability plot
10. DOE contour plot
Each of these plots will be presented with the following
format:
Purpose of the plot
Output of the plot
Definition of the plot
Motivation for the plot
An example of the plot using the defective springs data
A discussion of how to interpret the plot
Conclusions we can draw from the plot for the
defective springs data
Data set
Defective
springs
data
The plots presented in this section are demonstrated with a
data set from Box and Bisgaard (1987).
These data are from a 2
3
full factorial data set that contains
the following variables:
1. Response variable Y = percentage of springs without
cracks
2. Factor 1 = oven temperature (2 levels: 1450 and 1600
F)
3. Factor 2 = carbon concentration (2 levels: 0.5% and
0.7%)
4. Factor 3 = quench temperature (2 levels: 70 and 120 F)
Y X1 X2 X3
Percent Oven Carbon Quench
Acceptable Temperature Concentration
Temperature
--------------------------------------------------
--
67 -1 -1 -1
79 +1 -1 -1
61 -1 +1 -1
75 +1 +1 -1
59 -1 -1 +1
90 +1 -1 +1
52 -1 +1 +1
87 +1 +1 +1
5.5.9.1. Ordered data plot

Purpose The ordered data plot answers the following two questions:
1. What is the best setting (based on the data) for each of the k factors?
2. What is the most important factor?
In the above two questions, the terms "best" and "important" need more precise
definitions.
Settings may be declared as "best" in three different ways:
1. "best" with respect to the data;
2. "best" on average;
3. "best" with respect to predicted values from an adequate model.
In the worst case, each of the above three criteria may yield different "best settings".
If that occurs, then the three answers must be consolidated at the end of the 10-step
process.
The ordered data plot will yield best settings based on the first criteria (data). That is,
this technique yields those settings that correspond to the best response value, with
the best value dependent upon the project goals:
1. maximization of the response;
2. minimization of the response;
3. hitting a target for the response.
This, in turn, trivially yields the best response value:
1. maximization: the observed maximum data point;
2. minimization: the observed minimum data point;
3. target: the observed data value closest to the specified target.
With respect to the most "important" factor, this by default refers to the single factor
which causes the greatest change in the value of the response variable as we proceed
from the "-" setting to the "+" setting of the factor. In practice, if a factor has one
setting for the best and near-best response values and the opposite setting for the
worst and near-worst response values, then that factor is usually the most important
factor.
Output The output from the ordered data plot is:
1. Primary: Best setting for each of the k factors.
2. Secondary: The name of the most important factor.
Definition An ordered data plot is formed by:
Vertical Axis: The ordered (smallest to largest) raw response value for each of
the n runs in the experiment.
Horizontal Axis: The corresponding dummy run index (1 to n) with (at each
run) a designation of the corresponding settings (- or +) for each of the k
factors.
In essence, the ordered data plot may be viewed as a scatter plot of the ordered data
versus a single n-treatment consolidation factor.
Motivation To determine the best setting, an obvious place to start is the best response value.
What constitutes "best"? Are we trying to maximize the response, minimize the
response, or hit a specific target value? This non-statistical question must be
addressed and answered by the analyst. For example, if the project goal is ultimately
to achieve a large response, then the desired experimental goal is maximization. In
such a case, the analyst would note from the plot the largest response value and the
corresponding combination of the k-factor settings that yielded that best response.
Plot for
defective
springs
data
Applying the ordered response plot for the defective springs data set yields the
following plot.
How to
interpret
From the ordered data plot, we look for the following:
1. best settings;
2. most important factor.
Best Settings (Based on the Data):
At the best (highest or lowest or target) response value, what are the corresponding
settings for each of the k factors? This defines the best setting based on the raw data.
Most Important Factor:
For the best response point and for the nearby neighborhood of near-best response
points, which (if any) of the k factors has consistent settings? That is, for the subset
of response values that is best or near-best, do all of these values emanate from an
identical level of some factor?
Alternatively, for the best half of the data, does this half happen to result from some
factor with a common setting? If yes, then the factor that displays such consistency
is an excellent candidate for being declared the "most important factor". For a
balanced experimental design, when all of the best/near-best response values come
from one setting, it follows that all of the worst/near-worst response values will
come from the other setting of that factor. Hence that factor becomes "most
important".
At the bottom of the plot, step though each of the k factors and determine which
factor, if any, exhibits such behavior. This defines the "most important" factor.
Conclusions
for the
defective
springs
data
The application of the ordered data plot to the defective springs data set results in
the following conclusions:
1. Best Settings (Based on the Data):
(X
1
, X
2
, X
3
) = (+, -, +) = (+1, -1, +1) is the best setting since
1. the project goal is maximization of the percent acceptable springs;
2. Y = 90 is the largest observed response value; and
3. (X
1
, X
2
, X
3
) = (+, -, +) at Y = 90.
2. Most important factor:
X
1
is the most important factor since the four largest response values (90, 87,
79, and 75) have factor X
1
at +1, and the four smallest response values (52,
59, 61, and 67) have factor X
1
at -1.
5.5.9.2. DOE scatter plot

Purpose The DOE (design of experiments) scatter plot answers the following three questions:
1. What are the most important factors?
2. What is the best setting for each of these important factors?
3. What data points are outliers?
In the above questions, the terms "important", "best", and "outliers" need
clarification and specificity:
Important
A factor can be "important" if it leads to a significant shift in either the location or
the variation of the response variable as we go from the "-" setting to the "+" setting
of the factor. Both definitions are relevant and acceptable. The default definition of
"important" in engineering/scientific applications is a shift in location. Unless
specified otherwise, when a factor is claimed to be important, the implication is that
the factor caused a large location shift in the response.
Best
A factor setting is "best" if it results in a typical response that is closest, in location,
to the desired project goal (maximization, minimization, target). This desired project
goal is an engineering, not a statistical, question, and so the desired optimization
goal must be specified by the engineer.
Outlier
A data point is an "outlier" if it comes from a different probability distribution or
from a different deterministic model than the remainder of the data. A single outlier
in a data set can affect all effect estimates and so in turn can potentially invalidate
the factor rankings in terms of importance.
Given the above definitions, the DOE scatter plot is a useful early-step tool for
determining the important factors, best settings, and outliers. An alternate name for
the DOE scatter plot is "main effects plot".
Output The output for the DOE scatter plot is:
1. Primary: Identification of the important factors.
2. Secondary: Best setting for these factors and identification of outliers.
Definition The DOE scatter plot is formed by
Vertical Axis: The response (= the raw data) for a given setting (- or +) of a
factor for each of the k factors.
Horizontal Axis: The k factors, and the two settings (- and +) within each
factor.
Motivation The scatter plot is the primary data analysis tool for determining if and how a
response relates to another factor. Determining if such a relationship exists is a
necessary first step in converting statistical association to possible engineering
cause-and-effect. Looking at how the raw data change as a function of the different
levels of a factor is a fundamental step which, it may be argued, should never be
skipped in any data analysis.
From such a foundational plot, the analyst invariably extracts information dealing
with location shifts, variation shifts, and outliers. Such information may easily be
washed out by other "more advanced" quantitative or graphical procedures (even
computing and plotting means!). Hence there is motivation for the DOE scatter plot.
If we were interested in assessing the importance of a single factor, and since
"important" by default means shift in location, then the simple scatter plot is an ideal
tool. A large shift (with little data overlap) in the body of the data from the "-"
setting to the "+" setting of a given factor would imply that the factor is important.
A small shift (with much overlap) would imply the factor is not important.
The DOE scatter plot is actually a sequence of k such scatter plots with one scatter
plot for each factor.
Plot for
defective
springs
data
The DOE scatter plot for the defective springs data set is as follows.
How to
interpret
As discussed previously, the DOE scatter plot is used to look for the following:
1. Most Important Factors;
2. Best Settings of the Most Important Factors;
3. Outliers.
Each of these will be discussed in turn.
Most Important Factors:
For each of the k factors, as we go from the "-" setting to the "+" setting within the
factor, is there a location shift in the body of the data? If yes, then
1. Which factor has the biggest such data location shift (that is, has least data
overlap)? This defines the "most important factor".
2. Which factor has the next biggest shift (that is, has next least data overlap)?
This defines the "second most important factor".
3. Continue for the remaining factors.
In practice, the DOE scatter plot will typically only be able to discriminate the most
important factor (largest shift) and perhaps the second most important factor (next
largest shift). The degree of overlap in remaining factors is frequently too large to
ascertain with certainty the ranking for other factors.
Best Settings for the Most Important Factors:
For each of the most important factors, which setting ("-" or "+") yields the "best"
response?
In order to answer this question, the engineer must first define "best". This is done
with respect to the overall project goal in conjunction with the specific response
variable under study. For some experiments (e.g., maximizing the speed of a chip),
"best" means we are trying to maximize the response (speed). For other experiments
(e.g., semiconductor chip scrap), "best" means we are trying to minimize the
response (scrap). For yet other experiments (e.g., designing a resistor) "best" means
we are trying to hit a specific target (the specified resistance). Thus the definition of
"best" is an engineering precursor to the determination of best settings.
Suppose the analyst is attempting to maximize the response. In such a case, the
analyst would proceed as follows:
1. For factor 1, for what setting (- or +) is the body of the data higher?
2. For factor 2, for what setting (- or +) is the body of the data higher?
The resulting k-vector of best settings:
(x1best, x2best, ..., xkbest)
is thus theoretically obtained by looking at each factor individually in the DOE
scatter plot and choosing the setting (- or +) that has the body of data closest to the
desired optimal (maximal, minimal, target) response.
As indicated earlier, the DOE scatter plot will typically be able to estimate best
settings for only the first few important factors. Again, the degree of data overlap
precludes ascertaining best settings for the remaining factors. Other tools, such as
the DOE mean plot, will do a better job of determining such settings.
Outliers:
Do any data points stand apart from the bulk of the data? If so, then such values are
candidates for further investigation as outliers. For multiple outliers, it is of interest
to note if all such anomalous data cluster at the same setting for any of the various
factors. If so, then such settings become candidates for avoidance or inclusion,
depending on the nature (bad or good), of the outliers.
Conclusions
for the
defective
springs
data
The application of the DOE scatter plot to the defective springs data set results in the
following conclusions:
1. Most Important Factors:
1. X
1
(most important);
2. X
2
(of lesser importance);
3. X
3
(of least importance).
that is,
factor 1 definitely looks important;
factor 2 is a distant second;
factor 3 has too much overlap to be important with respect to location,
but is flagged for further investigation due to potential differences in
variation.
2. Best Settings:
(X
1
, X
2
, X
3
) = (+, -, -) = (+1, -1, -1)
3. Outliers: None detected.
5.5.9.3. DOE mean plot

Purpose The DOE (design of experiments) mean plot answers the following two questions:
1. What is the ranked list of factors (not including the interactions)? The ranking
is from the most important factor to least important factor.
2. What is the best setting for each of the k factors?
In the above two questions, the terms "important" and "best" need clarification and
specificity.
A factor can be important if it leads to a significant shift in the location of the
response variable as we go from the "-" setting of the factor to the "+" setting of the
factor. Alternatively, a factor can be important if it leads to a significant change in
variation (spread) as we go from the "-" to the "+" settings. Both definitions are
relevant and acceptable. The default definition of "important" in
engineering/scientific applications is the former (shift in location). Unless specified
to the contrary, when a factor is claimed to be important, the implication is that the
factor caused a large location shift in the response.
In this context, a factor setting is best if it results in a typical response that is closest
(in location) to the desired project goal (that is, a maximization, minimization, or
hitting a target). This desired project goal is an engineering, not a statistical,
question, and so the desired optimization goal must be overtly specified by the
engineer.
Given the above two definitions of important and best, the DOE mean plot is a
useful tool for determining the important factors and for determining the best
settings.
An alternate name for the DOE mean plot is the "main effects plot".
Output The output from the DOE mean plot is:
1. Primary: A ranked list of the factors (not including interactions) from most
important to least important.
2. Secondary: The best setting for each of the k factors.
Definition The DOE mean plot is formed by:
Vertical Axis: The mean response for a given setting ("-" or "+") of a factor,
for each of the k factors.
Horizontal Axis: The k factors and the two settings ("-" and "+") within each
factor.
Motivation If we were interested in assessing the importance of a single factor, and since
important, by default, means shift in location, and the average is the simplest
location estimator, a reasonable graphics tool to assess a single factor's importance
would be a simple mean plot. The vertical axis of such a plot would be the mean
response for each setting of the factor and the horizontal axis is the two settings of
the factor: "-" and "+" (-1 and +1). A large difference in the two means would imply
the factor is important while a small difference would imply the factor is not
important.
The DOE mean plot is actually a sequence of k such plots, with one mean plot for
each factor. To assist in comparability and relative importance, all of the mean plots
are on the same scale.
Plot for
defective
springs
data
Applying the DOE mean plot to the defective springs data yields the following plot.
How to
interpret
From the DOE mean plot, we look for the following:
1. A ranked list of factors from most important to least important.
2. The best settings for each factor (on average).
Ranked List of Factors--Most Important to Least Important:
For each of the k factors, as we go from the "-" setting to the "+" setting for the
factor, is there a shift in location of the average response?
If yes, we would like to identify the factor with the biggest shift (the "most
important factor"), the next biggest shift (the "second most important factor"), and
so on until all factors are accounted for.
Since we are only plotting the means and each factor has identical (-,+) = (-1,+1)
coded factor settings, the above simplifies to
1. What factor has the steepest line? This is the most important factor.
2. The next steepest line? This is the second most important factor.
This ranking of factors based on local means is the most important step in building
the definitive ranked list of factors as required in screening experiments.
Best Settings (on Average):
For each of the k factors, which setting (- or +) yields the "best" response?
In order to answer this, the engineer must first define "best". This is done with
respect to the overall project goal in conjunction with the specific response variable
under study. For some experiments, "best" means we are trying to maximize the
response (e.g., maximizing the speed of a chip). For other experiments, "best" means
we are trying to minimize the response (e.g., semiconductor chip scrap). For yet
other experiments, "best" means we are trying to hit a specific target (e.g., designing
a resistor to match a specified resistance). Thus the definition of "best" is a
precursor to the determination of best settings.
For example, suppose the analyst is attempting to maximize the response. In that
case, the analyst would proceed as follows:
1. For factor 1, what setting (- or +) has the largest average response?
2. For factor 2, what setting (- or +) has the largest average response?
The resulting k-vector of best settings:
is in general obtained by looking at each factor individually in the DOE mean plot
and choosing that setting (- or +) that has an average response closest to the desired
optimal (maximal, minimal, target) response.
This candidate for best settings is based on the averages. This k-vector of best
settings should be similar to that obtained from the DOE scatter plot, though the
DOE mean plot is easier to interpret.
Conclusions
for the
defective
springs
data
The application of the DOE mean plot to the defective springs data set results in the
1. Ranked list of factors (excluding interactions):
1. X
1
(most important). Qualitatively, this factor looks definitely
important.
2. X
2
(of lesser importantance). Qualitatively, this factor is a distant
second to X
1
.
3. X
3
(unimportant). Qualitatively, this factor appears to be unimportant.
2. Best settings (on average):
(X
1
, X
2
, X
3
) = (+, -, +) = (+1, -1, +1)
5.5.9.4. Interaction effects matrix plot

Purpose The interaction effects matrix plot is an extension of the DOE mean plot to include
both main effects and 2-factor interactions (the DOE mean plot focuses on main
effects only). The interaction effects matrix plot answers the following two
questions:
1. What is the ranked list of factors (including 2-factor interactions), ranked from
most important to least important; and
2. What is the best setting for each of the k factors?
For a k-factor experiment, the effect on the response could be due to main effects
and various interactions all the way up to k-term interactions. As the number of
factors, k, increases, the total number of interactions increases exponentially. The
total number of possible interactions of all orders = 2
k
- 1 - k. Thus for k = 3, the
total number of possible interactions = 4, but for k = 7 the total number of possible
interactions = 120.
In practice, the most important interactions are likely to be 2-factor interactions. The
total number of possible 2-factor interactions is
Thus for k = 3, the number of 2-factor interactions = 3, while for k = 7, the number
of 2-factor interactions = 21.
It is important to distinguish between the number of interactions that are active in a
given experiment versus the number of interactions that the analyst is capable of
making definitive conclusions about. The former depends only on the physics and
engineering of the problem. The latter depends on the number of factors, k, the
choice of the k factors, the constraints on the number of runs, n, and ultimately on
the experimental design that the analyst chooses to use. In short, the number of
possible interactions is not necessarily identical to the number of interactions that
we can detect.
Note that
1. with full factorial designs, we can uniquely estimate interactions of all orders;
2. with fractional factorial designs, we can uniquely estimate only some (or at
times no) interactions; the more fractionated the design, the fewer interactions
that we can estimate.
Output The output for the interaction effects matrix plot is
1. Primary: Ranked list of the factors (including 2-factor interactions) with the
factors are ranked from important to unimportant.
2. Secondary: Best setting for each of the k factors.
Definition The interaction effects matrix plot is an upper right-triangular matrix of mean plots
consisting of k main effects plots on the diagonal and k*(k-1)/2 2-factor interaction
effects plots on the off-diagonal.
In general, interactions are not the same as the usual (multiplicative) cross-products.
However, for the special case of 2-level designs coded as (-, +) = (-1, +1), the
interactions are identical to cross-products. By way of contrast, if the 2-level
designs are coded otherwise (e.g., the (1, 2) notation espoused by Taguchi and
others), then this equivalance is not true. Mathematically,
{-1, +1} x {-1, +1} => {-1, +1}
but
{1, 2} x {1, 2} => {1, 2, 4}
Thus, coding does make a difference. We recommend the use of the (-, +) coding.
It is remarkable that with the - and + coding, the 2-factor interactions are dealt with,
interpreted, and compared in the same way that the k main effects are handled. It is
thus natural to include both 2-factor interactions and main effects within the same
matrix plot for ease of comparison.
For the off-diagonal terms, the first construction step is to form the horizontal axis
values, which will be the derived values (also - and +) of the cross-product. For
example, the settings for the X
1
*X
2
interaction are derived by simple multiplication
from the data as shown below.
X
1
X
2
X
1
*X
2
- - +
+ - -
- + -
+ + +
Thus X
1
, X
2
, and X
1
*X
2
all form a closed (-, +) system. The advantage of the closed
system is that graphically interactions can be interpreted in the exact same fashion as
the k main effects.
After the entire X
1
*X
2
vector of settings has been formed in this way, the vertical
axis of the X
1
*X
2
interaction plot is formed:
1. the plot point above X
1
*X2 = "-" is simply the mean of all response values for
which X
1
*X
2
= "-"
2. the plot point above X
1
*X
2
= "+" is simply the mean of all response values for
which X
1
*X
2
= "+".
We form the plots for the remaining 2-factor interactions in a similar fashion.
All the mean plots, for both main effects and 2-factor interactions, have a common
scale to facilitate comparisons. Each mean plot has
1. Vertical Axis: The mean response for a given setting (- or +) of a given factor
or a given 2-factor interaction.
2. Horizontal Axis: The 2 settings (- and +) within each factor, or within each 2-
factor interaction.
3. Legend:
1. A tag (1, 2, ..., k, 12, 13, etc.), with 1 = X
1
, 2 = X
2
, ..., k = X
k
, 12 =
X
1
*X
2
, 13 = X
1
*X
3
, 35 = X
3
*X
5
, 123 = X
1
*X
2
*X
3
, etc.) which
identifies the particular mean plot; and
2. The least squares estimate of the factor (or 2-factor interaction) effect.
These effect estimates are large in magnitude for important factors and
near-zero in magnitude for unimportant factors.
In a later section, we discuss in detail the models associated with full and fractional
factorial 2-level designs. One such model representation is
Y = +
1
*X
1
+
2
*X
2
+
12
*X
1
*X
2
+ ... +
For factor variables coded with + and - settings, the

i
coefficient is one half of the
effect estimate due to factor X
i
. Thus, if we multiply the least-squares coefficients
by two, due to orthogonality, we obtain the simple difference of means at the +
setting and the - setting. This is true for the k main factors. It is also true for all two-
factor and multi-factor interactions.
Thus, visually, the difference in the mean values on the plot is identically the least
squares estimate for the effect. Large differences (steep lines) imply important
factors while small differences (flat lines) imply unimportant factors.
Motivation As discussed in detail above, the next logical step beyond main effects is displaying
2-factor interactions, and this plot matrix provides a convenient graphical tool for
examining the relative importance of main effects and 2-factor interactions in
concert. To do so, we make use of the striking aspect that in the context of 2-level
designs, the 2-factor interactions are identical to cross-products and the 2-factor
interaction effects can be interpreted and compared the same way as main effects.
Plot for
defective
springs
data
Constructing the interaction effects matrix plot for the defective springs data set
yields the following plot.
How to
interpret
From the interaction effects matrix, we can draw three important conclusions:
1. Important Factors (including 2-factor interactions);
2. Best Settings;
3. Confounding Structure (for fractional factorial designs).
We discuss each of these in turn.
1. Important factors (including 2-factor interactions):
Jointly compare the k main factors and the k*(k-1)/2 2-factor interactions. For
each of these subplots, as we go from the "-" setting to the "+" setting within
a subplot, is there a shift in location of the average data (yes/no)? Since all
subplots have a common (-1, +1) horizontal axis, questions involving shifts in
location translate into questions involving steepness of the mean lines (large
shifts imply steep mean lines while no shifts imply flat mean lines).
1. Identify the factor or 2-factor interaction that has the largest shift (based
on averages). This defines the "most important factor". The largest shift
is determined by the steepest line.
2. Identify the factor or 2-factor interaction that has the next largest shift
(based on averages). This defines the "second most important factor".
This shift is determined by the next steepest line.
This ranking of factors and 2-factor interactions based on local means is a
major step in building the definitive list of ranked factors as required for
screening experiments.
2. Best settings:
For each factor (of the k main factors along the diagonal), which setting (- or
+) yields the "best" (highest/lowest) average response?
Note that the experimenter has the ability to change settings for only the k
main factors, not for any 2-factor interactions. Although a setting of some 2-
factor interaction may yield a better average response than the alternative
setting for that same 2-factor interaction, the experimenter is unable to set a 2-
factor interaction setting in practice. That is to say, there is no "knob" on the
machine that controls 2-factor interactions; the "knobs" only control the
settings of the k main factors.
How then does this matrix of subplots serve as an improvement over the k
best settings that one would obtain from the DOE mean plot? There are two
common possibilities:
1. Steep Line:
For those main factors along the diagonal that have steep lines (that is,
are important), choose the best setting directly from the subplot. This
will be the same as the best setting derived from the DOE mean plot.
2. Flat line:
For those main factors along the diagonal that have flat lines (that is, are
unimportant), the naive conclusion to use either setting, perhaps giving
preference to the cheaper setting or the easier-to-implement setting,
may be unwittingly incorrect. In such a case, the use of the off-diagonal
2-factor interaction information from the interaction effects matrix is
critical for deducing the better setting for this nominally "unimportant"
factor.
To illustrate this, consider the following example:
Suppose the factor X
1
subplot is steep (important) with the best
setting for X
1
at "+".
Suppose the factor X
2
subplot is flat (unimportant) with both
settings yielding about the same mean response.
Then what setting should be used for X
2
? To answer this, consider the
following two cases:
1. Case 1. If the X
1
*X
2
interaction plot happens also to be flat
(unimportant), then choose either setting for X
2
based on cost or
ease.
2. Case 2. On the other hand, if the X
1
*X
2
interaction plot is steep
(important), then this dictates a prefered setting for X
2
not based
on cost or ease.
To be specific for case 2, if X
1
*X
2
is important, with X
1
*X
2
= "+" being
the better setting, and if X
1
is important, with X
1
= "+" being the better
setting, then this implies that the best setting for X
2
must be "+" (to
assure that X
1
*X2 (= +*+) will also be "+"). The reason for this is that
since we are already locked into X
1
= "+", and since X
1
*X
2
= "+" is
better, then the only way we can obtain X
1
*X
2
= "+" with X
1
= "+" is
for X
2
to be "+" (if X
2
were "-", then X
1
*X
2
with X
1
= "+" would yield
X
1
*X
2
= "-").
In general, if X
1
is important, X
1
*X
2
is important, and X
2
is not
important, then there are four distinct cases for deciding what the best
setting is for X
2
:
X
1
X
1
*X
2
=> X
2
+ + +
+ - -
- + -
- - +
By similar reasoning, examining each factor and pair of factors, we thus
arrive at a resulting vector of the k best settings:
This average-based k-vector should be compared with best settings k-
vectors obtained from previous steps (in particular, from step 1 in which
the best settings were drawn from the best data value).
When the average-based best settings and the data-based best settings
agree, we benefit from the increased confidence given our conclusions.
When the average-based best settings and the data-based best settings
disagree, then what settings should the analyst finally choose? Note that
in general the average-based settings and the data-based settings will
invariably be identical for all "important" factors. Factors that do differ
are virtually always "unimportant". Given such disagreement, the
analyst has three options:
1. Use the average-based settings for minor factors. This has the
advantage of a broader (average) base of support.
2. Use the data-based settings for minor factors. This has the
advantage of demonstrated local optimality.
3. Use the cheaper or more convenient settings for the local factor.
This has the advantage of practicality.
Thus the interaction effects matrix yields important information not only about
the ranked list of factors, but also about the best settings for each of the k
main factors. This matrix of subplots is one of the most important tools for the
experimenter in the analysis of 2-level screening designs.
3. Confounding Structure (for Fractional Factorial Designs)
When the interaction effects matrix is used to analyze 2-level fractional (as
opposed to full) factorial designs, important additional information can be
extracted from the matrix regarding confounding structure.
It is well-known that all fractional factorial designs have confounding, a
property whereby every estimated main effect is
confounded/contaminated/biased by some high-order interactions. The
practical effect of this is that the analyst is unsure of how much of the
estimated main effect is due to the main factor itself and how much is due to
some confounding interaction. Such contamination is the price that is paid by
examining k factors with a sample size n that is less than a full factorial n = 2
k
runs.
It is a "fundamental theorem" of the discipline of experimental design that for
a given number of factors k and a given number of runs n, some fractional
factorial designs are better than others. "Better" in this case means that the
intrinsic confounding that must exist in all fractional factorial designs has
been minimized by the choice of design. This minimization is done by
constructing the design so that the main effect confounding is pushed to as
high an order interaction as possible.
The rationale behind this is that in physical science and engineering systems it
has been found that the "likelihood" of high-order interactions being
significant is small (compared to the likelihood of main effects and 2-factor
interactions being significant). Given this, we would prefer that such
inescapable main effect confounding be with the highest order interaction
possible, and hence the bias to the estimated main effect be as small as
possible.
The worst designs are those in which the main effect confounding is with 2-
factor interactions. This may be dangerous because in physical/engineering
systems, it is quite common for Nature to have some real (and large) 2-factor
interactions. In such a case, the 2-factor interaction effect will be inseparably
entangled with some estimated main effect, and so the experiment will be
flawed in that
1. ambiguous estimated main effects and
2. an ambiguous list of ranked factors
will result.
If the number of factors, k, is large and the number of runs, n, is constrained to
be small, then confounding of main effects with 2-factor interactions is
unavoidable. For example, if we have k = 7 factors and can afford only n = 8
runs, then the corresponding 2-level fractional factorial design is a 2
7-4
which
necessarily will have main effects confounded with (3) 2-factor interactions.
This cannot be avoided.
On the other hand, situations arise in which 2-factor interaction confounding
with main effects results not from constraints on k or n, but on poor design
construction. For example, if we have k = 7 factors and can afford n = 16
runs, a poorly constructed design might have main effects counfounded with
2-factor interactions, but a well-constructed design with the same k = 7, n =
16 would have main effects confounded with 3-factor interactions but no 2-
factor interactions. Clearly, this latter design is preferable in terms of
minimizing main effect confounding/contamination/bias.
For those cases in which we do have main effects confounded with 2-factor
interactions, an important question arises:
For a particular main effect of interest, how do we know which 2-factor
interaction(s) confound/contaminate that main effect?
The usual answer to this question is by means of generator theory,
confounding tables, or alias charts. An alternate complementary approach is
given by the interaction effects matrix. In particular, if we are examining a 2-
level fractional factorial design and
1. if we are not sure that the design has main effects confounded with 2-
factor interactions, or
2. if we are sure that we have such 2-factor interaction confounding but
are not sure what effects are confounded,
then how can the interaction effects matrix be of assistance? The answer to
this question is that the confounding structure can be read directly from the
interaction effects matrix.
For example, for a 7-factor experiment, if, say, the factor X
3
is confounded
with the 2-factor interaction X
2
*X
5
, then
1. the appearance of the factor X
3
subplot and the appearance of the 2-
factor interaction X
2
*X
5
subplot will necessarily be identical, and
2. the value of the estimated main effect for X
3
(as given in the legend of
the main effect subplot) and the value of the estimated 2-factor
interaction effect for X
2
*X
5
(as given in the legend of the 2-factor
interaction subplot) will also necessarily be identical.
The above conditions are necessary, but not sufficient for the effects to be
confounded.
Hence, in the abscence of tabular descriptions (from your statistical software
program) of the confounding structure, the interaction effect matrix offers the
following graphical alternative for deducing confounding structure in
fractional factorial designs:
1. scan the main factors along the diagonal subplots and choose the subset
of factors that are "important".
2. For each of the "important" factors, scan all of the 2-factor interactions
and compare the main factor subplot and estimated effect with each 2-
factor interaction subplot and estimated effect.
3. If there is no match, this implies that the main effect is not confounded
with any 2-factor interaction.
4. If there is a match, this implies that the main effect may be confounded
with that 2-factor interaction.
5. If none of the main effects are confounded with any 2-factor
interactions, we can have high confidence in the integrity (non-
contamination) of our estimated main effects.
6. In practice, for highly-fractionated designs, each main effect may be
confounded with several 2-factor interactions. For example, for a 2
7-4
fractional factorial design, each main effect will be confounded with
three 2-factor interactions. These 1 + 3 = 4 identical subplots will be
blatantly obvious in the interaction effects matrix.
Finally, what happens in the case in which the design the main effects are not
confounded with 2-factor interactions (no diagonal subplot matches any off-
diagonal subplot). In such a case, does the interaction effects matrix offer any
useful further insight and information?
The answer to this question is yes because even though such designs have
main effects unconfounded with 2-factor interactions, it is fairly common for
such designs to have 2-factor interactions confounded with one another, and
on occasion it may be of interest to the analyst to understand that
confounding. A specific example of such a design is a 2
4-1
design formed
with X
4
settings = X
1
*X
2
*X
3
. In this case, the 2-factor-interaction
confounding structure may be deduced by comparing all of the 2-factor
interaction subplots (and effect estimates) with one another. Identical subplots
and effect estimates hint strongly that the two 2-factor interactions are
confounded. As before, such comparisons provide necessary (but not
sufficient) conditions for confounding. Most statistical software for analyzing
fractional factorial experiments will explicitly list the confounding structure.
Conclusions
for the
defective
springs
data
The application of the interaction effects matrix plot to the defective springs data set
results in the following conclusions:
1. Ranked list of factors (including 2-factor interactions):
1. X
1
(estimated effect = 23.0)
2. X
1
*X
3
3. X
2
(estimated effect = -5.0)
4. X
3
5. X
1
*X
2
6. X
2
*X
3
Factor 1 definitely looks important. The X
1
*X
3
interaction looks important.
Factor 2 is of lesser importance. All other factors and 2-factor interactions
appear to be unimportant.
2. Best Settings (on the average):
(X
1
, X
2
, X
3
) = (+, -, +) = (+1, -1, +1)
5.5.9.5. Block plot

5.5.9.5. Block plot
Purpose The block plot answers the following two general questions:
1. What are the important factors (including interactions)?
2. What are the best settings for these important factors?
The basic (single) block plot is a multifactor EDA technique to determine if a factor
is important and to ascertain if that importance is unconditional (robust) over all
settings of all other factors in the system. In an experimental design context, the
block plot is actually a sequence of block plots with one plot for each of the k
factors.
Due to the ability of the block plot to determine whether a factor is important over
all settings of all other factors, the block plot is also referred to as a DOE robustness
plot.
Output The block plot provides specific information on
1. Important factors (of the k factors and the 2-factor interactions); and
2. Best settings of the important factors.
Definition The block plot is a series of k basic block plots with each basic block plot for a main
effect. Each basic block plot asks the question as to whether that particular factor is
important:
1. The first block plot asks the question: "Is factor X
1
important?
2. The second block plot asks the question: "Is factor X
2
important?
The i-th basic block plot, which targets factor i and asks whether factor X
i
is
important, is formed by:
Vertical Axis: Response
Horizontal Axis: All 2
k-1
possible combinations of the (k-1) non-target factors
(that is, "robustness" factors). For example, for the block plot focusing on
factor X
1
from a 2
3
full factorial experiment, the horizontal axis will consist of
all 2
3-1
= 4 distinct combinations of factors X
2
and X
3
. We create this
robustness factors axis because we are interested in determining if X
1
is
5.5.9.5. Block plot
important robustly. That is, we are interested in whether X
1
is important not
only in a general/summary kind of way, but also whether the importance of X
is universally and consistently valid over each of the 2
3-1
= 4 combinations of
factors X
2
and X
3
. These four combinations are (X
2
, X
3
) = (+, +), (+, -), (-,
+), and (-, -). The robustness factors on the horizontal axis change from one
block plot to the next. For example, for the k = 3 factor case:
1. the block plot targeting X
1
will have robustness factors X
2
and X
3
;
2
1
and X
3
;
3
1
and X
2
.
Plot Character: The setting (- or +) for the target factor X
i
. Each point in a
block plot has an associated setting for the target factor X
i
. If X
i
= "-", the
corresponding plot point will be "-"; if X
i
= "+", the corresponding plot point
will be "+".
For a particular combination of robustness factor settings (horizontally), there will
be two points plotted above it (vertically):
1. one plot point for X
i
= "-"; and
2. the other plot point for X
i
= "+".
In a block plot, these two plot points are surrounded by a box (a block) to focus the
eye on the internal within-block differences as opposed to the distraction of the
external block-to-block differences. Internal block differences reflect on the
importance of the target factor (as desired). External block-to-block differences
reflect on the importance of various robustness factors, which is not of primary
interest.
Large within-block differences (that is, tall blocks) indicate a large local effect on
the response which, since all robustness factors are fixed for a given block, can only
be attributed to the target factor. This identifies an "important" target factor. Small
within-block differences (small blocks) indicate that the target factor X
i
is
unimportant.
For a given block plot, the specific question of interest is thus
Is the target factor X
i
important? That is, as we move within a block from the
target factor setting of "-" to the target factor setting of "+", does the response
variable value change by a large amount?
The height of the block reflects the "local" (that is, for that particular combination of
robustness factor settings) effect on the response due to a change in the target factor
settings. The "localized" estimate for the target factor effect for X
i
is in fact identical
to the difference in the response between the target factor X
i
at the "+" setting and at
the "-" setting. Each block height of a robustness plot is thus a localized estimate of
the target factor effect.
In summary, important factors will have both
1. consistently large block heights; and
2. consistent +/- sign arrangements
5.5.9.5. Block plot
where the "consistency" is over all settings of robustness factors. Less important
factors will have only one of these two properties. Unimportant factors will have
neither property.
Plot for
defective
springs
data
Applying the ordered response plot to the defective springs data set yields the
following plot.
How to
interpret
From the block plot, we are looking for the following:
1. Important factors (including 2-factor interactions);
2. Best settings for these factors.
We will discuss each of these in turn.
Important factors (including 2-factor interactions):
Look at each of the k block plots. Within a given block plot,
Are the corresponding block heights consistently large as we scan across the
within-plot robustness factor settings--yes/no; and are the within-block sign
patterns (+ above -, or - above +) consistent across all robustness factors
settings--yes/no?
To facilitate intercomparisons, all block plots have the same vertical axis scale.
Across such block plots,
1. Which plot has the consistently largest block heights, along with consistent
arrangement of within-block +'s and -'s? This defines the "most important
factor".
5.5.9.5. Block plot
2. Which plot has the consistently next-largest block heights, along with
consistent arrangement of within-block +'s and -'s? This defines the "second
most important factor".
This scanning and comparing of the k block plots easily leads to the identification of
the most important factors. This identification has the additional virtue over
previous steps in that it is robust. For a given important factor, the consistency of
block heights and sign arrangement across robustness factors gives additional
credence to the robust importance of that factor. The factor is important (the change
in the response will be large) irrespective of what settings the robustness factors
have. Having such information is both important and comforting.
Important Special Case; Large but Inconsistent:
What happens if the block heights are large but not consistent? Suppose, for
example, a 2
3
factorial experiment is being analyzed and the block plot focusing on
factor X
1
is being examined and interpreted so as to address the usual question of
whether factor X
1
is important.
Let us consider in some detail how such a block plot might appear. This X
1
block
plot will have 2
3-1
= 4 combinations of the robustness factors X
2
and X
3
along the
horizontal axis in the following order:
(X
2
, X
3
) = (+, +); (X
2
, X
3
) = (+, -); (X
2
, X
3
) = (-, +); (X
2
, X
3
) = (-, -).
If the block heights are consistently large (with "+" above "-" in each block) over
the four combinations of settings for X
2
and X
3
, as in
(X
2
, X
3
) block height (= local X
1
effect)
(+, +) 30
(+, -) 29
(-, +) 29
(-, -) 31
then from binomial considerations there is one chance in 2
4-1
= 1/8 12.5 % of the
the four local X
1
effects having the same sign (i.e., all positive or all negative). The
usual statistical cutoff of 5 % has not been achieved here, but the 12.5 % is
suggestive. Further, the consistency of the four X
1
effects (all near 30) is evidence of
a robustness of the X effect over the settings of the other two factors. In summary,
the above suggests:
1. Factor 1 is probably important (the issue of how large the effect has to be in
order to be considered important will be discussed in more detail in a later
section); and
2. The estimated factor 1 effect is about 30 units.
On the other hand, suppose the 4 block heights for factor 1 vary in the following
cyclic way:
5.5.9.5. Block plot
(X
2
, X
3
) block height (= local X
1
effect)
(+, +) 30
(+, -) 20
(-, +) 30
(-, -) 20
then how is this to be interpreted?
The key here to such interpretation is that the block plot is telling us that the
estimated X
1
effect is in fact at least 20 units, but not consistent. The effect is
changing, but it is changing in a structured way. The "trick" is to scan the X
2
and X
3
settings and deduce what that substructure is. Doing so from the above table, we see
that the estimated X
1
effect is 30
for point 1 (X
2
, X
3
) = (+, +) and
for point 3 (X
2
, X
3
) = (-, +)
and then the estimated X
1
effect drops 10 units to 20
for point 2 (X
2
, X
3
) = (+, -) and
for point 4 (X
2
, X
3
) = (-, -)
We thus deduce that the estimated X
1
effect is
1. 30 whenever X
3
= "+"
2. 20 whenever X
3
= "-"
When the factor X
1
effect is not consistent, but in fact changes depending on the
setting of factor X
3
, then definitionally that is said to be an "X
1
*X
3
interaction".
That is precisely the case here, and so our conclusions would be:
1. factor X
1
is probably important;
2. the estimated factor X
1
effect is 25 (the average of 30, 20, 30, and 20);
3. the X
1
*X
3
interaction is probably important;
4. the estimated X
1
*X
3
interaction is about 10 (the change in the factor X
1
effect
as X
3
changes = 30 - 20 = 10);
5. hence the X
1
*X
3
interaction is less important than the X
1
effect.
Note that we are using the term important in a qualitative sense here. More precise
determinations of importance in terms of statistical or engineering significance are
discussed in later sections.
The block plot gives us the structure and the detail to allow such conclusions to be
drawn and to be understood. It is a valuable adjunct to the previous analysis steps.
Best settings:
After identifying important factors, it is also of use to determine the best settings for
5.5.9.5. Block plot
these factors. As usual, best settings are determined for main effects only (since main
effects are all that the engineer can control). Best settings for interactions are not
done because the engineer has no direct way of controlling them.
In the block plot context, this determination of best factor settings is done simply by
noting which factor setting (+ or -) within each block is closest to that which the
engineer is ultimately trying to achieve. In the defective springs case, since the
response variable is percent acceptable springs, we are clearly trying to maximize
(as opposed to minimize, or hit a target) the response and the ideal optimum point is
100 %. Given this, we would look at the block plot of a given important factor and
note within each block which factor setting (+ or -) yields a data value closest to
100 % and then select that setting as the best for that factor.
From the defective springs block plots, we would thus conclude that
1. the best setting for factor 1 is +;
2. the best setting for factor 2 is -;
3. the best setting for factor 3 cannot be easily determined.
Conclusions
for the
defective
springs
data
In summary, applying the block plot to the defective springs data set results in the
1. Unranked list of important factors (including interactions):
X
1
is important;
X
2
is important;
X
1
*X
3
is important.
2. Best Settings:
(X
1
, X
2
, X
3
) = (+, -, ?) = (+1, -1, ?)
5.5.9.6. DOE Youden plot

Purpose The DOE (design of experiments) Youden plot answers the following question:
What are the important factors (including interactions)?
In its original interlab rendition, the Youden plot was a graphical technique
developed in the 1960's by Jack Youden of NIST for assessing between-lab biases
and within-lab variation problems in the context of interlab experimentation. In
particular, it was appropriate for the analysis of round-robin data when exactly two
materials, batches, etc. were used in the design.
In a design of experiments context, we borrow this duality emphasis and apply it to
2-level designs. The 2-component emphasis of the Youden plot makes it a natural to
be applied to such designs.
Output The DOE Youden plot provides specific information on
1. Ranked list of factors (including interactions); and
2. Separation of factors into two categories: important and unimportant.
The primary output from a DOE Youden plot is the ranked list of factors (out of the
k factors and interactions). For full factorial designs, interactions include the full
complement of interactions at all orders; for fractional factorial designs, interactions
include only some, and occasionally none, of the actual interactions. Further, the
DOE Youden plot yields information identifying which factors/interactions are
important and which are unimportant.
Definition The DOE Youden plot consists of the following:
Vertical Axis: Mean response at the "+" setting for each factor and each
interaction. For a given factor or interaction, n/2 response values will go into
computing the "+" mean.
Horizontal Axis: Mean response at the "-" setting for each factor and each
interaction. For a given factor or interaction, n/2 response values will go into
computing the "-" mean.
Plot Character: Factor/interaction identification for which
1 indicates factor X
1
;
2
;
...
12 indicates the 2-factor X
1
*X
2
interaction
1
*X
2
*X
3
interaction
etc.
In essence, the DOE Youden plot is a scatter plot of the "+" average responses
versus the "-" average responses. The plot will consist of n - 1 points with one point
for each factor and one point for each (available) interaction. Each point on the plot
is annotated to identify which factor or interaction is being represented.
Motivation Definitionally, if a factor is unimportant, the "+" average will be approximately the
same as the "-" average, and if a factor is important, the "+" average will be
considerably different from the "-" average. Hence a plot that compares the "+"
averages with the "-" averages directly seems potentially informative.
From the definition above, the DOE Youden plot is a scatter plot with the "+"
averages on the vertical axis and the "-" averages on the horizontal axis. Thus,
unimportant factors will tend to cluster in the middle of the plot and important
factors will tend to be far removed from the middle.
Because of an arithmetic identity which requires that the average of any
corresponding "+" and "-" means must equal the grand mean, all points on a DOE
Youden plot will lie on a -45 degree diagonal line. Or to put it another way, for each
factor
average (+) + average (-) = constant (with constant = grand mean)
So
average (+) = constant - average (-)
Therefore, the slope of the line is -1 and all points lie on the line. Important factors
will plot well-removed from the center because average (+) = average (-) at the
center.
Plot for
defective
springs
data
Applying the DOE Youden plot for the defective springs data set yields the
following plot.
How to
interpret
In the DOE Youden plot, we look for the following:
1. A ranked list of factors (including interactions). The intersecting dotted lines
at the center of the plot are the value of the grand mean on both the vertical
and horizontal axes. Scan the points along the negative-slope diagonal line
and note as to whether such points are clustered around the grand mean or are
displaced up or down the diagonal line.
1. Which point is farthest away from the center? This defines the "most
important" factor.
2. Which point is next farthest away from the center? This defines the
"second most important" factor.
3. Continue in a similar manner for the remaining points. The points
closest to the center define the "least important" factors.
2. Separation of factors into important/unimportant categories. Interpretationally,
if a factor is unimportant, the "+" average will be about the same as the "-"
average, so the plot of "+" vertically and "-" horizontally will be near the
grand mean of all n - 1 data points.
Conversely, if a factor is important, the "+" average will differ greatly from
the "-" average, and so the plot of "+" vertically and "-" horizontally will be
considerably displaced up into the top left quadrant or down into the bottom
right quadrant.
The separation of factors into important/unimportant categories is thus done by
answering the question:
Which points visually form a cluster around the center? (these define
the "unimportant factors"--all remaining factors are "important").
This ranked list of important factors derived from the DOE Youden plot is to be
compared with the ranked lists obtained from previous steps. Invariably, there will
be a large degree of consistency exhibited across all/most of the techniques.
Conclusions
for the
defective
springs
data
The application of the DOE Youden plot to the defective springs data set results in
1. Ranked list of factors (including interactions):
1. X
1
(most important)
2. X
1
*X
3
(next most important)
3. X
2
4. other factors are of lesser importance
2. Separation of factors into important/unimportant categories:
"Important": X
1
, X
1
*X
3
, and X
2
"Unimportant": the remainder
5.5.9.7. |Effects| plot

Purpose The |effects| plot answers the question:
Quantitatively, the question as to what is the estimated effect of a given factor or
interaction and what is its rank relative to other factors and interactions is answered
via the least squares estimation criterion (that is, forming effect estimates that
minimize the sum of the squared differences between the raw data and the fitted
values from such estimates). Based on such an estimation criterion, one could then
construct a tabular list of the factors and interactions ordered by the effect
magnitude.
The |effects| plot provides a graphical representation of these ordered estimates,
Pareto-style from largest to smallest.
The |effects| plot, as presented here, yields both of the above: the plot itself, and the
ranked list table. Further, the plot also presents auxiliary confounding information,
which is necessary in forming valid conclusions for fractional factorial designs.
Output The output of the |effects| plot is:
1. Primary: A ranked list of important effects (and interactions). For full factorial
designs, interactions include the full complement of interactions at all orders;
for fractional factorial designs, interactions include only some, and
occasionally none, of the actual interactions.
2. Secondary: Grouping of factors (and interactions) into two categories:
important and unimportant.
Definition The |effects| plot is formed by:
Vertical Axis: Ordered (largest to smallest) absolute value of the estimated
effects for the main factors and for (available) interactions. For n data points
(no replication), typically (n-1) effects will be estimated and the (n-1) |effects|
will be plotted.
Horizontal Axis : Factor/interaction identification:
1
;
2
;
...
1
*X
2
interaction
1
*X
2
*X
3
interaction,
etc.
Far right margin : Factor/interaction identification (built-in redundancy):
1
;
2
;
...
1
*X
2
interaction
1
*X
2
*X
3
interaction,
etc.
If the design is a fractional factorial,the confounding structure is provided for
main factors and 2-factor interactions.
Upper right table: Ranked (largest to smallest by magnitude) list of the least
squares estimates for the main effects and for (available) interactions.
As before, if the design is a fractional factorial, the confounding structure is
provided for main factors and 2-factor interactions.
The estimated effects that form the basis for the vertical axis are optimal in the least
squares sense. No other estimators exist that will yield a smaller sum of squared
deviations between the raw data and the fitted values based on these estimates.
For both the 2
k
full factorial designs and 2
k-p
fractional factorial designs, the form
for the least squares estimate of the factor i effect, the 2-factor interaction effect,
and the multi-factor interaction effect has the following simple form:
factor i effect = (+) - (-)
2-factor interaction effect = (+) - (-)
multi-factor interaction effect = (+) - (-)
with (+) denoting the average of all response values for which factor i (or the 2-
factor or multi-factor interaction) takes on a "+" value, and (-) denoting the
average of all response values for which factor i (or the 2-factor or multi-factor
interaction) takes on a "-" value.
The essence of the above simplification is that the 2-level full and fractional
factorial designs are all orthogonal in nature, and so all off-diagonal terms in the
least squares X'X matrix vanish.
Motivation Because of the difference-of-means definition of the least squares estimates, and
because of the fact that all factors (and interactions) are standardized by taking on
values of -1 and +1 (simplified to - and +), the resulting estimates are all on the
same scale. Therefore, comparing and ranking the estimates based on magnitude
makes eminently good sense.
Moreover, since the sign of each estimate is completely arbitrary and will reverse
depending on how the initial assignments were made (e.g., we could assign "-" to
treatment A and "+" to treatment B or just as easily assign "+" to treatment A and "-
" to treatment B), forming a ranking based on magnitudes (as opposed to signed
effects) is preferred.
Given that, the ultimate and definitive ranking of factor and interaction effects will
be made based on the ranked (magnitude) list of such least squares estimates. Such
rankings are given graphically, Pareto-style, within the plot; the rankings are given
quantitatively by the tableau in the upper right region of the plot. For the case when
we have fractional (versus full) factorial designs, the upper right tableau also gives
the confounding structure for whatever design was used.
If a factor is important, the "+" average will be considerably different from the "-"
average, and so the absolute value of the difference will be large. Conversely,
unimportant factors have small differences in the averages, and so the absolute value
will be small.
We choose to form a Pareto chart of such |effects|. In the Pareto chart, the largest
effects (most important factors) will be presented first (to the left) and then progress
down to the smallest effects (least important) factors to the right.
Plot for
defective
springs
data
Applying the |effects| plot to the defective springs data yields the following plot.
How to
interpret
From the |effects| plot, we look for the following:
1. The ranked list of factors (including interactions) is given by the left-to-right
order of the spikes. These spikes should be of decreasing height as we move
from left to right. Note the factor identifier associated with each of these bars.
2. Identify the important factors. Forming the ranked list of factors is important,
but is only half of the analysis. The second part of the analysis is to take the
ranking and "draw the (horizontal) line" in the list and on the graph so that
factors above the line are deemed "important while factors below the line are
deemed unimportant.
Since factor effects are frequently a continuum ranging from the very large
through the moderate and down to the very small, the separation of all such
factors into two groups (important and unimportant) may seem arbitrary and
severe. However, in practice, from both a research funding and a modeling
point of view, such a bifurcation is both common and necessary.
From an engineering research-funding point of view, one must frequently
focus on a subset of factors for future research, attention, and money, and
thereby necessarily set aside other factors from any further consideration.
From a model-building point of view, a final model either has a term in it or it
does not--there is no middle ground. Parsimonious models require in-or-out
decisions. It goes without saying that as soon as we have identified the
important factors, these are the factors that will comprise our (parsimonious)
good model, and those that are declared as unimportant will not be in the
model.
Given that, where does such a bifurcation line go?
There are four ways, each discussed in turn, to draw such a line:
1. Statistical significance;
2. Engineering significance;
3. Numerical significance; and
4. Pattern significance.
The ranked list and segregation of factors derived from the |effects| plot are to be
compared with the ranked list of factors obtained in previous steps. Invariably, there
will be a considerable degree of consistency exhibited across all of the techniques.
Conclusions
for the
defective
springs
data
The application of the |effects| plot to the defective springs data set results in the
1. X
1
(most important)
2. X
1
*X
3
3. X
2
Important: X
1
, X
1
*X
3
, and X
2
Unimportant: the remainder
5.5.9.7.1. Statistical significance

Formal
statistical
methods
Formal statistical methods to answer the question of
statistical significance commonly involve the use of
ANOVA (analysis of variance); and
t-based confidence intervals for the effects.
ANOVA The virtue of ANOVA is that it is a powerful, flexible tool
with many applications. The drawback of ANOVA is that
it is heavily quantitative and non-intuitive;
it must have an assumed underlying model; and
its validity depends on assumptions of a constant
error variance and normality of the errors.
t confidence
intervals
T confidence intervals for the effects, using the t-
distribution, are also heavily used for determining factor
significance. As part of the t approach, one first needs to
determine sd(effect), the standard deviation of an effect.
For 2-level full and fractional factorial designs, such a
standard deviation is related to , the standard deviation of
an observation under fixed conditions, via the formula:
which in turn leads to forming 95% confidence intervals
for an effect via
c * sd(effect)
for an appropriate multiple c (from the t distribution). Thus
in the context of the |effects| plot, "drawing the line" at c *
sd(effect) would serve to separate, as desired, the list of
effects into 2 domains:
significant (that is, important); and
not significant (that is, unimportant).
Estimating
sd(effect)
The key in the above approach is to determine an estimate
for sd(effect). Three statistical approaches are common:
1. Prior knowledge about :
If is known, we can compute sd(effect) from the
above expression and make use of a conservative
(normal-based) 95% confidence interval by drawing
the line at
This method is rarely used in practice because is
rarely known.
2. Replication in the experimental design:
Replication will allow to be estimated from the
data without depending on the correctness of a
deterministic model. This is a real benefit. On the
other hand, the downside of such replication is that it
increases the number of runs, time, and expense of
the experiment. If replication can be afforded, this
method should be used. In such a case, the analyst
separates important from unimportant terms by
drawing the line at
with t denoting the 97.5 percent point from the
appropriate Student's-t distribution.
3. Assume 3-factor interactions and higher are zero:
This approach "assumes away" all 3-factor
interactions and higher and uses the data pertaining
to these interactions to estimate . Specifically,
with h denoting the number of 3-factor interactions
and higher, and SSQ is the sum of squares for these
higher-order effects. The analyst separates important
from unimportant effects by drawing the line at
with t denoting the 97.5 percent point from the
appropriate (with h degrees of freedom) Student's-t
distribution.
This method warrants caution:
it involves an untestable assumption (that such
interactions = 0);
it can result in an estimate for sd(effect) based
on few terms (even a single term); and
it is virtually unusable for highly-fractionated
designs (since high-order interactions are not
directly estimable).
Non-
statistical
considerations
The above statistical methods can and should be used.
Additionally, the non-statistical considerations discussed in
the next few sections are frequently insightful in practice
and have their place in the EDA approach as advocated
here.
5.5.9.7.2. Engineering significance

5.5.9.7.2. Engineering significance
Engineering
cutoff
Draw the horizontal line on the chart at that value which you
as an engineer have declared beforehand as the engineering
cutoff. Any effect larger than this cutoff will be considered
as significant from an engineering point of view.
Specifying a
cutoff value
requires
non-
statistical
thinking,
but is
frequently
useful
This approach requires preliminary, data-free thinking on the
part of the analyst as to how big (= what number?) an effect
(any effect) must be before the analyst would "care" as an
engineer/scientist? In other words, in the units of the
response variable, how much would the response variable
have to change consistently before the analyst would say
"that's a big enough change for me from an engineering point
of view"? An engineering number, a cutoff value, is needed
here. This value is non-statistical; thie value must emanate
from the engineer's head.
If upon reflection the analyst does not have such a value in
mind, this "engineering significance" approach would be set
aside. From experience, it has been found that the
engineering soul-searching that goes into evoking such a
cutoff value is frequently useful and should be part of the
decision process, independent of statistical considerations, of
separating the effects into important/unimportant categories.
A rough
engineering
cutoff
In the absence of a known engineering cutoff, a rough cutoff
value is commonly 5 % or 10 % of the average (or current)
production response for the system. Thus, if a chemical
reaction production process is yielding a reaction rate of
about 70, then
5 % of 70 = 3. The engineer may declare any future effect
that causes an average change of 3 or more units in the
response (that is, any estimated effect whose magnitude
exceeds 3) to be "engineering significant". In the context of
the |effects| plot, the engineer would draw the line at a height
of 3 on the plot, and all effects that are above the line are
delared as significant and all below the line are declared not
significant.
5.5.9.7.3. Numerical significance

5.5.9.7.3. Numerical significance
10 % of
the largest
effect
Note the height of the largest bar (= the magnitude of the
largest effect). Declare as "significant" any effect that exceeds
10 % of the largest effect. The 10 % is arbitrary and has no
statistical (or engineering) basis, but it does have a "numeric"
basis in that it results in keeping the largest effect and any
effects that are within 90 % of the largest effect.
Apply with
caution
As with any rule-of-thumb, some caution should be used in
applying this critierion. Specifically, if the largest effect is in
fact not very large, this rule-of-thumb may not be useful.
5.5.9.7.4. Pattern significance

5.5.9.7.4. Pattern significance
Look for
L-shaped
pattern
The |effects| plot has a characteristic horizontally-elongated L-
shaped pattern. The vertical arm of the L consists of important
factors. The horizontal arm is comprised of unimportant
factors. If a factor is important, the bar height will be large
and succeeding bar heights may drop off considerably
(perhaps by 50 %)--such factors make up the left arm of the L.
On the other hand, if a factor is not important, its bar height
will tend to be small and near-zero--such factors make up the
bottom arm of the L. It is of interest to note where the kink is
in the L. Factors to the left of that kink are arguably declared
important while factors at the kink point and to the right of it
are declared unimportant.
Factor
labels
As a consequence of this "kinking", note the labels on the far
right margin of the plot. Factors to the left and above the kink
point tend to have far-right labels distinct and isolated. Factors
at, to the right, and below the kink point tend to have far right
labels that are overstruck and hard to read. A (rough) rule-of-
thumb would then be to declare as important those
factors/interactions whose far-right labels are easy to
distinguish, and to declare as unimportant those
factors/interactions whose far-right labels are overwritten and
hard to distinguish.
5.5.9.8. Half-normal probability plot

Purpose The half-normal probability plot answers the question:
Quantitatively, the estimated effect of a given main effect or interaction and its rank
relative to other main effects and interactions is given via least squares estimation
(that is, forming effect estimates that minimize the sum of the squared differences
between raw data and the fitted values from such estimates). Having such estimates
in hand, one could then construct a list of the main effects and interactions ordered
by the effect magnitude.
The half-normal probability plot is a graphical tool that uses these ordered estimated
effects to help assess which factors are important and which are unimportant.
A half-normal distribution is the distribution of the |X| with X having a normal
distribution.
Output The outputs from the half-normal probablity plot are
1. Primary: Grouping of factors and interactions into two categories: important
and unimportant. For full factorial designs, interactions include the full
complement of interactions of all orders; for fractional factorial designs,
interactions include only some, and occasionally none, of the actual
interactions (when they aren't estimable).
2. Secondary: Ranked list of factors and interactions from most important down
to least important.
Definition A half-normal probability plot is formed by
Vertical Axis: Ordered (largest to smallest) absolute value of the estimated
effects for the main factors and available interactions. If n data points (no
replication) have been collected, then typically (n-1) effects will be estimated
and the (n-1) |effects| will be plotted.
Horizontal Axis: (n-1) theoretical order statistic medians from a half-normal
distribution. These (n-1) values are not data-dependent. They depend only on
the half-normal distribution and the number of items plotted (= n-1). The
theoretical medians represent an "ideal" typical ordered data set that would
have been obtained from a random drawing of (n-1) samples from a half-
Far right margin : Factor/interaction identification:
1
;
2
;
...
1
*X
2
interaction
1
*X
2
*X
3
interaction,
etc.
If the design is a fractional factorial, the confounding structure is provided for
main effects and 2-factor interactions.
Motivation To provide a rationale for the half-normal probability plot, we first dicuss the
motivation for the normal probability plot (which also finds frequent use in these 2-
level designs).
The basis for the normal probability plot is the mathematical form for each (and all)
of the estimated effects. As discussed for the |effects| plot, the estimated effects are
the optimal least squares estimates. Because of the orthogonality of the 2
k
full
factorial and the 2
k-p
fractional factorial designs, all least squares estimators for
main effects and interactions simplify to the form:
estimated effect = (+) - (-)
with (+) the average of all response values for which the factor or interaction takes
on a "+" value, and where (-) is the average of all response values for which the
factor or interaction takes on a "-" value.
Under rather general conditions, the Central Limit Thereom allows that the
difference-of-sums form for the estimated effects tends to follow a normal
distribution (for a large enough sample size n) a normal distribution.
The question arises as to what normal distribution; that is, a normal distribution with
what mean and what standard deviation? Since all estimators have an identical form
(a difference of averages), the standard deviations, though unknown, will in fact be
the same under the assumption of constant . This is good in that it simplifies the
normality analysis.
As for the means, however, there will be differences from one effect to the next, and
these differences depend on whether a factor is unimportant or important.
Unimportant factors are those that have near-zero effects and important factors are
those whose effects are considerably removed from zero. Thus, unimportant effects
tend to have a normal distribution centered near zero while important effects tend
to have a normal distribution centered at their respective true large (but unknown)
effect values.
In the simplest experimental case, if the experiment were such that no factors were
important (that is, all effects were near zero), the (n-1) estimated effects would
behave like random drawings from a normal distribution centered at zero. We can
test for such normality (and hence test for a null-effect experiment) by using the
normal probability plot. Normal probability plots are easy to interpret. In simplest
terms:
if linear, then normal
If the normal probability plot of the (n-1) estimated effects is linear, this implies that
all of the true (unknown) effects are zero or near-zero. That is, no factor is
important.
On the other hand, if the truth behind the experiment is that there is exactly one
factor that was important (that is, significantly non-zero), and all remaining factors
are unimportant (that is, near-zero), then the normal probability plot of all (n-1)
effects is near-linear for the (n-2) unimportant factors and the remaining single
important factor would stand well off the line.
Similarly, if the experiment were such that some subset of factors were important
and all remaining factors were unimportant, then the normal probability plot of all
(n-1) effects would be near-linear for all unimportant factors with the remaining
important factors all well off the line.
In real life, with the number of important factors unknown, this suggests that one
could form a normal probability plot of the (n-1) estimated effects and draw a line
through those (unimportant) effects in the vicinity of zero. This identifies and
extracts all remaining effects off the line and declares them as important.
The above rationale and methodology works well in practice, with the net effect that
the normal probability plot of the effects is an important, commonly used and
successfully employed tool for identifying important factors in 2-level full and
factorial experiments. Following the lead of Cuthbert Daniel (1976), we augment the
methodology and arrive at a further improvement. Specifically, the sign of each
estimate is completely arbitrary and will reverse depending on how the initial
assignments were made (e.g., we could assign "-" to treatment A and "+" to
treatment B or just as easily assign "+" to treatment A and "-" to treatment B).
This arbitrariness is addressed by dealing with the effect magnitudes rather than the
signed effects. If the signed effects follow a normal distribution, the absolute values
of the effects follow a half-normal distribution.
In this new context, one tests for important versus unimportant factors by generating
a half-normal probability plot of the absolute value of the effects. As before,
linearity implies half-normality, which in turn implies all factors are unimportant.
More typically, however, the half-normal probability plot will be only partially
linear. Unimportant (that is, near-zero) effects manifest themselves as being near
zero and on a line while important (that is, large) effects manifest themselves by
being off the line and well-displaced from zero.
Plot for
defective
springs
data
The half-normal probability plot of the effects for the defectice springs data set is as
follows.
How to
interpret
From the half-normal probability plot, we look for the following:
1. Identifying Important Factors:
Determining the subset of important factors is the most important task of the
half-normal probability plot of |effects|. As discussed above, the estimated
|effect| of an unimportant factor will typically be on or close to a near-zero
line, while the estimated |effect| of an important factor will typically be
displaced well off the line.
The separation of factors into important/unimportant categories is thus done by
answering the question:
Which points on the half-normal probability plot of |effects| are large
and well-off the linear collection of points drawn in the vicinity of the
origin?
This line of unimportant factors typically encompasses the majority of the
points on the plot. The procedure consists, therefore, of the following:
1. identifying this line of near-zero (unimportant) factors; then
2. declaring the remaining off-line factors as important.
Note that the half-normal probability plot of |effects| and the |effects| plot
have the same vertical axis; namely, the ordered |effects|, so the following
discussion about right-margin factor identifiers is relevant to both plots. As a
consequence of the natural on-line/off-line segregation of the |effects| in half-
normal probability plots, factors off-line tend to have far-right labels that are
distinct and isolated while factors near the line tend to have far-right labels
that are overstruck and hard to read. The rough rule-of-thumb would then be
to declare as important those factors/interactions whose far-right labels are
easy to distinguish and to declare as unimportant those factors/interactions
whose far-right labels are overwritten and hard to distinguish.
2. Ranked List of Factors (including interactions):
This is a minor objective of the half-normal probability plot (it is better done
via the |effects| plot). To determine the ranked list of factors from a half-
normal probability plot, simply scan the vertical axis |effects|
1. Which |effect| is largest? Note the factor identifier associated with this
largest |effect| (this is the "most important factor").
2. Which |effect| is next in size? Note the factor identifier associated with
this next largest |effect| (this is the "second most important factor").
3. Continue for the remaining factors. In practice, the bottom end of the
ranked list (the unimportant factors) will be hard to extract because of
overstriking, but the top end of the ranked list (the important factors)
will be easy to determine.
In summary, it should be noted that since the signs of the estimated effects are
arbitrary, we recommend the use of the half-normal probability plot of |effects|
technique over the normal probability plot of the |effects|. These probability plots
are among the most commonly-employed EDA procedure for identification of
important factors in 2-level full and factorial designs. The half-normal probability
plot enjoys widespread usage across both "classical" and Taguchi camps. It
deservedly plays an important role in our recommended 10-step graphical procedure
for the analysis of 2-level designed experiments.
Conclusions
for the
defective
springs
data
The application of the half-normal probability plot to the defective springs data set
results in the following conclusions:
1. X
1
(most important)
2. X
1
*X
3
3. X
2
Important: X
1
, X
1
*X
3
, and X
2
Unimportant: the remainder
5.5.9.9. Cumulative residual standard deviation plot

Purpose The cumulative residual sd (standard deviation) plot answers the question:
What is a good model for the data?
The prior 8 steps in this analysis sequence addressed the two important goals:
1. Factors: determining the most important factors that affect the response, and
2. Settings: determining the best settings for these factors.
In addition to the above, a third goal is of interest:
3. Model: determining a model (that is, a prediction equation) that functionally
relates the observed response Y with the various main effects and interactions.
Such a function makes particular sense when all of the individual factors are
continuous and ordinal (such as temperature, pressure, humidity, concentration, etc.)
as opposed to any of the factors being discrete and non-ordinal (such as plant,
operator, catalyst, supplier).
In the continuous-factor case, the analyst could use such a function for the
following purposes.
1. Reproduction/Smoothing: predict the response at the observed design points.
2. Interpolation: predict what the response would be at (unobserved) regions
between the design points.
3. Extrapolation: predict what the response would be at (unobserved) regions
beyond the design points.
For the discrete-factor case, the methods developed below to arrive at such a
function still apply, and so the resulting model may be used for reproduction.
However, the interpolation and extrapolation aspects do not apply.
In modeling, we seek a function f in the k factors X
1
, X
2
, ..., X
k
such that the
predicted values
are "close" to the observed raw data values Y. To this end, two tasks exist:
1. Determine a good functional form f;
2. Determine good estimates for the coefficients in that function f.
For example, if we had two factors X
1
and X
2
, our goal would be to
1. determine some function f(X
1
,X
2
); and
2. estimate the parameters in f
such that the resulting model would yield predicted values that are as close as
possible to the observed response values Y. If the form f has been wisely chosen, a
good model will result and that model will have the characteristic that the
differences ("residuals" = Y - ) will be uniformly near zero. On the other hand, a
poor model (from a poor choice of the form f) will have the characteristic that some
or all of the residuals will be "large".
For a given model, a statistic that summarizes the quality of the fit via the typical
size of the n residuals is the residual standard deviation:
with p denoting the number of terms in the model (including the constant term) and
r denoting the ith residual. We are also assuming that the mean of the residuals is
zero, which will be the case for models with a constant term that are fit using least
squares.
If we have a good-fitting model, s
res
will be small. If we have a poor-fitting model,
s
res
will be large.
For a given data set, each proposed model has its own quality of fit, and hence its
own residual standard deviation. Clearly, the residual standard deviation is more of a
model-descriptor than a data-descriptor. Whereas "nature" creates the data, the
analyst creates the models. Theoretically, for the same data set, it is possible for the
analyst to propose an indefinitely large number of models.
In practice, however, an analyst usually forwards only a small, finite number of
plausible models for consideration. Each model will have its own residual standard
deviation. The cumulative residual standard deviation plot is simply a graphical
representation of this collection of residual standard deviations for various models.
The plot is beneficial in that
1. good models are distinguished from bad models;
2. simple good models are distinguished from complicated good models.
In summary, then, the cumulative residual standard deviation plot is a graphical tool
to help assess
1. which models are poor (least desirable); and
2. which models are good but complex (more desirable); and
3. which models are good and simple (most desirable).
Output The outputs from the cumulative residual standard deviation plot are
1. Primary: A good-fitting prediction equation consisting of an additive constant
plus the most important main effects and interactions.
2. Secondary: The residual standard deviation for this good-fitting model.
Definition A cumulative residual sd plot is formed by
1. Vertical Axis: Ordered (largest to smallest) residual standard deviations of a
sequence of progressively more complicated fitted models.
2. Horizontal Axis: Factor/interaction identification of the last term included into
the linear model:
1
;
2
;
...
1
*X
2
interaction
1
*X
2
*X
3
interaction
etc.
3. Far right margin: Factor/interaction identification (built-in redundancy):
1
;
2
;
...
1
*X
2
interaction
1
*X
2
*X
3
interaction
etc.
If the design is a fractional factorial, the confounding structure is provided for
main effects and 2-factor interactions.
The cumulative residual standard deviations plot is thus a Pareto-style, largest to
smallest, graphical summary of residual standard deviations for a selected series of
progressively more complicated linear models.
The plot shows, from left to right, a model with only a constant and the model then
augmented by including, one at a time, remaining factors and interactions. Each
factor and interaction is incorporated into the model in an additive (rather than in a
multiplicative or logarithmic or power, etc. fashion). At any stage, the ordering of
the next term to be added to the model is such that it will result in the maximal
decrease in the resulting residual standard deviation.
Motivation This section addresses the following questions:
1. What is a model?
2. How do we select a goodness-of-fit metric for a model?
3. How do we construct a good model?
4. How do we know when to stop adding terms?
5. What is the final form for the model?
6. What are the advantages of the linear model?
7. How do we use the model to generate predicted values?
8. How do we use the model beyond the data domain?
9. What is the best confirmation point for interpolation?
10. How do we use the model for interpolation?
11. How do we use the model for extrapolation?
Plot for
defective
springs
data
Applying the cumulative residual standard deviation plot to the defective springs
data set yields the following plot.
How to
interpret
As discussed in detail under question 4 in the Motivation section, the cumulative
residual standard deviation "curve" will characteristically decrease left to right as we
add more terms to the model. The incremental improvement (decrease) tends to be
large at the beginning when important factors are being added, but then the decrease
tends to be marginal at the end as unimportant factors are being added.
Including all terms would yield a perfect fit (residual standard deviation = 0) but
would also result in an unwieldy model. Including only the first term (the average)
would yield a simple model (only one term!) but typically will fit poorly. Although a
formal quantitative stopping rule can be developed based on statistical theory, a less-
rigorous (but good) alternative stopping rule that is graphical, easy to use, and
highly effective in practice is as follows:
Keep adding terms to the model until the curve's "elbow" is encountered. The
"elbow point" is that value in which there is a consistent, noticeably shallower
slope (decrease) in the curve. Include all terms up to (and including) the elbow
point (after all, each of these included terms decreased the residual standard
deviation by a large amount). Exclude any terms after the elbow point since
all such successive terms decreased the residual standard deviation so slowly
that the terms were "not worth the complication of keeping".
From the residual standard deviation plot for the defective springs data, we note the
following:
1. The residual standard deviation (rsd) for the "baseline" model
is s
res
= 13.7.
2. As we add the next term, X
1
, the rsd drops nearly 7 units (from 13.7 to 6.6).
3. If we add the term X
1
*X
3
, the rsd drops another 3 units (from 6.6 to 3.4).
4. If we add the term X
2
, the rsd drops another 2 units (from 3.4 to 1.5).
5. When the term X
3
is added, the reduction in the rsd (from about 1.5 to 1.3) is
negligible.
6. Thereafter to the end, the total reduction in the rsd is from only 1.3 to 0.
In step 5, note that when we have effects of equal magnitude (the X
3
effect is equal
to the X
1
*X
2
interaction effect), we prefer including a main effect before an
interaction effect and a lower-order interaction effect before a higher-order
interaction effect.
In this case, the "kink" in the residual standard deviation curve is at the X
2
term.
Prior to that, all added terms (including X
2
) reduced the rsd by a large amount (7,
then 3, then 2). After the addition of X
2
, the reduction in the rsd was small (all less
than 1): 0.2, then 0.8, then 0.5, then 0.
The final recommended model in this case thus involves p = 4 terms:
1. the average
2. factor X
1
3. the X
1
*X
3
interaction
4. factor X
2
The fitted model thus takes on the form
= average + B
1
*X
1
+ B
13
*X
1
*X
3
+ B
2
*X
2
The least-squares estimates for the coefficients in this model are
average = 71.25
B
1
= 11.5
B
13
= 5
B
2
= -2.5
The B
1
= 11.5, B
13
= 5, and B
2
= -2.5 least-squares values are, of course, one half of
the estimated effects E
1
= 23, E
13
= 10, and E
2
= -5. Effects, calculated as (+1) -
(-1), were previously derived in step 7 of the recommended 10-step DOE analysis
procedure.
The final fitted model is thus
= 71.25 + 11.5*X
1
+ 5*X
1
*X
3
- 2.5*X
2
Applying this prediction equation to the 8 design points yields: predicted values
that are close to the data Y, and residuals (Res = Y - ) that are close to zero:
X
1
X
2
X
3 Y Res
- - - 67 67.25 -0.25
+ - - 79 80.25 -1.25
- + - 61 62.25 -1.25
+ + - 75 75.25 -0.25
- - + 59 57.25 +1.75
+ - + 90 90.25 -0.25
- + + 52 52.25 -0.25
+ + + 87 85.25 +1.75
Computing the residual standard deviation:
with n = 8 data points, and p = 4 estimated coefficients (including the average)
yields
s
res
= 1.54 (or 1.5 if rounded to 1 decimal place)
The detailed s
res
= 1.54 calculation brings us full circle, for 1.54 is the value given
above the X
3
term on the cumulative residual standard deviation plot.
Conclusions
for the
defective
springs
data
The application of the Cumulative Residual Standard Deviation Plot to the defective
springs data set results in the following conclusions:
1. Good-fitting Parsimonious (constant + 3 terms) Model:
= 71.25 + 11.5*X
1
+ 5*X
1
*X
3
- 2.5*X
2
2. Residual Standard Deviation for this Model (as a measure of the goodness-of-
fit for the model):
s
res
= 1.54
5.5.9.9.1. Motivation: What is a Model?

5.5.9.9.1. Motivation: What is a Model?
Mathematical
models:
functional
form and
coefficients
A model is a mathematical function that relates the response
Y to the factors X
1
to X
k
. A model has a
1. functional form; and
2. coefficients.
An excellent and easy-to-use functional form that we find
particularly useful is a linear combination of the main
effects and the interactions (the selected model is a subset
of the full model and almost always a proper subset). The
coefficients in this linear model are easy to obtain via
application of the least squares estimation criterion
(regression). A given functional form with estimated
coefficients is referred to as a "fitted model" or a
"prediction equation".
Predicted
values and
residuals
For given settings of the factors X
1
to X
k
, a fitted model will
yield predicted values. For each (and every) setting of the
X
i
's, a "perfect-fit" model is one in which the predicted
values are identical to the observed responses Y at these
X
i
's. In other words, a perfect-fit model would yield a
vector of predicted values identical to the observed vector
of response values. For these same X
i
's, a "good-fitting"
model is one that yields predicted values "acceptably near",
but not necessarily identical to, the observed responses Y.
The residuals (= deviations = error) of a model are the
vector of differences (Y - ) between the responses and the
predicted values from the model. For a perfect-fit model,
the vector of residuals would be all zeros. For a good-fitting
model, the vector of residuals will be acceptably (from an
engineering point of view) close to zero.
5.5.9.9.2. Motivation: How do we Construct a Goodness-of-fit Metric for a Model?

5.5.9.9.2. Motivation: How do we Construct a
Goodness-of-fit Metric for a Model?
Motivation This question deals with the issue of how to construct a
metric, a statistic, that may be used to ascertain the quality of
the fitted model. The statistic should be such that for one range
of values, the implication is that the model is good, whereas
for another range of values, the implication is that the model
gives a poor fit.
Sum of
absolute
residuals
Since a model's adequacy is inversely related to the size of its
residuals, one obvious statistic is the sum of the absolute
residuals.
Clearly, for a fixed n,the smaller this sum is, the smaller are
the residuals, which implies the closer the predicted values are
to the raw data Y, and hence the better the fitted model. The
primary disadvantage of this statistic is that it may grow larger
simply as the sample size n grows larger.
Average
absolute
residual
A better metric that does not change (much) with increasing
sample size is the average absolute residual:
with n denoting the number of response values. Again, small
values for this statistic imply better-fitting models.
Square
root of the
average
squared
residual
An alternative, but similar, metric that has better statistical
properties is the square root of the average squared residual.
As with the previous statistic, the smaller this statistic, the
better the model.
5.5.9.9.2. Motivation: How do we Construct a Goodness-of-fit Metric for a Model?
Residual
standard
deviation
Our final metric, which is used directly in inferential statistics,
is the residual standard deviation
with p denoting the number of fitted coefficients in the model.
This statistic is the standard deviation of the residuals from a
given model. The smaller is this residual standard deviation,
the better fitting is the model. We shall use the residual
standard deviation as our metric of choice for evaluating and
comparing various proposed models.
5.5.9.9.3. Motivation: How do we Construct a Good Model?

5.5.9.9.3. Motivation: How do we Construct a
Good Model?
Models for
2
k
and 2
k-p
designs
Given that we have a statistic to measure the quality of a
model, any model, we move to the question of how to
construct reasonable models for fitting data from 2
k
and 2
k-p
designs.
Initial
simple
model
The simplest such proposed model is
that is, the response Y = a constant + random error. This
trivial model says that all of the factors (and interactions) are
in fact worthless for prediction and so the best-fit model is
one that consists of a simple horizontal straight line through
the body of the data. The least squares estimate for this
constant c in the above model is the sample mean . The
prediction equation for this model is thus
The predicted values for this fitted trivial model are thus
given by a vector consisting of the same value (namely )
throughout. The residual vector for this model will thus
simplify to simple deviations from the mean:
Since the number of fitted coefficients in this model is 1
(namely the constant c), the residual standard deviation is the
following:
which is of course the familiar, commonly employed sample
standard deviation. If the residual standard deviation for this
trivial model were "small enough", then we could terminate
the model-building process right there with no further
inclusion of terms. In practice, however, this trivial model
does not yield a residual standard deviation that is small
enough (because the common value will not be close
enough to some of the raw responses Y) and so the model
must be augmented--but how?
Next-step
model
The logical next-step proposed model will consist of the
above additive constant plus some term that will improve the
predicted values the most. This will equivalently reduce the
residuals the most and thus reduce the residual standard
deviation the most.
Using the
most
important
effects
As it turns out, it is a mathematical fact that the factor or
interaction that has the largest estimated effect
will necessarily, after being included in the model, yield the
"biggest bang for the buck" in terms of improving the
predicted values toward the response values Y. Hence at this
point the model-building process and the effect estimation
process merge.
In the previous steps in our analysis, we developed a ranked
list of factors and interactions. We thus have a ready-made
ordering of the terms that could be added, one at a time, to the
model. This ranked list of effects is precisely what we need to
cumulatively build more complicated, but better fitting,
models.
Step
through
the ranked
list of
factors
Our procedure will thus be to step through, one by one, the
ranked list of effects, cumulatively augmenting our current
model by the next term in the list, and then compute (for all n
design points) the predicted values, residuals, and residual
standard deviation. We continue this one-term-at-a-time
augmentation until the predicted values are acceptably close to
the observed responses Y (and hence the residuals and residual
standard deviation become acceptably close to zero).
Starting with the simple average, each cumulative model in
this iteration process will have its own associated residual
standard deviation. In practice, the iteration continues until the
residual standard deviations become sufficiently small.
Cumulative
residual
standard
deviation
plot
The cumulative residual standard deviation plot is a graphical
summary of the above model-building process. On the
horizontal axis is a series of terms (starting with the average,
and continuing on with various main effects and interactions).
After the average, the ordering of terms on the horizontal axis
is identical to the ordering of terms based on the half-normal
probability plot ranking based on effect magnitude.
On the vertical axis is the corresponding residual standard
deviation that results when the cumulative model has its
coefficients fitted via least squares, and then has its predicted
values, residuals, and residual standard deviations computed.
The first residual standard deviation (on the far left of the
cumulative residual standard deviation plot) is that which
results from the model consisting of
1. the average.
The second residual standard deviation plotted is from the
model consisting of
1. the average, plus
2. the term with the largest |effect|.
The third residual standard deviation plotted is from the model
consisting of
1. the average, plus
2. the term with the largest |effect|, plus
3. the term with the second largest |effect|.
and so forth.
5.5.9.9.4. Motivation: How do we Know When to Stop Adding Terms?

5.5.9.9.4. Motivation: How do we Know When
to Stop Adding Terms?
Cumulative
residual
standard
deviation
plot
typically
has a
hockey
stick
appearance
Proceeding left to right, as we add more terms to the model,
the cumulative residual standard deviation "curve" will
typically decrease. At the beginning (on the left), as we add
large-effect terms, the decrease from one residual standard
deviation to the next residual standard deviation will be large.
The incremental improvement (decrease) then tends to drop
off slightly. At some point the incremental improvement will
typically slacken off considerably. Appearance-wise, it is thus
very typical for such a curve to have a "hockey stick"
appearance:
1. starting with a series of large decrements between
successive residual standard deviations; then
2. hitting an elbow; then
3. having a series of gradual decrements thereafter.
Stopping
rule
The cumulative residual standard deviation plot provides a
visual answer to the question:
What is a good model?
by answering the related question:
When do we stop adding terms to the cumulative
model?
Graphically, the most common stopping rule for adding terms
is to cease immediately upon encountering the "elbow". We
include all terms up to and including the elbow point since
each of these terms decreased the residual standard deviation
by a large amount. However, we exclude any terms afterward
since these terms do not decrease the residual standard
deviation fast enough to warrant inclusion in the model.
5.5.9.9.5. Motivation: What is the Form of the Model?

5.5.9.9.5. Motivation: What is the Form of the
Model?
Models for
various
values of k
From the above discussion, we thus note and recommend a
form of the model that consists of an additive constant plus a
linear combination of main effects and interactions. What
then is the specific form for the linear combination?
The following are the full models for various values of k.
The selected final model will be a subset of the full model.
For the k = 1 factor case:
Y = f(X
1
) + = c + B
1
X
1
+
Y = f(X
1
,X
2
) +
= c + B
1
X
1
+ B
2
X
2
+
B
12
X
1
X
2
+
Y = f(X
1
,X
2
,X
3
) +
= c + B
1
X
1
+ B
2
X
2
+ B
3
X
3

+ B
12
X
1
X
2
+ B
13
X
1
X
3
+
B
23
X
2
X
3
+ B
123
X
1
X
2
X
2

+
and for the general k case:
Y = f(X
1
,X
2
, ..., X
k
) +
= c + (linear combination
of all main effects and all
interactions) +
Note that the model equations shown above include
coefficients that represent the change in Y for a one-unit
change in X
i
. To obtain an effect estimate, which represents a
5.5.9.9.5. Motivation: What is the Form of the Model?
two-unit change in X
i
if the levels of X
i
are +1 and -1, simply
multiply the coefficient by two.
Ordered
linear
combination
The listing above has the terms ordered with the main effects,
then the 2-factor interactions, then the 3-factor interactions,
etc. In practice, it is recommended that the terms be ordered
by importance (whether they be main effects or interactions).
Aside from providing a functional representation of the
response, models should help reinforce what is driving the
response, which such a re-ordering does. Thus for k = 2, if
factor 2 is most important, the 2-factor interaction is next in
importance, and factor 1 is least important, then it is
recommended that the above ordering of
Y = f(X
1
,X
2
) +
= c + B
1
X
1
+ B
2
X
2
+
B
12
X
1
X
2
+
be rewritten as
Y = f(X
1
,X
2
) +
= c + B
2
X
2
+ B
12
X
1
X
2
+
B
1
X
1
+
5.5.9.9.6. Motivation: What are the Advantages of the Linear Combinatoric Model?

5.5.9.9.6. Motivation: What are the Advantages
of the Linear Combinatoric Model?
Advantages:
perfect fit
and
comparable
coefficients
The linear model consisting of main effects and all
interactions has two advantages:
1. Perfect Fit: If we choose to include in the model all of
the main effects and all interactions (of all orders), then
the resulting least squares fitted model will have the
property that the predicted values will be identical to
the raw response values Y. We will illustrate this in the
next section.
2. Comparable Coefficients: Since the model fit has been
carried out in the coded factor (-1, +1) units rather
than the units of the original factor (temperature, time,
pressure, catalyst concentration, etc.), the factor
coefficients immediately become comparable to one
another, which serves as an immediate mechanism for
the scale-free ranking of the relative importance of the
factors.
Example To illustrate in detail the above latter point, suppose the (-1,
+1) factor X
1
is really a coding of temperature T with the
original temperature ranging from 300 to 350 degrees and the
(-1, +1) factor X
2
is really a coding of time t with the
original time ranging from 20 to 30 minutes. Given that, a
linear model in the original temperature T and time t would
yield coefficients whose magnitude depends on the
magnitude of T (300 to 350) and t (20 to 30), and whose
value would change if we decided to change the units of T
(e.g., from Fahrenheit degrees to Celsius degrees) and t (e.g.,
from minutes to seconds). All of this is avoided by carrying
out the fit not in the original units for T (300,350) and t (20,
30), but in the coded units of X
1
(-1, +1) and X
2
(-1, +1).
The resulting coefficients are unit-invariant, and thus the
coefficient magnitudes reflect the true contribution of the
factors and interactions without regard to the unit of
measurement.
Coding Such coding leads to no loss of generality since the coded
5.5.9.9.6. Motivation: What are the Advantages of the Linear Combinatoric Model?
does not
lead to loss
of
generality
factor may be expressed as a simple linear relation of the
original factor (X
1
to T, X
2
to t). The unit-invariant coded
coefficients may be easily transformed to unit-sensitive
original coefficients if so desired.
5.5.9.9.7. Motivation: How do we use the Model to Generate Predicted Values?

5.5.9.9.7. Motivation: How do we use the Model
to Generate Predicted Values?
Design matrix
with response
for two
factors
To illustrate the details as to how a model may be used for
prediction, let us consider a simple case and generalize
from it. Consider the simple Yates-order 2
2
full factorial
design in X
1
and X
2
, augmented with a response vector Y:
X
1
X
2
Y
- - 2
+ - 4
- + 6
+ + 8
Geometric
representation
This can be represented geometrically
Determining
the prediction
equation
For this case, we might consider the model
Y = c + B
1
*X
1
+ B
2
*X
2
+
B
12
*X
1
*X
2
+
From the above diagram, we may deduce that the estimated
factor effects are:
c = the average response =
(2 + 4 + 6 + 8) / 4 = 5
=
E
1
=
=
average change in Y as X
1
goes from -1 to +1
((4-2) + (8-6)) / 2 = (2 + 2) / 2 = 2
Note: the (4-2) is the change in Y (due to X
1
)
on the lower axis; the (8-6) is the change in Y
(due to X
1
) on the upper axis.
E
2
=
=
average change in Y as X
2
goes from -1 to +1
((6-2) + (8-4)) / 2 = (4 + 4) / 2 = 4
E
12
=
=
interaction = (the less obvious) average
change in Y as X
1
*X
2
goes from -1 to +1
((2-4) + (8-6)) / 2 = (-2 + 2) / 2 = 0
For factors coded using +1 and -1, the least-squares
estimate of a coefficient is one half of the effect estimate
(B
i
= E
i
/ 2), so the fitted model (that is, the prediction
equation) is
= 5 + 1*X
1
+ 2*X
2
+ 0*X
1
*X
2
or with the terms rearranged in descending order of
importance
= 5 + 2*X
2
+ X
1
Table of fitted
values
Substituting the values for the four design points into this
equation yields the following fitted values
X
1
X
2
Y
- - 2 2
+ - 4 4
- + 6 6
+ + 8 8
Perfect fit This is a perfect-fit model. Such perfect-fit models will
result anytime (in this orthogonal 2-level design family)
we include all main effects and all interactions.
Remarkably, this is true not only for k = 2 factors, but for
general k.
Residuals For a given model (any model), the difference between the
response value Y and the predicted value is referred to as
the "residual":
residual = Y -
The perfect-fit full-blown (all main factors and all
interactions of all orders) models will have all residuals
identically zero.
The perfect fit is a mathematical property that comes if we
choose to use the linear model with all possible terms.
Price for
perfect fit
What price is paid for this perfect fit? One price is that the
variance of is increased unnecessarily. In addition, we
have a non-parsimonious model. We must compute and
carry the average and the coefficients of all main effects
and all interactions. Including the average, there will in
general be 2
k
coefficients to fully describe the fitting of the
n = 2
k
points. This is very much akin to the Y = f(X)
polynomial fitting of n distinct points. It is well known that
this may be done "perfectly" by fitting a polynomial of
degree n-1. It is comforting to know that such perfection is
mathematically attainable, but in practice do we want to do
this all the time or even anytime? The answer is generally
"no" for two reasons:
1. Noise: It is very common that the response data Y has
noise (= error) in it. Do we want to go out of our
way to fit such noise? Or do we want our model to
filter out the noise and just fit the "signal"? For the
latter, fewer coefficients may be in order, in the same
spirit that we may forego a perfect-fitting (but
jagged) 11-th degree polynomial to 12 data points,
and opt out instead for an imperfect (but smoother)
3rd degree polynomial fit to the 12 points.
2. Parsimony: For full factorial designs, to fit the n = 2
k
points we would need to compute 2
k
coefficients.
We gain information by noting the magnitude and
sign of such coefficients, but numerically we have n
data values Y as input and n coefficients B as output,
and so no numerical reduction has been achieved.
We have simply used one set of n numbers (the data)
to obtain another set of n numbers (the coefficients).
Not all of these coefficients will be equally
important. At times that importance becomes clouded
by the sheer volume of the n = 2
k
coefficients.
Parsimony suggests that our result should be simpler
and more focused than our n starting points. Hence
fewer retained coefficients are called for.
The net result is that in practice we almost always give up
the perfect, but unwieldy, model for an imperfect, but
parsimonious, model.
Imperfect fit The above calculations illustrated the computation of
predicted values for the full model. On the other hand, as
discussed above, it will generally be convenient for signal
or parsimony purposes to deliberately omit some
unimportant factors. When the analyst chooses such a
model, we note that the methodology for computing
predicted values is precisely the same. In such a case,
however, the resulting predicted values will in general not
be identical to the original response values Y; that is, we no
longer obtain a perfect fit. Thus, linear models that omit
some terms will have virtually all non-zero residuals.
5.5.9.9.8. Motivation: How do we Use the Model Beyond the Data Domain?

5.5.9.9.8. Motivation: How do we Use the Model
Beyond the Data Domain?
Interpolation
and
extrapolation
The previous section illustrated how to compute predicted
values at the points included in the design. One of the
virtues of modeling is that the resulting prediction equation
is not restricted to the design data points. From the
prediction equation, predicted values can be computed
elsewhere and anywhere:
1. within the domain of the data (interpolation);
2. outside of the domain of the data (extrapolation).
In the hands of an expert scientist/engineer/analyst, the
ability to predict elsewhere is extremely valuable. Based on
the fitted model, we have the ability to compute predicted
values for the response at a large number of internal and
external points. Thus the analyst can go beyond the handful
of factor combinations at hand and can get a feel (typically
via subsequent contour plotting) as to what the nature of the
entire response surface is.
This added insight into the nature of the response is "free"
and is an incredibly important benefit of the entire model-
building exercise.
Predict with
caution
Can we be fooled and misled by such a mathematical and
computational exercise? After all, is not the only thing that
is "real" the data, and everything else artificial? The answer
is "yes", and so such interpolation/extrapolation is a double-
edged sword that must be wielded with care. The best
attitude, and especially for extrapolation, is that the derived
conclusions must be viewed with extra caution.
By construction, the recommended fitted models should be
good at the design points. If the full-blown model were
used, the fit will be perfect. If the full-blown model is
reduced just a bit, then the fit will still typically be quite
good. By continuity, one would expect perfection/goodness
at the design points would lead to goodness in the
immediate vicinity of the design points. However, such
local goodness does not guarantee that the derived model
will be good at some distance from the design points.
5.5.9.9.8. Motivation: How do we Use the Model Beyond the Data Domain?
Do
confirmation
runs
Modeling and prediction allow us to go beyond the data to
gain additional insights, but they must be done with great
caution. Interpolation is generally safer than extrapolation,
but mis-prediction, error, and misinterpretation are liable to
occur in either case.
The analyst should definitely perform the model-building
process and enjoy the ability to predict elsewhere, but the
analyst must always be prepared to validate the interpolated
and extrapolated predictions by collection of additional real,
confirmatory data. The general empirical model that we
recommend knows "nothing" about the engineering, physics,
or chemistry surrounding your particular measurement
problem, and although the model is the best generic model
available, it must nonetheless be confirmed by additional
data. Such additional data can be obtained pre-
experimentally or post-experimentally. If done pre-
experimentally, a recommended procedure for checking the
validity of the fitted model is to augment the usual 2
k
or 2
k-p
designs with additional points at the center of the design.
This is discussed in the next section.
Applies only
for
continuous
factors
Of course, all such discussion of interpolation and
extrapolation makes sense only in the context of continuous
ordinal factors such as temperature, time, pressure, size, etc.
Interpolation and extrapolation make no sense for discrete
non-ordinal factors such as supplier, operators, design types,
etc.
5.5.9.9.9. Motivation: What is the Best Confirmation Point for Interpolation?

5.5.9.9.9. Motivation: What is the Best
Confirmation Point for Interpolation?
Augment via
center point
For the usual continuous factor case, the best (most efficient
and highest leverage) additional model-validation point that
may be added to a 2
k
or 2
k-p
design is at the center point.
This center point augmentation "costs" the experimentalist
only one additional run.
Example For example, for the k = 2 factor (Temperature (300 to 350),
and time (20 to 30)) experiment discussed in the previous
sections, the usual 4-run 2
2
full factorial design may be
replaced by the following 5-run 2
2
full factorial design with
a center point.
X
1
X
2
Y
- - 2
+ - 4
- + 6
+ + 8
0 0
Predicted
value for the
center point
Since "-" stands for -1 and "+" stands for +1, it is natural to
code the center point as (0,0). Using the recommended
model
= 5 + 2*X
2
+ X
1
we can substitute 0 for X
1
and X
2
to generate the predicted
value of 5 for the confirmatory run.
Importance
of the
confirmatory
run
The importance of the confirmatory run cannot be
overstated. If the confirmatory run at the center point yields
a data value of, say, Y = 5.1, since the predicted value at the
center is 5 and we know the model is perfect at the corner
points, that would give the analyst a greater confidence that
the quality of the fitted model may extend over the entire
interior (interpolation) domain. On the other hand, if the
5.5.9.9.9. Motivation: What is the Best Confirmation Point for Interpolation?
confirmatory run yielded a center point data value quite
different (e.g., Y = 7.5) from the center point predicted value
of 5, then that would prompt the analyst to not trust the
fitted model even for interpolation purposes. Hence when
our factors are continuous, a single confirmatory run at the
center point helps immensely in assessing the range of trust
for our model.
Replicated
center points
In practice, this center point value frequently has two, or
even three or more, replications. This not only provides a
reference point for assessing the interpolative power of the
model at the center, but it also allows us to compute model-
free estimates of the natural error in the data. This in turn
allows us a more rigorous method for computing the
uncertainty for individual coefficients in the model and for
rigorously carrying out a lack-of-fit test for assessing
general model adequacy.
5.5.9.9.10. Motivation: How do we Use the Model for Interpolation?

5.5.9.9.10. Motivation: How do we Use the
Model for Interpolation?
Design table
in original
data units
As for the mechanics of interpolation itself, consider a
continuation of the prior k = 2 factor experiment. Suppose
temperature T ranges from 300 to 350 and time t ranges
from 20 to 30, and the analyst can afford n = 4 runs. A 2
2
full factorial design is run. Forming the coded temperature
as X
1
and the coded time as X
2
, we have the usual:
Temperature Time X
1
X
2
Y
300 20 - - 2
350 20 + - 4
300 30 - + 6
350 30 + + 8
Graphical
representation
Graphically the design and data are as follows:
Typical
interpolation
question
As before, from the data, the prediction equation is
= 5 + 2*X
2
+ X
1
We now pose the following typical interpolation question:
From the model, what is the predicted response at,
say, temperature = 310 and time = 26?
In short:
(T = 310, t = 26) = ?
To solve this problem, we first view the k = 2 design and
data graphically, and note (via an "X") as to where the
desired (T = 310, t = 26) interpolation point is:
Predicting the
response for
the
interpolated
point
The important next step is to convert the raw (in units of
the original factors T and t) interpolation point into a coded
(in units of X
1
and X
2
) interpolation point. From the graph
or otherwise, we note that a linear translation between T
and X
1
, and between t and X
2
yields
T = 300 => X
1
= -1
T = 350 => X
1
= +1
thus
X
1
= 0 is at T = 325
|-------------|-------------|
-1 ? 0 +1
300 310 325 350

which in turn implies that
T = 310 => X
1
= -0.6
Similarly,
t = 20 => X
2
= -1
t = 30 => X
2
= +1
therefore,
X
2
= 0 is at t = 25
|-------------|-------------|
-1 0 ? +1
20 25 26 30

thus
t = 26 => X
2
= +0.2
Substituting X
1
= -0.6 and X
2
= +0.2 into the prediction
equation
= 5 + 2*X
2
+ X
1
yields a predicted value of 4.8.
Graphical
representation
of response
value for
interpolated
data point
Thus
5.5.9.9.11. Motivation: How do we Use the Model for Extrapolation?
http://www.itl.nist.gov/div898/handbook/pri/section5/pri599b.htm[6/27/2012 2:25:44 PM]

Model for Extrapolation?
Graphical
representation
of
extrapolation
Extrapolation is performed similarly to interpolation. For
example, the predicted value at temperature T = 375 and
time t = 28 is indicated by the "X":
and is computed by substituting the values X
1
= +2.0
(T=375) and X
2
= +0.8 (t=28) into the prediction equation
= 5 + 2*X
2
+ X
1
yielding a predicted value of 8.6. Thus we have
Pseudo-data The predicted value from the modeling effort may be
viewed as pseudo-data, data obtained without the
experimental effort. Such "free" data can add tremendously
to the insight via the application of graphical techniques (in
particular, the contour plots and can add significant insight
and understanding as to the nature of the response surface
relating Y to the X's.
But, again, a final word of caution: the "pseudo data" that
results from the modeling process is exactly that, pseudo-
data. It is not real data, and so the model and the model's
predicted values must be validated by additional
confirmatory (real) data points. A more balanced approach
is that:
Models may be trusted as "real" [that is, to generate
predicted values and contour curves], but must
always be verified [that is, by the addition of
confirmatory data points].
The rule of thumb is thus to take advantage of the available
and recommended model-building mechanics for these 2-
level designs, but do treat the resulting derived model with
an equal dose of both optimism and caution.
Summary In summary, the motivation for model building is that it
gives us insight into the nature of the response surface
along with the ability to do interpolation and extrapolation;
further, the motivation for the use of the cumulative
residual standard deviation plot is that it serves as an easy-
to-interpret tool for determining a good and parsimonious
model.
http://www.itl.nist.gov/div898/handbook/pri/section5/pri599c.htm[6/27/2012 2:25:45 PM]

Model for Extrapolation?
Graphical
representation
of
extrapolation
Extrapolation is performed similarly to interpolation. For
example, the predicted value at temperature T = 375 and
time t = 28 is indicated by the "X":
and is computed by substituting the values X1 = +2.0
(T=375) and X2 = +0.8 (t=28) into the prediction equation
yielding a predicted value of 8.6. Thus we have
Pseudo-data The predicted value from the modeling effort may be
viewed as pseudo-data, data obtained without the
experimental effort. Such "free" data can add tremendously
to the insight via the application of graphical techniques (in
particular, the contour plots and can add significant insight
and understanding as to the nature of the response surface
relating Y to the X's.
But, again, a final word of caution: the "pseudo data" that
results from the modeling process is exactly that, pseudo-
data. It is not real data, and so the model and the model's
predicted values must be validated by additional
confirmatory (real) data points. A more balanced approach
is that:
Models may be trusted as "real" [that is, to generate
predicted values and contour curves], but must
always be verified [that is, by the addition of
confirmatory data points].
The rule of thumb is thus to take advantage of the available
and recommended model-building mechanics for these 2-
level designs, but do treat the resulting derived model with
an equal dose of both optimism and caution.
Summary In summary, the motivation for model building is that it
gives us insight into the nature of the response surface
along with the ability to do interpolation and extrapolation;
further, the motivation for the use of the cumulative
residual standard deviation plot is that it serves as an easy-
to-interpret tool for determining a good and parsimonious
model.
5.5.9.10. DOE contour plot

Purpose The DOE contour plot answers the question:
Where else could we have run the experiment to optimize the response?
Prior steps in this analysis have suggested the best setting for each of the k factors.
These best settings may have been derived from
1. Data: which of the n design points yielded the best response, and what were
the settings of that design point, or from
2. Averages: what setting of each factor yielded the best response "on the
average".
This 10th (and last) step in the analysis sequence goes beyond the limitations of the
n data points already chosen in the design and replaces the data-limited question
"From among the n data points, what was the best setting?"
to a region-related question:
"In general, what should the settings have been to optimize the response?"
Output The outputs from the DOE contour plot are
1. Primary: Best setting (X
10
, X
20
, ..., X
k0
) for each of the k factors. This derived
setting should yield an optimal response.
2. Secondary: Insight into the nature of the response surface and the
importance/unimportance of interactions.
Definition A DOE contour plot is formed by
Vertical Axis: The second most important factor in the experiment.
Horizontal Axis: The most important factor in the experiment.
More specifically, the DOE contour plot is constructed and utilized via the following
7 steps:
1. Axes
2. Contour Curves
3. Optimal Response Value
4. Best Corner
5. Steepest Ascent/Descent
6. Optimal Curve
7. Optimal Setting
with
1. Axes: Choose the two most important factors in the experiment as the two
axes on the plot.
2. Contour Curves: Based on the fitted model and the best data settings for all of
the remaining factors, draw contour curves involving the two dominant
factors. This yields a graphical representation of the response surface. The
details for constructing linear contour curves are given in a later section.
3. Optimal Value: Identify the theoretical value of the response that constitutes
"best." In particular, what value would we like to have seen for the response?
4. Best "Corner": The contour plot will have four "corners" for the two most
important factors X
i
and X
j
: (X
i
, X
j
) = (-, -), (-, +), (+, -), and (+, +). From the
data, identify which of these four corners yields the highest average response
.
5. Steepest Ascent/Descent: From this optimum corner point, and based on the
nature of the contour lines near that corner, step out in the direction of steepest
ascent (if maximizing) or steepest descent (if minimizing).
6. Optimal Curve: Identify the curve on the contour plot that corresponds to the
ideal optimal value.
7. Optimal Setting: Determine where the steepest ascent/descent line intersects
the optimum contour curve. This point represents our "best guess" as to where
we could have run our experiment so as to obtain the desired optimal
response.
Motivation In addition to increasing insight, most experiments have a goal of optimizing the
response. That is, of determining a setting (X
10
, X
20
, ..., X
k0
) for which the response
is optimized.
The tool of choice to address this goal is the DOE contour plot. For a pair of factors
X
i
and X
j
, the DOE contour plot is a 2-dimensional representation of the 3-
dimensional Y = f(X
i
, X
j
) response surface. The position and spacing of the
isocurves on the DOE contour plot are an easily interpreted reflection of the nature
of the surface.
In terms of the construction of the DOE contour plot, there are three aspects of note:
1. Pairs of Factors: A DOE contour plot necessarily has two axes (only); hence
only two out of the k factors can be represented on this plot. All other factors
must be set at a fixed value (their optimum settings as determined by the
ordered data plot, the DOE mean plot, and the interaction effects matrix plot).
2. Most Important Factor Pair: Many DOE contour plots are possible. For an
experiment with k factors, there are
possible contour plots. For example, for k = 4 factors there are 6 possible
contour plots: X
1
and X
2
, X
1
and X
3
, X
1
and X
4
, X
2
and X
3
, X
2
and X
4
, and X
3
and X
4
. In practice, we usually generate only one contour plot involving the
two most important factors.
3. Main Effects Only: The contour plot axes involve main effects only, not
interactions. The rationale for this is that the "deliverable" for this step is k
settings, a best setting for each of the k factors. These k factors are real and
can be controlled, and so optimal settings can be used in production.
Interactions are of a different nature as there is no "knob on the machine" by
which an interaction may be set to -, or to +. Hence the candidates for the
axes on contour plots are main effects only--no interactions.
In summary, the motivation for the DOE contour plot is that it is an easy-to-use
graphic that provides insight as to the nature of the response surface, and provides a
specific answer to the question "Where (else) should we have collected the data so
to have optimized the response?".
Plot for
defective
springs
data
Applying the DOE contour plot for the defective springs data set yields the
following plot.
How to
interpret
From the DOE contour plot for the defective springs data, we note the following
regarding the 7 framework issues:
Axes
Contour curves
Optimal response value
Optimal response curve
Best corner
Steepest Ascent/Descent
Optimal setting
Conclusions
for the
defective
springs
data
The application of the DOE contour plot to the defective springs data set results in
1. Optimal settings for the "next" run:
Coded : (X
1
, X
2
, X
3
) = (+1.5, +1.0, +1.3)
Uncoded: (OT, CC, QT) = (1637.5, 0.7, 127.5)
2. Nature of the response surface:
The X
1
*X
3
interaction is important, hence the effect of factor X
1
will change
depending on the setting of factor X
3
.
5.5.9.10.1. How to Interpret: Axes
http://www.itl.nist.gov/div898/handbook/pri/section5/pri59a1.htm[6/27/2012 2:25:48 PM]

What factors
go on the two
axes?
For this first item, we choose the two most important
factors in the experiment as the plot axes.
These are determined from the ranked list of important
factors as discussed in the previous steps. In particular, the
|effects| plot includes a ranked factor table. For the
defective springs data, that ranked list consists of
Factor/Interaction Effect Estimate
X
1
23
X
1
*X
3
10
X
2
-5
X
3
1.5
X
1
*X
2
1.5
X
1
*X
2
*X
3
0.5
X
2
*X
3
0
Possible
choices
In general, the two axes of the contour plot could consist of
X
1
and X
2
,
X
1
and X
3
, or
X
2
and X
3
.
In this case, since X
1
is the top item in the ranked list, with
an estimated effect of 23, X
1
is the most important factor
and so will occupy the horizontal axis of the contour plot.
The admissible list thus reduces to
X
1
and X
2
, or
X
1
and X
3
.
To decide between these two pairs, we look to the second
item in the ranked list. This is the interaction term X
1
*X
3
,
with an estimated effect of 10. Since interactions are not
allowed as contour plot axes, X
1
*X
3
must be set aside. On
the other hand, the components of this interaction (X
1
and
X
3
) are not to be set aside. Since X
1
has already been
identified as one axis in the contour plot, this suggests that
the other component (X
3
) be used as the second axis. We
do so. Note that X
3
itself does not need to be important (in
fact, it is noted that X
3
is ranked fourth in the listed table
with a value of 1.5).
In summary then, for this example the contour plot axes
are:
Horizontal Axis: X
1

Vertical Axis: X
3
Four cases
for
recommended
choice of
axes
Other cases can be more complicated. In general, the
recommended rule for selecting the two plot axes is that
they be drawn from the first two items in the ranked list of
factors. The following four cases cover most situations in
practice:
Case 1:
1. Item 1 is a main effect (e.g., X
3
)
2. Item 2 is another main effect (e.g., X
5
)
Recommended choice:
1. Horizontal axis: item 1 (e.g., X
3
);
2. Vertical axis: item 2 (e.g., X
5
).
Case 2:
3
)
2. Item 2 is a (common-element) interaction (e.g.,
X
3
*X
4
)
Recommended choice:
3
);
2. Vertical axis: the remaining component in item
2 (e.g., X
4
).
Case 3:
3
)
2. Item 2 is a (non-common-element) interaction
(e.g., X
2
*X
4
)
Recommended choice:
3
);
2. Vertical axis: either component in item 2 (e.g.,
X
2
, or X
4
), but preferably the one with the
largest individual effect (thus scan the rest of
the ranked factors and if the X
2
|effect| > X
4
|effect|, choose X
2
; otherwise choose X
4
).
Case 4:
1. Item 1 is a (2-factor) interaction (e.g., X
2
*X
4
)
2. Item 2 is anything
Recommended choice:
1. Horizontal axis: component 1 from the item 1
interaction (e.g., X
2
);
2. Horizontal axis: component 2 from the item 1
interaction (e.g., X
4
).
5.5.9.10.2. How to Interpret: Contour Curves

Non-linear
appearance
of contour
curves
implies
strong
interaction
Based on the fitted model (cumulative residual standard
deviation plot) and the best data settings for all of the
remaining factors, we draw contour curves involving the two
dominant factors. This yields a graphical representation of
the response surface.
Before delving into the details as to how the contour lines
were generated, let us first note as to what insight can be
gained regarding the general nature of the response surface.
For the defective springs data, the dominant characteristic of
the contour plot is the non-linear (fan-shaped, in this case)
appearance. Such non-linearity implies a strong X
1
*X
3
interaction effect. If the X
1
*X
3
interaction were small, the
contour plot would consist of a series of near-parallel lines.
Such is decidedly not the case here.
Constructing
the contour
curves
As for the details of the construction of the contour plot, we
draw on the model-fitting results that were achieved in the
cumulative residual standard deviation plot. In that step, we
derived the following good-fitting prediction equation:
= 71.25 + 11.5*X
1
+ 5*X
1
*X
3
- 2.5*X
2
The contour plot has axes of X
1
and X
3
. X
2
is not included
and so a fixed value of X
2
must be assigned. The response
variable is the percentage of acceptable springs, so we are
attempting to maximize the response. From the ordered data
plot, the main effects plot, and the interaction effects matrix
plot of the general analysis sequence, we saw that the best
setting for factor X
2
was "-". The best observed response
data value (Y = 90) was achieved with the run (X
1
, X
2
, X
3
) =
(+, -, +), which has X
2
= "-". Also, the average response for
X
2
= "-" was 73 while the average response for X
2
= "+" was
68. We thus set X
2
= -1 in the prediction equation to obtain
= 71.25 + 11.5*X
1
+ 5*X
1
*X
3
- 2.5*(-1)
= 73.75 + 11.5*X
1
+ 5*X
1
*X
3
This equation involves only X
1
and X
3
and is immediately
usable for the X
1
and X
3
contour plot. The raw response
values in the data ranged from 52 to 90. The response
implies that the theoretical worst is Y = 0 and the theoretical
best is Y = 100.
To generate the contour curve for, say, Y = 70, we solve
70 = 73.75 + 11.5*X
1
+ 5*X
1
*X
3
by rearranging the equation in X
3
(the vertical axis) as a
function of X
1
(the horizontal axis). By substituting various
values of X
1
into the rearranged equation, the above
equation generates the desired response curve for Y = 70.
We do so similarly for contour curves for any desired
response value Y.
Values for
X1
For these X
3
= g(X
1
) equations, what values should be used
for X
1
? Since X
1
is coded in the range -1 to +1, we
recommend expanding the horizontal axis to -2 to +2 to
allow extrapolation. In practice, for the DOE contour plot
generated previously, we chose to generate X
1
values from -
2, at increments of 0.05, up to +2. For most data sets, this
gives a smooth enough curve for proper interpretation.
Values for Y What values should be used for Y? Since the total theoretical
range for the response Y (= percent acceptable springs) is 0
% to 100 %, we chose to generate contour curves starting
with 0, at increments of 5, and ending with 100. We thus
generated 21 contour curves. Many of these curves did not
appear since they were beyond the -2 to +2 plot range for
the X
1
and X
3
factors.
Summary In summary, the contour plot curves are generated by
making use of the (rearranged) previously derived prediction
equation. For the defective springs data, the appearance of
the contour plot implied a strong X
1
*X
3
interaction.
5.5.9.10.3. How to Interpret: Optimal Response Value

5.5.9.10.3. How to Interpret: Optimal Response
Value
Need to
define
"best"
We need to identify the theoretical value of the response that
would constitute "best". What value would we like to have
seen for the response?
For example, if the response variable in a chemical experiment
is percent reacted, then the ideal theoretical optimum would be
100 %. If the response variable in a manufacturing experiment
is amount of waste, then the ideal theoretical optimum would
be zero. If the response variable in a flow experiment is the
fuel flow rate in an engine, then the ideal theoretical optimum
(as dictated by engine specifications) may be a specific value
(e.g., 175 cc/sec). In any event, for the experiment at hand,
select a number that represents the ideal response value.
Optimal
value for
this
example
For the defective springs data, the response (percentage of
acceptable springs) ranged from Y = 52 to 90. The
theoretically worst value would be 0 (= no springs are
acceptable), and the theoretically best value would be 100 (100
% of the springs are acceptable). Since we are trying to
maximize the response, the selected optimal value is 100.
5.5.9.10.4. How to Interpret: Best Corner

Four
corners
representing
2 levels for
2 factors
The contour plot will have four "corners" (two factors times
two settings per factor) for the two most important factors X
i
and X
j
: (X
i
, X
j
) = (-, -), (-, +), (+, -), or (+, +). Which of
these four corners yields the highest average response ?
That is, what is the "best corner"?
Use the raw
data
This is done by using the raw data, extracting out the two
"axes factors", computing the average response at each of the
four corners, then choosing the corner with the best average.
For the defective springs data, the raw data were
X
1
X
2
X
3
Y
- - - 67
+ - - 79
- + - 61
+ + - 75
- - + 59
+ - + 90
- + + 52
+ + + 87
The two plot axes are X
1
and X
3
and so the relevant raw data
collapses to
X
1
X
3
Y
- - 67
+ - 79
- - 61
+ - 75
- + 59
+ + 90
- + 52
+ + 87
Averages which yields averages
X
1
X
3
Y
- - (67 + 61)/2 = 64
+ - (79 + 75)/2 = 77
- + (59 + 52)/2 = 55.5
+ + (90 + 87)/2 = 88.5
These four average values for the corners are annotated on
the plot. The best (highest) of these values is 88.5. This
comes from the (+, +) upper right corner. We conclude that
for the defective springs data the best corner is (+, +).
5.5.9.10.5. How to Interpret: Steepest Ascent/Descent

5.5.9.10.5. How to Interpret: Steepest
Ascent/Descent
Start at
optimum
corner point
From the optimum corner point, based on the nature of the
contour surface at that corner, step out in the direction of
steepest ascent (if maximizing) or steepest descent (if
minimizing).
Defective
springs
example
Since our goal for the defective springs problem is to
maximize the response, we seek the path of steepest ascent.
Our starting point is the best corner (the upper right corner
(+, +)), which has an average response value of 88.5. The
contour lines for this plot have increments of 5 units. As we
move from left to right across the contour plot, the contour
lines go from low to high response values. In the plot, we
have drawn the maximum contour level, 105, as a thick line.
For easier identification, we have also drawn the contour
level of 90 as thick line. This contour level of 90 is
immediately to the right of the best corner
Conclusions
on steepest
ascent for
defective
springs
example
The nature of the contour curves in the vicinity of (+, +)
suggests a path of steepest ascent
1. in the "northeast" direction
2. about 30 degrees above the horizontal.
5.5.9.10.6. How to Interpret: Optimal Curve

5.5.9.10.6. How to Interpret: Optimal Curve
Corresponds
to ideal
optimum
value
The optimal curve is the curve on the contour plot that
corresponds to the ideal optimum value.
Defective
springs
example
For the defective springs data, we search for the Y = 100
contour curve. As determined in the steepest ascent/descent
section, the Y = 90 curve is immediately outside the (+, +)
point. The next curve to the right is the Y = 95 curve, and
the next curve beyond that is the Y = 100 curve. This is the
optimal response curve.
5.5.9.10.7. How to Interpret: Optimal Setting

Optimal
setting
The "near-point" optimality setting is the intersection of the
steepest-ascent line with the optimal setting curve.
Theoretically, any (X
1
, X
3
) setting along the optimal curve
would generate the desired response of Y = 100. In practice,
however, this is true only if our estimated contour surface is
identical to "nature's" response surface. In reality, the plotted
contour curves are truth estimates based on the available
(and "noisy") n = 8 data values. We are confident of the
contour curves in the vicinity of the data points (the four
corner points on the chart), but as we move away from the
corner points, our confidence in the contour curves
decreases. Thus the point on the Y = 100 optimal response
curve that is "most likely" to be valid is the one that is
closest to a corner point. Our objective then is to locate that
"near-point".
Defective
springs
example
In terms of the defective springs contour plot, we draw a line
from the best corner, (+, +), outward and perpendicular to
the Y = 90, Y = 95, and Y = 100 contour curves. The Y = 100
intersection yields the "nearest point" on the optimal
response curve.
Having done so, it is of interest to note the coordinates of
that optimal setting. In this case, from the graph, that setting
is (in coded units) approximately at
(X
1
= 1.5, X
3
= 1.3)
Table of
coded and
uncoded
factors
With the determination of this setting, we have thus, in
theory, formally completed our original task. In practice,
however, more needs to be done. We need to know "What is
this optimal setting, not just in the coded units, but also in
the original (uncoded) units"? That is, what does (X
1
=1.5,
X
3
=1.3) correspond to in the units of the original data?
To deduce his, we need to refer back to the original
(uncoded) factors in this problem. They were:
Coded
Factor
Uncoded Factor
X
1
OT: Oven Temperature
X
2
CC: Carbon Concentration
X
3
QT: Quench Temperature
Uncoded
and coded
factor
settings
These factors had settings-- what were the settings of the
coded and uncoded factors? From the original description of
the problem, the uncoded factor settings were:
1. Oven Temperature (1450 and 1600 degrees)
2. Carbon Concentration (0.5 % and 0.7 %)
3. Quench Temperature (70 and 120 degrees)
with the usual settings for the corresponding coded factors:
1. X
1
(-1, +1)
2. X
2
(-1, +1)
3. X
3
(-1, +1)
Diagram To determine the corresponding setting for (X
1
=1.5,
X
3
=1.3), we thus refer to the following diagram, which
mimics a scatter plot of response averages--oven
temperature (OT) on the horizontal axis and quench
temperature (QT) on the vertical axis:
The "X" on the chart represents the "near point" setting on
the optimal curve.
Optimal
setting for
X
1
(oven
To determine what "X" is in uncoded units, we note (from
the graph) that a linear transformation between OT and X
1
as defined by
temperature)
OT = 1450 => X
1
= -1
OT = 1600 => X
1
= +1
yields
X
1
= 0 being at OT = (1450 + 1600) / 2 = 1525
thus
|-------------|-------------|
X1: -1 0 +1
OT: 1450 1525 1600
and so X
1
= +2, say, would be at oven temperature OT =
1675:
|-------------|-------------|---------
----|
X1: -1 0 +1
+2
OT: 1450 1525 1600
1675
and hence the optimal X
1
setting of 1.5 must be at
OT = 1600 + 0.5*(1675-1600) = 1637.5
Optimal
setting for
X
3
(quench
temperature)
Similarly, from the graph we note that a linear
transformation between quench temperature QT and coded
factor X
3
as specified by
QT = 70 => X
3
= -1
QT = 120 => X
3
= +1
yields
X
3
= 0 being at QT = (70 + 120) / 2 = 95
as in
|-------------|-------------|
X3: -1 0 +1
QT: 70 95 120
and so X
3
= +2, say, would be quench temperature = 145:
|-------------|-------------|------------
-|
X3: -1 0 +1
+2
QT: 70 95 120
145
Hence, the optimal X
3
setting of 1.3 must be at
QT = 120 + 0.3*(145-120)
QT = 127.5
Summary of
optimal
settings
In summary, the optimal setting is
coded : (X
1
= +1.5, X
3
= +1.3)
uncoded: (OT = 1637.5 degrees, QT = 127.5 degrees)
and finally, including the best setting of the fixed X
2
factor
(carbon concentration CC) of X
2
= -1 (CC = 0.5 %), we thus
have the final, complete recommended optimal settings for
all three factors:
coded : (X
1
= +1.5, X
2
= -1.0, X
3
= +1.3)
uncoded: (OT = 1637.5, CC = 0.7 %, QT = 127.5)
If we were to run another experiment, this is the point (based
on the data) that we would set oven temperature, carbon
concentration, and quench temperature with the hope/goal of
achieving 100 % acceptable springs.
Options for
next step
In practice, we could either
1. collect a single data point (if money and time are an
issue) at this recommended setting and see how close
to 100 % we achieve, or
2. collect two, or preferably three, (if money and time are
less of an issue) replicates at the center point
(recommended setting).
3. if money and time are not an issue, run a 2
2
full
factorial design with center point. The design is
centered on the optimal setting (X
1
= +1, 5, X
3
= +1.3)
with one overlapping new corner point at (X
1
= +1, X
3
= +1) and with new corner points at (X
1
, X
3
) = (+1,
+1), (+2, +1), (+1, +1.6), (+2, +1.6). Of these four
new corner points, the point (+1, +1) has the
advantage that it overlaps with a corner point of the
original design.
5.6. Case Studies

5.6. Case Studies
Contents The purpose of this section is to illustrate the analysis of
designed experiments with data collected from experiments
run at the National Institute of Standards and Technology and
SEMATECH.
1. Eddy current probe sensitivity study
2. Sonoluminescent light intensity study
5.6.1. Eddy Current Probe Sensitivity Case Study

5.6. Case Studies
5.6.1. Eddy Current Probe Sensitivity Case
Study
Analysis of
a 2
3
Full
Factorial
Design
This case study demonstrates the analysis of a 2
3
full factorial
design.
The analysis for this case study is based on the EDA approach
discussed in an earlier section.
Contents The case study is divided into the following sections:
1. Background and data
2. Initial plots/main effects
3. Interaction effects
4. Main and interaction effects: block plots
5. Estimate main and interaction effects
6. Modeling and prediction equations
7. Intermediate conclusions
8. Important factors and parsimonious prediction
9. Validate the fitted model
10. Using the model
11. Conclusions and next step
12. Work this example yourself

5.6. Case Studies
Background The data for this case study is a subset of a study performed
by Capobianco, Splett, and Iyer. Capobianco was a member
of the NIST Electromagnetics Division and Splett and Iyer
were members of the NIST Statistical Engineering Division
at the time of this study.
The goal of this project is to develop a nondestructive
portable device for detecting cracks and fractures in metals. A
primary application would be the detection of defects in
airplane wings. The internal mechanism of the detector would
be for sensing crack-induced changes in the detector's
electromagnetic field, which would in turn result in changes
in the impedance level of the detector. This change of
impedance is termed "sensitivity" and it is a sub-goal of this
experiment to maximize such sensitivity as the detector is
moved from an unflawed region to a flawed region on the
metal.
Statistical
Goals
The case study illustrates the analysis of a 2
3
full factorial
experimental design. The specific statistical goals of the
experiment are:
1. Determine the important factors that affect sensitivity.
2. Determine the settings that maximize sensitivity.
3. Determine a predicition equation that functionally
relates sensitivity to various factors.
Data Used
in the
Analysis
There were three detector wiring component factors under
consideration:
1. X1 = Number of wire turns
2. X2 = Wire winding distance
3. X3 = Wire gauge
Since the maximum number of runs that could be afforded
timewise and costwise in this experiment was n = 10, a 2
3
full factoral experiment (involving n = 8 runs) was chosen.
With an eye to the usual monotonicity assumption for two-
level factorial designs, the selected settings for the three
factors were as follows:
1. X1 = Number of wire turns : -1 = 90, +1 = 180
2. X2 = Wire winding distance: -1 = 0.38, +1 = 1.14
3. X3 = Wire gauge : -1 = 40, +1 = 48
The experiment was run with the eight settings executed in
random order. The following data resulted.
Y X1 X2 X3
Probe Number Winding Wire Run
Impedance of Turns Distance Gauge Sequence
-------------------------------------------------
1.70 -1 -1 -1 2
4.57 +1 -1 -1 8
0.55 -1 +1 -1 3
3.39 +1 +1 -1 6
1.51 -1 -1 +1 7
4.59 +1 -1 +1 1
0.67 -1 +1 +1 4
4.29 +1 +1 +1 5
Note that the independent variables are coded as +1 and -1.
These represent the low and high settings for the levels of
each variable. Factorial designs often have two levels for
each factor (independent variable) with the levels being
coded as -1 and +1. This is a scaling of the data that can
simplify the analysis. If desired, these scaled values can be
converted back to the original units of the data for
presentation.
5.6.1.2. Initial Plots/Main Effects

5.6. Case Studies
Plot the
Data:
Ordered
Data Plot
The first step in the analysis is to generate an ordered data plot.
Conclusions
from the
Ordered
Data Plot
We can make the following conclusions based on the ordered data plot.
1. Important Factors: The four highest response values have X1 = + while the four
lowest response values have X1 = -. This implies X1 is the most important factor.
When X1 = -, the - values of X2 are higher than the + values of X2. Similarly, when
X1 = +, the - values of X2 are higher than the + values of X2. This implies X2 is
important, but less so than X1. There is no clear pattern for X3.
2. Best Settings: In this experiment, we are using the device as a detector, and so high
sensitivities are desirable. Given this, our first pass at best settings yields (X1 = +1,
X2 = -1, X3 = either).
Plot the
Data: DOE
Scatter Plot
The next step in the analysis is to generate a DOE scatter plot.
Conclusions
from the
DOE
Scatter Plot
We can make the following conclusions based on the DOE scatter plot.
1. Important Factors: X1 (Number of Turns) is clearly important. When X1 = -1, all four
senstivities are low, and when X1 = +1, all four sensitivities are high. X2 (Winding
Distance) is less important. The four sensitivities for X2 = -1 are slightly higher, as a
group, than the four sensitivities for X2 = +1. X3 (Wire Gauge) does not appear to be
important at all. The sensitivity is about the same (on the average) regardless of the
settings for X3.
2. Best Settings: In this experiment, we are using the device as a detector, so high
sensitivities are desirable. Given this, our first pass at best settings yields (X1 = +1,
X2 = -1, X3 = either).
3. There does not appear to be any significant outliers.
Check for
Main
Effects:
DOE Mean
Plot
One of the primary questions is: what are the most important factors? The ordered data plot
and the DOE scatter plot provide useful summary plots of the data. Both of these plots
indicated that X1 is clearly important, X2 is somewhat important, and X3 is probably not
important.
The DOE mean plot shows the main effects. This provides probably the easiest to interpret
indication of the important factors.
Conclusions
from the
DOE Mean
Plot
The DOE mean plot (or main effects plot) reaffirms the ordering of the DOE scatter plot, but
additional information is gleaned because the eyeball distance between the mean values
gives an approximation to the least-squares estimate of the factor effects.
We can make the following conclusions from the DOE mean plot.
1. Important Factors:
X1 (effect = large: about 3 ohms)
X2 (effect = moderate: about -1 ohm)
X3 (effect = small: about 1/4 ohm)
2. Best Settings: As before, choose the factor settings that (on the average) maximize the
sensitivity:
(X1,X2,X3) = (+,-,+)
Comparison
of Plots
All of these plots are used primarily to detect the most important factors. Because it plots a
summary statistic rather than the raw data, the DOE mean plot shows the main effects most
clearly. However, it is still recommended to generate either the ordered data plot or the DOE
scatter plot (or both). Since these plot the raw data, they can sometimes reveal features of
the data that might be masked by the DOE mean plot.
5.6.1.3. Interaction Effects

5.6. Case Studies
Check for
Interaction
Effects:
DOE
Interaction
Plot
In addition to the main effects, it is also important to check for interaction effects, especially
two-factor interaction effects. The DOE interaction effects plot is an effective tool for this.
The effects on the plot represent the change in sensitivity from low to high levels of the
factors.
Conclusions
from the
We can make the following conclusions from the DOE interaction effects plot.
DOE
Interaction
Effects Plot
1. Important Factors: Looking for the plots that have the steepest lines (that is, largest
effects), we note that:
X1 (number of turns) is the most important effect: estimated effect = -3.1025;
X2 (winding distance) is next most important: estimated effect = -0.8675;
X3 (wire gauge) is relatively unimportant;
All three two-factor interactions are relatively unimporant.
2. Best Settings: As with the main effects plot, the best settings to maximize the
sensitivity are
(X1,X2,X3) = (+1,-1,+1)
but with the X3 setting of +1 mattering little.
5.6.1.4. Main and Interaction Effects: Block Plots

5.6. Case Studies
Block Plots Block plots are a useful adjunct to the DOE mean plot and the DOE interaction effects plot to confirm the importance of factors,
to establish the robustness of main effect conclusions, and to determine the existence of interactions. Specifically,
1. The first plot below answers the question: Is X1 important? If X1 is important, is this importance robust over all four
settings of X2 and X3?
2. The second plot below answers the question: Is X2 important? If X2 is important, is this importance robust over all four
3. The third plot below answers the question: Is X3 important? If X3 is important, is this importance robust over all four
For block plots, it is the height of the bars that is important, not the relative positioning of each bar. Hence we focus on the size
and internals of the blocks, not "where" the blocks are one relative to another.
Conclusions
from the
Block Plots
Recall that the block plot will access factor importance by the degree of consistency (robustness) of the factor effect over a
variety of conditions. In this light, we can make the following conclusions from the block plots.
1. Relative Importance of Factors: All of the bar heights in plot 1 (turns) are greater than the bar heights in plots 2 and 3.
Hence, X1 is more important than X2 and X3.
2. Statistical Significance: In plot 1, looking at the levels within each bar, we note that the response for level 2 is higher than
level 1 in each of the four bars. By chance, this happens with probability 1/(2
4
) = 1/16 = 6.25 %. Hence, X1 is near-
statistically significant at the 5 % level. Similarly, for plot 2, level 1 is greater than level 2 for all four bars. Hence, X2 is
near-statistically significant. For X3, there is no consistent ordering within all four bars, and hence X3 is not statistically
significant. Rigorously speaking then, X1 and X2 are not statistically significant (since 6.25 % is not < 5 %); on the other
hand such near-significance is suggestive to the analyst that such factors may in fact be important, and hence warrant
further attention.
Note that the usual method for determining statistical significance is to perform an analysis of variance (ANOVA).
ANOVA is based on normality assumptions. If these normality assumptions are valid, then ANOVA methods are the most
powerful method for determining statistical signficance. The advantage of the block-plot method is that it is based on less
rigorous assumptions than ANOVA. At an exploratory stage, it is useful to know that our conclusions regarding important
factors are valid under a wide range of assumptions.
3. Interactions: For X1, the four bars do not change height in any systematic way and hence there is no evidence of X1
interacting with either X2 or X3. Similarly, there is no evidence of interactions for X2.
5.6.1.5. Estimate Main and Interaction Effects

5.6. Case Studies
5.6.1.5. Estimate Main and Interaction Effects
Effects
Estimation
Although the effect estimates were given on the DOE
interaction plot on a previous page, we also display them in
tabular form.
The full model for the 2
3
factorial design is
Data from factorial designs with two levels can be analyzed
using least-squares regression. The regresson coefficients
represent the change per one unit of the factor variable, the
effects shown on the interaction plot represent changes
between high and low factor levels so they are twice as large
as the regression coefficients.
Effect
Estimates
The parameter estimates from a least-squares regression
analysis for the full model are shown below.
Effect Estimate
------ --------
Mean 2.65875
X1 1.55125
X2 -0.43375
X3 0.10625
X1*X2 0.06375
X1*X3 0.12375
X2*X3 0.14875
X1*X2*X3 0.07125
Because we fit the full model to the data, there are no degrees
of freedom for error and no significance tests are available.
If we sort the effects from largest to smallest (excluding the
mean), the four most important factors are: X1 (number of
turns), X2 (winding distance), X2*X3 (winding distance by
wire gauge interaction), and X1*X3 (number of turns by wire
gauge interaction).
5.6.1.6. Modeling and Prediction Equations

5.6. Case Studies
Parameter
Estimates
Don't
Change as
Additional
Terms
Added
In most cases of least-squares fitting, the model coefficient
estimates for previously added terms change depending on
what was successively added. For example, the estimate for
the X1 coefficient might change depending on whether or not
an X2 term was included in the model. This is not the case
when the design is orthogonal, as is this 2
3
full factorial
design. In such a case, the estimates for the previously
included terms do not change as additional terms are added.
This means the list of effect estimates in section 5.6.1.5 serves
as the least-squares coefficient estimates for progressively
more complicated models.
Default
Model:
Grand
Mean
If none of the factors are important, the prediction equation
defaults to the mean of all the response values (the overall or
grand mean). That is,
For our example, the default model has a grand mean of
2.65875 with a residual standard deviation (a measure of
goodness of fit) of 1.74106 ohms.
Possible
Prediction
Equations
We add effects to the default model in decreasing order of
absolute magnitude and compute the residual standard
deviation after adding each effect. The prediction equations
and their residual standard deviations are shown below.

Residual
Model Terms
Std. Dev.
---------------------------------------------------
-- ---------
Mean + X1
0.57272
Mean + X1 + X2
0.30429
Mean + X1 + X2 + X2*X3
0.26737
Mean + X1 + X2 + X2*X3 + X1*X3
0.23341
Mean + X1 + X2 + X2*X3 + X1*X3 + X3
0.19121
Mean + X1 + X2 + X2*X3 + X1*X3 + X3 + X1*X2*X3
0.18031
Mean + X1 + X2 + X2*X3 + X1*X3 + X3 + X1*X2*X3 +
X1*X2 NA
Note that the full model is a perfect fit to the data.
5.6.1.7. Intermediate Conclusions

5.6. Case Studies
Important
Factors
Taking stock from all of the graphical and quantitative
analyses of the previous sections, we conclude that X1
(number of turns) is the most important engineering factor
affecting sensitivity, followed by X2 (wire distance) as next in
importance, followed then by some less important interactions
and X3 (wire gauge).
Best
Settings
Also, from the various analyses, we conclude that the best
design settings (on the average) for a high-sensitivity detector
are
(X1,X2,X3) = (+,-,+)
that is
number of turns = 180,
winding distance = 0.38, and
wire gauge = 48.
Can We
Extract
More
From the
Data?
Thus, in a very real sense, the analysis is complete. We have
achieved the two most important stated goals of the
experiment:
1. gaining insight into the most important factors, and
2. ascertaining the optimal production settings.
On the other hand, more information can be squeezed from the
data, and that is what this section and the remaining sections
address.
1. First of all, we focus on the problem of taking the
ranked list of factors and objectively ascertaining which
factors are "important" versus "unimportant".
2. In a parallel fashion, we use the subset of important
factors derived above to form a "final" prediction
equation that is good (that is, having a sufficiently small
residual standard deviation) while being parsimonious
(having a small number of terms), compared to the full
model, which is perfect (having a residual standard
deviation = 0, that is, the predicted values = the raw
data), but is unduly complicated (consisting of a
constant + 7 terms).
5.6.1.8. Important Factors and Parsimonious Prediction

5.6. Case Studies
Identify
Important
Factors
The two problems discussed in the previous section (important factors and a parsimonious
model) will be handled in parallel since determination of one yields the other. In regard to
the "important factors", our immediate goal is to take the full subset of seven main effects
and interactions and extract a subset that we will declare as "important", with the
complementary subset being "unimportant". Seven criteria are discussed in detail in section
1.3.5.18.2 in Chapter 1. The relevant criteria will be applied here. These criteria are not all
equally important, nor will they yield identical subsets, in which case a consensus subset or
a weighted consensus subset must be extracted.
Criteria for
Including
Terms in
the Model
The criteria that we can use in determining whether to keep a factor in the model can be
summarized as follows.
1. Effects: Engineering Significance
2. Effects: 90 % Numerical Significance
3. Effects: Statistical Significance
4. Effects: Normal Probability Plot
5. Averages: Youden Plot
The first four criteria focus on effect estimates with three numerical criteria and one
graphical criterion. The fifth criterion focuses on averages. We discuss each of these criteria
in detail in the following sections.
The last section summarizes the conclusions based on all of the criteria.
Effects:
Engineering
Significance
The minimum engineering significant difference is defined as
where is the absolute value of the parameter estimate (i.e., the effect) and is the
minimum engineering significant difference. That is, declare a factor as "important" if the
effect is greater than some a priori declared engineering difference. We use a rough rule-of-
thumb of keeping only those factors whose effect is greater than 10 % of the current
production average. In this case, let's say that the average detector has a sensitivity of 1.25
ohms. This suggests that we would declare all factors whose effect is greater than 10 % of
1.25 ohms = 0.125 ohms to be significant from an engineering point of view.
Based on this minimum engineering-significant-difference criterion, we conclude to keep
two terms: X1 (1.55125) and X2 (-0.43375).
Effects: 90
%
Numerical
The 90 % numerical significance criterion is defined as
Significance
That is, declare a factor as important if it exceeds 10 % of the largest effect. For the current
case study, the largest effect is from X1 (1.55125 ohms), and so 10 % of that is 0.155 ohms.
This suggests keeping all factors whose effects exceed 0.155 ohms.
Based on the 90 % numerical criterion, we would keep two terms: X1 (1.55125) and X2 (-
0.43375). The X2*X3 term, (0.14875), is just under the cutoff.
Effects:
Statistical
Significance
Statistical significance is defined as
That is, declare a factor as "important" if its effect is more than 2 standard deviations away
from 0 (0, by definition, meaning "no effect"). The difficulty with this is that in order to
invoke this rule we need the is the standard deviation of an observation.
For the eddy current case study, ignoring three-factor and higher interactions leads to an
estimate of based on omitting only a single term: the X1*X2*X3 interaction.
Thus for our example, if one assumes that the three-factor interaction is nil and hence
represents a single drawing from a population centered at zero, an estimate of the standard
deviation of an effect is simply the estimate of the interaction effect (0.07125). Two such
effect standard deviations is 0.1425. This rule becomes to keep all > 0.1425. This results
in keeping three terms: X1 (1.55125), X2 (-0.43375), and X1*X2 (0.14875).
Effects:
Probability
Plot
The normal probability plot can be used to identify important factors. The following graph
shows the normal probability plot of the effects.
The normal probablity plot clearly shows two factors displaced off the line, and we see that
those two factors are X1 and X2. Thus, we would keep X1 (1.55125) and X2 (-0.43375).
Effects:
Youden Plot
A DOE Youden plot can be used in the following way. A factor is "important" if it is
displaced away from the central-tendency bunch in a Youden plot of high and low averages.
For our example, the Youden plot clearly shows a cluster of points near the grand average
(2.65875) with two displaced points above (X1) and below (X2). Based on the Youden plot,
we keep two factors: X1 (1.55125) and X2 (-0.43375).
Conclusions In summary, the criterion for specifying "important" factors yielded the following:
1. Effects, Engineering Significant: X1 X2
2. Effects, Numerically Significant: X1 X2 (X2*X3 is borderline)
3. Effects, Statistically Significant: X1 X2 X2*X3
4. Effects, Normal Probability Plot: X1 X2
5. Averages, Youden Plot: X1 X2
All the criteria select X1 and X2. One also includes the X2*X3 interaction term (and it is
borderline for another criteria).
We thus declare the following consensus:
1. Important Factors: X1 and X2
2. Parsimonious Prediction Equation:
(with a residual standard deviation of 0.30429 ohms)
5.6.1.9. Validate the Fitted Model

5.6. Case Studies
Model
Validation
In the Important Factors and Parsimonious Prediction section, we selected the following model
The residual standard deviation for this model is 0.30429.
The next step is to validate the model. The primary method of model validation is graphical residual analysis; that is, through an
assortment of plots of the differences between the observed data Y and the predicted value from the model. For example, the
design point (-1, -1, -1) has an observed data point (from the Background and data section) of Y = 1.70, while the predicted value
from the above fitted model for this design point is
which leads to the residual 0.15875.
Table of
Residuals
If the model fits well, should be near Y for all eight design points. Hence the eight residuals should all be near zero. The eight
predicted values and residuals for the model with these data are:
X1 X2 X3 Observed Predicted Residual
----------------------------------------------
-1 -1 -1 1.70 1.54125 0.15875
+1 -1 -1 4.57 4.64375 -0.07375
-1 +1 -1 0.55 0.67375 -0.12375
+1 +1 -1 3.39 3.77625 -0.38625
-1 -1 +1 1.51 1.54125 -0.03125
+1 -1 +1 4.59 4.64375 -0.05375
-1 +1 +1 0.67 0.67375 -0.00375
+1 +1 +1 4.29 3.77625 0.51375
Residual
Standard
Deviation
What is the magnitude of the typical residual? There are several ways to compute this, but the statistically optimal measure is the
residual standard deviation:
with r
i
denoting the ith residual, N = 8 is the number of observations, and P = 3 is the number of fitted parameters. From the
table of prediction equations, the residual standard deviation is 0.30429.
How Should
Residuals
Behave?
If the prediction equation is adequate, the residuals from that equation should behave like random drawings (typically from an
approximately normal distribution), and should, since presumably random, have no structural relationship with any factor. This
includes any and all potential terms (X1, X2, X3, X1*X2, X1*X3, X2*X3, X1*X2*X3).
Further, if the model is adequate and complete, the residuals should have no structural relationship with any other variables that
may have been recorded. In particular, this includes the run sequence (time), which is really serving as a surrogate for any
physical or environmental variable correlated with time. Ideally, all such residual scatter plots should appear structureless. Any
scatter plot that exhibits structure suggests that the factor should have been formally included as part of the prediction equation.
Validating the prediction equation thus means that we do a final check as to whether any other variables may have been
inadvertently left out of the prediction equation, including variables drifting with time.
The graphical residual analysis thus consists of scatter plots of the residuals versus all three factors and four interactions (all such
plots should be structureless), a scatter plot of the residuals versus run sequence (which also should be structureless), and a
normal probability plot of the residuals (which should be near linear). We present such plots below.
Residual
Plots
The first plot is a normal probability plot of the residuals. The second plot is a run sequence plot of the residuals. The remaining
plots show the residuals plotted against each of the factors and each of the interaction terms.
Conclusions We make the following conclusions based on the above plots.
1. Main Effects and Interactions: The X1 and X2 scatter plots are "flat" (as they must be since X1 and X2 were explicitly
included in the model). The X3 plot shows some structure as does the X1*X3, the X2*X3, and the X1*X2*X3 plots. The
X1*X2 plot shows little structure. The net effect is that the relative ordering of these scatter plots is very much in
agreement (again, as it must be) with the relative ordering of the "unimportant" factors. From the table of effects and the
X2*X3 residual plot, the third most influential term to be added to the model would be X2*X3. In effect, these plots offer a
higher-resolution confirmation of the ordering of effects. On the other hand, none of these other factors "passed" the
criteria given in the previous section, and so these factors, suggestively influential as they might be, are still not influential
enough to be added to the model.
2. Time Drift: The run sequence scatter plot is random. Hence there does not appear to be a drift either from time, or from
any factor (e.g., temperature, humidity, pressure, etc.) possibly correlated with time.
3. Normality: The normal probability plot of the eight residuals has some trend, which suggests that additional terms might be
added. On the other hand, the correlation coefficient of the 8 ordered residuals and the eight theoretical normal N(0,1)
order statistic medians (which define the two axes of the plot) has the value 0.934, which is well within acceptable (5 %)
limits of the normal probability plot correlation coefficient test for normality. Thus, the plot is not so non-linear as to reject
normality.
In summary, therefore, we accept the fitted model
as a parsimonious, but good, representation of the sensitivity phenomenon under study.
5.6.1.10. Using the Fitted Model

5.6. Case Studies
Model
Provides
Additional
Insight
Although deriving the fitted model was not the primary purpose of the study, it does have
two benefits in terms of additional insight:
1. Global prediction
2. Global determination of best settings
Global
Prediction
How does one predict the response at points other than those used in the experiment? The
prediction equation yields good results at the eight combinations of coded -1 and +1 values
for the three factors:
1. X1 = Number of turns = 90 and 180
2. X2 = Winding distance = 0.38 and 1.14
3. X3 = Wire gauge = 40 and 48
What, however, would one expect the detector to yield at target settings of, say,
1. Number of turns = 150
2. Winding distance = 0.50
3. Wire gauge = 46
Based on the fitted equation, we first translate the target values into coded target values as
follows:
coded target = -1 + 2*(target-low)/(high-low)
Hence the coded target values are
1. X1 = -1 + 2*(150-90)/(180-90) = 0.333333
2. X2 = -1 + 2*(0.50-0.38)/(1.14-0.38) = -0.684211
3. X3 = -1 + 2*(46-40)/(48-40) = 0.5000
Thus the raw data
(Number of turns, Winding distance, Wire gauge) = (150, 0.50, 46)
translates into the coded
(X1, X2, X3) = (0.333333, -0.684211, 0.50000)
on the -1 to +1 scale.
Inserting these coded values into the fitted equation yields, as desired, a predicted value of
= 2.65875 + 1.55125(0.333333) - 0.43375(-0.684211) = 3.47261
The above procedure can be carried out for any values of turns, distance, and gauge. This is
subject to the usual cautions that equations that are good near the data point vertices may not
necessarily be good everywhere in the factor space. Interpolation is a bit safer than
extrapolation, but it is not guaranteed to provide good results, of course. One would feel
more comfortable about interpolation (as in our example) if additional data had been
collected at the center point and the center point data turned out to be in good agreement
with predicted values at the center point based on the fitted model. In our case, we had no
such data and so the sobering truth is that the user of the equation is assuming something in
which the data set as given is not capable of suggesting one way or the other. Given that
assumption, we have demonstrated how one may cautiously but insightfully generate
predicted values that go well beyond our limited original data set of eight points.
Global
Determination
of Best
Settings
In order to determine the best settings for the factors, we can use a DOE contour plot. The
DOE contour plot is generated for the two most significant factors and shows the value of
the response variable at the vertices (i.e, the -1 and +1 settings for the factor variables) and
indicates the direction that maximizes (or minimizes) the response variable. If you have
more than two significant factors, you can generate a series of DOE contour plots with each
one using two of the important factors.
DOE Contour
Plot
The following is the DOE contour plot of the number of turns and the winding distance.
The maximum value of the response variable (eddy current) corresponds to X1 (number of
turns) equal to -1 and X2 (winding distance) equal to +1. The lower right corner of the
contour plot corresponds to the direction that maximizes the response variable. This
information can be used in planning the next phase of the experiment.
5.6.1.11. Conclusions and Next Step

5.6. Case Studies
5.6.1.11. Conclusions and Next Step
Conclusions The goals of this case study were:
1. Determine the most important factors.
2. Determine the best settings for the factors.
3. Determine a good prediction equation for the data.
The various plots and analysis showed that the number of
turns (X1) and the winding distance (X2) were the most
important factors and a good prediction equation for the data
is:
The DOE contour plot gave us the best settings for the factors
(X1 = -1 and X2 = 1).
Next Step Full and fractional designs are typically used to identify the
most important factors. In some applications, this is
sufficient and no further experimentation is performed. In
other applications, it is desired to maximize (or minimize) the
response variable. This typically involves the use of response
surface designs. The DOE contour plot can provide guidance
on the settings to use for the factor variables in this next
phase of the experiment.
This is a common sequence for designed experiments in
engineering and scientific applications. Note the iterative
nature of this approach. That is, you typically do not design
one large experiment to answer all your questions. Rather,
you run a series of smaller experiments. The initial
experiment or experiments are used to identify the important
factors. Once these factors are identified, follow-up
experiments can be run to fine tune the optimal settings (in
terms of maximizing/minimizing the response variable) for
these most important factors.
For this particular case study, a response surface design was
not used.

5.6. Case Studies
View
Dataplot
Macro for
this Case
Study
case study description on the previous page using Dataplot. It
window, and the Data Sheet window. Across the top of the
main windows are menus for executing Dataplot commands.
Data Analysis Steps
Results and
Conclusions
more detailed
columns of numbers
into Dataplot:
variables Y, X1, X2,
and X3.
2. Plot the main effects.
1. Ordered data plot.
2. DOE scatter plot.
3. DOE mean plot.
shows factor 1
clearly
important, factor 2
somewhat
important.
2. DOE scatter plot
shows significant
differences for
factors 1 and 2.
3. DOE mean plot
shows significant
differences in
means for factors
1 and 2.
3. Plots for interaction effects
effects matrix plot.
1. The DOE
interaction effects
matrix
plot does not
show any major
interaction
effects.
4. Block plots for main and interaction
effects
1. The block plots
show that the
factor 1 and
factor 2 effects
are consistent
over all
combinations of
the other
factors.
5. Estimate main and interaction effects
1. Perform a Yates fit to estimate
the
main effects and interaction
effects.
1. The Yates
analysis shows that
the
factor 1 and
factor 2 main effects
are significant,
and the interaction
for factors 2 and
3 is at the
boundary of
statistical
significance.
6. Model selection
1. Generate half-normal
probability plots of the effects.
2. Generate a Youden plot of the
effects.
1. The probability
plot indicates
that the model
should include
main effects for
factors 1 and 2.
2. The Youden plot
indicates
that the model
should include
main effects for
factors 1 and 2.
7. Model validation
1. Compute residuals and predicted
values
from the partial model suggested
by
the Yates analysis.
1. Check the link
for the
values of the
residual and
predicted values.
2. Generate residual plots to
validate
the model.
2. The residual
plots do not
indicate any
major problems
with the model
using main
effects for
factors 1 and 2.
8. DOE contour plot
1. Generate a DOE contour plot using
factors 1 and 2.
1. The DOE contour
plot shows
X1 = -1 and X2 =
+1 to be the
best settings.
5.6.2. Sonoluminescent Light Intensity Case Study

5.6. Case Studies
5.6.2. Sonoluminescent Light Intensity Case
Study
Analysis of
a 2
7-3
Fractional
Factorial
Design
7-3
fractional
factorial design. The purpose of the study is to optimize
sonoluminescent light intensity.
The case study is based on the EDA approach to experimental
design discussed in an earlier section.
Contents The case study is divided into the following sections:
1. Background and data
2. Initial plots/main effects
3. Interaction effects
4. Main and interaction effects: block plots
5. Important Factors: Youden plot
6. Important Factors: |effects| plot
7. Important Factors: half-normal probability plot
8. Cumulative Residual SD plot
9. Next step: DOE contour plot
10. Summary of conclusions
11. Work this example yourself

5.6. Case Studies
Background
and
Motivation
Sonoluminescence is the process of turning sound energy into
light. An ultrasonic horn is used to resonate a bubble of air in
a medium, usually water. The bubble is ultrasonically
compressed and then collapses to light-emitting plasma.
In the general physics community, sonoluminescence studies
are being carried out to characterize it, to understand it, and
to uncover its practical uses. An unanswered question in the
community is whether sonoluminescence may be used for
cold fusion.
NIST's motive for sonoluminescent investigations is to assess
its suitability for the dissolution of physical samples, which is
needed in the production of homogeneous Standard
Reference Materials (SRMs). It is believed that maximal
dissolution coincides with maximal energy and maximal light
intensity. The ultimate motivation for striving for maximal
dissolution is that this allows improved determination of
alpha-and beta-emitting radionuclides in such samples.
The objectives of the NIST experiment were to determine the
important factors that affect sonoluminescent light intensity
and to ascertain optimal settings of such factors that will
predictably achieve high intensities. An original list of 49
factors was reduced, based on physics reasons, to the
following seven factors: molarity (amount of solute), solute
type, pH, gas type in the water, water depth, horn depth, and
flask clamping.
Time restrictions caused the experiment to be about one
month, which in turn translated into an upper limit of roughly
20 runs. A 7-factor, 2-level fractional factorial design
(Resolution IV) was constructed and run. The factor level
settings are given below.
Eva Wilcox and Ken Inn of the NIST Physics Laboratory
conducted this experiment during 1999. Jim Filliben of the
NIST Statistical Engineering Division performed the analysis
of the experimental data.
Response
Variable,
Factor
Variables,
and Factor-
Level
Settings
This experiment utilizes the following response and factor
variables.
1. Response Variable (Y) = The sonoluminescent light
intensity.
2. Factor 1 (X1) = Molarity (amount of Solute). The
coding is -1 for 0.10 mol and +1 for 0.33 mol.
3. Factor 2 (X2) = Solute type. The coding is -1 for sugar
and +1 for glycerol.
4. Factor 3 (X3) = pH. The coding is -1 for 3 and +1 for
11.
5. Factor 4 (X4) = Gas type in water. The coding is -1 for
helium and +1 for air.
6. Factor 5 (X5) = Water depth. The coding is -1 for half
and +1 for full.
7. Factor 6 (X6) = Horn depth. The coding is -1 for 5 mm
and +1 for 10 mm.
8. Factor 7 (X7) = Flask clamping. The coding is -1 for
unclamped and +1 for clamped.
This data set has 16 observations. It is a 2
7-3
design with no
center points.
Goal of the
Experiment
7-3
fractional
factorial experimental design. The goals of this case study
are:
1. Determine the important factors that affect the
sonoluminescent light intensity. Specifically, we are
trying to maximize this intensity.
2. Determine the best settings of the seven factors so as to
maximize the sonoluminescent light intensity.
Data Used
in the
Analysis
The following are the data used for this analysis. This data set
is given in Yates order.
Y X1 X2 X3 X4 X5
X6 X7
Light Solute Gas Water
Horn Flask
Intensity Molarity type pH Type Depth
Depth Clamping
---------------------------------------------------
---------------
80.6 -1.0 -1.0 -1.0 -1.0 -1.0
-1.0 -1.0
66.1 1.0 -1.0 -1.0 -1.0 -1.0
1.0 1.0
59.1 -1.0 1.0 -1.0 -1.0 1.0
-1.0 1.0
68.9 1.0 1.0 -1.0 -1.0 1.0
1.0 -1.0
75.1 -1.0 -1.0 1.0 -1.0 1.0
1.0 1.0
373.8 1.0 -1.0 1.0 -1.0 1.0
-1.0 -1.0
66.8 -1.0 1.0 1.0 -1.0 -1.0
1.0 -1.0
79.6 1.0 1.0 1.0 -1.0 -1.0
-1.0 1.0
114.3 -1.0 -1.0 -1.0 1.0 1.0
1.0 -1.0
84.1 1.0 -1.0 -1.0 1.0 1.0
-1.0 1.0
68.4 -1.0 1.0 -1.0 1.0 -1.0
1.0 1.0
88.1 1.0 1.0 -1.0 1.0 -1.0
-1.0 -1.0
78.1 -1.0 -1.0 1.0 1.0 -1.0
-1.0 1.0
327.2 1.0 -1.0 1.0 1.0 -1.0
1.0 -1.0
77.6 -1.0 1.0 1.0 1.0 1.0
-1.0 -1.0
61.9 1.0 1.0 1.0 1.0 1.0
1.0 1.0

5.6. Case Studies
Plot the
Data:
Ordered
Data Plot
The first step in the analysis is to generate an ordered data plot.
Conclusions
from the
Ordered
Data Plot
We can make the following conclusions based on the ordered data plot.
1. Two points clearly stand out. The first 13 points lie in the 50 to 100 range, the
next point is greater than 100, and the last two points are greater than 300.
2. Important Factors: For these two highest points, factors X1, X2, X3, and X7
have the same value (namely, +, -, +, -, respectively) while X4, X5, and X6
have differing values. We conclude that X1, X2, X3, and X7 are potentially
important factors, while X4, X5, and X6 are not.
3. Best Settings: Our first pass makes use of the settings at the observed
maximum (Y = 373.8). The settings for this maximum are (+, -, +, -, +, -, -).
Plot the The next step in the analysis is to generate a DOE scatter plot.
Data: DOE
Scatter Plot
Conclusions
from the
DOE
Scatter Plot
We can make the following conclusions based on the DOE scatter plot.
1. Important Factors: Again, two points dominate the plot. For X1, X2, X3, and
X7, these two points emanate from the same setting, (+, -, +, -), while for X4,
X5, and X6 they emanate from different settings. We conclude that X1, X2, X3,
and X7 are potentially important, while X4, X5, and X6 are probably not
important.
2. Best Settings: Our first pass at best settings yields (X1 = +, X2 = -, X3 = +, X4
= either, X5 = either, X6 = either, X7 = -).
Check for
Main
Effects:
DOE Mean
Plot
The DOE mean plot is generated to more clearly show the main effects:
Conclusions
from the
DOE Mean
Plot
We can make the following conclusions from the DOE mean plot.
1. Important Factors:
X2 (effect = large: about -80)
X7 (effect = large: about -80)
X1 (effect = large: about 70)
X3 (effect = large: about 65)
X6 (effect = small: about -10)
X5 (effect = small: between 5 and 10)
X4 (effect = small: less than 5)
2. Best Settings: Here we step through each factor, one by one, and choose the
setting that yields the highest average for the sonoluminescent light intensity:
(X1,X2,X3,X4,X5,X6,X7) = (+,-,+,+,+,-,-)
Comparison
of Plots
All of the above three plots are used primarily to determine the most important
factors. Because it plots a summary statistic rather than the raw data, the DOE mean
plot shows the ordering of the main effects most clearly. However, it is still
recommended to generate either the ordered data plot or the DOE scatter plot (or
both). Since these plot the raw data, they can sometimes reveal features of the data
that might be masked by the DOE mean plot.
In this case, the ordered data plot and the DOE scatter plot clearly show two
dominant points. This feature would not be obvious if we had generated only the
DOE mean plot.
Interpretation-wise, the most important factor X2 (solute) will, on the average,
change the light intensity by about 80 units regardless of the settings of the other
factors. The other factors are interpreted similarly.
In terms of the best settings, note that the ordered data plot, based on the maximum
response value, yielded
+, -, +, -, +, -, -
Note that a consensus best value, with "." indicating a setting for which the three
plots disagree, would be
+, -, +, ., +, -, -
Note that the factor for which the settings disagree, X4, invariably defines itself as
an "unimportant" factor.

5.6. Case Studies
Check for
Interaction
Effects:
DOE
Interaction
Plot
In addition to the main effects, it is also important to check for interaction effects,
especially 2-factor interaction effects. The DOE interaction effects plot is an
effective tool for this.
Conclusions
from the
DOE
Interaction
Effects Plot
We make the following conclusions from the DOE interaction effects plot.
1. Important Factors: Looking for the plots that have the steepest lines (that is,
the largest effects), and noting that the legends on each subplot give the
estimated effect, we have that
The diagonal plots are the main effects. The important factors are: X2,
X7, X1, and X3. These four factors have |effect| > 60. The remaining
three factors have |effect| < 10.
The off-diagonal plots are the 2-factor interaction effects. Of the 21 2-
factor interactions, 9 are nominally important, but they fall into three
groups of three:
X1*X3, X4*X6, X2*X7 (effect = 70)
X2*X3, X4*X5, X1*X7 (effect approximately 63.5)
X1*X2, X5*X6, X3*X7 (effect = -59.6)
All remaining 2-factor interactions are small having an |effect| < 20. A
virtue of the interaction effects matrix plot is that the confounding
structure of this Resolution IV design can be read off the plot. In this
case, the fact that X1*X3, X4*X6, and X2*X7 all have effect estimates
identical to 70 is not a mathematical coincidence. It is a reflection of the
fact that for this design, the three 2-factor interactions are confounded.
This is also true for the other two sets of three (X2*X3, X4*X5, X1*X7,
and X1*X2, X5*X6, X3*X7).
2. Best Settings: Reading down the diagonal plots, we select, as before, the best
settings "on the average":
(X1,X2,X3,X4,X5,X6,X7) = (+,-,+,+,+,-,-)
For the more important factors (X1, X2, X3, X7), we note that the best settings
(+, -, +, -) are consistent with the best settings for the 2-factor interactions
(cross-products):
X1: +, X2: - with X1*X2: -
X1: +, X3: + with X1*X3: +
X1: +, X7: - with X1*X7: -
X2: -, X3: + with X2*X3: -
X2: -, X7: - with X2*X7: +
X3: +, X7: - with X3*X7: -

5.6. Case Studies
Block Plots Block plots are a useful adjunct to the DOE mean plot and the DOE interaction
effects plot to confirm the importance of factors, to establish the robustness of main
effect conclusions, and to determine the existence of interactions.
For block plots, it is the height of the bars that is important, not the relative
positioning of each bar. Hence we focus on the size and internal signs of the blocks,
not "where" the blocks are relative to each other.
We note in passing that for a fractional factorial design, we cannot display all
combinations of the six remaining factors. We have arbitrarily chosen two
robustness factors, which yields four blocks for comparison.
Conclusions
from the
Block Plots
We can make the following conclusions from the block plots.
1. Relative Importance of Factors: Because of the expanded vertical axis, due to
the two "outliers", the block plot is not particularly revealing. Block plots
based on alternatively scaled data (e.g., LOG(Y)) would be more informative.
5.6.2.5. Important Factors: Youden Plot

5.6. Case Studies
Purpose The DOE Youden plot is used to distinguish between important and unimportant
factors.
Sample
Youden Plot
Conclusions
from the
Youden plot
We can make the following conclusions from the Youden plot.
1. In the upper left corner are the interaction term X1*X3 and the main effects X1
and X3.
2. In the lower right corner are the main effects X2 and X7 and the interaction
terms X2*X3 and X1*X2.
3. The remaining terms are clustered in the center, which indicates that such
effects have averages that are similar (and hence the effects are near zero), and
so such effects are relatively unimportant.
4. On the far right of the plot, the confounding structure is given (e.g., 13:
13+27+46), which suggests that the information on X1*X3 (on the plot) must
be tempered with the fact that X1*X3 is confounded with X2*X7 and X4*X6.
5.6.2.6. Important Factors: |Effects| Plot

5.6. Case Studies
Purpose The |effects| plot displays the results of a Yates analysis in both a tabular and a
graphical format. It is used to distinguish between important and unimportant
effects.
Sample
|Effects|
Plot
Conclusions
from the
|effects| plot
We can make the following conclusions from the |effects| plot.
1. A ranked list of main effects and interaction terms is:
X2
X7
X1*X3 (confounded with X2*X7 and X4*X6)
X1
X3
X6
X5
X1*X2*X4 (confounded with other 3-factor interactions)
X4
2. From the graph, there is a clear dividing line between the first seven effects
(all |effect| > 50) and the last eight effects (all |effect| < 20). This suggests we
retain the first seven terms as "important" and discard the remaining as
"unimportant".
3. Again, the confounding structure on the right reminds us that, for example, the
nominal effect size of 70.0125 for X1*X3 (molarity*pH) can come from an
X1*X3 interaction, an X2*X7 (solute*clamping) interaction, an X4*X6
(gas*horn depth) interaction, or any mixture of the three interactions.
5.6.2.7. Important Factors: Half-Normal Probability Plot

5.6. Case Studies
Purpose The half-normal probability plot is used to distinguish between important and
unimportant effects.
Sample
Half-
Normal
Probability
Plot
Conclusions
from the
Half-
Normal
Probability
Plot
We can make the following conclusions from the half-normal probability plot.
1. The points in the plot divide into two clear clusters:
An upper cluster (|effect| > 60).
A lower cluster (|effect| < 20).
2. The upper cluster contains the effects:
X2, X7, X1*X3 (and confounding), X1, X3, X2*X3 (and confounding),
X1*X2 (and confounding)
These effects should definitely be considered important.
3. The remaining effects lie on a line and form a lower cluster. These effects are
declared relatively unimportant.
4. The effect id's and the confounding structure are given on the far right (e.g.,
13:13+27+46).
5.6.2.8. Cumulative Residual Standard Deviation Plot

5.6. Case Studies
Purpose The cumulative residual standard deviation plot is used to identify the best
(parsimonious) model.
Sample
Cumulative
Residual
Standard
Deviation
Plot
Conclusions
from the
Cumulative
Residual SD
Plot
We can make the following conclusions from the cumulative residual standard
deviation plot.
1. The baseline model consisting only of the average ( = 110.6063) has a high
residual standard deviation (95).
2. The cumulative residual standard deviation shows a significant and steady
decrease as the following terms are added to the average: X2, X7, X1*X3, X1,
X3, X2*X3, and X1*X2. Including these terms reduces the cumulative residual
standard deviation from approximately 95 to approximately 17.
3. Exclude from the model any term after X1*X2 as the decrease in the residual
standard deviation becomes relatively small.
4. From the |effects| plot, we see that the average is 110.6063, the estimated X2
effect is -78.6126, and so on. (The model coefficients are one half of the
effect estimates.) We use this to from the following prediction equation:
Note that X1*X3 is confounded with X2*X7 and X4*X6, X1*X5 is confounded
with X2*X6 and X4*X7, and X1*X2 is confounded with X3*X7 and X5*X6.
From the above graph, we see that the residual standard deviation for this
model is approximately 17.
5.6.2.9. Next Step: DOE Contour Plot

5.6. Case Studies
Purpose The DOE contour plot is used to determine the best factor settings for the two most
important factors in the next iteration of the experiment.
From the previous plots, we identified X2 (solute) and X7 (horn depth) as the two
most important factors.
Sample
DOE
Contour
Plot
Conclusions
from the
DOE
Contour
Plot
We can make the following conclusions from the DOE contour plot.
1. The best (high light intensity) setting for X2 is "-" and the best setting for X7
is "-". This combination yields an average response of approximately 224. The
next highest average response from any other combination of these factors is
only 76.
2. The non-linear nature of the contour lines implies that the X2*X7 interaction is
important.
3. On the left side of the plot from top to bottom, the contour lines start at 0,
increment by 50 and stop at 400. On the bottom of the plot from right to left,
the contour lines start at 0, increment by 50 and stop at 400.
To achieve a light intensity of, say 400, this suggests an extrapolated best
setting of (X2, X7) = (-2,-2).
4. Such extrapolation only makes sense if X2 and X7 are continuous factors.
Such is not the case here. In this example, X2 is solute (-1 = sugar and +1 =
glycerol) and X7 is flask clamping (-1 is unclamped and +1 is clamped). Both
factors are discrete, and so extrapolated settings are not possible.
5.6.2.10. Summary of Conclusions

5.6. Case Studies
Most
Important
Factors
The primary goal of this experiment was to identify the most
important factors in maximizing the sonoluminescent light
intensity.
Based on the preceding graphical analysis, we make the
following conclusions.
Four factors and three groups of 2-factor interactions are
important. A rank-order listing of factors is:
1. X2: Solute (effect = -78.6)
2. X7: Clamping (effect = -78.1)
3. X1*X3 (Molarity*pH) or
X2*X7 (Solute*Clamping)
(effect = 70.0)
4. X1: Molarity (effect = 66.2)
5. X3: pH (effect = 63.5)
6. X2*X3 (Solute*pH) or
X4*X5 (Gas*Water Depth)
X1*X7 (Molarity*Clamping)
(effect = -63.5)
7. X1*X2 (Molarity*Solute) or
X3*X7 (Ph*Clamping)
(effect = -59.6)
Thus, of the seven factors and 21 2-factor interactions, it
was found that four factors and at most seven 2-factor
interactions seem important, with the remaining three
factors and 14 interactions apparently being
unimportant.
Best
Settings
The best settings to maximize sonoluminescent light intensity
are
X1 (Molarity) + (0.33 mol)
X2 (Solute) - (sugar)
X3 (pH) + (11)
X4 (Gas) . (either)
X5 (Water Depth) + (full)
X6 (Horn Depth) - (5 mm)
X7 (Clamping) - (unclamped)
with the X1, X2, X3, and X7 settings especially important.

5.6. Case Studies
View
Dataplot
Macro for
this Case
Study
window, and the Data Sheet window. Across the top of the
Data Analysis Steps
Results and
Conclusions
more detailed

1. You have read 8
columns of numbers
into Dataplot:
variables Y, X1, X2,
X3, X4, X5, X6,
and X7.
2. Plot the main effects.
1. Ordered data plot.
2. DOE scatter plot.
3. DOE mean plot.
shows 2 points
that stand out.
Potential
important factors
are X1, X2, X3,
and X7.
2. DOE scatter plot
identifies X1, X2,
X3, and X7 as
important factors.
3. DOE mean plot
identifies X1, X2,
X3, and X7 as
important factors.
3. Plots for interaction effects
effects plot.
1. The DOE
interaction effects
plot shows
several important
interaction
effects.
4. Block plots for main and interaction
effects
1. The block plots
are not
particularly
helpful in
this case.
5. Youden plot to identify important
factors
1. Generate a Youden plot.
1. The Youden plot
identifies
X1, X2, X3, and
X7 as important
factors. It also
identifies a
number of
important
interactions
(X1*X3, X1*X2,
X2*X3).
6. |Effects| plot to identify important
factors
1. Generate |effects| plot.
1. The |effects|
plot identifies
X2, X7, X1*X3,
X1, X3, X2*X3,
and X1*X2 as
important factors
and interactions.
7. Half-normal probability plot to
identify important factors
1. Generate half-normal probability
plot.
1. The half-normal
probability plot
identifies X2,
X7, X1*X3, X1, X3,
X2*X3, and X1*X2
as important
factors and
interactions.
8. Cumulative residual standard
deviation plot
1. Generate a cumulative residual
standard deviation plot.
1. The cumulative
residual standard
deviation plot
results in a model
with 4 main
effects and 3 2-
factor
interactions.
9. DOE contour plot
1. Generate a DOE contour plot using
factors 2 and 7.
1. The DOE contour
plot shows
X2 = -1 and X7 =
-1 to be the
best settings.
5.7. A Glossary of DOE Terminology

Definitions
for key
DOE
terms
This page gives definitions and information for many of the
basic terms used in DOE.
Alias: When the estimate of an effect also
includes the influence of one or more other effects
(usually high order interactions) the effects are
said to be aliased (see confounding). For example,
if the estimate of effect D in a four factor
experiment actually estimates (D + ABC), then
the main effect D is aliased with the 3-way
interaction ABC. Note: This causes no difficulty
when the higher order interaction is either non-
existent or insignificant.
Analysis of Variance (ANOVA): A
mathematical process for separating the variability
of a group of observations into assignable causes
and setting up various significance tests.
Balanced Design: An experimental design
where all cells (i.e. treatment combinations) have
the same number of observations.
Blocking: A schedule for conducting treatment
combinations in an experimental study such that
any effects on the experimental results due to a
known change in raw materials, operators,
machines, etc., become concentrated in the levels
of the blocking variable. Note: the reason for
blocking is to isolate a systematic effect and
prevent it from obscuring the main effects.
Blocking is achieved by restricting randomization.
Center Points: Points at the center value of all
factor ranges.
Coding Factor Levels: Transforming the scale of
measurement for a factor so that the high value
becomes +1 and the low value becomes -1 (see
scaling). After coding all factors in a 2-level full
factorial experiment, the design matrix has all
orthogonal columns.
Coding is a simple linear transformation of the
original measurement scale. If the "high" value is
X
h
and the "low" value is X
L
(in the original
scale), then the scaling transformation takes any
original X value and converts it to (X - a)/b,
where
a = (X
h
+ X
L
)/2 and b = ( X
h
-X
L
)/2.
To go back to the original measurement scale, just
take the coded value and multiply it by "b" and
add "a" or, X = b(coded value) + a.
As an example, if the factor is temperature and
the high setting is 65
o
C and the low setting is
55
o
C, then a = (65 + 55)/2 = 60 and b = (65 -
55)/2 = 5. The center point (where the coded
value is 0) has a temperature of 5(0) + 60 =
60
o
C.
Comparative Designs: A design aimed at
making conclusions about one a priori important
factor, possibly in the presence of one or more
other "nuisance" factors.
Confounding: A confounding design is one
where some treatment effects (main or
interactions) are estimated by the same linear
combination of the experimental observations as
some blocking effects. In this case, the treatment
effect and the blocking effect are said to be
confounded. Confounding is also used as a
general term to indicate that the value of a main
effect estimate comes from both the main effect
itself and also contamination or bias from higher
order interactions. Note: Confounding designs
naturally arise when full factorial designs have to
be run in blocks and the block size is smaller than
the number of different treatment combinations.
They also occur whenever a fractional factorial
design is chosen instead of a full factorial design.
Crossed Factors: See factors below.
Design: A set of experimental runs which
allows you to fit a particular model and estimate
your desired effects.
Design Matrix: A matrix description of an
experiment that is useful for constructing and
analyzing experiments.
Effect: How changing the settings of a factor
changes the response. The effect of a single factor
is also called a main effect. Note: For a factor A
with two levels, scaled so that low = -1 and high
= +1, the effect of A is estimated by subtracting
the average response when A is -1 from the
average response when A = +1 and dividing the
result by 2 (division by 2 is needed because the -1
level is 2 scaled units away from the +1 level).
Error: Unexplained variation in a collection of
observations. Note: DOE's typically require
understanding of both random error and lack of fit
error.
Experimental Unit: The entity to which a
specific treatment combination is applied. Note:
an experimental unit can be a
PC board
silicon wafer
tray of components simultaneously treated
individual agricultural plants
plot of land
automotive transmissions
etc.
Factors: Process inputs an investigator
manipulates to cause a change in the output. Some
factors cannot be controlled by the experimenter
but may effect the responses. If their effect is
significant, these uncontrolled factors should be
measured and used in the data analysis. Note: The
inputs can be discrete or continuous.
Crossed Factors: Two factors are crossed if
every level of one occurs with every level
of the other in the experiment.
Nested Factors: A factor "A" is nested
within another factor "B" if the levels or
values of "A" are different for every level
or value of "B". Note: Nested factors or
effects have a hierarchical relationship.
Fixed Effect: An effect associated with an input
variable that has a limited number of levels or in
which only a limited number of levels are of
interest to the experimenter.
I nteractions: Occurs when the effect of one
factor on a response depends on the level of
another factor(s).
Lack of Fit Error: Error that occurs when the
analysis omits one or more important terms or
factors from the process model. Note: Including
replication in a DOE allows separation of
experimental error into its components: lack of fit
and random (pure) error.
Model: Mathematical relationship which relates
changes in a given response to changes in one or
more factors.
Nested Factors: See factors above.
Orthogonality: Two vectors of the same length
are orthogonal if the sum of the products of their
corresponding elements is 0. Note: An
experimental design is orthogonal if the effects of
any factor balance out (sum to zero) across the
effects of the other factors.
Random Effect: An effect associated with input
variables chosen at random from a population
having a large or infinite number of possible
values.
Random error: Error that occurs due to natural
variation in the process. Note: Random error is
typically assumed to be normally distributed with
zero mean and a constant variance. Note: Random
error is also called experimental error.
Randomization: A schedule for allocating
treatment material and for conducting treatment
combinations in a DOE such that the conditions in
one run neither depend on the conditions of the
previous run nor predict the conditions in the
subsequent runs. Note: The importance of
randomization cannot be over stressed.
Randomization is necessary for conclusions drawn
from the experiment to be correct, unambiguous
and defensible.
Replication: Performing the same treatment
combination more than once. Note: Including
replication allows an estimate of the random error
independent of any lack of fit error.
Resolution: A term which describes the degree
to which estimated main effects are aliased (or
confounded) with estimated 2-level interactions,
3-level interactions, etc. In general, the resolution
of a design is one more than the smallest order
interaction that some main effect is confounded
(aliased) with. If some main effects are
confounded with some 2-level interactions, the
resolution is 3. Note: Full factorial designs have
no confounding and are said to have resolution
"infinity". For most practical purposes, a
resolution 5 design is excellent and a resolution 4
design may be adequate. Resolution 3 designs are
useful as economical screening designs.
Responses: The output(s) of a process.
Sometimes called dependent variable(s).
Response Surface Designs: A DOE that fully
explores the process window and models the
responses. Note: These designs are most effective
when there are less than 5 factors. Quadratic
models are used for response surface designs and
at least three levels of every factor are needed in
the design.
Rotatability: A design is rotatable if the
variance of the predicted response at any point x
depends only on the distance of x from the design
center point. A design with this property can be
rotated around its center point without changing
the prediction variance at x. Note: Rotatability is a
desirable property for response surface designs
(i.e. quadratic model designs).
Scaling Factor Levels: Transforming factor
levels so that the high value becomes +1 and the
low value becomes -1.
Screening Designs: A DOE that identifies
which of many factors have a significant effect on
the response. Note: Typically screening designs
have more than 5 factors.
Treatment: A treatment is a specific
combination of factor levels whose effect is to be
compared with other treatments.
Treatment Combination: The combination of
the settings of several factors in a given
experimental trial. Also known as a run.
Variance Components: Partitioning of the
overall variation into assignable components.

5.8. References

5.8. References
Chapter
specific
references
Bisgaard, S. and Steinberg, D. M., (1997), "The Design
and Analysis of 2
k-p
Prototype Experiments,"
Technometrics, 39, 1, 52-62.
Box, G. E. P., and Draper, N. R., (1987), Empirical Model
Building and Response Surfaces, John Wiley & Sons, New
York, NY.
Box, G. E. P., and Hunter, J. S., (1954), "A Confidence
Region for the Solution of a Set of Simultaneous
Equations with an Application to Experimental Design,"
Biometrika, 41, 190-199
Box, G. E. P., and Wilson, K. B., (1951), "On the
Experimental Attainment of Optimum Conditions,"
Journal of the Royal Statistical Society, Series B, 13, 1-45.
Capobianco, T. E., Splett, J. D. and Iyer, H. K., "Eddy
Current Probe Sensitivity as a Function of Coil
Construction Parameters." Research in Nondesructive
Evaluation, Vol. 2, pp. 169-186, December, 1990.
Cornell, J. A., (1990), Experiments with Mixtures:
Designs, Models, and the Analysis of Mixture Data, John
Wiley & Sons, New York, NY.
Del Castillo, E., (1996), "Multiresponse Optimization
Confidence Regions," Journal of Quality Technology, 28,
1, 61-70.
Derringer, G., and Suich, R., (1980), "Simultaneous
Optimization of Several Response Variables," Journal of
Quality Technology, 12, 4, 214-219.
Draper, N.R., (1963), "Ridge Analysis of Response
Surfaces," Technometrics, 5, 469-479.
Hoerl, A. E., (1959), "Optimum Solution of Many
Variables Equations," Chemical Engineering Progress, 55,
67-78.
Hoerl, A. E., (1964), "Ridge Analysis," Chemical
Engineering Symposium Series, 60, 67-77.
5.8. References
Khuri, A. I., and Cornell, J. A., (1987), Response Surfaces,
Marcel Dekker, New York, NY.
Mee, R. W., and Peralta, M. (2000), "Semifolding 2
k-p
Designs," Technometrics, 42, 2, p122.
Miller, A. (1997), "Strip-Plot Configuration of Fractional
Factorials," Technometrics, 39, 2, p153.
Myers, R. H., and Montgomery, D. C., (1995), Response
Surface Methodology: Process and Product Optimization
Using Designed Experiments, John Wiley & Sons, New
York, NY.
Ryan, Thomas P., (2000), Statistical Methods for Quality
Improvement, John Wiley & Sons, New York, NY.
Taguchi, G. and Konishi, S., (1987), Orthogonal Arrays
and Linear Graphs, Dearborn, MI, ASI press.
Well Known
General
References
Box, G. E. P., Hunter, W. G., and Hunter, S. J. (1978),
Statistics for Experimenters, John Wiley & Sons, Inc.,
New York, NY.
Diamond, W. J., (1989), Practical Experimental Designs,
Second Ed., Van Nostrand Reinhold, NY.
John, P. W. M., (1971), Statistical Design and Analysis of
Experiments, SIAM Classics in Applied Mathematics,
Philadelphia, PA.
Milliken, G. A., and Johnson, D. E., (1984), Analysis of
Messy Data, Vol. 1, Van Nostrand Reinhold, NY.
Montgomery, D. C., (2000), Design and Analysis of
Experiments, Fifth Edition, John Wiley & Sons, New
York, NY.
Case studies
for different
industries
Snee, R. D., Hare, L. B., and Trout, J. R.(1985),
Experiments in Industry. Design, Analysis and
Interpretation of Results, Milwaukee, WI, American
Society for Quality.
Case studies
in Process
Improvement,
including
DOE, in the
Semiconductor
Industry
Czitrom, V., and Spagon, P. D., (1997), Statistical Case
Studies for Industrial process Improvement, Philadelphia,
PA, ASA-SIAM Series on Statistics and Applied
Probability.
Software to
design and
analyze
In addition to the extensive design and analysis
documentation and routines in Dataplot, there are many
other good commercial DOE packages. This Chapter
5.8. References
experiments showed examples using "JMP" (by the SAS Institute, 100
SAS CampusDrive, Cary, North Carolina 27513-9905), as
an illustration of a good commercial package.
6. Process or Product Monitoring and Control
http://www.itl.nist.gov/div898/handbook/pmc/pmc.htm[6/27/2012 2:37:20 PM]

This chapter presents techniques for monitoring and controlling processes
and signaling when corrective actions are necessary.
1. Introduction
1. History
2. Process Control Techniques
3. Process Control
4. "Out of Control"
5. "In Control" but Unacceptable
6. Process Capability
2. Test Product for Acceptability
1. Acceptance Sampling
2. Kinds of Sampling Plans
3. Choosing a Single Sampling
Plan
4. Double Sampling Plans
5. Multiple Sampling Plans
6. Sequential Sampling Plans
7. Skip Lot Sampling Plans
3. Univariate and Multivariate
Control Charts
1. Control Charts
2. Variables Control Charts
3. Attributes Control Charts
4. Multivariate Control charts
4. Time Series Models
1. Definitions, Applications and
Techniques
2. Moving Average or
Smoothing Techniques
3. Exponential Smoothing
4. Univariate Time Series
Models
5. Multivariate Time Series
Models
5. Tutorials
1. What do we mean by
"Normal" data?
2. What to do when data are non-
normal
3. Elements of Matrix Algebra
4. Elements of Multivariate
Analysis
5. Principal Components
6. Case Study
1. Lithography Process Data
2. Box-Jenkins Modeling
Example
Detailed Table of Contents
References
http://www.itl.nist.gov/div898/handbook/pmc/pmc_d.htm[6/27/2012 2:35:29 PM]

6. Process or Product Monitoring and Control - Detailed Table of
Contents [6.]
1. How did Statistical Quality Control Begin? [6.1.1.]
2. What are Process Control Techniques? [6.1.2.]
3. What is Process Control? [6.1.3.]
4. What to do if the process is "Out of Control"? [6.1.4.]
5. What to do if "In Control" but Unacceptable? [6.1.5.]
6. What is Process Capability? [6.1.6.]
2. Test Product for Acceptability: Lot Acceptance Sampling [6.2.]
1. What is Acceptance Sampling? [6.2.1.]
2. What kinds of Lot Acceptance Sampling Plans (LASPs) are there? [6.2.2.]
3. How do you Choose a Single Sampling Plan? [6.2.3.]
1. Choosing a Sampling Plan: MIL Standard 105D [6.2.3.1.]
2. Choosing a Sampling Plan with a given OC Curve [6.2.3.2.]
4. What is Double Sampling? [6.2.4.]
5. What is Multiple Sampling? [6.2.5.]
6. What is a Sequential Sampling Plan? [6.2.6.]
7. What is Skip Lot Sampling? [6.2.7.]
3. Univariate and Multivariate Control Charts [6.3.]
1. What are Control Charts? [6.3.1.]
2. What are Variables Control Charts? [6.3.2.]
1. Shewhart X-bar and R and S Control Charts [6.3.2.1.]
2. Individuals Control Charts [6.3.2.2.]
3. Cusum Control Charts [6.3.2.3.]
1. Cusum Average Run Length [6.3.2.3.1.]
4. EWMA Control Charts [6.3.2.4.]
3. What are Attributes Control Charts? [6.3.3.]
1. Counts Control Charts [6.3.3.1.]
2. Proportions Control Charts [6.3.3.2.]
4. What are Multivariate Control Charts? [6.3.4.]
1. Hotelling Control Charts [6.3.4.1.]
2. Principal Components Control Charts [6.3.4.2.]
3. Multivariate EWMA Charts [6.3.4.3.]
4. Introduction to Time Series Analysis [6.4.]
1. Definitions, Applications and Techniques [6.4.1.]
2. What are Moving Average or Smoothing Techniques? [6.4.2.]
1. Single Moving Average [6.4.2.1.]
2. Centered Moving Average [6.4.2.2.]
3. What is Exponential Smoothing? [6.4.3.]
1. Single Exponential Smoothing [6.4.3.1.]
2. Forecasting with Single Exponential Smoothing [6.4.3.2.]
3. Double Exponential Smoothing [6.4.3.3.]
4. Forecasting with Double Exponential Smoothing(LASP) [6.4.3.4.]
5. Triple Exponential Smoothing [6.4.3.5.]
6. Example of Triple Exponential Smoothing [6.4.3.6.]
7. Exponential Smoothing Summary [6.4.3.7.]
4. Univariate Time Series Models [6.4.4.]
1. Sample Data Sets [6.4.4.1.]
1. Data Set of Monthly CO2 Concentrations [6.4.4.1.1.]
2. Data Set of Southern Oscillations [6.4.4.1.2.]
2. Stationarity [6.4.4.2.]
3. Seasonality [6.4.4.3.]
1. Seasonal Subseries Plot [6.4.4.3.1.]
4. Common Approaches to Univariate Time Series [6.4.4.4.]
5. Box-Jenkins Models [6.4.4.5.]
6. Box-Jenkins Model Identification [6.4.4.6.]
1. Model Identification for Southern Oscillations Data [6.4.4.6.1.]
2. Model Identification for the CO
2
Concentrations Data [6.4.4.6.2.]
3. Partial Autocorrelation Plot [6.4.4.6.3.]
7. Box-Jenkins Model Estimation [6.4.4.7.]
8. Box-Jenkins Model Diagnostics [6.4.4.8.]
1. Box-Ljung Test [6.4.4.8.1.]
9. Example of Univariate Box-Jenkins Analysis [6.4.4.9.]
10. Box-Jenkins Analysis on Seasonal Data [6.4.4.10.]
5. Multivariate Time Series Models [6.4.5.]
1. Example of Multivariate Time Series Analysis [6.4.5.1.]
5. Tutorials [6.5.]
1. What do we mean by "Normal" data? [6.5.1.]
2. What do we do when data are "Non-normal"? [6.5.2.]
3. Elements of Matrix Algebra [6.5.3.]
1. Numerical Examples [6.5.3.1.]
2. Determinant and Eigenstructure [6.5.3.2.]
4. Elements of Multivariate Analysis [6.5.4.]
1. Mean Vector and Covariance Matrix [6.5.4.1.]
2. The Multivariate Normal Distribution [6.5.4.2.]
3. Hotelling's T squared [6.5.4.3.]
1. T
2
Chart for Subgroup Averages -- Phase I [6.5.4.3.1.]
2. T
2
Chart for Subgroup Averages -- Phase II [6.5.4.3.2.]
3. Chart for Individual Observations -- Phase I [6.5.4.3.3.]
4. Chart for Individual Observations -- Phase II [6.5.4.3.4.]
5. Charts for Controlling Multivariate Variability [6.5.4.3.5.]
6. Constructing Multivariate Charts [6.5.4.3.6.]
5. Principal Components [6.5.5.]
1. Properties of Principal Components [6.5.5.1.]
2. Numerical Example [6.5.5.2.]
6. Case Studies in Process Monitoring [6.6.]
1. Lithography Process [6.6.1.]
2. Graphical Representation of the Data [6.6.1.2.]
3. Subgroup Analysis [6.6.1.3.]
4. Shewhart Control Chart [6.6.1.4.]
2. Aerosol Particle Size [6.6.2.]
2. Model Identification [6.6.2.2.]
3. Model Estimation [6.6.2.3.]
4. Model Validation [6.6.2.4.]
6.1. Introduction
http://www.itl.nist.gov/div898/handbook/pmc/section1/pmc1.htm[6/27/2012 2:35:35 PM]

6.1. Introduction
Contents
of Section
This section discusses the basic concepts of statistical process
control, quality control and process capability.

1. How did Statistical Quality Control Begin?
2. What are Process Control Techniques?
3. What is Process Control?
4. What to do if the process is "Out of
Control"?
5. What to do if "In Control" but
Unacceptable?
6. What is Process Capability?

6.1.1. How did Statistical Quality Control Begin?

6.1. Introduction
6.1.1. How did Statistical Quality Control
Begin?
Historical
perspective
Quality Control has been with us for a long time. How
long? It is safe to say that when manufacturing began and
competition accompanied manufacturing, consumers would
compare and choose the most attractive product (barring a
monopoly of course). If manufacturer A discovered that
manufacturer B's profits soared, the former tried to
improve his/her offerings, probably by improving the
quality of the output, and/or lowering the price.
Improvement of quality did not necessarily stop with the
product - but also included the process used for making the
product.
The process was held in high esteem, as manifested by the
medieval guilds of the Middle Ages. These guilds
mandated long periods of training for apprentices, and
those who were aiming to become master craftsmen had to
demonstrate evidence of their ability. Such procedures
were, in general, aimed at the maintenance and
improvement of the quality of the process.
In modern times we have professional societies,
governmental regulatory bodies such as the Food and Drug
Administration, factory inspection, etc., aimed at assuring
the quality of products sold to consumers. Quality Control
has thus had a long history.
Science of
statistics is
fairly recent
On the other hand, statistical quality control is
comparatively new. The science of statistics itself goes
back only two to three centuries. And its greatest
developments have taken place during the 20th century.
The earlier applications were made in astronomy and
physics and in the biological and social sciences. It was not
until the 1920s that statistical theory began to be applied
effectively to quality control as a result of the development
of sampling theory.
The concept of
quality
control in
manufacturing
was first
The first to apply the newly discovered statistical methods
to the problem of quality control was Walter A. Shewhart
of the Bell Telephone Laboratories. He issued a
memorandum on May 16, 1924 that featured a sketch of a
modern control chart.
6.1.1. How did Statistical Quality Control Begin?
advanced by
Walter
Shewhart
Shewhart kept improving and working on this scheme, and
in 1931 he published a book on statistical quality control,
"Economic Control of Quality of Manufactured Product",
published by Van Nostrand in New York. This book set the
tone for subsequent applications of statistical methods to
process control.
Contributions
of Dodge and
Romig to
sampling
inspection
Two other Bell Labs statisticians, H.F. Dodge and H.G.
Romig spearheaded efforts in applying statistical theory to
sampling inspection. The work of these three pioneers
constitutes much of what nowadays comprises the theory
of statistical quality and control. There is much more to
say about the history of statistical quality control and the
interested reader is invited to peruse one or more of the
references. A very good summary of the historical
background of SQC is found in chapter 1 of "Quality
Control and Industrial Statistics", by Acheson J. Duncan.
See also Juran (1997).
6.1.2. What are Process Control Techniques?

6.1. Introduction
Statistical Process Control (SPC)
Typical
process
control
techniques
There are many ways to implement process control. Key
monitoring and investigating tools include:
Histograms
Check Sheets
Pareto Charts
Cause and Effect Diagrams
Defect Concentration Diagrams
Scatter Diagrams
Control Charts
All these are described in Montgomery (2000). This chapter
will focus (Section 3) on control chart methods, specifically:
Classical Shewhart Control charts,
Cumulative Sum (CUSUM) charts
Exponentially Weighted Moving Average (EWMA)
charts
Multivariate control charts
Underlying
concepts
The underlying concept of statistical process control is based
on a comparison of what is happening today with what
happened previously. We take a snapshot of how the process
typically performs or build a model of how we think the
process will perform and calculate control limits for the
expected measurements of the output of the process. Then we
collect data from the process and compare the data to the
control limits. The majority of measurements should fall
within the control limits. Measurements that fall outside the
control limits are examined to see if they belong to the same
population as our initial snapshot or model. Stated differently,
we use historical data to compute the initial control limits.
Then the data are compared against these initial limits. Points
that fall outside of the limits are investigated and, perhaps,
some will later be discarded. If so, the limits would be
recomputed and the process repeated. This is referred to as
Phase I. Real-time process monitoring, using the limits from
the end of Phase I, is Phase II.
Statistical Quality Control (SQC)
Tools of
statistical
quality
control
Several techniques can be used to investigate the product for
defects or defective pieces after all processing is complete.
Typical tools of SQC (described in section 2) are:
Lot Acceptance sampling plans
Skip lot sampling plans
Military (MIL) Standard sampling plans
Underlying
concepts of
statistical
quality
control
The purpose of statistical quality control is to ensure, in a cost
efficient manner, that the product shipped to customers meets
their specifications. Inspecting every product is costly and
inefficient, but the consequences of shipping non conforming
product can be significant in terms of customer dissatisfaction.
Statistical Quality Control is the process of inspecting enough
product from given lots to probabilistically ensure a specified
quality level.
6.1.3. What is Process Control?

6.1. Introduction
6.1.3. What is Process Control?
Two types
of
intervention
are
possible --
one is
based on
engineering
judgment
and the
other is
automated
Process Control is the active changing of the process based on
the results of process monitoring. Once the process
monitoring tools have detected an out-of-control situation,
the person responsible for the process makes a change to
bring the process back into control.
1. Out-of-control Action Plans (OCAPS) detail the action
to be taken once an out-of-control situation is detected.
A specific flowchart, that leads the process engineer
through the corrective procedure, may be provided for
each unique process.
2. Advanced Process Control Loops are automated
changes to the process that are programmed to correct
for the size of the out-of-control measurement.
6.1.4. What to do if the process is "Out of Control"?

6.1. Introduction
6.1.4. What to do if the process is "Out of
Control"?
Reactions
to out-of-
control
conditions
If the process is out-of-control, the process engineer looks for
an assignable cause by following the out-of-control action
plan (OCAP) associated with the control chart. Out-of-control
refers to rejecting the assumption that the current data are from
the same population as the data used to create the initial
control chart limits.
For classical Shewhart charts, a set of rules called the Western
Electric Rules (WECO Rules) and a set of trend rules often are
used to determine out-of-control.
6.1.5. What to do if "In Control" but Unacceptable?

6.1. Introduction
6.1.5. What to do if "In Control" but
Unacceptable?
In control
means
process is
predictable
"In Control" only means that the process is predictable in a
statistical sense. What do you do if the process is in
control but the average level is too high or too low or the
variability is unacceptable?
Process
improvement
techniques
Process improvement techniques such as
experiments
calibration
re-analysis of historical database
can be initiated to put the process on target or reduce the
variability.
Process
must be
stable
Note that the process must be stable before it can be
centered at a target value or its overall variation can be
reduced.
6.1.6. What is Process Capability?

6.1. Introduction
Process capability compares the output of an in-control process to the specification
limits by using capability indices. The comparison is made by forming the ratio of
the spread between the process specifications (the specification "width") to the
spread of the process values, as measured by 6 process standard deviation units (the
process "width").
Process Capability Indices
A process
capability
index uses
both the
process
variability
and the
process
specifications
to determine
whether the
process is
"capable"
We are often required to compare the output of a stable process with the process
specifications and make a statement about how well the process meets specification.
To do this we compare the natural variability of a stable process with the process
specification limits.
A process where almost all the measurements fall inside the specification limits is a
capable process. This can be represented pictorially by the plot below:

There are several statistics that can be used to measure the capability of a process:
C
p
, C
pk
, C
pm
.
Most capability indices estimates are valid only if the sample size used is 'large
enough'. Large enough is generally thought to be about 50 independent data values.
The C
p
, C
pk
, and C
pm
statistics assume that the population of data values is normally
distributed. Assuming a two-sided specification, if and are the mean and
standard deviation, respectively, of the normal data and USL, LSL, and T are the
upper and lower specification limits and the target value, respectively, then the
population capability indices are defined as follows:
Definitions of
various
process
capability
indices
Sample
estimates of
capability
indices
Sample estimators for these indices are given below. (Estimators are indicated with
a "hat" over them).
The estimator for C
pk
can also be expressed as C
pk
= C
p
(1-k), where k is a scaled
distance between the midpoint of the specification range, m, and the process mean,
.
Denote the midpoint of the specification range by m = (USL+LSL)/2. The distance
between the process mean, , and the optimum, which is m, is - m, where
. The scaled distance is
(the absolute sign takes care of the case when ). To determine the
estimated value, , we estimate by . Note that .
The estimator for the C
p
index, adjusted by the k factor, is
Since , it follows that .
Plot showing
C
p
for varying
process
widths
To get an idea of the value of the C
p
statistic for varying process widths, consider
the following plot
This can be expressed numerically by the table below:
Translating
capability into
"rejects"
USL - LSL 6 8 10 12
C
p
1.00 1.33 1.66 2.00
Rejects .27% 64 ppm .6 ppm 2 ppb
% of spec used 100 75 60 50
where ppm = parts per million and ppb = parts per billion. Note that the reject
figures are based on the assumption that the distribution is centered at .
We have discussed the situation with two spec. limits, the USL and LSL. This is
known as the bilateral or two-sided case. There are many cases where only the
lower or upper specifications are used. Using one spec limit is called unilateral or
one-sided. The corresponding capability indices are
One-sided
specifications
and the
corresponding
capability
indices
and
where and are the process mean and standard deviation, respectively.
Estimators of C
pu
and C
pl
are obtained by replacing and by and s,
respectively. The following relationship holds
C
p
= (C
pu
+ C
pl
) /2.
This can be represented pictorially by
Note that we also can write:
C
pk
= min {C
pl
, C
pu
}.
Confidence Limits For Capability Indices
Confidence
intervals for
indices
Assuming normally distributed process data, the distribution of the sample
follows from a Chi-square distribution and and have distributions related
to the non-central t distribution. Fortunately, approximate confidence limits related
to the normal distribution have been derived. Various approximations to the
distribution of have been proposed, including those given by Bissell (1990),
and we will use a normal approximation.
The resulting formulas for confidence limits are given below:
100(1- )% Confidence Limits for C
p
where
= degrees of freedom.
Confidence
Intervals for
C
pu
and C
pl
Approximate 100(1- )% confidence limits for C
pu
with sample size n are:
with z denoting the percent point function of the standard normal distribution. If is
not known, set it to .
Limits for C
pl
are obtained by replacing by .
Confidence
Interval for
C
pk
Zhang et al. (1990) derived the exact variance for the estimator of C
pk
as well as an
approximation for large n. The reference paper is Zhang, Stenback and Wardrop
(1990), "Interval Estimation of the process capability index", Communications in
Statistics: Theory and Methods, 19(21), 4455-4470.
The variance is obtained as follows:
Let
Then
Their approximation is given by:
where
The following approximation is commonly used in practice
It is important to note that the sample size should be at least 25 before these
approximations are valid. In general, however, we need n 100 for capability
studies. Another point to observe is that variations are not negligible due to the
randomness of capability indices.
Capability Index Example
An example For a certain process the USL = 20 and the LSL = 8. The observed process average,
= 16, and the standard deviation, s = 2. From this we obtain
This means that the process is capable as long as it is located at the midpoint, m =
(USL + LSL)/2 = 14.
But it doesn't, since = 16. The factor is found by
and
We would like to have at least 1.0, so this is not a good process. If possible,
reduce the variability or/and center the process. We can compute the and
From this we see that the , which is the smallest of the above indices, is 0.6667.
Note that the formula is the algebraic equivalent of the min{
, } definition.
What happens if the process is not approximately normally distributed?
What you can The indices that we considered thus far are based on normality of the process
do with non-
normal data
distribution. This poses a problem when the process distribution is not normal.
Without going into the specifics, we can list some remedies.
1. Transform the data so that they become approximately normal. A popular
transformation is the Box-Cox transformation
2. Use or develop another set of indices, that apply to nonnormal distributions.
One statistic is called C
npk
(for non-parametric C
pk
). Its estimator is calculated
by
where p(0.995) is the 99.5th percentile of the data and p(.005) is the 0.5th
percentile of the data.
For additional information on nonnormal distributions, see Johnson and Kotz
(1993).
There is, of course, much more that can be said about the case of nonnormal data.
However, if a Box-Cox transformation can be successfully performed, one is
encouraged to use it.
6.2. Test Product for Acceptability: Lot Acceptance Sampling

6.2. Test Product for Acceptability: Lot
Acceptance Sampling
This section describes how to make decisions on a lot-by-lot
basis whether to accept a lot as likely to meet requirements or
reject the lot as likely to have too many defective units.
Contents
of section
2
This section consists of the following topics.
1. What is Acceptance Sampling?
2. What kinds of Lot Acceptance Sampling Plans (LASPs)
are there?
3. How do you Choose a Single Sampling Plan?
1. Choosing a Sampling Plan: MIL Standard 105D
2. Choosing a Sampling Plan with a given OC
Curve
4. What is Double Sampling?
5. What is Multiple Sampling?
6. What is a Sequential Sampling Plan?
7. What is Skip Lot Sampling?
6.2.1. What is Acceptance Sampling?

Contributions
of Dodge and
Romig to
acceptance
sampling
Acceptance sampling is an important field of statistical
quality control that was popularized by Dodge and Romig
and originally applied by the U.S. military to the testing of
bullets during World War II. If every bullet was tested in
advance, no bullets would be left to ship. If, on the other
hand, none were tested, malfunctions might occur in the
field of battle, with potentially disastrous results.
Definintion
of Lot
Acceptance
Sampling
Dodge reasoned that a sample should be picked at random
from the lot, and on the basis of information that was
yielded by the sample, a decision should be made regarding
the disposition of the lot. In general, the decision is either to
accept or reject the lot. This process is called Lot
Acceptance Sampling or just Acceptance Sampling.
"Attributes"
(i.e., defect
counting)
will be
assumed
Acceptance sampling is "the middle of the road" approach
between no inspection and 100% inspection. There are two
major classifications of acceptance plans: by attributes ("go,
no-go") and by variables. The attribute case is the most
common for acceptance sampling, and will be assumed for
the rest of this section.
Important
point
A point to remember is that the main purpose of acceptance
sampling is to decide whether or not the lot is likely to be
acceptable, not to estimate the quality of the lot.
Scenarios
leading to
acceptance
sampling
Acceptance sampling is employed when one or several of
the following hold:
Testing is destructive
The cost of 100% inspection is very high
100% inspection takes too long
Acceptance
Quality
Control and
Acceptance
Sampling
It was pointed out by Harold Dodge in 1969 that
Acceptance Quality Control is not the same as Acceptance
Sampling. The latter depends on specific sampling plans,
which when implemented indicate the conditions for
acceptance or rejection of the immediate lot that is being
inspected. The former may be implemented in the form of
an Acceptance Control Chart. The control limits for the
Acceptance Control Chart are computed using the
specification limits and the standard deviation of what is
being monitored (see Ryan, 2000 for details).
An
observation
by Harold
Dodge
In 1942, Dodge stated:
"....basically the "acceptance quality control" system that
was developed encompasses the concept of protecting the
consumer from getting unacceptable defective product, and
encouraging the producer in the use of process quality
control by: varying the quantity and severity of acceptance
inspections in direct relation to the importance of the
characteristics inspected, and in the inverse relation to the
goodness of the quality level as indication by those
inspections."
To reiterate the difference in these two approaches:
acceptance sampling plans are one-shot deals, which
essentially test short-run effects. Quality control is of the
long-run variety, and is part of a well-designed system for
lot acceptance.
An
observation
by Ed
Schilling
Schilling (1989) said:
"An individual sampling plan has much the effect of a lone
sniper, while the sampling plan scheme can provide a
fusillade in the battle for quality improvement."
Control of
product
quality using
acceptance
control
charts
According to the ISO standard on acceptance control charts
(ISO 7966, 1993), an acceptance control chart combines
consideration of control implications with elements of
acceptance sampling. It is an appropriate tool for helping to
make decisions with respect to process acceptance. The
difference between acceptance sampling approaches and
acceptance control charts is the emphasis on process
acceptability rather than on product disposition decisions.
6.2.2. What kinds of Lot Acceptance Sampling Plans (LASPs) are there?

6.2.2. What kinds of Lot Acceptance Sampling
Plans (LASPs) are there?
LASP is a
sampling
scheme
and a set
of rules
A lot acceptance sampling plan (LASP) is a sampling scheme
and a set of rules for making decisions. The decision, based
on counting the number of defectives in a sample, can be to
accept the lot, reject the lot, or even, for multiple or sequential
sampling schemes, to take another sample and then repeat the
decision process.
Types of
acceptance
plans to
choose
from
LASPs fall into the following categories:
Single sampling plans:. One sample of items is
selected at random from a lot and the disposition of the
lot is determined from the resulting information. These
plans are usually denoted as (n,c) plans for a sample
size n, where the lot is rejected if there are more than c
defectives. These are the most common (and easiest)
plans to use although not the most efficient in terms of
average number of samples needed.
Double sampling plans: After the first sample is tested,
there are three possibilities:
1. Accept the lot
2. Reject the lot
3. No decision
If the outcome is (3), and a second sample is taken, the
procedure is to combine the results of both samples and
make a final decision based on that information.
Multiple sampling plans: This is an extension of the
double sampling plans where more than two samples are
needed to reach a conclusion. The advantage of multiple
sampling is smaller sample sizes.
Sequential sampling plans: . This is the ultimate
extension of multiple sampling where items are selected
from a lot one at a time and after inspection of each
item a decision is made to accept or reject the lot or
select another unit.
Skip lot sampling plans:. Skip lot sampling means that
only a fraction of the submitted lots are inspected.
Definitions
of basic
Acceptance
Sampling
terms
Deriving a plan, within one of the categories listed above, is
discussed in the pages that follow. All derivations depend on
the properties you want the plan to have. These are described
using the following terms:
Acceptable Quality Level (AQL): The AQL is a percent
defective that is the base line requirement for the quality
of the producer's product. The producer would like to
design a sampling plan such that there is a high
probability of accepting a lot that has a defect level less
than or equal to the AQL.
Lot Tolerance Percent Defective (LTPD): The LTPD is
a designated high defect level that would be
unacceptable to the consumer. The consumer would like
the sampling plan to have a low probability of
accepting a lot with a defect level as high as the LTPD.
Type I Error (Producer's Risk): This is the probability,
for a given (n,c) sampling plan, of rejecting a lot that
has a defect level equal to the AQL. The producer
suffers when this occurs, because a lot with acceptable
quality was rejected. The symbol is commonly used
for the Type I error and typical values for range from
0.2 to 0.01.
Type I I Error (Consumer's Risk): This is the
probability, for a given (n,c) sampling plan, of
accepting a lot with a defect level equal to the LTPD.
The consumer suffers when this occurs, because a lot
with unacceptable quality was accepted. The symbol
is commonly used for the Type II error and typical
values range from 0.2 to 0.01.
Operating Characteristic (OC) Curve: This curve plots
the probability of accepting the lot (Y-axis) versus the
lot fraction or percent defectives (X-axis). The OC
curve is the primary tool for displaying and
investigating the properties of a LASP.
Average Outgoing Quality (AOQ): A common
procedure, when sampling and testing is non-
destructive, is to 100% inspect rejected lots and replace
all defectives with good units. In this case, all rejected
lots are made perfect and the only defects left are those
in lots that were accepted. AOQ's refer to the long term
defect level for this combined LASP and 100%
inspection of rejected lots process. If all lots come in
with a defect level of exactly p, and the OC curve for
the chosen (n,c) LASP indicates a probability p
a
of
accepting such a lot, over the long run the AOQ can
easily be shown to be:
where N is the lot size.
Average Outgoing Quality Level (AOQL): A plot of the
AOQ (Y-axis) versus the incoming lot p (X-axis) will
start at 0 for p = 0, and return to 0 for p = 1 (where
every lot is 100% inspected and rectified). In between,
it will rise to a maximum. This maximum, which is the
worst possible long term AOQ, is called the AOQL.
Average Total I nspection (ATI ): When rejected lots are
100% inspected, it is easy to calculate the ATI if lots
come consistently with a defect level of p. For a LASP
(n,c) with a probability p
a
of accepting a lot with defect
level p, we have
ATI = n + (1 - p
a
) (N - n)
where N is the lot size.
Average Sample Number (ASN): For a single sampling
LASP (n,c) we know each and every lot has a sample
of size n taken and inspected or tested. For double,
multiple and sequential LASP's, the amount of sampling
varies depending on the number of defects observed.
For any given double, multiple or sequential plan, a
long term ASN can be calculated assuming all lots come
in with a defect level of p. A plot of the ASN, versus the
incoming defect level p, describes the sampling
efficiency of a given LASP scheme.
The final
choice is a
tradeoff
decision
Making a final choice between single or multiple sampling
plans that have acceptable properties is a matter of deciding
whether the average sampling savings gained by the various
multiple sampling plans justifies the additional complexity of
these plans and the uncertainty of not knowing how much
sampling and inspection will be done on a day-by-day basis.
6.2.3. How do you Choose a Single Sampling Plan?

6.2.3. How do you Choose a Single Sampling
Plan?
Two
methods
for
choosing a
single
sample
acceptance
plan
A single sampling plan, as previously defined, is specified by
the pair of numbers (n,c). The sample size is n, and the lot is
rejected if there are more than c defectives in the sample;
otherwise the lot is accepted.
There are two widely used ways of picking (n,c):
1. Use tables (such as MIL STD 105D) that focus on either
the AQL or the LTPD desired.
2. Specify 2 desired points on the OC curve and solve for
the (n,c) that uniquely determines an OC curve going
through these points.
The next two pages describe these methods in detail.
6.2.3.1. Choosing a Sampling Plan: MIL Standard 105D

6.2.3.1. Choosing a Sampling Plan: MIL
Standard 105D
The AQL or
Acceptable
Quality
Level is the
baseline
requirement
Sampling plans are typically set up with reference to an
acceptable quality level, or AQL . The AQL is the base line
requirement for the quality of the producer's product. The
producer would like to design a sampling plan such that the
OC curve yields a high probability of acceptance at the AQL.
On the other side of the OC curve, the consumer wishes to be
protected from accepting poor quality from the producer. So
the consumer establishes a criterion, the lot tolerance percent
defective or LTPD . Here the idea is to only accept poor
quality product with a very low probability. Mil. Std. plans
have been used for over 50 years to achieve these goals.
The U.S. Department of Defense Military Standard 105E
Military
Standard
105E
sampling
plan
Standard military sampling procedures for inspection by
attributes were developed during World War II. Army
Ordnance tables and procedures were generated in the early
1940's and these grew into the Army Service Forces tables.
At the end of the war, the Navy also worked on a set of
tables. In the meanwhile, the Statistical Research Group at
Columbia University performed research and outputted many
outstanding results on attribute sampling plans.
These three streams combined in 1950 into a standard called
Mil. Std. 105A. It has since been modified from time to time
and issued as 105B, 195C and 105D. Mil. Std. 105D was
issued by the U.S. government in 1963. It was adopted in
1971 by the American National Standards Institute as ANSI
Standard Z1.4 and in 1974 it was adopted (with minor
changes) by the International Organization for
Standardization as ISO Std. 2859. The latest revision is Mil.
Std 105E and was issued in 1989.
These three similar standards are continuously being updated
and revised, but the basic tables remain the same. Thus the
discussion that follows of the germane aspects of Mil. Std.
105E also applies to the other two standards.
Description of Mil. Std. 105D
Military
Standard
105D
sampling
plan
This document is essentially a set of individual plans,
organized in a system of sampling schemes. A sampling
scheme consists of a combination of a normal sampling plan,
a tightened sampling plan, and a reduced sampling plan plus
rules for switching from one to the other.
AQL is
foundation
of standard
The foundation of the Standard is the acceptable quality level
or AQL. In the following scenario, a certain military agency,
called the Consumer from here on, wants to purchase a
particular product from a supplier, called the Producer from
here on.
In applying the Mil. Std. 105D it is expected that there is
perfect agreement between Producer and Consumer regarding
what the AQL is for a given product characteristic. It is
understood by both parties that the Producer will be
submitting for inspection a number of lots whose quality
level is typically as good as specified by the Consumer.
Continued quality is assured by the acceptance or rejection of
lots following a particular sampling plan and also by
providing for a shift to another, tighter sampling plan, when
there is evidence that the Producer's product does not meet
the agreed-upon AQL.
Standard
offers 3
types of
sampling
plans
Mil. Std. 105E offers three types of sampling plans: single,
double and multiple plans. The choice is, in general, up to the
inspectors.
Because of the three possible selections, the standard does
not give a sample size, but rather a sample code letter. This,
together with the decision of the type of plan yields the
specific sampling plan to be used.
Inspection
level
In addition to an initial decision on an AQL it is also
necessary to decide on an "inspection level". This determines
the relationship between the lot size and the sample size. The
standard offers three general and four special levels.
Steps in the
standard
The steps in the use of the standard can be summarized as
follows:
1. Decide on the AQL.
2. Decide on the inspection level.
3. Determine the lot size.
4. Enter the table to find sample size code letter.
5. Decide on type of sampling to be used.
6. Enter proper table to find the plan to be used.
7. Begin with normal inspection, follow the switching
rules and the rule for stopping the inspection (if
needed).
Additional There is much more that can be said about Mil. Std. 105E,
information (and 105D). The interested reader is referred to references
such as (Montgomery (2000), Schilling, tables 11-2 to 11-17,
and Duncan, pages 214 - 248).
There is also (currently) a web site developed by Galit
Shmueli that will develop sampling plans interactively with
the user, according to Military Standard 105E (ANSI/ASQC
Z1.4, ISO 2859) Tables.
6.2.3.2. Choosing a Sampling Plan with a given OC Curve

6.2.3.2. Choosing a Sampling Plan with a given OC
Curve
Sample
OC
curve
We start by looking at a typical OC curve. The OC curve for a (52 ,3)
sampling plan is shown below.
Number of
defectives is
approximately
binomial
It is instructive to show how the points on this curve are
obtained, once we have a sampling plan (n,c) - later we
will demonstrate how a sampling plan (n,c) is obtained.
We assume that the lot size N is very large, as compared to
the sample size n, so that removing the sample doesn't
significantly change the remainder of the lot, no matter how
many defects are in the sample. Then the distribution of the
number of defectives, d, in a random sample of n items is
approximately binomial with parameters n and p, where p
is the fraction of defectives per lot.
The probability of observing exactly d defectives is given
by
The binomial
distribution
The probability of acceptance is the probability that d, the
number of defectives, is less than or equal to c, the accept
number. This means that
Sample table
for Pa, Pd
using the
binomial
distribution
Using this formula with n = 52 and c=3 and p = .01, .02,
...,.12 we find
P
a
P
d
.998 .01
.980 .02
.930 .03
.845 .04
.739 .05
.620 .06
.502 .07
.394 .08
.300 .09
.223 .10
.162 .11
.115 .12
Solving for (n,c)
Equations for
calculating a
sampling plan
with a given
OC curve
In order to design a sampling plan with a specified OC
curve one needs two designated points. Let us design a
sampling plan such that the probability of acceptance is 1-
for lots with fraction defective p
1
and the probability of
acceptance is for lots with fraction defective p
2
. Typical
choices for these points are: p
1
is the AQL, p
2
is the LTPD
and , are the Producer's Risk (Type I error) and
Consumer's Risk (Type II error), respectively.
If we are willing to assume that binomial sampling is valid,
then the sample size n, and the acceptance number c are the
solution to
These two simultaneous equations are nonlinear so there is
no simple, direct solution. There are however a number of
iterative techniques available that give approximate
solutions so that composition of a computer program poses
few problems.
Average Outgoing Quality (AOQ)
Calculating
AOQ's
We can also calculate the AOQ for a (n,c) sampling plan,
provided rejected lots are 100% inspected and defectives
are replaced with good parts.
Assume all lots come in with exactly a p
0
proportion of
defectives. After screening a rejected lot, the final fraction
defectives will be zero for that lot. However, accepted lots
have fraction defectivep
0
. Therefore, the outgoing lots
from the inspection stations are a mixture of lots with
fractions defective p
0
and 0. Assuming the lot size is N, we
have.
For example, let N = 10000, n = 52, c = 3, and p, the
quality of incoming lots, = 0.03. Now at p = 0.03, we glean
from the OC curve table that p
a
= 0.930 and
AOQ = (.930)(.03)(10000-52) / 10000 = 0.02775.
Sample table
of AOQ
versus p
Setting p = .01, .02, ..., .12, we can generate the following
table
AOQ p
.0010 .01
.0196 .02
.0278 .03
.0338 .04
.0369 .05
.0372 .06
.0351 .07
.0315 .08
.0270 .09
.0223 .10
.0178 .11
.0138 .12
Sample plot
of AOQ
versus p
A plot of the AOQ versus p is given below.
Interpretation
of AOQ plot
From examining this curve we observe that when the
incoming quality is very good (very small fraction of
defectives coming in), then the outgoing quality is also very
good (very small fraction of defectives going out). When
the incoming lot quality is very bad, most of the lots are
rejected and then inspected. The "duds" are eliminated or
replaced by good ones, so that the quality of the outgoing
lots, the AOQ, becomes very good. In between these
extremes, the AOQ rises, reaches a maximum, and then
drops.
The maximum ordinate on the AOQ curve represents the
worst possible quality that results from the rectifying
inspection program. It is called the average outgoing
quality limit, (AOQL ).
From the table we see that the AOQL = 0.0372 at p = .06
for the above example.
One final remark: if N >> n, then the AOQ ~ p
a
p .
The Average Total Inspection (ATI)
Calculating
the Average
Total
Inspection
What is the total amount of inspection when rejected lots
are screened?
If all lots contain zero defectives, no lot will be rejected.
If all items are defective, all lots will be inspected, and the
amount to be inspected is N.
Finally, if the lot quality is 0 < p < 1, the average amount
of inspection per lot will vary between the sample size n,
and the lot size N.
Let the quality of the lot be p and the probability of lot
acceptance be p
a
, then the ATI per lot is
ATI = n + (1 - p
a
) (N - n)
For example, let N = 10000, n = 52, c = 3, and p = .03 We
know from the OC table that p
a
= 0.930. Then ATI = 52 +
(1-.930) (10000 - 52) = 753. (Note that while 0.930 was
rounded to three decimal places, 753 was obtained using
more decimal places.)
Sample table
of ATI versus
p
Setting p= .01, .02, ....14 generates the following table
ATI P
70 .01
253 .02
753 .03
1584 .04
2655 .05
3836 .06
5007 .07
6083 .08
7012 .09
7779 .10
8388 .11
8854 .12
9201 .13
9453 .14
Plot of ATI
versus p
A plot of ATI versus p, the Incoming Lot Quality (ILQ) is
given below.
6.2.4. What is Double Sampling?

Double Sampling Plans
How double
sampling
plans work
Double and multiple sampling plans were invented to give a
questionable lot another chance. For example, if in double sampling
the results of the first sample are not conclusive with regard to
accepting or rejecting, a second sample is taken. Application of
double sampling requires that a first sample of size n
1
is taken at
random from the (large) lot. The number of defectives is then
counted and compared to the first sample's acceptance number a
1
and rejection number r
1
. Denote the number of defectives in sample
1 by d
1
and in sample 2 by d
2
, then:
If d
1
a
1
, the lot is accepted.
If d
1
r
1
, the lot is rejected.
If a
1
< d
1
< r
1
, a second sample is taken.
If a second sample of size n
2
is taken, the number of defectives, d
2
,
is counted. The total number of defectives is D
2
= d
1
+ d
2
. Now this
is compared to the acceptance number a
2
and the rejection number
r
2
of sample 2. In double sampling, r
2
= a
2
+ 1 to ensure a decision
on the sample.
If D
2
a
2
, the lot is accepted.
If D
2
r
2
, the lot is rejected.
Design of a Double Sampling Plan
Design of a
double
sampling
plan
The parameters required to construct the OC curve are similar to the
single sample case. The two points of interest are (p
1
, 1- ) and (p
2
,
, where p
1
is the lot fraction defective for plan 1 and p
2
is the lot
fraction defective for plan 2. As far as the respective sample sizes are
concerned, the second sample size must be equal to, or an even
multiple of, the first sample size.
There exist a variety of tables that assist the user in constructing
double and multiple sampling plans. The index to these tables is the
p
2
/p
1
ratio, where p
2
> p
1
. One set of tables, taken from the Army
Chemical Corps Engineering Agency for = .05 and = .10, is
given below:
Tables for n
1
= n
2
accept approximation values
R = numbers of pn
1
for
p
2
/p
1
c
1
c
2
P = .95 P = .10
11.90 0 1 0.21 2.50
7.54 1 2 0.52 3.92
6.79 0 2 0.43 2.96
5.39 1 3 0.76 4.11
4.65 2 4 1.16 5.39
4.25 1 4 1.04 4.42
3.88 2 5 1.43 5.55
3.63 3 6 1.87 6.78
3.38 2 6 1.72 5.82
3.21 3 7 2.15 6.91
3.09 4 8 2.62 8.10
2.85 4 9 2.90 8.26
2.60 5 11 3.68 9.56
2.44 5 12 4.00 9.77
2.32 5 13 4.35 10.08
2.22 5 14 4.70 10.45
2.12 5 16 5.39 11.41
Tables for n
2
= 2n
1
accept approximation values
R = numbers of pn
1
for
p
2
/p
1
c
1
c
2
P = .95 P = .10
14.50 0 1 0.16 2.32
8.07 0 2 0.30 2.42
6.48 1 3 0.60 3.89
5.39 0 3 0.49 2.64
5.09 0 4 0.77 3.92
4.31 1 4 0.68 2.93
4.19 0 5 0.96 4.02
3.60 1 6 1.16 4.17
3.26 1 8 1.68 5.47
2.96 2 10 2.27 6.72
2.77 3 11 2.46 6.82
2.62 4 13 3.07 8.05
2.46 4 14 3.29 8.11
2.21 3 15 3.41 7.55
1.97 4 20 4.75 9.35
1.74 6 30 7.45 12.96
Example
Example of
a double
sampling
plan
We wish to construct a double sampling plan according to
p
1
= 0.01 = 0.05 p
2
= 0.05 = 0.10 and n
1
= n
2
The plans in the corresponding table are indexed on the ratio
R = p
2
/p
1
= 5
We find the row whose R is closet to 5. This is the 5th row (R =
4.65). This gives c
1
= 2 and c
2
= 4. The value of n
1
is determined
from either of the two columns labeled pn
1
.
The left holds constant at 0.05 (P = 0.95 = 1 - ) and the right
holds constant at 0.10. (P = 0.10). Then holding constant we
find pn
1
= 1.16 so n
1
= 1.16/p
1
= 116. And, holding constant we
find pn
1
= 5.39, so n
1
= 5.39/p
2
= 108. Thus the desired sampling
plan is
n
1
= 108 c
1
= 2 n
2
= 108 c
2
= 4
If we opt for n
2
= 2n
1
, and follow the same procedure using the
appropriate table, the plan is:
n
1
= 77 c
1
= 1 n
2
= 154 c
2
= 4
The first plan needs less samples if the number of defectives in
sample 1 is greater than 2, while the second plan needs less samples
if the number of defectives in sample 1 is less than 2.
ASN Curve for a Double Sampling Plan
Construction
of the ASN
curve
Since when using a double sampling plan the sample size depends on
whether or not a second sample is required, an important
consideration for this kind of sampling is the Average Sample
Number (ASN) curve. This curve plots the ASN versus p', the true
fraction defective in an incoming lot.
We will illustrate how to calculate the ASN curve with an example.
Consider a double-sampling plan n
1
= 50, c
1
= 2, n
2
= 100, c
2
= 6,
where n
1
is the sample size for plan 1, with accept number c
1
, and
n
2
, c
2
, are the sample size and accept number, respectively, for plan
2.
Let p' = .06. Then the probability of acceptance on the first sample,
which is the chance of getting two or less defectives, is .416 (using
binomial tables). The probability of rejection on the second sample,
which is the chance of getting more than six defectives, is (1-.971) =
.029. The probability of making a decision on the first sample is
.445, equal to the sum of .416 and .029. With complete inspection of
the second sample, the average size sample is equal to the size of the
first sample times the probability that there will be only one sample
plus the size of the combined samples times the probability that a
second sample will be necessary. For the sampling plan under
consideration, the ASN with complete inspection of the second
sample for a p' of .06 is
50(.445) + 150(.555) = 106
The general formula for an average sample number curve of a
double-sampling plan with complete inspection of the second sample
is
ASN = n
1
P
1
+ (n
1
+ n
2
)(1 - P
1
) = n
1
+ n
2
(1 - P
1
)
where P
1
is the probability of a decision on the first sample. The
graph below shows a plot of the ASN versus p'.
The ASN
curve for a
double
sampling
plan
6.2.5. What is Multiple Sampling?

6.2.5. What is Multiple Sampling?
Multiple
Sampling
is an
extension
of the
double
sampling
concept
Multiple sampling is an extension of double sampling. It
involves inspection of 1 to k successive samples as required to
reach an ultimate decision.
Mil-Std 105D suggests k = 7 is a good number. Multiple
sampling plans are usually presented in tabular form:
Procedure
for
multiple
sampling
The procedure commences with taking a random sample of
size n
1
from a large lot of size N and counting the number of
defectives, d
1
.
if d
1
a
1
the lot is accepted.
if d
1
r
1
the lot is rejected.
if a
1
< d
1
< r
1
, another sample is taken.
If subsequent samples are required, the first sample procedure
is repeated sample by sample. For each sample, the total
number of defectives found at any stage, say stage i, is
This is compared with the acceptance number a
i
and the
rejection number r
i
for that stage until a decision is made.
Sometimes acceptance is not allowed at the early stages of
multiple sampling; however, rejection can occur at any stage.
Efficiency
measured
by the
ASN
Efficiency for a multiple sampling scheme is measured by the
average sample number (ASN) required for a given Type I and
Type II set of errors. The number of samples needed when
following a multiple sampling scheme may vary from trial to
trial, and the ASN represents the average of what might
happen over many trials with a fixed incoming defect level.
6.2.6. What is a Sequential Sampling Plan?

Sequential
Sampling
Sequential sampling is different from single, double or
multiple sampling. Here one takes a sequence of samples from
a lot. How many total samples looked at is a function of the
results of the sampling process.
Item-by-
item and
group
sequential
sampling
The sequence can be one sample at a time, and then the
sampling process is usually called item-by-item sequential
sampling. One can also select sample sizes greater than one, in
which case the process is referred to as group sequential
sampling. Item-by-item is more popular so we concentrate on
it. The operation of such a plan is illustrated below:
Diagram of
item-by-
item
sampling
Description
of
sequentail
sampling
graph
The cumulative observed number of defectives is plotted on
the graph. For each point, the x-axis is the total number of
items thus far selected, and the y-axis is the total number of
observed defectives. If the plotted point falls within the parallel
lines the process continues by drawing another sample. As
soon as a point falls on or above the upper line, the lot is
rejected. And when a point falls on or below the lower line,
the lot is accepted. The process can theoretically last until the
lot is 100% inspected. However, as a rule of thumb,
sequential-sampling plans are truncated after the number
inspected reaches three times the number that would have been
inspected using a corresponding single sampling plan.
Equations
for the
limit lines
The equations for the two limit lines are functions of the
parameters p
1
, , p
2
, and .

where

Instead of using the graph to determine the fate of the lot, one
can resort to generating tables (with the help of a computer
program).
Example of
a
sequential
sampling
plan
As an example, let p
1
= .01, p
2
= .10, = .05, = .10. The
resulting equations are

Both acceptance numbers and rejection numbers must be
integers. The acceptance number is the next integer less than
or equal to x
a
and the rejection number is the next integer
greater than or equal to x
r
. Thus for n = 1, the acceptance
number = -1, which is impossible, and the rejection number =
2, which is also impossible. For n = 24, the acceptance number
is 0 and the rejection number = 3.
The results for n =1, 2, 3... 26 are tabulated below.
n
inspect
n
accept
n
reject
n
inspect
n
accept
n
reject
1 x x 14 x 2
2 x 2 15 x 2
3 x 2 16 x 3
4 x 2 17 x 3
5 x 2 18 x 3
6 x 2 19 x 3
7 x 2 20 x 3
8 x 2 21 x 3
9 x 2 22 x 3
10 x 2 23 x 3
11 x 2 24 0 3
12 x 2 25 0 3
13 x 2 26 0 3
So, for n = 24 the acceptance number is 0 and the rejection
number is 3. The "x" means that acceptance or rejection is not
possible.
Other sequential plans are given below.
n
inspect
n
accept
n
reject
49 1 3
58 1 4
74 2 4
83 2 5
100 3 5
109 3 6
The corresponding single sampling plan is (52,2) and double
sampling plan is (21,0), (21,1).
Efficiency
measured
by ASN
Efficiency for a sequential sampling scheme is measured by
the average sample number (ASN) required for a given Type I
and Type II set of errors. The number of samples needed when
following a sequential sampling scheme may vary from trial to
trial, and the ASN represents the average of what might happen
over many trials with a fixed incoming defect level. Good
software for designing sequential sampling schemes will
calculate the ASN curve as a function of the incoming defect
level.
6.2.7. What is Skip Lot Sampling?

Skip Lot
Sampling
Skip Lot sampling means that only a fraction of the submitted lots
are inspected. This mode of sampling is of the cost-saving variety in
terms of time and effort. However skip-lot sampling should only be
used when it has been demonstrated that the quality of the submitted
product is very good.
Implementation
of skip-lot
sampling plan
A skip-lot sampling plan is implemented as follows:
1. Design a single sampling plan by specifying the alpha and beta
risks and the consumer/producer's risks. This plan is called
"the reference sampling plan".
2. Start with normal lot-by-lot inspection, using the reference
plan.
3. When a pre-specified number, i, of consecutive lots are
accepted, switch to inspecting only a fraction f of the lots. The
selection of the members of that fraction is done at random.
4. When a lot is rejected return to normal inspection.
The f and i
parameters
The parameters f and i are essential to calculating the probability of
acceptance for a skip-lot sampling plan. In this scheme, i, called the
clearance number, is a positive integer and the sampling fraction f is
such that 0 < f < 1. Hence, when f = 1 there is no longer skip-lot
sampling. The calculation of the acceptance probability for the skip-
lot sampling plan is performed via the following formula
where P is the probability of accepting a lot with a given proportion
of incoming defectives p, from the OC curve of the single sampling
plan.
The following relationships hold:
for a given i, the smaller is f, the greater is P
a

for a given f, the smaller is i, the greater is P
a
Illustration of
a skip lot
sampling plan
An illustration of a a skip-lot sampling plan is given below.
ASN of skip-lot
sampling plan
An important property of skip-lot sampling plans is the average
sample number (ASN ). The ASN of a skip-lot sampling plan is
ASN
skip-lot
= (F)(ASN
reference
)
where F is defined by
Therefore, since 0 < F < 1, it follows that the ASN of skip-lot
sampling is smaller than the ASN of the reference sampling plan.
In summary, skip-lot sampling is preferred when the quality of the
submitted lots is excellent and the supplier can demonstrate a proven
track record.
6.3. Univariate and Multivariate Control Charts

Contents
of section
3
Control charts in this section are classified and described
according to three general types: variables, attributes and
multivariate.
1. What are Control Charts?
2. What are Variables Control Charts?
1. Shewhart X bar and R and S Control Charts
2. Individuals Control Charts
3. Cusum Control Charts
1. Cusum Average Run Length
4. EWMA Control Charts
3. What are Attributes Control Charts?
1. Counts Control Charts
2. Proportions Control Charts
4. What are Multivariate Control Charts?
1. Hotelling Control Charts
2. Principal Components Control Charts
3. Multivariate EWMA Charts

6.3.1. What are Control Charts?

Comparison of
univariate and
multivariate
control data
Control charts are used to routinely monitor quality.
Depending on the number of process characteristics to be
monitored, there are two basic types of control charts. The
first, referred to as a univariate control chart, is a graphical
display (chart) of one quality characteristic. The second,
referred to as a multivariate control chart, is a graphical
display of a statistic that summarizes or represents more than
one quality characteristic.
Characteristics
of control
charts
If a single quality characteristic has been measured or
computed from a sample, the control chart shows the value
of the quality characteristic versus the sample number or
versus time. In general, the chart contains a center line that
represents the mean value for the in-control process. Two
other horizontal lines, called the upper control limit (UCL)
and the lower control limit (LCL), are also shown on the
chart. These control limits are chosen so that almost all of the
data points will fall within these limits as long as the process
remains in-control. The figure below illustrates this.
Chart
demonstrating
basis of
control chart
Why control
charts "work"
The control limits as pictured in the graph might be .001
probability limits. If so, and if chance causes alone were
present, the probability of a point falling above the upper
limit would be one out of a thousand, and similarly, a point
falling below the lower limit would be one out of a thousand.
We would be searching for an assignable cause if a point
would fall outside these limits. Where we put these limits
will determine the risk of undertaking such a search when in
reality there is no assignable cause for variation.
Since two out of a thousand is a very small risk, the 0.001
limits may be said to give practical assurances that, if a point
falls outside these limits, the variation was caused be an
assignable cause. It must be noted that two out of one
thousand is a purely arbitrary number. There is no reason
why it could not have been set to one out a hundred or even
larger. The decision would depend on the amount of risk the
management of the quality control program is willing to take.
In general (in the world of quality control) it is customary to
use limits that approximate the 0.002 standard.
Letting X denote the value of a process characteristic, if the
system of chance causes generates a variation in X that
follows the normal distribution, the 0.001 probability limits
will be very close to the 3 limits. From normal tables we
glean that the 3 in one direction is 0.00135, or in both
directions 0.0027. For normal distributions, therefore, the 3
limits are the practical equivalent of 0.001 probability limits.
Plus or minus
"3 sigma"
limits are
typical
In the U.S., whether X is normally distributed or not, it is an
acceptable practice to base the control limits upon a multiple
of the standard deviation. Usually this multiple is 3 and thus
the limits are called 3-sigma limits. This term is used
whether the standard deviation is the universe or population
parameter, or some estimate thereof, or simply a "standard
value" for control chart purposes. It should be inferred from
the context what standard deviation is involved. (Note that in
the U.K., statisticians generally prefer to adhere to
probability limits.)
If the underlying distribution is skewed, say in the positive
direction, the 3-sigma limit will fall short of the upper 0.001
limit, while the lower 3-sigma limit will fall below the 0.001
limit. This situation means that the risk of looking for
assignable causes of positive variation when none exists will
be greater than one out of a thousand. But the risk of
searching for an assignable cause of negative variation, when
none exists, will be reduced. The net result, however, will be
an increase in the risk of a chance variation beyond the
control limits. How much this risk will be increased will
depend on the degree of skewness.
If variation in quality follows a Poisson distribution, for
example, for which np = .8, the risk of exceeding the upper
limit by chance would be raised by the use of 3-sigma limits
from 0.001 to 0.009 and the lower limit reduces from 0.001
to 0. For a Poisson distribution the mean and variance both
equal np. Hence the upper 3-sigma limit is 0.8 + 3 sqrt(.8) =
3.48 and the lower limit = 0 (here sqrt denotes "square root").
For np = .8 the probability of getting more than 3 successes =
0.009.
Strategies for
dealing with
out-of-control
findings
If a data point falls outside the control limits, we assume that
the process is probably out of control and that an
investigation is warranted to find and eliminate the cause or
causes.
Does this mean that when all points fall within the limits, the
process is in control? Not necessarily. If the plot looks non-
random, that is, if the points exhibit some form of systematic
behavior, there is still something wrong. For example, if the
first 25 of 30 points fall above the center line and the last 5
fall below the center line, we would wish to know why this is
so. Statistical methods to detect sequences or nonrandom
patterns can be applied to the interpretation of control charts.
To be sure, "in control" implies that all points are between
the control limits and they form a random pattern.
6.3.2. What are Variables Control Charts?

During the 1920's, Dr. Walter A. Shewhart proposed a general
model for control charts as follows:
Shewhart
Control
Charts for
variables
Let w be a sample statistic that measures some continuously
varying quality characteristic of interest (e.g., thickness), and
suppose that the mean of w is
w
, with a standard deviation of
w
. Then the center line, the UCL and the LCL are
UCL =
w
+ k
w

Center Line =
w

LCL =
w
- k
w
where k is the distance of the control limits from the center
line, expressed in terms of standard deviation units. When k is
set to 3, we speak of 3-sigma control charts.
Historically, k = 3 has become an accepted standard in
industry.
The centerline is the process mean, which in general is
unknown. We replace it with a target or the average of all the
data. The quantity that we plot is the sample average, . The
chart is called the chart.
We also have to deal with the fact that is, in general,
unknown. Here we replace
w
with a given standard value, or
we estimate it by a function of the average standard deviation.
This is obtained by averaging the individual standard
deviations that we calculated from each of m preliminary (or
present) samples, each of size n. This function will be
discussed shortly.
It is equally important to examine the standard deviations in
ascertaining whether the process is in control. There is,
unfortunately, a slight problem involved when we work with
the usual estimator of . The following discussion will
illustrate this.
Sample
Variance
If
2
is the unknown variance of a probability distribution,
then an unbiased estimator of
2
is the sample variance
However, s, the sample standard deviation is not an unbiased
estimator of . If the underlying distribution is normal, then s
actually estimates c
4
, where c
4
is a constant that depends on
the sample size n. This constant is tabulated in most text books
on statistical quality control and may be calculated using
C
4
factor
To compute this we need a non-integer factorial, which is
defined for n/2 as follows:
Fractional
Factorials
For example, let n = 7. Then n/2 = 7/2 = 3.5 and
With this definition the reader should have no problem
verifying that the c
4
factor for n = 10 is .9727.
Mean and
standard
deviation of
the
estimators
So the mean or expected value of the sample standard
deviation is c
4
.
The standard deviation of the sample standard deviation is
What are the differences between control limits and
specification limits ?
Control
limits vs.
specifications
Control Limits are used to determine if the process is in a state
of statistical control (i.e., is producing consistent output).
Specification Limits are used to determine if the product will
function in the intended fashion.
How many data points are needed to set up a control chart?
How many
samples are
needed?
Shewhart gave the following rule of thumb:
"It has also been observed that a person would
seldom if ever be justified in concluding that a
state of statistical control of a given repetitive
operation or production process has been reached
until he had obtained, under presumably the same
essential conditions, a sequence of not less than
twenty five samples of size four that are in
control."
It is important to note that control chart properties, such as
false alarm probabilities, are generally given under the
assumption that the parameters, such as and , are known.
When the control limits are not computed from a large amount
of data, the actual properties might be quite different from
what is assumed (see, e.g., Quesenberry, 1993).
When do we recalculate control limits?
When do we
recalculate
control
limits?
Since a control chart "compares" the current performance of
the process characteristic to the past performance of this
characteristic, changing the control limits frequently would
negate any usefulness.
So, only change your control limits if you have a valid,
compelling reason for doing so. Some examples of reasons:
When you have at least 30 more data points to add to the
chart and there have been no known changes to the
process
- you get a better estimate of the variability
If a major process change occurs and affects the way
your process runs.
If a known, preventable act changes the way the tool or
process would behave (power goes out, consumable is
corrupted or bad quality, etc.)
What are the WECO rules for signaling "Out of Control"?
General
rules for
detecting out
of control or
non-random
situaltions
WECO stands for Western Electric Company Rules

Any Point Above +3 Sigma
--------------------------------------------- +3 LIMIT
2 Out of the Last 3 Points Above +2 Sigma
--------------------------------------------- +2 LIMIT
4 Out of the Last 5 Points Above +1 Sigma
--------------------------------------------- +1 LIMIT
8 Consecutive Points on This Side of Control Line
=================================== CENTER
LINE
8 Consecutive Points on This Side of Control Line
--------------------------------------------- -1 LIMIT
4 Out of the Last 5 Points Below - 1 Sigma
---------------------------------------------- -2 LIMIT
2 Out of the Last 3 Points Below -2 Sigma
--------------------------------------------- -3 LIMIT
Any Point Below -3 Sigma
Trend
Rules:
6 in a row trending up or down. 14 in a row
alternating up and down
WECO rules
based on
probabilities
The WECO rules are based on probability. We know that, for a
normal distribution, the probability of encountering a point
outside 3 is 0.3%. This is a rare event. Therefore, if we
observe a point outside the control limits, we conclude the
process has shifted and is unstable. Similarly, we can identify
other events that are equally rare and use them as flags for
instability. The probability of observing two points out of three
in a row between 2 and 3 and the probability of observing
four points out of five in a row between 1 and 2 are also
about 0.3%.
WECO rules
increase
false alarms
Note: While the WECO rules increase a Shewhart chart's
sensitivity to trends or drifts in the mean, there is a severe
downside to adding the WECO rules to an ordinary Shewhart
control chart that the user should understand. When following
the standard Shewhart "out of control" rule (i.e., signal if and
only if you see a point beyond the plus or minus 3 sigma
control limits) you will have "false alarms" every 371 points on
the average (see the description of Average Run Length or
ARL on the next page). Adding the WECO rules increases the
frequency of false alarms to about once in every 91.75 points,
on the average (see Champ and Woodall, 1987). The user has
to decide whether this price is worth paying (some users add
the WECO rules, but take them "less seriously" in terms of the
effort put into troubleshooting activities when out of control
signals occur).
With this background, the next page will describe how to
construct Shewhart variables control charts.
6.3.2.1. Shewhart X-bar and R and S Control Charts

6.3.2.1. Shewhart X-bar and R and S Control
Charts
and S Charts
and S
Shewhart
Control
Charts
We begin with and s charts. We should use the s chart first
to determine if the distribution for the process characteristic is
stable.
Let us consider the case where we have to estimate by
analyzing past data. Suppose we have m preliminary samples
at our disposition, each of size n, and let s
i
be the standard
deviation of the ith sample. Then the average of the m
standard deviations is
Control
Limits for
and S
Control
Charts
We make use of the factor c
4
described on the previous page.
The statistic is an unbiased estimator of . Therefore, the
parameters of the S chart would be
Similarly, the parameters of the chart would be
, the "grand" mean is the average of all the observations.
It is often convenient to plot the and s charts on one page.
and R Control Charts
and R
control
charts
If the sample size is relatively small (say equal to or less than
10), we can use the range instead of the standard deviation of
a sample to construct control charts on and the range, R.
The range of a sample is simply the difference between the
largest and smallest observation.
There is a statistical relationship (Patnaik, 1946) between the
mean range for data from a normal distribution and , the
standard deviation of that distribution. This relationship
depends only on the sample size, n. The mean of R is d
2
,
where the value of d
2
is also a function of n. An estimator of
is therefore R /d
2
.
Armed with this background we can now develop the and R
control chart.
Let R
1
, R
2
, ..., R
k
, be the range of k samples. The average
range is
Then an estimate of can be computed as
control
charts
So, if we use (or a given target) as an estimator of and
/d
2
as an estimator of , then the parameters of the chart
are
The simplest way to describe the limits is to define the factor
and the construction of the becomes
The factor A
2
depends only on n, and is tabled below.
The R chart
R control
charts
This chart controls the process variability since the sample
range is related to the process standard deviation. The center
line of the R chart is the average range.
To compute the control limits we need an estimate of the true,
but unknown standard deviation W = R/ . This can be found
from the distribution of W = R/ (assuming that the items that
we measure follow a normal distribution). The standard
deviation of W is d
3
, and is a known function of the sample
size, n. It is tabulated in many textbooks on statistical quality
control.
Therefore since R = W , the standard deviation of R is
R
=
d
3
. But since the true is unknown, we may estimate
R
by
As a result, the parameters of the R chart with the customary
3-sigma control limits are
As was the case with the control chart parameters for the
subgroup averages, defining another set of factors will ease the
computations, namely:
D
3
= 1 - 3 d
3
/ d
2
and D
4
= 1 + 3 d
3
/ d
2
. These yield
The factors D
3
and D
4
depend only on n, and are tabled
below.

Factors for Calculating Limits for and R Charts
n
A
2
D
3
D
4
2 1.880 0 3.267
3 1.023 0 2.575
4 0.729 0 2.282
5 0.577 0 2.115
6 0.483 0 2.004
7 0.419 0.076 1.924
8 0.373 0.136 1.864
9 0.337 0.184 1.816
10 0.308 0.223 1.777
In general, the range approach is quite satisfactory for sample
sizes up to around 10. For larger sample sizes, using subgroup
standard deviations is preferable. For small sample sizes, the
relative efficiency of using the range approach as opposed to
using standard deviations is shown in the following table.
Efficiency
of R versus
S/c
4
n Relative
Efficiency
2 1.000
3 0.992
4 0.975
5 0.955
6 0.930
10 0.850
A typical sample size is 4 or 5, so not much is lost by using
the range for such sample sizes.
Time To Detection or Average Run Length (ARL)
Waiting
time to
signal "out
of control"
Two important questions when dealing with control charts are:
1. How often will there be false alarms where we look for
an assignable cause but nothing has changed?
2. How quickly will we detect certain kinds of systematic
changes, such as mean shifts?
The ARL tells us, for a given situation, how long on the
average we will plot successive control charts points before we
detect a point beyond the control limits.
For an chart, with no change in the process, we wait on the
average 1/p points before a false alarm takes place, with p
denoting the probability of an observation plotting outside the
control limits. For a normal distribution, p = .0027 and the
ARL is approximately 371.
A table comparing Shewhart chart ARL's to Cumulative
Sum (CUSUM) ARL's for various mean shifts is given later in
this section.
There is also (currently) a web site developed by Galit
Shmueli that will do ARL calculations interactively with the
user, for Shewhart charts with or without additional (Western
Electric) rules added.
6.3.2.2. Individuals Control Charts

Samples are Individual Measurements
Moving
range used
to derive
upper and
lower limits
Control charts for individual measurements, e.g., the sample size =
1, use the moving range of two successive observations to measure
the process variability.
The moving range is defined as
which is the absolute value of the first difference (e.g., the
difference between two consecutive data points) of the data.
Analogous to the Shewhart control chart, one can plot both the data
(which are the individuals) and the moving range.
Individuals
control
limits for an
observation
For the control chart for individual measurements, the lines plotted
are:
where is the average of all the individuals and is the
average of all the moving ranges of two observations. Keep in mind
that either or both averages may be replaced by a standard or
target, if available. (Note that 1.128 is the value of d
2
for n = 2).
Example of
moving
range
The following example illustrates the control chart for individual
observations. A new process was studied in order to monitor flow
rate. The first 10 batches resulted in
Batch
Number
Flowrate
x
Moving
Range
MR
1 49.6
2 47.6 2.0
3 49.9 2.3
4 51.3 1.4
5 47.8 3.5
6 51.2 3.4
7 52.6 1.4
8 52.4 0.2
9 53.6 1.2
10 52.1 1.5

= 50.81 =
1.8778
Limits for
the moving
range chart
This yields the parameters below.
Example of
individuals
chart
The control chart is given below
The process is in control, since none of the plotted points fall
outside either the UCL or LCL.
Alternative
for
constructing
individuals
control
Note: Another way to construct the individuals chart is by using
the standard deviation. Then we can obtain the chart from
chart
It is preferable to have the limits computed this way for the start of
Phase 2.
6.3.2.3. Cusum Control Charts

6.3.2.3. CUSUM Control Charts
CUSUM is
an efficient
alternative
to
Shewhart
procedures
CUSUM charts, while not as intuitive and simple to operate as
Shewhart charts, have been shown to be more efficient in
detecting small shifts in the mean of a process. In particular,
analyzing ARL's for CUSUM control charts shows that they
are better than Shewhart control charts when it is desired to
detect shifts in the mean that are 2 sigma or less.
CUSUM works as follows: Let us collect m samples, each of
size n, and compute the mean of each sample. Then the
cumulative sum (CUSUM) control chart is formed by plotting
one of the following quantities:
Definition
of
cumulative
sum against the sample number m, where is the estimate of the
in-control mean and is the known (or estimated) standard
deviation of the sample means. The choice of which of these
two quantities is plotted is usually determined by the statistical
software package. In either case, as long as the process
remains in control centered at , the CUSUM plot will show
variation in a random pattern centered about zero. If the
process mean shifts upward, the charted CUSUM points will
eventually drift upwards, and vice versa if the process mean
decreases.
V-Mask
used to
determine
if process
is out of
control
A visual procedure proposed by Barnard in 1959, known as
the V-Mask, is sometimes used to determine whether a process
is out of control. More often, the tabular form of the V-Mask
is preferred. The tabular form is illustrated later in this section.
A V-Mask is an overlay shape in the form of a V on its side
that is superimposed on the graph of the cumulative sums. The
origin point of the V-Mask (see diagram below) is placed on
top of the latest cumulative sum point and past points are
examined to see if any fall above or below the sides of the V.
As long as all the previous points lie between the sides of the
V, the process is in control. Otherwise (even if one point lies
outside) the process is suspected of being out of control.
Sample V-
Mask
demonstrating
an out of
control
process
Interpretation
of the V-
Mask on the
plot
In the diagram above, the V-Mask shows an out of control
situation because of the point that lies above the upper arm.
By sliding the V-Mask backwards so that the origin point
covers other cumulative sum data points, we can determine
the first point that signaled an out-of-control situation. This
is useful for diagnosing what might have caused the process
to go out of control.
From the diagram it is clear that the behavior of the V-Mask
is determined by the distance k (which is the slope of the
lower arm) and the rise distance h. These are the design
parameters of the V-Mask. Note that we could also specify
d and the vertex angle (or, as is more common in the
literature, = 1/2 of the vertex angle) as the design
parameters, and we would end up with the same V-Mask.
In practice, designing and manually constructing a V-Mask
is a complicated procedure. A CUSUM spreadsheet style
procedure shown below is more practical, unless you have
statistical software that automates the V-Mask
methodology. Before describing the spreadsheet approach,
we will look briefly at an example of a V-Mask in graph
form.
V-Mask
Example
An example will be used to illustrate the construction and
application of a V-Mask. The 20 data points
324.925, 324.675, 324.725, 324.350, 325.350, 325.225,
324.125, 324.525, 325.225, 324.600, 324.625, 325.150,
328.325, 327.250, 327.825, 328.500, 326.675, 327.775,
326.875, 328.350
are each the average of samples of size 4 taken from a
process that has an estimated mean of 325. Based on
process data, the process standard deviation is 1.27 and
therefore the sample means have a standard deviation of
1.27/(4
1/2
) = 0.635.
We can design a V-Mask using h and k or we can use an
alpha and beta design approach. For the latter approach we
must specify
: the probability of a false alarm, i.e., concluding
that a shift in the process has occurred, while in fact it
did not,
: the the probability of not detecting that a shift in
the process mean has, in fact, occurred, and
(delta): the amount of shift in the process mean that
we wish to detect, expressed as a multiple of the
standard deviation of the data points (which are the
sample means).
Note: Technically, and are calculated in terms of one
sequential trial where we monitor S
m
until we have either
an out-of-control signal or S
m
returns to the starting point
(and the monitoring begins, in effect, all over again).
The values of h and k are related to , , and based on the
following equations (adapted from Montgomery, 2000).
In our example we choose = 0.0027 (equivalent to the
plus or minus 3 sigma criteria used in a standard Shewhart
chart), and = 0.01. Finally, we decide we want to quickly
detect a shift as large as 1 sigma, which sets = 1.
CUSUM
Chart with
V-Mask
When the V-Mask is placed over the last data point, the
mask clearly indicates an out of control situation.
CUSUM
chart after
moving V-
Mask to first
out of control
point
We next move the V-Mask and back to the first point that
indicated the process was out of control. This is point
number 14, as shown below.
Rule of
thumb for
choosing h
and k
Note: A general rule of thumb (Montgomery) if one chooses
to design with the h and k approach, instead of the and
method illustrated above, is to choose k to be half the shift
(0.5 in our example) and h to be around 4 or 5.
For more information on CUSUM chart design, see Woodall
and Adams (1993).
Tabular or Spreadsheet Form of the V-Mask
A
spreadsheet
approach to
CUSUM
monitoring
Most users of CUSUM procedures prefer tabular charts over
the V-Mask. The V-Mask is actually a carry-over of the pre-
computer era. The tabular method can be quickly
implemented by standard spreadsheet software.
To generate the tabular form we use the h and k parameters
expressed in the original data units. It is also possible to use
sigma units.
The following quantities are calculated:
S
hi
(i) = max(0, S
hi
(i-1) + x
i
- - k)
S
lo
(i) = max(0, S
lo
(i-1) + - k - x
i
) )
where S
hi
(0) and S
lo
(0) are 0. When either S
hi
(i) or S
lo
(i)
exceeds h, the process is out of control.
Example of
spreadsheet
calculations
We will construct a CUSUM tabular chart for the example
described above. For this example, the parameter are h =
4.1959 and k = 0.3175. Using these design values, the tabular
form of the example is
h k
325 4.1959 0.3175
Increase
in mean
Decrease
in mean

Group x x-
325
x-325-k S
hi
325-k-x S
lo
CUSUM
1 324.93 -
0.07
-0.39 0.00 -0.24 0.00 -0.007
2 324.68 -
0.32
-0.64 0.00 0.01 0.01 -0.40
3 324.73 -
0.27
-0.59 0.00 -0.04 0.00 -0.67
4 324.35 -
0.65
-0.97 0.00 0.33 0.33 -1.32
5 325.35 0.35 0.03 0.03 -0.67 0.00 -0.97
6 325.23 0.23 -0.09 0.00 -0.54 0.00 -0.75
7 324.13 -
0.88
-1.19 0.00 0.56 0.56 -1.62
8 324.53 -
0.48
-0.79 0.00 0.16 0.72 -2.10
9 325.23 0.23 -0.09 0.00 0.54 0.17 -1.87
10 324.60 -
0.40
-0.72 0.00 0.08 0.25 -2.27
11 324.63 -
0.38
-0.69 0.00 0.06 0.31 -2.65
12 325.15 0.15 -0.17 0.00 0.47 0.00 -2.50
13 328.33 3.32 3.01 3.01 -3.64 0.00 0.83
14 327.25 2.25 1.93 4.94* -0.57 0.00 3.08
15 327.83 2.82 2.51 7.45* -3.14 0.00 5.90
16 328.50 3.50 3.18 10.63* -3.82 0.00 9.40
17 326.68 1.68 1.36 11.99* -1.99 0.00 11.08
18 327.78 2.77 2.46 14.44* -3.09 0.00 13.85
19 326.88 1.88 1.56 16.00* -2.19 0.00 15.73
20 328.35 3.35 3.03 19.04* -3.67 0.00 19.08
* = out of control signal
6.3.2.3.1. Cusum Average Run Length

The Average Run Length of Cumulative Sum Control
Charts
The ARL of
CUSUM
The operation of obtaining samples to use with a cumulative sum
(CUSUM) control chart consists of taking samples of size n and
plotting the cumulative sums
versus the sample number r, where is the sample mean and k is a
reference value.
In practice, k might be set equal to ( +
1
)/2, where is the
estimated in-control mean, which is sometimes known as the
acceptable quality level, and
1
is referred to as the rejectable
quality level.
If the distance between a plotted point and the lowest previous point
is equal to or greater than h, one concludes that the process mean has
shifted (increased).
h is decision
limit
Hence, h is referred to as the decision limit. Thus the sample size n,
reference value k, and decision limit h are the parameters required
for operating a one-sided CUSUM chart. If one has to control both
positive and negative deviations, as is usually the case, two one-
sided charts are used, with respective values k
1,
k
2
, (k
1
> k
2
) and
respective decision limits h and -h.
Standardizing
shift in mean
and decision
limit
The shift in the mean can be expressed as - k. If we are dealing
with normally distributed measurements, we can standardize this
shift by
Similarly, the decision limit can be standardized by
Determination
of the ARL,
given h and k
The average run length (ARL) at a given quality level is the average
number of samples (subgroups) taken before an action signal is
given. The standardized parameters k
s
and h
s
together with the
sample size n are usually selected to yield approximate ARL's L
0
and
L
1
at acceptable and rejectable quality levels
0
and
1
respectively.
We would like to see a high ARL, L
0
, when the process is on target,
(i.e. in control), and a low ARL, L
1
, when the process mean shifts to
an unsatisfactory level.
In order to determine the parameters of a CUSUM chart, the
acceptable and rejectable quality levels along with the desired
respective ARL ' s are usually specified. The design parameters can
then be obtained by a number of ways. Unfortunately, the
calculations of the ARL for CUSUM charts are quite involved.
There are several nomographs available from different sources that
can be utilized to find the ARL's when the standardized h and k are
given. Some of the nomographs solve the unpleasant integral
equations that form the basis of the exact solutions, using an
approximation of Systems of Linear Algebraic Equations (SLAE).
This Handbook used a computer program that furnished the required
ARL's given the standardized h and k. An example is given below:
Example of
finding ARL's
given the
standardized
h and k
mean shift Shewart
(k = .5) 4 5
0 336 930 371.00
.25 74.2 140 281.14
.5 26.6 30.0 155.22
.75 13.3 17.0 81.22
1.00 8.38 10.4 44.0
1.50 4.75 5.75 14.97
2.00 3.34 4.01 6.30
2.50 2.62 3.11 3.24
3.00 2.19 2.57 2.00
4.00 1.71 2.01 1.19
Using the
table
If k = .5, then the shift of the mean (in multiples of the standard
deviation of the mean) is obtained by adding .5 to the first column.
For example to detect a mean shift of 1 sigma at h = 4, the ARL =
8.38. (at first column entry of .5).
The last column of the table contains the ARL's for a Shewhart
control chart at selected mean shifts. The ARL for Shewhart = 1/p,
where p is the probability for a point to fall outside established
control limits. Thus, for 3-sigma control limits and assuming
normality, the probability to exceed the upper control limit = .00135
and to fall below the lower control limit is also .00135 and their sum
= .0027. (These numbers come from standard normal distribution
tables or computer programs, setting z = 3). Then the ARL = 1/.0027
= 370.37. This says that when a process is in control one expects an
out-of-control signal (false alarm) each 371 runs.
ARL if a 1
sigma shift
has occurred
When the means shifts up by 1 sigma, then the distance between the
upper control limit and the shifted mean is 2 sigma (instead of 3 ).
Entering normal distribution tables with z = 2 yields a probability of
p = .02275 to exceed this value. The distance between the shifted
mean and the lower limit is now 4 sigma and the probability of <
-4 is only .000032 and can be ignored. The ARL is 1 / .02275 =
43.96 .
Shewhart is
better for
detecting
large shifts,
CUSUM is
faster for
small shifts
The conclusion can be drawn that the Shewhart chart is superior for
detecting large shifts and the CUSUM scheme is faster for small
shifts. The break-even point is a function of h, as the table shows.
6.3.2.4. EWMA Control Charts

EWMA
statistic
The Exponentially Weighted Moving Average (EWMA) is a
statistic for monitoring the process that averages the data in a
way that gives less and less weight to data as they are further
removed in time.
Comparison
of Shewhart
control
chart and
EWMA
control
chart
techniques
For the Shewhart chart control technique, the decision
regarding the state of control of the process at any time, t,
depends solely on the most recent measurement from the
process and, of course, the degree of 'trueness' of the
estimates of the control limits from historical data. For the
EWMA control technique, the decision depends on the
EWMA statistic, which is an exponentially weighted average
of all prior data, including the most recent measurement.
By the choice of weighting factor, , the EWMA control
procedure can be made sensitive to a small or gradual drift in
the process, whereas the Shewhart control procedure can only
react when the last data point is outside a control limit.
Definition
of EWMA
The statistic that is calculated is:
EWMA
t
= Y
t
+ ( 1- ) EWMA
t-1
for t = 1, 2, ..., n.
where
EWMA
0
is the mean of historical data (target)
Y
t
is the observation at time t
n is the number of observations to be monitored
including EWMA
0
0 < 1 is a constant that determines the depth of
memory of the EWMA.
The equation is due to Roberts (1959).
Choice of
weighting
factor
The parameter determines the rate at which 'older' data
enter into the calculation of the EWMA statistic. A value of
= 1 implies that only the most recent measurement influences
the EWMA (degrades to Shewhart chart). Thus, a large value
of = 1 gives more weight to recent data and less weight to
older data; a small value of gives more weight to older
data. The value of is usually set between 0.2 and 0.3
(Hunter) although this choice is somewhat arbitrary. Lucas
and Saccucci (1990) give tables that help the user select .
Variance of
EWMA
statistic
The estimated variance of the EWMA statistic is
approximately
s
2
ewma
= ( /(2- )) s
2
when t is not small, where s is the standard deviation
calculated from the historical data.
Definition
of control
limits for
EWMA
The center line for the control chart is the target value or
EWMA
0
. The control limits are:
UCL = EWMA
0
+ ks
ewma

LCL = EWMA
0
- ks
ewma
where the factor k is either set equal 3 or chosen using the
Lucas and Saccucci (1990) tables. The data are assumed to
be independent and these tables also assume a normal
population.
As with all control procedures, the EWMA procedure
depends on a database of measurements that are truly
representative of the process. Once the mean value and
standard deviation have been calculated from this database,
the process can enter the monitoring stage, provided the
process was in control when the data were collected. If not,
then the usual Phase 1 work would have to be completed
first.
Example of
calculation
of
parameters
for an
EWMA
control
chart
To illustrate the construction of an EWMA control chart,
consider a process with the following parameters calculated
from historical data:
EWMA
0
= 50
s = 2.0539
with chosen to be 0.3 so that / (2- ) = .3 / 1.7 = 0.1765
and the square root = 0.4201. The control limits are given by
UCL = 50 + 3 (0.4201)(2.0539) = 52.5884
LCL = 50 - 3 (0.4201) (2.0539) = 47.4115
Sample
data
Consider the following data consisting of 20 points where 1 -
10 are on the top row from left to right and 11-20 are on the
bottom row from left to right:
52.0 47.0 53.0 49.3 50.1 47.0
51.0 50.1 51.2 50.5 49.6 47.6
49.9 51.3 47.8 51.2 52.6 52.4
53.6 52.1
EWMA
statistics for
sample data
These data represent control measurements from the process
which is to be monitored using the EWMA control chart
technique. The corresponding EWMA statistics that are
computed from this data set are:
50.00 50.60 49.52 50.56 50.18
50.16 49.12 49.75 49.85 50.26
50.33 50.11 49.36 49.52 50.05
49.34 49.92 50.73 51.23 51.94
Sample
EWMA
plot
The control chart is given below.
Interpretation
of EWMA
control chart
The red dots are the raw data; the jagged line is the EWMA
statistic over time. The chart tells us that the process is in
control because all EWMA
t
lie between the control limits.
However, there seems to be a trend upwards for the last 5
periods.
6.3.3. What are Attributes Control Charts?

Attributes
data arise
when
classifying
or counting
observations
The Shewhart control chart plots quality characteristics that
can be measured and expressed numerically. We measure
weight, height, position, thickness, etc. If we cannot
represent a particular quality characteristic numerically, or if
it is impractical to do so, we then often resort to using a
quality characteristic to sort or classify an item that is
inspected into one of two "buckets".
An example of a common quality characteristic classification
would be designating units as "conforming units" or
"nonconforming units". Another quality characteristic criteria
would be sorting units into "non defective" and "defective"
categories. Quality characteristics of that type are called
attributes.
Note that there is a difference between "nonconforming to an
engineering specification" and "defective" -- a
nonconforming unit may function just fine and be, in fact,
not defective at all, while a part can be "in spec" and not
fucntion as desired (i.e., be defective).
Examples of quality characteristics that are attributes are the
number of failures in a production run, the proportion of
malfunctioning wafers in a lot, the number of people eating
in the cafeteria on a given day, etc.
Types of
attribute
control
charts
Control charts dealing with the number of defects or
nonconformities are called c charts (for count).
Control charts dealing with the proportion or fraction of
defective product are called p charts (for proportion).
There is another chart which handles defects per unit, called
the u chart (for unit). This applies when we wish to work
with the average number of nonconformities per unit of
product.
For additional references, see Woodall (1997) which reviews
papers showing examples of attribute control charting,
including examples from semiconductor manufacturing such
as those examining the spatial depencence of defects.
6.3.3.1. Counts Control Charts

Defective
items vs
individual
defects
The literature differentiates between defect and defective,
which is the same as differentiating between
nonconformity and nonconforming units. This may sound
like splitting hairs, but in the interest of clarity let's try to
unravel this man-made mystery.
Consider a wafer with a number of chips on it. The wafer
is referred to as an "item of a product". The chip may be
referred to as "a specific point". There exist certain
specifications for the wafers. When a particular wafer (e.g.,
the item of the product) does not meet at least one of the
specifications, it is classified as a nonconforming item.
Furthermore, each chip, (e.g., the specific point) at which a
specification is not met becomes a defect or
nonconformity.
So, a nonconforming or defective item contains at least
one defect or nonconformity. It should be pointed out that
a wafer can contain several defects but still be classified as
conforming. For example, the defects may be located at
noncritical positions on the wafer. If, on the other hand, the
number of the so-called "unimportant" defects becomes
alarmingly large, an investigation of the production of
these wafers is warranted.
Control charts involving counts can be either for the total
number of nonconformities (defects) for the sample of
inspected units, or for the average number of defects per
inspection unit.
Poisson
approximation
for numbers
or counts of
defects
Let us consider an assembled product such as a
microcomputer. The opportunity for the occurrence of any
given defect may be quite large. However, the probability
of occurrence of a defect in any one arbitrarily chosen spot
is likely to be very small. In such a case, the incidence of
defects might be modeled by a Poisson distribution.
Actually, the Poisson distribution is an approximation of
the binomial distribution and applies well in this capacity
according to the following rule of thumb:
The sample size n should be equal to or larger
than 20 and the probability of a single success,
p, should be smaller than or equal to .05. If n
100, the approximation is excellent if np is
also 10.
Illustrate
Poisson
approximation
to binomial
To illustrate the use of the Poisson distribution as an
approximation of a binomial distribution, consider the
following comparison: Let p, the probability of a single
success in n = 200 trials, be .025.
Find the probability of exactly 3 successes. If we assume
that p remains constant then the solution follows the
binomial distribution rules, that is:
By the Poisson approximation we have
and
The inspection
unit
Before the control chart parameters are defined there is one
more definition: the inspection unit. We shall count the
number of defects that occur in a so-called inspection unit.
More often than not, an inspection unit is a single unit or
item of product; for example, a wafer. However,
sometimes the inspection unit could consist of five wafers,
or ten wafers and so on. The size of the inspection units
may depend on the recording facility, measuring
equipment, operators, etc.
Suppose that defects occur in a given inspection unit
according to the Poisson distribution, with parameter c
(often denoted by np or the Greek letter ). In other words
Control charts
for counts,
using the
Poisson
distribution
where x is the number of defects and c > 0 is the parameter
of the Poisson distribution. It is known that both the mean
and the variance of this distribution are equal to c. Then
the k-sigma control chart is
If the LCL comes out negative, then there is no lower
control limit. This control scheme assumes that a standard
value for c is available. If this is not the case then c may
be estimated as the average of the number of defects in a
preliminary sample of inspection units, call it . Usually k
is set to 3 by many practioners.
Control chart
example using
counts
An example may help to illustrate the construction of
control limits for counts data. We are inspecting 25
successive wafers, each containing 100 chips. Here the
wafer is the inspection unit. The observed number of
defects are
Wafer Number Wafer Number
Number of Defects Number of Defects
1 16 14 16
2 14 15 15
3 28 16 13
4 16 17 14
5 12 18 16
6 20 19 11
7 10 20 20
8 12 21 11
9 10 22 19
10 17 23 16
11 19 24 31
12 17 25 13
13 14
From this table we have
Sample
counts
control
Control Chart for Counts
chart
Transforming Poisson Data
Normal
approximation
to Poisson is
adequate
when the
mean of the
Poisson is at
least 5
We have seen that the 3-sigma limits for a c chart, where c
represents the number of nonconformities, are given by
where it is assumed that the normal approximation to the
Poisson distribution holds, hence the symmetry of the
control limits. It is shown in the literature that the normal
approximation to the Poisson is adequate when the mean of
the Poisson is at least 5. When applied to the c chart this
implies that the mean of the defects should be at least 5.
This requirement will often be met in practice, but still,
when the mean is smaller than 9 (solving the above
equation) there will be no lower control limit.
Let the mean be 10. Then the lower control limit = 0.513.
However, P(c = 0) = .000045, using the Poisson formula.
This is only 1/30 of the assumed area of .00135. So one
has to raise the lower limit so as to get as close as possible
to .00135. From Poisson tables or computer software we
find that P(1) = .0005 and P(2) = .0027, so the lower limit
should actually be 2 or 3.
Transforming
count data
into
approximately
normal data
To avoid this type of problem, we may resort to a
transformation that makes the transformed data match the
normal distribution better. One such transformation
described by Ryan (2000) is
which is, for a large sample, approximately normally
distributed with mean = 2 and variace = 1, where is
the mean of the Poisson distribution.
Similar transformations have been proposed by Anscombe
(1948) and Freeman and Tukey (1950). When applied to a
c chart these are
The repspective control limits are
While using transformations may result in meaningful
control limits, one has to bear in mind that the user is now
working with data on a different scale than the original
measurements. There is another way to remedy the
problem of symmetric limits applied to non symmetric
cases, and that is to use probability limits. These can be
obtained from tables given by Molina (1973). This allows
the user to work with data on the original scale, but they
require special tables to obtain the limits. Of course,
software might be used instead.
Warning for
highly skewed
distributions
Note: In general, it is not a good idea to use 3-sigma limits
for distributions that are highly skewed (see Ryan and
Schwertman (1997) for more about the possibly extreme
consequences of doing this).
6.3.3.2. Proportions Control Charts

p is the
fraction
defective in
a lot or
population
The proportion or fraction nonconforming (defective) in a
population is defined as the ratio of the number of
nonconforming items in the population to the total number of
items in that population. The item under consideration may
have one or more quality characteristics that are inspected
simultaneously. If at least one of the characteristics does not
conform to standard, the item is classified as nonconforming.
The fraction or proportion can be expressed as a decimal, or,
when multiplied by 100, as a percent. The underlying
statistical principles for a control chart for proportion
nonconforming are based on the binomial distribution.
Let us suppose that the production process operates in a stable
manner, such that the probability that a given unit will not
conform to specifications is p. Furthermore, we assume that
successive units produced are independent. Under these
conditions, each unit that is produced is a realization of a
Bernoulli random variable with parameter p. If a random
sample of n units of product is selected and if D is the number
of units that are nonconforming, the D follows a binomial
distribution with parameters n and p
The
binomial
distribution
model for
number of
defectives
in a sample
The mean of D is np and the variance is np(1-p). The sample
proportion nonconforming is the ratio of the number of
nonconforming units in the sample, D, to the sample size n,
The mean and variance of this estimator are
and
This background is sufficient to develop the control chart for
proportion or fraction nonconforming. The chart is called the
p-chart.
p control
charts for
lot
proportion
defective
If the true fraction conforming p is known (or a standard
value is given), then the center line and control limits of the
fraction nonconforming control chart is
When the process fraction (proportion) p is not known, it
must be estimated from the available data. This is
accomplished by selecting m preliminary samples, each of
size n. If there are D
i
defectives in sample i, the fraction
nonconforming in sample i is
and the average of these individuals sample fractions is
The is used instead of p in the control chart setup.
Example of
a p-chart
A numerical example will now be given to illustrate the above
mentioned principles. The location of chips on a wafer is
measured on 30 wafers.
On each wafer 50 chips are measured and a defective is
defined whenever a misregistration, in terms of horizontal
and/or vertical distances from the center, is recorded. The
results are
Sample Fraction Sample Fraction Sample Fraction
Number Defectives Number Defectives Number Defectives
1 .24 11 .10 21 .40
2 .30 12 .12 22 .36
3 .16 13 .34 23 .48
4 .20 14 .24 24 .30
5 .08 15 .44 25 .18
6 .14 16 .16 26 .24
7 .32 17 .20 27 .14
8 .18 18 .10 28 .26
9 .28 19 .26 29 .18
10 .20 20 .22 30 .12
Sample
proportions
control
chart
The corresponding control chart is given below:
6.3.4. What are Multivariate Control Charts?

Multivariate
control
charts and
Hotelling's
T
2
statistic
It is a fact of life that most data are naturally multivariate.
Hotelling in 1947 introduced a statistic which uniquely lends
itself to plotting multivariate observations. This statistic,
appropriately named Hotelling's T
2
, is a scalar that combines
information from the dispersion and mean of several variables.
Due to the fact that computations are laborious and fairly
complex and require some knowledge of matrix algebra,
acceptance of multivariate control charts by industry was slow
and hesitant.
Multivariate
control
charts now
more
accessible
Nowadays, modern computers in general and the PC in
particular have made complex calculations accessible and
during the last decade, multivariate control charts were given
more attention. In fact, the multivariate charts which display the
Hotelling T
2
statistic became so popular that they sometimes
are called Shewhart charts as well (e.g., Crosier, 1988),
although Shewhart had nothing to do with them.
Hotelling
charts for
both means
and
dispersion
As in the univariate case, when data are grouped, the T
2
chart
can be paired with a chart that displays a measure of variability
within the subgroups for all the analyzed characteristics. The
combined T
2
and (dispersion) charts are thus a multivariate
counterpart of the univariate and S (or and R) charts.
Hotelling
mean and
dispersion
control
charts
An example of a Hotelling T
2
and pair of charts is given
below:
Interpretation
of sample
Hotelling
control
charts
Each chart represents 14 consecutive measurements on the
means of four variables. The T
2
chart for means indicates an
out-of-control state for groups 1,2 and 9-11. The T
2
d
chart for
dispersions indicate that groups 10, 13 and 14 are also out of
control. The interpretation is that the multivariate system is
suspect. To find an assignable cause, one has to resort to the
individual univariate control charts or some other univariate
procedure that should accompany this multivariate chart.
Additional
discussion
For more details and examples see the next page and also
Tutorials, section 5, subsections 4.3, 4.3.1 and 4.3.2. An
introduction to Elements of multivariate analysis is also given
in the Tutorials.
6.3.4.1. Hotelling Control Charts

Definition
of
Hotelling's
T
2
"distance"
statistic
The Hotelling T
2
distance is a measure that accounts for the
covariance structure of a multivariate normal distribution. It
was proposed by Harold Hotelling in 1947 and is called
Hotelling T
2
. It may be thought of as the multivariate
counterpart of the Student's-t statistic.
The T
2
distance is a constant multiplied by a quadratic form.
This quadratic form is obtained by multiplying the following
three quantities:
1. The vector of deviations between the
observations and the mean m, which is
expressed by (X-m)',
2. The inverse of the covariance matrix, S
-1
,
3. The vector of deviations, (X-m).
It should be mentioned that for independent variables, the
covariance matrix is a diagonal matrix and T
2
becomes
proportional to the sum of squared standardized variables.
In general, the higher the T
2
value, the more distant is the
observation from the mean. The formula for computing the T
2
is:
The constant c is the sample size from which the covariance
matrix was estimated.
T
2
readily
graphable
The T
2
distances lend themselves readily to graphical
displays and as a result the T
2
-chart is the most popular
among the multivariate control charts.
Estimation of the Mean and Covariance Matrix
Mean and
Covariance
matrices
Let X
1
,...X
n
be n p-dimensional vectors of observations that
are sampled independently from N
p
(m, ) with p < n-1, with
the covariance matrix of X. The observed mean vector
and the sample dispersion matrix
are the unbiased estimators of m and , respectively.
Additional
discussion
See Tutorials (section 5), subsections 4.3, 4.3.1 and 4.3.2 for
more details and examples. An introduction to Elements of
multivariate analysis is also given in the Tutorials.
6.3.4.2. Principal Components Control Charts

Problems
with T
2
charts
Although the T
2
chart is the most popular, easiest to use and
interpret method for handling multivariate process data, and
is beginning to be widely accepted by quality engineers and
operators, it is not a panacea. First, unlike the univariate case,
the scale of the values displayed on the chart is not related to
the scales of any of the monitored variables. Secondly, when
the T
2
statistic exceeds the upper control limit (UCL), the
user does not know which particular variable(s) caused the
out-of-control signal.
Run
univariate
charts
along with
the
multivariate
ones
With respect to scaling, we strongly advise to run individual
univariate charts in tandem with the multivariate chart. This
will also help in honing in on the culprit(s) that might have
caused the signal. However, individual univariate charts
cannot explain situations that are a result of some problems
in the covariance or correlation between the variables. This is
why a dispersion chart must also be used.
Another
way to
monitor
multivariate
data:
Principal
Components
control
charts
Another way to analyze the data is to use principal
components. For each multivariate measurement (or
observation), the principal components are linear
combinations of the standardized p variables (to standardize
subtract their respective targets and divide by their standard
deviations). The principal components have two important
advantages:
1. the new variables are uncorrelated (or almost)
2. very often, a few (sometimes 1 or 2) principal
components may capture most of the variability in the
data so that we do not have to use all of the p principal
components for control.
Eigenvalues Unfortunately, there is one big disadvantage: The identity of
the original variables is lost! However, in some cases the
specific linear combinations corresponding to the principal
components with the largest eigenvalues may yield
meaningful measurement units. What is being used in control
charts are the principal factors.
A principal factor is the principal component divided by the
square root of its eigenvalue.
Additional
discussion
More details and examples are given in the Tutorials (section
5).
6.3.4.3. Multivariate EWMA Charts

Multivariate EWMA Control Chart
Univariate
EWMA
model
The model for a univariate EWMA chart is given by:
where Z
i
is the ith EWMA, X
i
is the the ith observation, Z
0
is
the average from the historical data, and 0 < 1.
Multivariate
EWMA
model
In the multivariate case, one can extend this formula to
where Z
i
is the ith EWMA vector, X
i
is the the ith
observation vector i = 1, 2, ..., n, Z
0
is the vector of variable
values from the historical data, is the diag(
1
,
2
, ... ,
p
)
which is a diagonal matrix with
1
,
2
, ... ,
p
on the main
diagonal, and p is the number of variables; that is the number
of elements in each vector.
Illustration of
multivariate
EWMA
The following illustration may clarify this. There are p
variables and each variable contains n observations. The
input data matrix looks like:
The quantity to be plotted on the control chart is
Simplification It has been shown (Lowry et al., 1992) that the (k,l)th
element of the covariance matrix of the ith EWMA, , is
where is the (k,l)th element of , the covariance matrix
of the X's.
If
1
=
2
= ... =
p
= , then the above expression simplifies
to
where is the covariance matrix of the input data.
Further
simplification
There is a further simplification. When i becomes large, the
covariance matrix may be expressed as:
The question is "What is large?". When we examine the
formula with the 2i in it, we observe that when 2i becomes
sufficiently large such that (1 - )
2i
becomes almost zero,
then we can use the simplified formula.
Table for
selected
values of
and i
The following table gives the values of (1- )
2i
for selected
values of and i.
2i
1 - 4 6 8 10 12 20 30 40 50
.9 .656 .531 .430 .349 .282 .122 .042 .015 .005
.8 .410 .262 .168 .107 .069 .012 .001 .000 .000
.7 .240 .118 .058 .028 .014 .001 .000 .000 .000
.6 .130 .047 .017 .006 .002 .000 .000 .000 .000
.5 .063 .016 .004 .001 .000 .000 .000 .000 .000
.4 .026 .004 .001 .000 .000 .000 .000 .000 .000
.3 .008 .001 .000 .000 .000 .000 .000 .000 .000
.2 .002 .000 .000 .000 .000 .000 .000 .000 .000
.1 .000 .000 .000 .000 .000 .000 .000 .000 .000
Simplified
formuala not
required
It should be pointed out that a well-meaning computer
program does not have to adhere to the simplified formula,
and potential inaccuracies for low values for and i can thus
be avoided.
MEWMA
computer
output for
the Lowry
data
Here is an example of the application of an MEWMA control
chart. To faciltate comparison with existing literature, we used
data from Lowry et al. The data were simulated from a
bivariate normal distribution with unit variances and a
correlation coefficient of 0.5. The value for = .10 and the
values for were obtained by the equation given above. The
covariance of the MEWMA vectors was obtained by using the
non-simplified equation. That means that for each MEWMA
control statistic, the computer computed a covariance matrix,
where i = 1, 2, ...10. The results of the computer routine are:
*****************************************************
* Multi-Variate EWMA Control Chart
*
*****************************************************
DATA SERIES MEWMA Vector MEWMA
1 2 1 2
STATISTIC
-1.190 0.590 -0.119 0.059 2.1886
0.120 0.900 -0.095 0.143 2.0697
-1.690 0.400 -0.255 0.169 4.8365
0.300 0.460 -0.199 0.198 3.4158
0.890 -0.750 -0.090 0.103 0.7089
0.820 0.980 0.001 0.191 0.9268
-0.300 2.280 -0.029 0.400 4.0018
0.630 1.750 0.037 0.535 6.1657
1.560 1.580 0.189 0.639 7.8554
1.460 3.050 0.316 0.880 14.4158
VEC XBAR MSE Lamda
1 .260 1.200 0.100
2 1.124 1.774 0.100
The UCL = 5.938 for = .05. Smaller choices of are also
used.
Sample
MEWMA
plot
The following is the plot of the above MEWMA.
6.4. Introduction to Time Series Analysis

Time
series
methods
take into
account
possible
internal
structure
in the data
Time series data often arise when monitoring industrial
processes or tracking corporate business metrics. The essential
difference between modeling data via time series methods or
using the process monitoring methods discussed earlier in this
chapter is the following:
Time series analysis accounts for the fact that
data points taken over time may have an internal
structure (such as autocorrelation, trend or
seasonal variation) that should be accounted for.
This section will give a brief overview of some of the more
widely used techniques in the rich and rapidly growing field
of time series modeling and analysis.
Contents
for this
section
Areas covered are:
1. Definitions, Applications and Techniques
2. What are Moving Average or Smoothing
Techniques?
1. Single Moving Average
2. Centered Moving Average
3. What is Exponential Smoothing?
1. Single Exponential Smoothing
2. Forecasting with Single Exponential
Smoothing
3. Double Exponential Smoothing
4. Forecasting with Double Exponential
Smoothing
5. Triple Exponential Smoothing
6. Example of Triple Exponential
Smoothing
7. Exponential Smoothing Summary
4. Univariate Time Series Models
1. Sample Data Sets
2. Stationarity
3. Seasonality
4. Common Approaches
5. Box-Jenkins Approach
6. Box-Jenkins Model Identification
7. Box-Jenkins Model Estimation
8. Box-Jenkins Model Validation
9. Example of Univariate Box-Jenkins
Analysis
10. Box-Jenkins Model Analysis on
Seasonal Data
5. Multivariate Time Series Models
1. Example of Multivariate Time Series
Analysis
6.4.1. Definitions, Applications and Techniques

Definition Definition of Time Series: An ordered sequence of values of a
variable at equally spaced time intervals.
Time
series
occur
frequently
when
looking at
industrial
data
Applications: The usage of time series models is twofold:
Obtain an understanding of the underlying forces and
structure that produced the observed data
Fit a model and proceed to forecasting, monitoring or
even feedback and feedforward control.
Time Series Analysis is used for many applications such as:
Economic Forecasting
Sales Forecasting
Budgetary Analysis
Stock Market Analysis
Yield Projections
Process and Quality Control
Inventory Studies
Workload Projections
Utility Studies
Census Analysis
and many, many more...
There are
many
methods
used to
model and
forecast
time series
Techniques: The fitting of time series models can be an
ambitious undertaking. There are many methods of model
fitting including the following:
Box-Jenkins ARIMA models
Box-Jenkins Multivariate Models
Holt-Winters Exponential Smoothing (single, double,
triple)
The user's application and preference will decide the selection
of the appropriate technique. It is beyond the realm and
intention of the authors of this handbook to cover all these
methods. The overview presented here will start by looking at
some basic smoothing techniques:
Averaging Methods
Exponential Smoothing Techniques.
Later in this section we will discuss the Box-Jenkins modeling
methods and Multivariate Time Series.
6.4.2. What are Moving Average or Smoothing Techniques?

6.4.2. What are Moving Average or Smoothing
Techniques?
Smoothing
data
removes
random
variation
and shows
trends and
cyclic
components
Inherent in the collection of data taken over time is some
form of random variation. There exist methods for reducing
of canceling the effect due to random variation. An often-
used technique in industry is "smoothing". This technique,
when properly applied, reveals more clearly the underlying
trend, seasonal and cyclic components.
There are two distinct groups of smoothing methods
Averaging Methods
Exponential Smoothing Methods
Taking
averages is
the simplest
way to
smooth data
We will first investigate some averaging methods, such as the
"simple" average of all past data.
A manager of a warehouse wants to know how much a
typical supplier delivers in 1000 dollar units. He/she takes a
sample of 12 suppliers, at random, obtaining the following
results:
Supplier Amount Supplier Amount
1 9 7 11
2 8 8 7
3 9 9 13
4 12 10 9
5 9 11 11
6 12 12 10
The computed mean or average of the data = 10. The
manager decides to use this as the estimate for expenditure of
a typical supplier.
Is this a good or bad estimate?
Mean
squared
error is a
way to
judge how
We shall compute the "mean squared error":
The "error" = true amount spent minus the estimated
amount.
The "error squared" is the error above, squared.
good a
model is
The "SSE" is the sum of the squared errors.
The "MSE" is the mean of the squared errors.
MSE results
for example
The results are:
Error and Squared Errors
The estimate = 10
Supplier $ Error
Error
Squared
1 9 -1 1
2 8 -2 4
3 9 -1 1
4 12 2 4
5 9 -1 1
6 12 2 4
7 11 1 1
8 7 -3 9
9 13 3 9
10 9 -1 1
11 11 1 1
12 10 0 0
The SSE = 36 and the MSE = 36/12 = 3.
Table of
MSE results
for example
using
different
estimates
So how good was the estimator for the amount spent for each
supplier? Let us compare the estimate (10) with the
following estimates: 7, 9, and 12. That is, we estimate that
each supplier will spend $7, or $9 or $12.
Performing the same calculations we arrive at:
Estimator 7 9 10 12
SSE 144 48 36 84
MSE 12 4 3 7
The estimator with the smallest MSE is the best. It can be
shown mathematically that the estimator that minimizes the
MSE for a set of random data is the mean.
Table
showing
squared
error for the
mean for
sample data
Next we will examine the mean to see how well it predicts
net income over time.
The next table gives the income before taxes of a PC
manufacturer between 1985 and 1994.
Squared
Year $ (millions) Mean Error Error
1985 46.163 48.776 -2.613 6.828
1986 46.998 48.776 -1.778 3.161
1987 47.816 48.776 -0.960 0.922
1988 48.311 48.776 -0.465 0.216
1989 48.758 48.776 -0.018 0.000
1990 49.164 48.776 0.388 0.151
1991 49.548 48.776 0.772 0.596
1992 48.915 48.776 1.139 1.297
1993 50.315 48.776 1.539 2.369
1994 50.768 48.776 1.992 3.968
The MSE = 1.9508.
The mean is
not a good
estimator
when there
are trends
The question arises: can we use the mean to forecast income
if we suspect a trend? A look at the graph below shows
clearly that we should not do this.
Average
weighs all
past
observations
equally
In summary, we state that
1. The "simple" average or mean of all past observations
is only a useful estimate for forecasting when there are
no trends. If there are trends, use different estimates
that take the trend into account.
2. The average "weighs" all past observations equally. For
example, the average of the values 3, 4, 5 is 4. We
know, of course, that an average is computed by adding
all the values and dividing the sum by the number of
values. Another way of computing the average is by
adding each value divided by the number of values, or
3/3 + 4/3 + 5/3 = 1 + 1.3333 + 1.6667 = 4.
The multiplier 1/3 is called the weight. In general:
The are the weights and of course they sum to 1.
6.4.2.1. Single Moving Average

Taking a
moving
average is
a
smoothing
process
An alternative way to summarize the past data is to compute
the mean of successive smaller sets of numbers of past data as
follows:
Recall the set of numbers 9, 8, 9, 12, 9, 12, 11, 7,
13, 9, 11, 10 which were the dollar amount of 12
suppliers selected at random. Let us set M, the
size of the "smaller set" equal to 3. Then the
average of the first 3 numbers is: (9 + 8 + 9) / 3
= 8.667.
This is called "smoothing" (i.e., some form of averaging). This
smoothing process is continued by advancing one period and
calculating the next average of three numbers, dropping the
first number.
Moving
average
example
The next table summarizes the process, which is referred to as
Moving Averaging. The general expression for the moving
average is
M
t
= [ X
t
+ X
t-1
+ ... + X
t-N+1
] / N
Results of Moving Average
Supplier $ MA Error Error squared
1 9
2 8
3 9 8.667 0.333 0.111
4 12 9.667 2.333 5.444
5 9 10.000 -1.000 1.000
6 12 11.000 1.000 1.000
7 11 10.667 0.333 0.111
8 7 10.000 -3.000 9.000
9 13 10.333 2.667 7.111
10 9 9.667 -0.667 0.444
11 11 11.000 0 0
12 10 10.000 0 0
The MSE = 2.018 as compared to 3 in the previous case.
6.4.2.2. Centered Moving Average

When
computing
a running
moving
average,
placing the
average in
the middle
time
period
makes
sense
In the previous example we computed the average of the first
3 time periods and placed it next to period 3. We could have
placed the average in the middle of the time interval of three
periods, that is, next to period 2. This works well with odd
time periods, but not so good for even time periods. So where
would we place the first moving average when M = 4?
Technically, the Moving Average would fall at t = 2.5, 3.5, ...
To avoid this problem we smooth the MA's using M = 2. Thus
we smooth the smoothed values!
If we
average an
even
number of
terms, we
need to
smooth the
smoothed
values
The following table shows the results using M = 4.
Interim Steps
Period Value MA Centered
1 9
1.5
2 8
2.5 9.5
3 9 9.5
3.5 9.5
4 12 10.0
4.5 10.5
5 9 10.750
5.5 11.0
6 12
6.5
7 9
Final table This is the final table:
Period Value Centered MA
1 9
2 8
3 9 9.5
4 12 10.0
5 9 10.75
6 12
7 11
Double Moving Averages for a Linear Trend
Process
Moving
averages
are still
not able to
handle
significant
trends
when
forecasting
Unfortunately, neither the mean of all data nor the moving
average of the most recent M values, when used as forecasts
for the next period, are able to cope with a significant trend.
There exists a variation on the MA procedure that often does a
better job of handling trend. It is called Double Moving
Averages for a Linear Trend Process. It calculates a second
moving average from the original moving average, using the
same value for M. As soon as both single and double moving
averages are available, a computer routine uses these averages
to compute a slope and intercept, and then forecasts one or
more periods ahead.
6.4.3. What is Exponential Smoothing?

Exponential
smoothing
schemes
weight past
observations
using
exponentially
decreasing
weights
This is a very popular scheme to produce a smoothed Time
Series. Whereas in Single Moving Averages the past
observations are weighted equally, Exponential Smoothing
assigns exponentially decreasing weights as the observation
get older.
In other words, recent observations are given relatively
more weight in forecasting than the older observations.
In the case of moving averages, the weights assigned to the
observations are the same and are equal to 1/N. In
exponential smoothing, however, there are one or more
smoothing parameters to be determined (or estimated) and
these choices determine the weights assigned to the
observations.
Single, double and triple Exponential Smoothing will be
described in this section.
6.4.3.1. Single Exponential Smoothing

Exponential
smoothing
weights past
observations
with
exponentially
decreasing
weights to
forecast
future values
This smoothing scheme begins by setting S
2
to y
1
, where S
i
stands for smoothed observation or EWMA, and y stands for
the original observation. The subscripts refer to the time
periods, 1, 2, ..., n. For the third period, S
3
= y
2
+ (1- )
S
2
; and so on. There is no S
1
; the smoothed series starts with
the smoothed version of the second observation.
For any time period t, the smoothed value S
t
is found by
computing
This is the basic equation of exponential smoothing and the
constant or parameter is called the smoothing constant.
Note: There is an alternative approach to exponential
smoothing that replaces y
t-1
in the basic equation with y
t
, the
current observation. That formulation, due to Roberts (1959),
is described in the section on EWMA control charts. The
formulation here follows Hunter (1986).
Setting the first EWMA
The first
forecast is
very
important
The initial EWMA plays an important role in computing all
the subsequent EWMA's. Setting S
2
to y
1
is one method of
initialization. Another way is to set it to the target of the
process.
Still another possibility would be to average the first four or
five observations.
It can also be shown that the smaller the value of , the
more important is the selection of the initial EWMA. The
user would be wise to try a few methods, (assuming that the
software has them available) before finalizing the settings.
Why is it called "Exponential"?
Expand basic Let us expand the basic equation by first substituting for S
equation
t-1
in the basic equation to obtain
S
t
= y
t-1
+ (1- ) [ y
t-2
+ (1- ) S
t-2
]
= y
t-1
+ (1- ) y
t-2
+ (1- )
2
S
t-2
Summation
formula for
basic
equation
By substituting for S
t-2
, then for S
t-3
, and so forth, until we
reach S
2
(which is just y
1
), it can be shown that the
expanding equation can be written as:
Expanded
equation for
S
5
For example, the expanded equation for the smoothed value
S
5
is:
Illustrates
exponential
behavior
This illustrates the exponential behavior. The weights, (1-
)
t
decrease geometrically, and their sum is unity as shown
below, using a property of geometric series:
From the last formula we can see that the summation term
shows that the contribution to the smoothed value S
t
becomes
less at each consecutive time period.
Example
for = .3
Let = .3. Observe that the weights (1- )
t
decrease
exponentially (geometrically) with time.
Value weight
last y
1
.2100
y
2
.1470
y
3
.1029
y
4
.0720
What is the "best" value for ?
How do
you choose
The speed at which the older responses are dampened
(smoothed) is a function of the value of . When is close to
the weight
parameter?
1, dampening is quick and when is close to 0, dampening is
slow. This is illustrated in the table below:
---------------> towards past observations
(1- ) (1- )
2
(1- )
3
(1- )
4
.9 .1 .01 .001 .0001
.5 .5 .25 .125 .0625
.1 .9 .81 .729 .6561
We choose the best value for so the value which results in
the smallest MSE.
Example Let us illustrate this principle with an example. Consider the
following data set consisting of 12 observations taken over
time:
Time
y
t S ( =.1) Error
Error
squared
1 71
2 70 71 -1.00 1.00
3 69 70.9 -1.90 3.61
4 68 70.71 -2.71 7.34
5 64 70.44 -6.44 41.47
6 65 69.80 -4.80 23.04
7 72 69.32 2.68 7.18
8 78 69.58 8.42 70.90
9 75 70.43 4.57 20.88
10 75 70.88 4.12 16.97
11 75 71.29 3.71 13.76
12 70 71.67 -1.67 2.79
The sum of the squared errors (SSE) = 208.94. The mean of
the squared errors (MSE) is the SSE /11 = 19.0.
Calculate
for
different
values of
The MSE was again calculated for = .5 and turned out to be
16.29, so in this case we would prefer an of .5. Can we do
better? We could apply the proven trial-and-error method.
This is an iterative procedure beginning with a range of
between .1 and .9. We determine the best initial choice for
and then search between - and + . We could repeat
this perhaps one more time to find the best to 3 decimal
places.
Nonlinear
optimizers
can be
But there are better search methods, such as the Marquardt
procedure. This is a nonlinear optimizer that minimizes the
sum of squares of residuals. In general, most well designed
used statistical software programs should be able to find the value
of that minimizes the MSE.
Sample
plot
showing
smoothed
data for 2
values of
6.4.3.2. Forecasting with Single Exponential Smoothing

6.4.3.2. Forecasting with Single Exponential
Smoothing
Forecasting Formula
Forecasting
the next point
The forecasting formula is the basic equation
New forecast
is previous
forecast plus
an error
adjustment
This can be written as:
where
t
is the forecast error (actual - forecast) for period t.
In other words, the new forecast is the old one plus an
adjustment for the error that occurred in the last forecast.
Bootstrapping of Forecasts
Bootstrapping
forecasts
What happens if you wish to forecast from some origin,
usually the last data point, and no actual observations are
available? In this situation we have to modify the formula to
become:
where y
origin
remains constant. This technique is known as
bootstrapping.
Example of Bootstrapping
Example The last data point in the previous example was 70 and its
forecast (smoothed value S) was 71.7. Since we do have the
data point and the forecast available, we can calculate the
next forecast using the regular formula

= .1(70) + .9(71.7) = 71.5 ( = .1)
But for the next forecast we have no data point (observation).
So now we compute:
S
t+2
=. 1(70) + .9(71.5 )= 71.35
Comparison between bootstrap and regular forecasting
Table
comparing
two methods
The following table displays the comparison between the two
methods:
Period Bootstrap
forecast
Data Single Smoothing
Forecast
13 71.50 75 71.5
14 71.35 75 71.9
15 71.21 74 72.2
16 71.09 78 72.4
17 70.98 86 73.0
Single Exponential Smoothing with Trend
Single Smoothing (short for single exponential smoothing) is
not very good when there is a trend. The single coefficient
is not enough.
Sample data
set with trend
Let us demonstrate this with the following data set smoothed
with an of 0.3:
Data Fit
6.4
5.6 6.4
7.8 6.2
8.8 6.7
11.0 7.3
11.6 8.4
16.7 9.4
15.3 11.6
21.6 12.7
22.4 15.4
Plot
demonstrating
The resulting graph looks like:
inadequacy of
single
exponential
smoothing
when there is
trend
6.4.3.3. Double Exponential Smoothing

Double
exponential
smoothing
uses two
constants
and is better
at handling
trends
As was previously observed, Single Smoothing does not
excel in following the data when there is a trend. This
situation can be improved by the introduction of a second
equation with a second constant, , which must be chosen in
conjunction with .
Here are the two equations associated with Double
Exponential Smoothing:
Note that the current value of the series is used to calculate
its smoothed value replacement in double exponential
smoothing.
Initial Values
Several
methods to
choose the
initial
values
As in the case for single smoothing, there are a variety of
schemes to set initial values for S
t
and b
t
in double
smoothing.
S
1
is in general set to y
1
. Here are three suggestions for b
1
:
b
1
= y
2
- y
1
b
1
= [(y
2
- y
1
) + (y
3
- y
2
) + (y
4
- y
3
)]/3
b
1
= (y
n
- y
1
)/(n - 1)
Comments
Meaning of
the
smoothing
equations
The first smoothing equation adjusts S
t
directly for the trend
of the previous period, b
t-1
, by adding it to the last smoothed
value, S
t-1
. This helps to eliminate the lag and brings S
t
to
the appropriate base of the current value.
The second smoothing equation then updates the trend,
which is expressed as the difference between the last two
values. The equation is similar to the basic form of single
smoothing, but here applied to the updating of the trend.
Non-linear
optimization
techniques
can be used
The values for and can be obtained via non-linear
optimization techniques, such as the Marquardt Algorithm.
6.4.3.4. Forecasting with Double Exponential Smoothing(LASP)

6.4.3.4. Forecasting with Double Exponential
Smoothing(LASP)
Forecasting
formula
The one-period-ahead forecast is given by:
F
t+1
= S
t
+ b
t
The m-periods-ahead forecast is given by:
F
t+m
= S
t
+ mb
t
Example
Example Consider once more the data set:
6.4, 5.6, 7.8, 8.8, 11, 11.6, 16.7, 15.3, 21.6, 22.4.
Now we will fit a double smoothing model with = .3623
and = 1.0. These are the estimates that result in the lowest
possible MSE when comparing the orignal series to one step
ahead at a time forecasts (since this version of double
exponential smoothing uses the current series value to
calculate a smoothed value, the smoothed series cannot be
used to determine an with minimum MSE). The chosen
starting values are S
1
= y
1
= 6.4 and b
1
= ((y
2
- y
1
) + (y
3
-
y
2
) + (y
4
- y
3
))/3 = 0.8.
For comparison's sake we also fit a single smoothing model
with = 0.977 (this results in the lowest MSE for single
exponential smoothing).
The MSE for double smoothing is 3.7024.
The MSE for single smoothing is 8.8867.
Forecasting
results for
the example
The smoothed results for the example are:
Data Double Single
6.4 6.4
5.6 6.6 (Forecast = 7.2) 6.4
7.8 7.2 (Forecast = 6.8) 5.6
8.8 8.1 (Forecast = 7.8) 7.8
11.0 9.8 (Forecast = 9.1) 8.8
11.6 11.5 (Forecast = 11.4) 10.9
16.7 14.5 (Forecast = 13.2) 11.6
15.3 16.7 (Forecast = 17.4) 16.6
21.6 19.9 (Forecast = 18.9) 15.3
22.4 22.8 (Forecast = 23.1) 21.5
Comparison of Forecasts
Table
showing
single and
double
exponential
smoothing
forecasts
To see how each method predicts the future, we computed the
first five forecasts from the last observation as follows:
Period Single Double
11 22.4 25.8
12 22.4 28.7
13 22.4 31.7
14 22.4 34.6
15 22.4 37.6
Plot
comparing
single and
double
exponential
smoothing
forecasts
A plot of these results (using the forecasted double smoothing
values) is very enlightening.
This graph indicates that double smoothing follows the data
much closer than single smoothing. Furthermore, for
forecasting single smoothing cannot do better than projecting
a straight horizontal line, which is not very likely to occur in
reality. So in this case double smoothing is preferred.
Plot
comparing
double
exponential
smoothing
and
Finally, let us compare double smoothing with linear
regression:
regression
forecasts
This is an interesting picture. Both techniques follow the data
in similar fashion, but the regression line is more
conservative. That is, there is a slower increase with the
regression line than with double smoothing.
Selection of
technique
depends on
the
forecaster
The selection of the technique depends on the forecaster. If it
is desired to portray the growth process in a more aggressive
manner, then one selects double smoothing. Otherwise,
regression may be preferable. It should be noted that in linear
regression "time" functions as the independent variable.
Chapter 4 discusses the basics of linear regression, and the
details of regression estimation.
6.4.3.5. Triple Exponential Smoothing

What happens if the data show trend and seasonality?
To handle
seasonality,
we have to
add a third
parameter
In this case double smoothing will not work. We now introduce a
third equation to take care of seasonality (sometimes called
periodicity). The resulting set of equations is called the "Holt-
Winters" (HW) method after the names of the inventors.
The basic equations for their method are given by:
where
y is the observation
S is the smoothed observation
b is the trend factor
I is the seasonal index
F is the forecast at m periods ahead
t is an index denoting a time period
and , , and are constants that must be estimated in such a
way that the MSE of the error is minimized. This is best left to a
good software package.
Complete
season
needed
To initialize the HW method we need at least one complete
season's data to determine initial estimates of the seasonal indices
I
t-L
.
L periods
in a season
A complete season's data consists of L periods. And we need to
estimate the trend factor from one period to the next. To
accomplish this, it is advisable to use two complete seasons; that
is, 2L periods.
Initial values for the trend factor
How to get
initial
estimates
for trend
and
seasonality
parameters
The general formula to estimate the initial trend is given by
Initial values for the Seasonal Indices
As we will see in the example, we work with data that consist of
6 years with 4 periods (that is, 4 quarters) per year. Then
Step 1:
compute
yearly
averages
Step 1: Compute the averages of each of the 6 years
Step 2:
divide by
yearly
averages
Step 2: Divide the observations by the appropriate yearly mean
1 2 3 4 5 6
y
1
/A
1
y
5
/A
2
y
9
/A
3
y
13
/A
4
y
17
/A
5
y
21
/A
6
y
2
/A
1
y
6
/A
2
y
10
/A
3
y
14
/A
4
y
18
/A
5
y
22
/A
6
y
3
/A
1
y
7
/A
2
y
11
/A
3
y
15
/A
4
y
19
/A
5
y
23
/A
6
y
4
/A
1
y
8
/A
2
y
12
/A
3
y
16
/A
4
y
20
/A
5
y
24
/A
6
Step 3:
form
seasonal
indices
Step 3: Now the seasonal indices are formed by computing the
average of each row. Thus the initial seasonal indices
(symbolically) are:
I
1
= ( y
1
/A
1
+ y
5
/A
2
+ y
9
/A
3
+ y
13
/A
4
+ y
17
/A
5
+
y
21
/A
6
)/6
I
2
= ( y
2
/A
1
+ y
6
/A
2
+ y
10
/A
3
+ y
14
/A
4
+ y
18
/A
5
+
y
22
/A
6
)/6
I
3
= ( y
3
/A
1
+ y
7
/A
2
+ y
11
/A
3
+ y
15
/A
4
+ y
19
/A
5
+
y
22
/A
6
)/6
I
4
= ( y
4
/A
1
+ y
8
/A
2
+ y
12
/A
3
+ y
16
/A
4
+ y
20
/A
5
+
y
24
/A
6
)/6
We now know the algebra behind the computation of the initial
estimates.
The next page contains an example of triple exponential
smoothing.
The case of the Zero Coefficients
Zero
coefficients
for trend
and
seasonality
parameters
Sometimes it happens that a computer program for triple
exponential smoothing outputs a final coefficient for trend ( ) or
for seasonality ( ) of zero. Or worse, both are outputted as zero!
Does this indicate that there is no trend and/or no seasonality?
Of course not! It only means that the initial values for trend
and/or seasonality were right on the money. No updating was
necessary in order to arrive at the lowest possible MSE. We
should inspect the updating formulas to verify this.
6.4.3.6. Example of Triple Exponential Smoothing

6.4.3.6. Example of Triple Exponential
Smoothing
Example
comparing
single,
double,
triple
exponential
smoothing
This example shows comparison of single, double and triple
exponential smoothing for a data set.
The following data set represents 24 observations. These are
six years of quarterly data (each year = 4 quarters).
Table
showing the
data for the
example
Quarter Period Sales Quarter Period Sales
90 1 1 362 93 1 13 544
2 2 385 2 14 582
3 3 432 3 15 681
4 4 341 4 16 557
91 1 5 382 94 1 17 628
2 6 409 2 18 707
3 7 498 3 19 773
4 8 387 4 20 592
92 1 9 473 95 1 21 627
2 10 513 2 22 725
3 11 582 3 23 854
4 12 474 4 24 661
Plot of raw
data with
single,
double, and
triple
exponential
forecasts
Plot of raw
data with
triple
exponential
forecasts
Actual Time Series with forecasts
Comparison
of MSE's
Comparison of MSE's
MSE

demand

trend

seasonality
6906 .4694
5054 .1086 1.000
936 1.000 1.000
520 .7556 0.000 .9837
The updating coefficients were chosen by a computer
program such that the MSE for each of the methods was
minimized.
Example of the computation of the Initial Trend
Computation
of initial
trend
The data set consists of quarterly sales data. The season is 1
year and since there are 4 quarters per year, L = 4. Using the
formula we obtain:
Example of the computation of the Initial Seasonal
Indices
Table of
initial
seasonal
indices
1 2 3 4 5 6
1 362 382 473 544 628 627
2 385 409 513 582 707 725
3 432 498 582 681 773 854
4 341 387 474 557 592 661
380 419 510.5 591 675 716.75
In this example we used the full 6 years of data. Other
schemes may use only 3, or some other number of years.
There are also a number of ways to compute initial estimates.
6.4.3.7. Exponential Smoothing Summary

6.4.3.7. Exponential Smoothing Summary
Summary
Exponential
smoothing
has proven
to be a
useful
technique
Exponential smoothing has proven through the years to be
very useful in many forecasting situations. It was first
suggested by C.C. Holt in 1957 and was meant to be used for
non-seasonal time series showing no trend. He later offered a
procedure (1958) that does handle trends. Winters(1965)
generalized the method to include seasonality, hence the
name "Holt-Winters Method".
Holt-
Winters has
3 updating
equations
The Holt-Winters Method has 3 updating equations, each
with a constant that ranges from 0 to 1. The equations are
intended to give more weight to recent observations and less
weights to observations further in the past.
These weights are geometrically decreasing by a constant
ratio.
The HW procedure can be made fully automatic by user-
friendly software.
6.4.4. Univariate Time Series Models

Univariate
Time
Series
The term "univariate time series" refers to a time series that
consists of single (scalar) observations recorded sequentially
over equal time increments. Some examples are monthly CO
2
concentrations and southern oscillations to predict el nino
effects.
Although a univariate time series data set is usually given as a
single column of numbers, time is in fact an implicit variable
in the time series. If the data are equi-spaced, the time
variable, or index, does not need to be explicitly given. The
time variable may sometimes be explicitly used for plotting
the series. However, it is not used in the time series model
itself.
The analysis of time series where the data are not collected in
equal time increments is beyond the scope of this handbook.
Contents 1. Sample Data Sets
2. Stationarity
3. Seasonality
4. Common Approaches
5. Box-Jenkins Approach
6. Box-Jenkins Model Identification
7. Box-Jenkins Model Estimation
8. Box-Jenkins Model Validation
9. SEMPLOT Sample Output for a Box-Jenkins Analysis
10. SEMPLOT Sample Output for a Box-Jenkins Analysis
with Seasonality
6.4.4.1. Sample Data Sets

Sample
Data Sets
The following two data sets are used as examples in the text
for this section.
1. Monthly mean CO
2
concentrations.
2. Southern oscillations.
6.4.4.1.1. Data Set of Monthly CO2 Concentrations

6.4.4.1.1. Data Set of Monthly CO2
Concentrations
Source and
Background
This data set contains selected monthly mean CO2
concentrations at the Mauna Loa Observatory from 1974 to
1987. The CO2 concentrations were measured by the
continuous infrared analyser of the Geophysical Monitoring
for Climatic Change division of NOAA's Air Resources
Laboratory. The selection has been for an approximation of
'background conditions'. See Thoning et al., "Atmospheric
Carbon Dioxide at Mauna Loa Observatory: II Analysis of
the NOAA/GMCC Data 1974-1985", Journal of Geophysical
Research (submitted) for details.
This dataset was received from Jim Elkins of NOAA in 1988.
Data Each line contains the CO2 concentration (mixing ratio in dry
air, expressed in the WMO X85 mole fraction scale,
maintained by the Scripps Institution of Oceanography). In
addition, it contains the year, month, and a numeric value for
the combined month and year. This combined date is useful
for plotting purposes.
CO2 Year&Month Year Month
--------------------------------------------------
333.13 1974.38 1974 5
332.09 1974.46 1974 6
331.10 1974.54 1974 7
329.14 1974.63 1974 8
327.36 1974.71 1974 9
327.29 1974.79 1974 10
328.23 1974.88 1974 11
329.55 1974.96 1974 12

330.62 1975.04 1975 1
331.40 1975.13 1975 2
331.87 1975.21 1975 3
333.18 1975.29 1975 4
333.92 1975.38 1975 5
333.43 1975.46 1975 6
331.85 1975.54 1975 7
330.01 1975.63 1975 8
328.51 1975.71 1975 9
328.41 1975.79 1975 10
329.25 1975.88 1975 11
330.97 1975.96 1975 12

331.60 1976.04 1976 1
332.60 1976.13 1976 2
333.57 1976.21 1976 3
334.72 1976.29 1976 4
334.68 1976.38 1976 5
334.17 1976.46 1976 6
332.96 1976.54 1976 7
330.80 1976.63 1976 8
328.98 1976.71 1976 9
328.57 1976.79 1976 10
330.20 1976.88 1976 11
331.58 1976.96 1976 12

332.67 1977.04 1977 1
333.17 1977.13 1977 2
334.86 1977.21 1977 3
336.07 1977.29 1977 4
336.82 1977.38 1977 5
336.12 1977.46 1977 6
334.81 1977.54 1977 7
332.56 1977.63 1977 8
331.30 1977.71 1977 9
331.22 1977.79 1977 10
332.37 1977.88 1977 11
333.49 1977.96 1977 12

334.71 1978.04 1978 1
335.23 1978.13 1978 2
336.54 1978.21 1978 3
337.79 1978.29 1978 4
337.95 1978.38 1978 5
338.00 1978.46 1978 6
336.37 1978.54 1978 7
334.47 1978.63 1978 8
332.46 1978.71 1978 9
332.29 1978.79 1978 10
333.76 1978.88 1978 11
334.80 1978.96 1978 12

336.00 1979.04 1979 1
336.63 1979.13 1979 2
337.93 1979.21 1979 3
338.95 1979.29 1979 4
339.05 1979.38 1979 5
339.27 1979.46 1979 6
337.64 1979.54 1979 7
335.68 1979.63 1979 8
333.77 1979.71 1979 9
334.09 1979.79 1979 10
335.29 1979.88 1979 11
336.76 1979.96 1979 12

337.77 1980.04 1980 1
338.26 1980.13 1980 2
340.10 1980.21 1980 3
340.88 1980.29 1980 4
341.47 1980.38 1980 5
341.31 1980.46 1980 6
339.41 1980.54 1980 7
337.74 1980.63 1980 8
336.07 1980.71 1980 9
336.07 1980.79 1980 10
337.22 1980.88 1980 11
338.38 1980.96 1980 12

339.32 1981.04 1981 1
340.41 1981.13 1981 2
341.69 1981.21 1981 3
342.51 1981.29 1981 4
343.02 1981.38 1981 5
342.54 1981.46 1981 6
340.88 1981.54 1981 7
338.75 1981.63 1981 8
337.05 1981.71 1981 9
337.13 1981.79 1981 10
338.45 1981.88 1981 11
339.85 1981.96 1981 12

340.90 1982.04 1982 1
341.70 1982.13 1982 2
342.70 1982.21 1982 3
343.65 1982.29 1982 4
344.28 1982.38 1982 5
343.42 1982.46 1982 6
342.02 1982.54 1982 7
339.97 1982.63 1982 8
337.84 1982.71 1982 9
338.00 1982.79 1982 10
339.20 1982.88 1982 11
340.63 1982.96 1982 12

341.41 1983.04 1983 1
342.68 1983.13 1983 2
343.04 1983.21 1983 3
345.27 1983.29 1983 4
345.92 1983.38 1983 5
345.40 1983.46 1983 6
344.16 1983.54 1983 7
342.11 1983.63 1983 8
340.11 1983.71 1983 9
340.15 1983.79 1983 10
341.38 1983.88 1983 11
343.02 1983.96 1983 12

343.87 1984.04 1984 1
344.59 1984.13 1984 2
345.11 1984.21 1984 3
347.07 1984.29 1984 4
347.38 1984.38 1984 5
346.78 1984.46 1984 6
344.96 1984.54 1984 7
342.71 1984.63 1984 8
340.86 1984.71 1984 9
341.13 1984.79 1984 10
342.84 1984.88 1984 11
344.32 1984.96 1984 12

344.88 1985.04 1985 1
345.62 1985.13 1985 2
347.23 1985.21 1985 3
347.62 1985.29 1985 4
348.53 1985.38 1985 5
347.87 1985.46 1985 6
346.00 1985.54 1985 7
343.86 1985.63 1985 8
342.55 1985.71 1985 9
342.57 1985.79 1985 10
344.11 1985.88 1985 11
345.49 1985.96 1985 12

346.04 1986.04 1986 1
346.70 1986.13 1986 2
347.38 1986.21 1986 3
349.38 1986.29 1986 4
349.93 1986.38 1986 5
349.26 1986.46 1986 6
347.44 1986.54 1986 7
345.55 1986.63 1986 8
344.21 1986.71 1986 9
343.67 1986.79 1986 10
345.09 1986.88 1986 11
346.27 1986.96 1986 12

347.33 1987.04 1987 1
347.82 1987.13 1987 2
349.29 1987.21 1987 3
350.91 1987.29 1987 4
351.71 1987.38 1987 5
350.94 1987.46 1987 6
349.10 1987.54 1987 7
346.77 1987.63 1987 8
345.73 1987.71 1987 9
6.4.4.1.2. Data Set of Southern Oscillations

Source and
Background
The southern oscillation is defined as the barametric pressure
difference between Tahiti and the Darwin Islands at sea level.
The southern oscillation is a predictor of el nino which in
turn is thought to be a driver of world-wide weather.
Specifically, repeated southern oscillation values less than -1
typically defines an el nino. Note: the decimal values in the
second column of the data given below are obtained as
(month number - 0.5)/12.
Data
Southern
Oscillation Year + fraction Year Month
----------------------------------------------
-0.7 1955.04 1955 1
1.3 1955.13 1955 2
0.1 1955.21 1955 3
-0.9 1955.29 1955 4
0.8 1955.38 1955 5
1.6 1955.46 1955 6
1.7 1955.54 1955 7
1.4 1955.63 1955 8
1.4 1955.71 1955 9
1.5 1955.79 1955 10
1.4 1955.88 1955 11
0.9 1955.96 1955 12

1.2 1956.04 1956 1
1.1 1956.13 1956 2
0.9 1956.21 1956 3
1.1 1956.29 1956 4
1.4 1956.38 1956 5
1.2 1956.46 1956 6
1.1 1956.54 1956 7
1.0 1956.63 1956 8
0.0 1956.71 1956 9
1.9 1956.79 1956 10
0.1 1956.88 1956 11
0.9 1956.96 1956 12

0.4 1957.04 1957 1
-0.4 1957.13 1957 2
-0.4 1957.21 1957 3
0.0 1957.29 1957 4
-1.1 1957.38 1957 5
-0.4 1957.46 1957 6
0.1 1957.54 1957 7
-1.1 1957.63 1957 8
-1.0 1957.71 1957 9
-0.1 1957.79 1957 10
-1.2 1957.88 1957 11
-0.5 1957.96 1957 12

-1.9 1958.04 1958 1
-0.7 1958.13 1958 2
-0.3 1958.21 1958 3
0.1 1958.29 1958 4
-1.3 1958.38 1958 5
-0.3 1958.46 1958 6
0.3 1958.54 1958 7
0.7 1958.63 1958 8
-0.4 1958.71 1958 9
-0.4 1958.79 1958 10
-0.6 1958.88 1958 11
-0.8 1958.96 1958 12

-0.9 1959.04 1959 1
-1.5 1959.13 1959 2
0.8 1959.21 1959 3
0.2 1959.29 1959 4
0.2 1959.38 1959 5
-0.9 1959.46 1959 6
-0.5 1959.54 1959 7
-0.6 1959.63 1959 8
0.0 1959.71 1959 9
0.3 1959.79 1959 10
0.9 1959.88 1959 11
0.8 1959.96 1959 12

0.0 1960.04 1960 1
-0.2 1960.13 1960 2
0.5 1960.21 1960 3
0.9 1960.29 1960 4
0.2 1960.38 1960 5
-0.5 1960.46 1960 6
0.4 1960.54 1960 7
0.5 1960.63 1960 8
0.7 1960.71 1960 9
-0.1 1960.79 1960 10
0.6 1960.88 1960 11
0.7 1960.96 1960 12

-0.4 1961.04 1961 1
0.5 1961.13 1961 2
-2.6 1961.21 1961 3
1.1 1961.29 1961 4
0.2 1961.38 1961 5
-0.4 1961.46 1961 6
0.1 1961.54 1961 7
-0.3 1961.63 1961 8
0.0 1961.71 1961 9
-0.8 1961.79 1961 10
0.7 1961.88 1961 11
1.4 1961.96 1961 12

1.7 1962.04 1962 1
-0.5 1962.13 1962 2
-0.4 1962.21 1962 3
0.0 1962.29 1962 4
1.2 1962.38 1962 5
0.5 1962.46 1962 6
-0.1 1962.54 1962 7
0.3 1962.63 1962 8
0.5 1962.71 1962 9
0.9 1962.79 1962 10
0.2 1962.88 1962 11
0.0 1962.96 1962 12

0.8 1963.04 1963 1
0.3 1963.13 1963 2
0.6 1963.21 1963 3
0.9 1963.29 1963 4
0.0 1963.38 1963 5
-1.5 1963.46 1963 6
-0.3 1963.54 1963 7
-0.4 1963.63 1963 8
-0.7 1963.71 1963 9
-1.6 1963.79 1963 10
-1.0 1963.88 1963 11
-1.4 1963.96 1963 12

-0.5 1964.04 1964 1
-0.2 1964.13 1964 2
0.6 1964.21 1964 3
1.7 1964.29 1964 4
-0.2 1964.38 1964 5
0.7 1964.46 1964 6
0.5 1964.54 1964 7
1.4 1964.63 1964 8
1.3 1964.71 1964 9
1.3 1964.79 1964 10
0.0 1964.88 1964 11
-0.5 1964.96 1964 12

-0.5 1965.04 1965 1
0.0 1965.13 1965 2
0.2 1965.21 1965 3
-1.1 1965.29 1965 4
0.0 1965.38 1965 5
-1.5 1965.46 1965 6
-2.3 1965.54 1965 7
-1.3 1965.63 1965 8
-1.4 1965.71 1965 9
-1.2 1965.79 1965 10
-1.8 1965.88 1965 11
0.0 1965.96 1965 12

-1.4 1966.04 1966 1
-0.5 1966.13 1966 2
-1.6 1966.21 1966 3
-0.7 1966.29 1966 4
-0.6 1966.38 1966 5
0.0 1966.46 1966 6
-0.1 1966.54 1966 7
0.3 1966.63 1966 8
-0.3 1966.71 1966 9
-0.3 1966.79 1966 10
-0.1 1966.88 1966 11
-0.5 1966.96 1966 12

1.5 1967.04 1967 1
1.2 1967.13 1967 2
0.8 1967.21 1967 3
-0.2 1967.29 1967 4
-0.4 1967.38 1967 5
0.6 1967.46 1967 6
0.0 1967.54 1967 7
0.4 1967.63 1967 8
0.5 1967.71 1967 9
-0.2 1967.79 1967 10
-0.7 1967.88 1967 11
-0.7 1967.96 1967 12

0.5 1968.04 1968 1
0.8 1968.13 1968 2
-0.5 1968.21 1968 3
-0.3 1968.29 1968 4
1.2 1968.38 1968 5
1.4 1968.46 1968 6
0.6 1968.54 1968 7
-0.1 1968.63 1968 8
-0.3 1968.71 1968 9
-0.3 1968.79 1968 10
-0.4 1968.88 1968 11
0.0 1968.96 1968 12

-1.4 1969.04 1969 1
0.8 1969.13 1969 2
-0.1 1969.21 1969 3
-0.8 1969.29 1969 4
-0.8 1969.38 1969 5
-0.2 1969.46 1969 6
-0.7 1969.54 1969 7
-0.6 1969.63 1969 8
-1.0 1969.71 1969 9
-1.4 1969.79 1969 10
-0.1 1969.88 1969 11
0.3 1969.96 1969 12

-1.2 1970.04 1970 1
-1.2 1970.13 1970 2
0.0 1970.21 1970 3
-0.5 1970.29 1970 4
0.1 1970.38 1970 5
1.1 1970.46 1970 6
-0.6 1970.54 1970 7
0.3 1970.63 1970 8
1.2 1970.71 1970 9
0.8 1970.79 1970 10
1.8 1970.88 1970 11
1.8 1970.96 1970 12

0.2 1971.04 1971 1
1.4 1971.13 1971 2
2.0 1971.21 1971 3
2.6 1971.29 1971 4
0.9 1971.38 1971 5
0.2 1971.46 1971 6
0.1 1971.54 1971 7
1.4 1971.63 1971 8
1.5 1971.71 1971 9
1.8 1971.79 1971 10
0.5 1971.88 1971 11
0.1 1971.96 1971 12

0.3 1972.04 1972 1
0.6 1972.13 1972 2
0.1 1972.21 1972 3
-0.5 1972.29 1972 4
-2.1 1972.38 1972 5
-1.7 1972.46 1972 6
-1.9 1972.54 1972 7
-1.1 1972.63 1972 8
-1.5 1972.71 1972 9
-1.1 1972.79 1972 10
-0.4 1972.88 1972 11
-1.5 1972.96 1972 12

-0.4 1973.04 1973 1
-1.5 1973.13 1973 2
0.2 1973.21 1973 3
-0.4 1973.29 1973 4
0.3 1973.38 1973 5
1.2 1973.46 1973 6
0.5 1973.54 1973 7
1.2 1973.63 1973 8
1.3 1973.71 1973 9
0.6 1973.79 1973 10
2.9 1973.88 1973 11
1.7 1973.96 1973 12

2.2 1974.04 1974 1
1.5 1974.13 1974 2
2.1 1974.21 1974 3
1.3 1974.29 1974 4
1.3 1974.38 1974 5
0.1 1974.46 1974 6
1.2 1974.54 1974 7
0.5 1974.63 1974 8
1.1 1974.71 1974 9
0.8 1974.79 1974 10
-0.4 1974.88 1974 11
0.0 1974.96 1974 12

-0.6 1975.04 1975 1
0.4 1975.13 1975 2
1.1 1975.21 1975 3
1.5 1975.29 1975 4
0.5 1975.38 1975 5
1.7 1975.46 1975 6
2.1 1975.54 1975 7
2.0 1975.63 1975 8
2.2 1975.71 1975 9
1.7 1975.79 1975 10
1.3 1975.88 1975 11
2.0 1975.96 1975 12

1.2 1976.04 1976 1
1.2 1976.13 1976 2
1.3 1976.21 1976 3
0.2 1976.29 1976 4
0.6 1976.38 1976 5
-0.1 1976.46 1976 6
-1.2 1976.54 1976 7
-1.5 1976.63 1976 8
-1.2 1976.71 1976 9
0.2 1976.79 1976 10
0.7 1976.88 1976 11
-0.5 1976.96 1976 12

-0.5 1977.04 1977 1
0.8 1977.13 1977 2
-1.2 1977.21 1977 3
-1.3 1977.29 1977 4
-1.1 1977.38 1977 5
-2.3 1977.46 1977 6
-1.5 1977.54 1977 7
-1.4 1977.63 1977 8
-0.9 1977.71 1977 9
-1.4 1977.79 1977 10
-1.6 1977.88 1977 11
-1.3 1977.96 1977 12

-0.5 1978.04 1978 1
-2.6 1978.13 1978 2
-0.8 1978.21 1978 3
-0.9 1978.29 1978 4
1.3 1978.38 1978 5
0.4 1978.46 1978 6
0.4 1978.54 1978 7
0.1 1978.63 1978 8
0.0 1978.71 1978 9
-0.8 1978.79 1978 10
-0.1 1978.88 1978 11
-0.2 1978.96 1978 12

-0.5 1979.04 1979 1
0.6 1979.13 1979 2
-0.5 1979.21 1979 3
-0.7 1979.29 1979 4
0.5 1979.38 1979 5
0.6 1979.46 1979 6
1.3 1979.54 1979 7
-0.7 1979.63 1979 8
0.1 1979.71 1979 9
-0.4 1979.79 1979 10
-0.6 1979.88 1979 11
-0.9 1979.96 1979 12

0.3 1980.04 1980 1
0.0 1980.13 1980 2
-1.1 1980.21 1980 3
-1.7 1980.29 1980 4
-0.3 1980.38 1980 5
-0.7 1980.46 1980 6
-0.2 1980.54 1980 7
-0.1 1980.63 1980 8
-0.5 1980.71 1980 9
-0.3 1980.79 1980 10
-0.5 1980.88 1980 11
-0.2 1980.96 1980 12

0.3 1981.04 1981 1
-0.5 1981.13 1981 2
-2.0 1981.21 1981 3
-0.6 1981.29 1981 4
0.8 1981.38 1981 5
1.6 1981.46 1981 6
0.8 1981.54 1981 7
0.4 1981.63 1981 8
0.3 1981.71 1981 9
-0.7 1981.79 1981 10
0.1 1981.88 1981 11
0.4 1981.96 1981 12

1.0 1982.04 1982 1
0.0 1982.13 1982 2
0.0 1982.21 1982 3
-0.1 1982.29 1982 4
-0.6 1982.38 1982 5
-2.5 1982.46 1982 6
-2.0 1982.54 1982 7
-2.7 1982.63 1982 8
-1.9 1982.71 1982 9
-2.2 1982.79 1982 10
-3.2 1982.88 1982 11
-2.5 1982.96 1982 12

-3.4 1983.04 1983 1
-3.5 1983.13 1983 2
-3.2 1983.21 1983 3
-2.1 1983.29 1983 4
0.9 1983.38 1983 5
-0.5 1983.46 1983 6
-0.9 1983.54 1983 7
-0.4 1983.63 1983 8
0.9 1983.71 1983 9
0.3 1983.79 1983 10
-0.1 1983.88 1983 11
-0.1 1983.96 1983 12

0.0 1984.04 1984 1
0.4 1984.13 1984 2
-0.8 1984.21 1984 3
0.4 1984.29 1984 4
0.0 1984.38 1984 5
-1.2 1984.46 1984 6
0.0 1984.54 1984 7
0.1 1984.63 1984 8
0.1 1984.71 1984 9
-0.6 1984.79 1984 10
0.3 1984.88 1984 11
-0.3 1984.96 1984 12

-0.5 1985.04 1985 1
0.8 1985.13 1985 2
0.2 1985.21 1985 3
1.4 1985.29 1985 4
-0.2 1985.38 1985 5
-1.4 1985.46 1985 6
-0.3 1985.54 1985 7
0.7 1985.63 1985 8
0.0 1985.71 1985 9
-0.8 1985.79 1985 10
-0.4 1985.88 1985 11
0.1 1985.96 1985 12

0.8 1986.04 1986 1
-1.2 1986.13 1986 2
-0.1 1986.21 1986 3
0.1 1986.29 1986 4
-0.6 1986.38 1986 5
1.0 1986.46 1986 6
0.1 1986.54 1986 7
-0.9 1986.63 1986 8
-0.5 1986.71 1986 9
0.6 1986.79 1986 10
-1.6 1986.88 1986 11
-1.6 1986.96 1986 12

-0.7 1987.04 1987 1
-1.4 1987.13 1987 2
-2.0 1987.21 1987 3
-2.7 1987.29 1987 4
-2.0 1987.38 1987 5
-2.7 1987.46 1987 6
-1.8 1987.54 1987 7
-1.7 1987.63 1987 8
-1.1 1987.71 1987 9
-0.7 1987.79 1987 10
-0.1 1987.88 1987 11
-0.6 1987.96 1987 12

-0.3 1988.04 1988 1
-0.6 1988.13 1988 2
0.1 1988.21 1988 3
0.0 1988.29 1988 4
1.1 1988.38 1988 5
-0.3 1988.46 1988 6
1.1 1988.54 1988 7
1.4 1988.63 1988 8
1.9 1988.71 1988 9
1.5 1988.79 1988 10
1.9 1988.88 1988 11
1.1 1988.96 1988 12

1.5 1989.04 1989 1
1.1 1989.13 1989 2
0.6 1989.21 1989 3
1.6 1989.29 1989 4
1.2 1989.38 1989 5
0.5 1989.46 1989 6
0.8 1989.54 1989 7
-0.8 1989.63 1989 8
0.6 1989.71 1989 9
0.6 1989.79 1989 10
-0.4 1989.88 1989 11
-0.7 1989.96 1989 12

-0.2 1990.04 1990 1
-2.4 1990.13 1990 2
-1.2 1990.21 1990 3
0.0 1990.29 1990 4
1.1 1990.38 1990 5
0.0 1990.46 1990 6
0.5 1990.54 1990 7
-0.5 1990.63 1990 8
-0.8 1990.71 1990 9
0.1 1990.79 1990 10
-0.7 1990.88 1990 11
-0.4 1990.96 1990 12

0.6 1991.04 1991 1
-0.1 1991.13 1991 2
-1.4 1991.21 1991 3
-1.0 1991.29 1991 4
-1.5 1991.38 1991 5
-0.5 1991.46 1991 6
-0.2 1991.54 1991 7
-0.9 1991.63 1991 8
-1.8 1991.71 1991 9
-1.5 1991.79 1991 10
-0.8 1991.88 1991 11
-2.3 1991.96 1991 12

-3.4 1992.04 1992 1
-1.4 1992.13 1992 2
-3.0 1992.21 1992 3
-1.4 1992.29 1992 4
0.0 1992.38 1992 5
-1.2 1992.46 1992 6
-0.8 1992.54 1992 7
0.0 1992.63 1992 8
0.0 1992.71 1992 9
-1.9 1992.79 1992 10
-0.9 1992.88 1992 11
-1.1 1992.96 1992 12
6.4.4.2. Stationarity

Stationarity A common assumption in many time series techniques is
that the data are stationary.
A stationary process has the property that the mean,
variance and autocorrelation structure do not change over
time. Stationarity can be defined in precise mathematical
terms, but for our purpose we mean a flat looking series,
without trend, constant variance over time, a constant
autocorrelation structure over time and no periodic
fluctuations (seasonality).
For practical purposes, stationarity can usually be
determined from a run sequence plot.
Transformations
to Achieve
Stationarity
If the time series is not stationary, we can often transform
it to stationarity with one of the following techniques.
1. We can difference the data. That is, given the series
Z
t
, we create the new series
The differenced data will contain one less point than
the original data. Although you can difference the
data more than once, one difference is usually
sufficient.
2. If the data contain a trend, we can fit some type of
curve to the data and then model the residuals from
that fit. Since the purpose of the fit is to simply
remove long term trend, a simple fit, such as a
straight line, is typically used.
3. For non-constant variance, taking the logarithm or
square root of the series may stabilize the variance.
For negative data, you can add a suitable constant
to make all the data positive before applying the
transformation. This constant can then be subtracted
from the model to obtain predicted (i.e., the fitted)
values and forecasts for future points.
The above techniques are intended to generate series with
constant location and scale. Although seasonality also
violates stationarity, this is usually explicitly incorporated
into the time series model.
Example The following plots are from a data set of monthly CO2
concentrations.
Run Sequence
Plot
The initial run sequence plot of the data indicates a rising
trend. A visual inspection of this plot indicates that a
simple linear fit should be sufficient to remove this
upward trend.
This plot also shows periodical behavior. This is discussed
in the next section.
Linear Trend
Removed
This plot contains the residuals from a linear fit to the
original data. After removing the linear trend, the run
sequence plot indicates that the data have a constant
location and variance, although the pattern of the residuals
shows that the data depart from the model in a systematic
way.
6.4.4.3. Seasonality

Seasonality Many time series display seasonality. By seasonality, we
mean periodic fluctuations. For example, retail sales tend to
peak for the Christmas season and then decline after the
holidays. So time series of retail sales will typically show
increasing sales from September through December and
declining sales in January and February.
Seasonality is quite common in economic time series. It is less
common in engineering and scientific data.
If seasonality is present, it must be incorporated into the time
series model. In this section, we discuss techniques for
detecting seasonality. We defer modeling of seasonality until
later sections.
Detecting
Seasonality
he following graphical techniques can be used to detect
seasonality.
1. A run sequence plot will often show seasonality.
2. A seasonal subseries plot is a specialized technique for
showing seasonality.
3. Multiple box plots can be used as an alternative to the
seasonal subseries plot to detect seasonality.
4. The autocorrelation plot can help identify seasonality.
Examples of each of these plots will be shown below.
The run sequence plot is a recommended first step for
analyzing any time series. Although seasonality can
sometimes be indicated with this plot, seasonality is shown
more clearly by the seasonal subseries plot or the box plot.
The seasonal subseries plot does an excellent job of showing
both the seasonal differences (between group patterns) and
also the within-group patterns. The box plot shows the
seasonal difference (between group patterns) quite well, but it
does not show within group patterns. However, for large data
sets, the box plot is usually easier to read than the seasonal
subseries plot.
Both the seasonal subseries plot and the box plot assume that
the seasonal periods are known. In most cases, the analyst will
in fact know this. For example, for monthly data, the period is
12 since there are 12 months in a year. However, if the period
is not known, the autocorrelation plot can help. If there is
significant seasonality, the autocorrelation plot should show
spikes at lags equal to the period. For example, for monthly
data, if there is a seasonality effect, we would expect to see
significant peaks at lag 12, 24, 36, and so on (although the
intensity may decrease the further out we go).
Example
without
Seasonality
The following plots are from a data set of southern
oscillations for predicting el nino.
Run
Sequence
Plot
No obvious periodic patterns are apparent in the run sequence
plot.
Seasonal
Subseries
Plot
The means for each month are relatively close and show no
obvious pattern.
Box Plot
As with the seasonal subseries plot, no obvious seasonal
pattern is apparent.
Due to the rather large number of observations, the box plot
shows the difference between months better than the seasonal
subseries plot.
Example
with
Seasonality
The following plots are from a data set of monthly CO2
concentrations. A linear trend has been removed from these
data.
Run
Sequence
Plot
This plot shows periodic behavior. However, it is difficult to
determine the nature of the seasonality from this plot.
Seasonal
Subseries
Plot
The seasonal subseries plot shows the seasonal pattern more
clearly. In this case, the CO
2
concentrations are at a minimun
in September and October. From there, steadily the
concentrations increase until June and then begin declining
until September.
Box Plot
As with the seasonal subseries plot, the seasonal pattern is
quite evident in the box plot.
6.4.4.3.1. Seasonal Subseries Plot

Purpose Seasonal subseries plots (Cleveland 1993) are a tool for
detecting seasonality in a time series.
This plot is only useful if the period of the seasonality is
already known. In many cases, this will in fact be known. For
example, monthly data typically has a period of 12.
If the period is not known, an autocorrelation plot or spectral
plot can be used to determine it.
Sample
Plot
This seasonal subseries plot containing monthly data of CO
2
concentrations reveals a strong seasonality pattern. The CO
2
concentrations peak in May, steadily decrease through
September, and then begin rising again until the May peak.
This plot allows you to detect both between group and within
group patterns.
If there is a large number of observations, then a box plot may
be preferable.
Definition Seasonal subseries plots are formed by
Vertical Response variable
axis:
Horizontal
axis:
Time ordered by season. For example, with
monthly data, all the January values are
plotted (in chronological order), then all the
February values, and so on.
In addition, a reference line is drawn at the group means.
The user must specify the length of the seasonal pattern before
generating this plot. In most cases, the analyst will know this
from the context of the problem and data collection.
Sometimes the series will need to be detrended before
generating the plot, as was the case for the CO
2
data.
Questions The seasonal subseries plot can provide answers to the
1. Do the data exhibit a seasonal pattern?
2. What is the nature of the seasonality?
3. Is there a within-group pattern (e.g., do January and
July exhibit similar patterns)?
4. Are there any outliers once seasonality has been
accounted for?
Importance It is important to know when analyzing a time series if there is
a significant seasonality effect. The seasonal subseries plot is
an excellent tool for determining if there is a seasonal pattern.
Related
Techniques
Box Plot
Run Sequence Plot
Software Seasonal subseries plots are available in a few general purpose
statistical software programs. It may possible to write macros
to generate this plot in most statistical software programs that
do not provide it directly. Seasonal subseries plots can be
generated using both Dataplot code and R code.
6.4.4.4. Common Approaches to Univariate Time Series

6.4.4.4. Common Approaches to Univariate Time
Series
There are a number of approaches to modeling time series.
We outline a few of the most common approaches below.
Trend,
Seasonal,
Residual
Decompositions
One approach is to decompose the time series into a trend,
seasonal, and residual component.
Triple exponential smoothing is an example of this approach.
Another example, called seasonal loess, is based on locally
weighted least squares and is discussed by Cleveland (1993).
We do not discuss seasonal loess in this handbook.
Frequency
Based Methods
Another approach, commonly used in scientific and
engineering applications, is to analyze the series in the
frequency domain. An example of this approach in modeling
a sinusoidal type data set is shown in the beam deflection
case study. The spectral plot is the primary tool for the
frequency analysis of time series.
Detailed discussions of frequency-based methods are
included in Bloomfield (1976), Jenkins and Watts (1968),
and Chatfield (1996).
Autoregressive
(AR) Models
A common approach for modeling univariate time series is
the autoregressive (AR) model:
where X
t
is the time series, A
t
is white noise, and
with denoting the process mean.
An autoregressive model is simply a linear regression of the
current value of the series against one or more prior values
of the series. The value of p is called the order of the AR
model.
AR models can be analyzed with one of various methods,
6.4.4.4. Common Approaches to Univariate Time Series
including standard linear least squares techniques. They also
have a straightforward interpretation.
Moving
Average (MA)
Models
Another common approach for modeling univariate time
series models is the moving average (MA) model:
where X
t
is the time series, is the mean of the series, A
t-i
are white noise, and
1
, ... ,
q
are the parameters of the
model. The value of q is called the order of the MA model.
That is, a moving average model is conceptually a linear
regression of the current value of the series against the white
noise or random shocks of one or more prior values of the
series. The random shocks at each point are assumed to come
from the same distribution, typically a normal distribution,
with location at zero and constant scale. The distinction in
this model is that these random shocks are propogated to
future values of the time series. Fitting the MA estimates is
more complicated than with AR models because the error
terms are not observable. This means that iterative non-linear
fitting procedures need to be used in place of linear least
squares. MA models also have a less obvious interpretation
than AR models.
Sometimes the ACF and PACF will suggest that a MA
model would be a better model choice and sometimes both
AR and MA terms should be used in the same model (see
Section 6.4.4.5).
Note, however, that the error terms after the model is fit
should be independent and follow the standard assumptions
for a univariate process.
Box-Jenkins
Approach
Box and Jenkins popularized an approach that combines the
moving average and the autoregressive approaches in the
book "Time Series Analysis: Forecasting and Control" (Box,
Jenkins, and Reinsel, 1994).
Although both autoregressive and moving average
approaches were already known (and were originally
investigated by Yule), the contribution of Box and Jenkins
was in developing a systematic methodology for identifying
and estimating models that could incorporate both
approaches. This makes Box-Jenkins models a powerful
class of models. The next several sections will discuss these
models in detail.
6.4.4.5. Box-Jenkins Models

Box-
Jenkins
Approach
The Box-Jenkins ARMA model is a combination of the AR
and MA models (described on the previous page):
where the terms in the equation have the same meaning as
given for the AR and MA model.
Comments
on Box-
Jenkins
Model
A couple of notes on this model.
1. The Box-Jenkins model assumes that the time series is
stationary. Box and Jenkins recommend differencing
non-stationary series one or more times to achieve
stationarity. Doing so produces an ARIMA model, with
the "I" standing for "Integrated".
2. Some formulations transform the series by subtracting
the mean of the series from each data point. This yields
a series with a mean of zero. Whether you need to do
this or not is dependent on the software you use to
estimate the model.
3. Box-Jenkins models can be extended to include
seasonal autoregressive and seasonal moving average
terms. Although this complicates the notation and
mathematics of the model, the underlying concepts for
terms are similar to the non-seasonal autoregressive and
moving average terms.
4. The most general Box-Jenkins model includes
difference operators, autoregressive terms, moving
average terms, seasonal difference operators, seasonal
autoregressive terms, and seasonal moving average
terms. As with modeling in general, however, only
necessary terms should be included in the model. Those
interested in the mathematical details can consult Box,
Jenkins and Reisel (1994), Chatfield (1996), or
Brockwell and Davis (2002).
Stages in
Box-
Jenkins
Modeling
There are three primary stages in building a Box-Jenkins time
series model.
1. Model Identification
2. Model Estimation
3. Model Validation
Remarks The following remarks regarding Box-Jenkins models should
be noted.
1. Box-Jenkins models are quite flexible due to the
inclusion of both autoregressive and moving average
terms.
2. Based on the Wold decomposition thereom (not
discussed in the Handbook), a stationary process can be
approximated by an ARMA model. In practice, finding
that approximation may not be easy.
3. Chatfield (1996) recommends decomposition methods
for series in which the trend and seasonal components
are dominant.
4. Building good ARIMA models generally requires more
experience than commonly used statistical methods such
as regression.
Sufficiently
Long
Series
Required
Typically, effective fitting of Box-Jenkins models requires at
least a moderately long series. Chatfield (1996) recommends
at least 50 observations. Many others would recommend at
least 100 observations.
6.4.4.6. Box-Jenkins Model Identification

Stationarity
and Seasonality
The first step in developing a Box-Jenkins model is to
determine if the series is stationary and if there is any
significant seasonality that needs to be modeled.
Detecting
stationarity
Stationarity can be assessed from a run sequence plot.
The run sequence plot should show constant location and
scale. It can also be detected from an autocorrelation plot.
Specifically, non-stationarity is often indicated by an
autocorrelation plot with very slow decay.
Detecting
seasonality
Seasonality (or periodicity) can usually be assessed from
an autocorrelation plot, a seasonal subseries plot, or a
spectral plot.
Differencing to
achieve
stationarity
Box and Jenkins recommend the differencing approach to
achieve stationarity. However, fitting a curve and
subtracting the fitted values from the original data can
also be used in the context of Box-Jenkins models.
Seasonal
differencing
At the model identification stage, our goal is to detect
seasonality, if it exists, and to identify the order for the
terms. For many series, the period is known and a single
seasonality term is sufficient. For example, for monthly
data we would typically include either a seasonal AR 12
term or a seasonal MA 12 term. For Box-Jenkins models,
we do not explicitly remove seasonality before fitting the
model. Instead, we include the order of the seasonal terms
in the model specification to the ARIMA estimation
software. However, it may be helpful to apply a seasonal
difference to the data and regenerate the autocorrelation
and partial autocorrelation plots. This may help in the
model idenfitication of the non-seasonal component of
the model. In some cases, the seasonal differencing may
remove most or all of the seasonality effect.
Identify p and q Once stationarity and seasonality have been addressed,
the next step is to identify the order (i.e., the p and q) of
the autoregressive and moving average terms.
Autocorrelation The primary tools for doing this are the autocorrelation
and Partial
Autocorrelation
Plots
plot and the partial autocorrelation plot. The sample
autocorrelation plot and the sample partial autocorrelation
plot are compared to the theoretical behavior of these
plots when the order is known.
Order of
Autoregressive
Process (p)
Specifically, for an AR(1) process, the sample
autocorrelation function should have an exponentially
decreasing appearance. However, higher-order AR
processes are often a mixture of exponentially decreasing
and damped sinusoidal components.
For higher-order autoregressive processes, the sample
autocorrelation needs to be supplemented with a partial
autocorrelation plot. The partial autocorrelation of an
AR(p) process becomes zero at lag p+1 and greater, so
we examine the sample partial autocorrelation function to
see if there is evidence of a departure from zero. This is
usually determined by placing a 95% confidence interval
on the sample partial autocorrelation plot (most software
programs that generate sample autocorrelation plots will
also plot this confidence interval). If the software
program does not generate the confidence band, it is
approximately , with N denoting the sample
size.
Order of
Moving
Average
Process (q)
The autocorrelation function of a MA(q) process becomes
zero at lag q+1 and greater, so we examine the sample
autocorrelation function to see where it essentially
becomes zero. We do this by placing the 95% confidence
interval for the sample autocorrelation function on the
sample autocorrelation plot. Most software that can
generate the autocorrelation plot can also generate this
confidence interval.
The sample partial autocorrelation function is generally
not helpful for identifying the order of the moving
average process.
Shape of
Autocorrelation
Function
The following table summarizes how we use the sample
autocorrelation function for model identification.
SHAPE INDICATED MODEL
Exponential,
decaying to
zero
Autoregressive model. Use the
partial autocorrelation plot to
identify the order of the
autoregressive model.
Alternating
positive and
negative,
decaying to
zero
Autoregressive model. Use the
partial autocorrelation plot to
help identify the order.
One or more
spikes, rest are
essentially
zero
Moving average model, order
identified by where plot becomes
zero.
Decay, starting
after a few
lags
Mixed autoregressive and
moving average model.
All zero or
close to zero
Data is essentially random.
High values at
fixed intervals
Include seasonal autoregressive
term.
No decay to
zero
Series is not stationary.
Mixed Models
Difficult to
Identify
In practice, the sample autocorrelation and partial
autocorrelation functions are random variables and will
not give the same picture as the theoretical functions.
This makes the model identification more difficult. In
particular, mixed models can be particularly difficult to
identify.
Although experience is helpful, developing good models
using these sample plots can involve much trial and error.
For this reason, in recent years information-based criteria
such as FPE (Final Prediction Error) and AIC (Aikake
Information Criterion) and others have been preferred
and used. These techniques can help automate the model
identification process. These techniques require computer
software to use. Fortunately, these techniques are
available in many commerical statistical software
programs that provide ARIMA modeling capabilities.
For additional information on these techniques, see
Brockwell and Davis (1987, 2002).
Examples We show a typical series of plots for performing the
initial model identification for
1. the southern oscillations data and
2. the CO
2
monthly concentrations data.
6.4.4.6.1. Model Identification for Southern Oscillations Data

6.4.4.6.1. Model Identification for Southern
Oscillations Data
Example for
Southern
Oscillations
We show typical series of plots for the initial model
identification stages of Box-Jenkins modeling for two
different examples.
The first example is for the southern oscillations data set.
We start with the run sequence plot and seasonal subseries
plot to determine if we need to address stationarity and
seasonality.
Run Sequence
Plot
The run sequence plot indicates stationarity.
Seasonal
Subseries Plot
The seasonal subseries plot indicates that there is no
significant seasonality.
Since the above plots show that this series does not
exhibit any significant non-stationarity or seasonality, we
generate the autocorrelation and partial autocorrelation
plots of the raw data.
Autocorrelation
Plot
The autocorrelation plot shows a mixture of exponentially
decaying and damped sinusoidal components. This
indicates that an autoregressive model, with order greater
than one, may be appropriate for these data. The partial
autocorrelation plot should be examined to determine the
order.
Partial
Autocorrelation
Plot
The partial autocorrelation plot suggests that an AR(2)
model might be appropriate.
In summary, our intial attempt would be to fit an AR(2)
model with no seasonal terms and no differencing or trend
removal. Model validation should be performed before
accepting this as a final model.
6.4.4.6.2. Model Identification for the CO2 Concentrations Data

6.4.4.6.2. Model Identification for the CO
2
Concentrations Data
Example for
Monthly CO
2
Concentrations
The second example is for the monthly CO
2
concentrations data set. As before, we start with the run
sequence plot to check for stationarity.
Run Sequence
Plot
The initial run sequence plot of the data indicates a rising
trend. A visual inspection of this plot indicates that a
simple linear fit should be sufficient to remove this
upward trend.
Linear Trend
Removed
This plot contains the residuals from a linear fit to the
original data. After removing the linear trend, the run
sequence plot indicates that the data have a constant
location and variance, which implies stationarity.
However, the plot does show seasonality. We generate an
autocorrelation plot to help determine the period followed
by a seasonal subseries plot.
Autocorrelation
Plot
The autocorrelation plot shows an alternating pattern of
positive and negative spikes. It also shows a repeating
pattern every 12 lags, which indicates a seasonality effect.
The two connected lines on the autocorrelation plot are
95% and 99% confidence intervals for statistical
significance of the autocorrelations.
Seasonal
Subseries Plot
A significant seasonal pattern is obvious in this plot, so
we need to include seasonal terms in fitting a Box-Jenkins
model. Since this is monthly data, we would typically
include either a lag 12 seasonal autoregressive and/or
moving average term.
To help identify the non-seasonal components, we will
take a seasonal difference of 12 and generate the
autocorrelation plot on the seasonally differenced data.
Autocorrelation
Plot for
Seasonally
Differenced
Data
This autocorrelation plot shows a mixture of exponential
decay and a damped sinusoidal pattern. This indicates that
an AR model, with order greater than one, may be
appropriate. We generate a partial autocorrelation plot to
help identify the order.
Partial
Autocorrelation
Plot of
Seasonally
Differenced
Data
The partial autocorrelation plot suggests that an AR(2)
model might be appropriate since the partial
autocorrelation becomes zero after the second lag. The lag
12 is also significant, indicating some remaining
seasonality.
In summary, our intial attempt would be to fit an AR(2)
model with a seasonal AR(12) term on the data with a
linear trend line removed. We could try the model both
with and without seasonal differencing applied. Model
validation should be performed before accepting this as a
final model.
6.4.4.6.3. Partial Autocorrelation Plot

Purpose:
Model
Identification
for Box-
Jenkins
Models
Partial autocorrelation plots (Box and Jenkins, pp. 64-65,
1970) are a commonly used tool for model identification in
Box-Jenkins models.
The partial autocorrelation at lag k is the autocorrelation
between X
t
and X
t-k
that is not accounted for by lags 1
through k-1.
There are algorithms, not discussed here, for computing the
partial autocorrelation based on the sample autocorrelations.
See (Box, Jenkins, and Reinsel 1970) or (Brockwell, 1991)
for the mathematical details.
Specifically, partial autocorrelations are useful in
identifying the order of an autoregressive model. The partial
autocorrelation of an AR(p) process is zero at lag p+1 and
greater. If the sample autocorrelation plot indicates that an
AR model may be appropriate, then the sample partial
autocorrelation plot is examined to help identify the order.
We look for the point on the plot where the partial
autocorrelations essentially become zero. Placing a 95%
confidence interval for statistical significance is helpful for
this purpose.
The approximate 95% confidence interval for the partial
autocorrelations are at .
Sample Plot
This partial autocorrelation plot shows clear statistical
significance for lags 1 and 2 (lag 0 is always 1). The next
few lags are at the borderline of statistical significance. If
the autocorrelation plot indicates that an AR model is
appropriate, we could start our modeling with an AR(2)
model. We might compare this with an AR(3) model.
Definition Partial autocorrelation plots are formed by
Vertical axis: Partial autocorrelation coefficient at
lag h.
Horizontal
axis:
Time lag h (h = 0, 1, 2, 3, ...).
In addition, 95% confidence interval bands are typically
included on the plot.
Questions The partial autocorrelation plot can help provide answers to
the following questions:
1. Is an AR model appropriate for the data?
2. If an AR model is appropriate, what order should we
use?
Related
Techniques
Run Sequence Plot
Spectral Plot
Case Study The partial autocorrelation plot is demonstrated in the Negiz
data case study.
Software Partial autocorrelation plots are available in many general
6.4.4.7. Box-Jenkins Model Estimation

6.4.4.7. Box-Jenkins Model Estimation
Use
Software
Estimating the parameters for the Box-Jenkins models is a
quite complicated non-linear estimation problem. For this
reason, the parameter estimation should be left to a high
quality software program that fits Box-Jenkins models.
Fortunately, many commerical statistical software programs
now fit Box-Jenkins models.
Approaches The main approaches to fitting Box-Jenkins models are non-
linear least squares and maximum likelihood estimation.
Maximum likelihood estimation is generally the preferred
technique. The likelihood equations for the full Box-Jenkins
model are complicated and are not included here. See
(Brockwell and Davis, 1991) for the mathematical details.
Model
Estimation
Example
The Negiz case study shows an example of the Box-Jenkins
model-fitting.
6.4.4.8. Box-Jenkins Model Diagnostics

6.4.4.8. Box-Jenkins Model Diagnostics
Assumptions
for a Stable
Univariate
Process
Model diagnostics for Box-Jenkins models is similar to
model validation for non-linear least squares fitting.
That is, the error term A
t
is assumed to follow the
assumptions for a stationary univariate process. The residuals
should be white noise (or independent when their
distributions are normal) drawings from a fixed distribution
with a constant mean and variance. If the Box-Jenkins model
is a good model for the data, the residuals should satisfy
these assumptions.
If these assumptions are not satisfied, we need to fit a more
appropriate model. That is, we go back to the model
identification step and try to develop a better model.
Hopefully the analysis of the residuals can provide some
clues as to a more appropriate model.
4-Plot of
Residuals
As discussed in the EDA chapter, one way to assess if the
residuals from the Box-Jenkins model follow the
assumptions is to generate a 4-plot of the residuals and an
autocorrelation plot of the residuals. One could also look at
the value of the Box-Ljung (1978) statistic.
An example of analyzing the residuals from a Box-Jenkins
model is given in the Negiz data case study.
Information Technology Laboratory Homepage
http://www.nist.gov/itl/[6/27/2012 2:36:39 PM]
Topics
Biometrics
Computer Security
Math
Networking
Statistics
Software
Usability
Divisions
Applied and Computational
Mathematics
Advanced Network Technologies
Computer Security
Information Access
Software and Systems
Statistical Engineering
Staff Directory
News
New Quantum Computing Algorithm Could Simulate Giant
Particle Accelerators
NIST Releases Final Version of Revised Bluetooth Security Guide
NIST Will Hold Big Data Workshop June 13-14
>> see all news ...
Conferences and Events
ICT Supply Chain Risk Management (SCRM) Workshop -
POSTPONED
July 11-12, 2012
Revised Draft FIPS 2012 Workshop
July 25, 2012
Programs/Projects
Complex Systems
Pervasive Information Technology
Virtual Measurements
Health Information Technology
Voting
Quantum
Cloud Computing

Contact
Charles H. Romine, Director
James St. Pierre, Deputy Director
General Information:
301-975-2900 Telephone
301-975-2378 Facsimile
100 Bureau Drive, M/S 8900
Gaithersburg, MD 20899-8900
Sign Up for ITL Newsletters &
Bulletins:

NIST Home > Information Technology Laboratory Homepage
NIST Time NIST Home About NIST Contact Us A-Z Site Index

Popular Links
Computer Security Resource Center
Biometrics Resource Center
ITL Brochure - January 2012
NIST Voting Activities
Federal Processing Information
Standards (FIPS)
ANSI Accredited Standards
Development
Current ITL Newsletter
ITL Bulletins
ITL SURF Program
Comments received in Response to
Internet Policy Task Force
Cybersecurity Green Paper
NIST Special Publication 500 Series
Comments received in Response to
Models To Advance Voluntary
Corporate Notification to
Consumers Regarding the Illicit Use
of Computer Equipment by Botnets
and Related Malware Federal
Register Notice

NIST Digital Library of Mathematical Functions
Receives IT Award
1 2 3 4
About ITL Publications Topic/Subject Areas Products/Services News/Multimedia Programs/Projects
Enter email address Go
Search
Information Technology Laboratory Homepage
http://www.nist.gov/itl/[6/27/2012 2:36:39 PM]
Sign Up for NIST E-mail alerts:

The National Institute of Standards and Technology (NIST) is an agency of the U.S. Department of Commerce.
Privacy Policy / Security Notice / Accessibility Statement / Disclaimer / Freedom of Information Act (FOIA) /
Environmental Policy Statement / No Fear Act Policy / NIST Information Quality Standards /
Scientific Integrity Summary
Date created: August 24, 2010 | Last updated: May 23, 2012 Contact: Webmaster
Enter email address
Go
6.4.4.9. Example of Univariate Box-Jenkins Analysis

6.4.4.9. Example of Univariate Box-Jenkins
Analysis
Series F We analyze the series F data set in Box, Jenkins, and Reinsel,
1994. A plot of the 70 raw data points is shown below.
The data do not appear to have a seasonal component or a
noticeable trend. (The stationarity of the series was verified
by fitting a straight line to the data versus time period. The
slope was not found to be significantly different from zero
(p-value = 0.2).)
Model
Identification
We compute the autocorrelation function (ACF) of the data
for the first 35 lags to determine the type of model to fit to
the data. We list the numeric results and plot the ACF (along
with 95 % confidence limits) versus the lag number.
Lag ACF
0 1.000000000
1 -0.389878319
2 0.304394082
3 -0.165554717
4 0.070719321
5 -0.097039288
6 -0.047057692
7 0.035373112
8 -0.043458199
9 -0.004796162
10 0.014393137
11 0.109917200
12 -0.068778492
13 0.148034489
14 0.035768581
15 -0.006677806
16 0.173004275
17 -0.111342583
18 0.019970791
19 -0.047349722
20 0.016136806
21 0.022279561
22 -0.078710582
23 -0.009577413
24 -0.073114034
25 -0.019503289
26 0.041465024
27 -0.022134370
28 0.088887299
29 0.016247148
30 0.003946351
31 0.004584069
32 -0.024782198
33 -0.025905040
34 -0.062879966
35 0.026101117
The ACF values alternate in sign and decay quickly after lag
2, indicating that an AR(2) model is appropriate for the data.
Model
Fitting
We fit an AR(2) model to the data.
The model fitting results are shown below.
Source Estimate Standard Error
------ -------- --------------
1
-0.3198 0.1202
2
0.1797 0.1202
= 51.1286
Test randomness of residuals:
Standardized Runs Statistic Z = 0.4887, p-value =
0.625
Forecasting Using our AR(2) model, we forcast values six time periods
into the future.
Period Prediction Standard Error
71 60.6405 10.9479
72 43.0317 11.4941
73 55.4274 11.9015
74 48.2987 12.0108
75 52.8061 12.0585
76 50.0835 12.0751
The "historical" data and forecasted values (with 90 %
confidence limits) are shown in the graph below.
6.4.4.10. Box-Jenkins Analysis on Seasonal Data
http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc44a.htm[6/27/2012 2:36:41 PM]

Series G This example illustrates a Box-Jenkins time series analysis
for seasonal data using the series G data set in Box, Jenkins,
and Reinsel, 1994. A plot of the 144 observations is shown
below.
Non-constant variance can be removed by performing a
natural log transformation.
Next, we remove trend in the series by taking first
differences. The resulting series is shown below.
Analyzing
Autocorrelation
Plot for
Seasonality
To identify an appropriate model, we plot the ACF of the
time series.
If very large autocorrelations are observed at lags spaced n
periods apart, for example at lags 12 and 24, then there is
evidence of periodicity. That effect should be removed, since
the objective of the identification stage is to reduce the
autocorrelation throughout. So if simple differencing is not
enough, try seasonal differencing at a selected period, such
as 4, 6, or 12. In our example, the seasonal period is 12.
A plot of Series G after taking the natural log, first
differencing, and seasonal differencing is shown below.
The number of seasonal terms is rarely more than one. If you
know the shape of your forecast function, or you wish to
assign a particular shape to the forecast function, you can
select the appropriate number of terms for seasonal AR or
seasonal MA models.
The book by Box and Jenkins, Time Series Analysis
Forecasting and Control (the later edition is Box, Jenkins
and Reinsel, 1994) has a discussion on these forecast
functions on pages 326 - 328. Again, if you have only a faint
notion, but you do know that there was a trend upwards
before differencing, pick a seasonal MA term and see what
comes out in the diagnostics.
An ACF plot of the seasonal and first differenced natural log
of series G is shown below.
The plot has a few spikes, but most autocorrelations are near
zero, indicating that a seasonal MA(1) model is appropriate.
Model Fitting We fit an MA(1) model to the data.
The model fitting results are shown below.
Seasonal
Estimate MA(1) MA(1)
-------- ------- -------
Parameter -0.4018 -0.5569
Standard Error 0.0896 0.0731
Log likelihood = 244.7
AIC = -483.4
Test the randomness of the residuals up to 30 lags using the
Box-Ljung test. Recall that the degrees of freedom for the
critical region must be adjusted to account for two estimated
parameters.
H
0
: The residuals are random.
H
a
: The residuals are not random.
Test statistic: Q = 29.4935
Degrees of freedom: h = 30 - 2 = 28
Critical value:
2
1-,h
= 41.3371
0
if Q > 41.3371
Since the null hypothesis of the Box-Ljung test is not
rejected we conclude that the fitted model is adequate.
Forecasting Using our seasonal MA(1) model, we forcast values 12
periods into the future and compute 90 % confidence limits.
Lower Upper
Period Limit Forecast Limit
------ -------- -------- --------
145 424.0234 450.7261 478.4649
146 396.7861 426.0042 456.7577
147 442.5731 479.3298 518.4399
148 451.3902 492.7365 537.1454
149 463.3034 509.3982 559.3245
150 527.3754 583.7383 645.2544
151 601.9371 670.4625 745.7830
152 595.7602 667.5274 746.9323
153 495.7137 558.5657 628.5389
154 439.1900 497.5430 562.8899
155 377.7598 430.1618 489.1730
156 417.3149 477.5643 545.7760
6.4.5. Multivariate Time Series Models

If each time
series
observation
is a vector
of numbers,
you can
model them
using a
multivariate
form of the
Box-Jenkins
model
The multivariate form of the Box-Jenkins univariate models
is sometimes called the ARMAV model, for AutoRegressive
Moving Average Vector or simply vector ARMA process.
The ARMAV model for a stationary multivariate time series,
with a zero mean vector, represented by
is of the form
where
x
t
and a
t
are n x 1 column vectors with a
t
representing
multivariate white noise
are n x n matrices for autoregressive and moving
average parameters
E[a
t
] = 0
where
a
is the dispersion or covariance matrix of a
t
As an example, for a bivariate series with n = 2, p = 2, and q
= 1, the ARMAV(2,1) model is:
with
Estimation
of
parameters
and
covariance
matrix
difficult
The estimation of the matrix parameters and covariance
matrix is complicated and very difficult without computer
software. The estimation of the Moving Average matrices is
especially an ordeal. If we opt to ignore the MA
component(s) we are left with the ARV model given by:
where
x
t
is a vector of observations, x
1t
, x
2t
, ... , x
nt
at time t
a
t
is a vector of white noise, a
1t
, a
2t
, ... , a
nt
at time t

is a n x n matrix of autoregressive parameters
E[a
t
] = 0
where
a
is the dispersion or covariance matrix
A model with p autoregressive matrix parameters is an
ARV(p) model or a vector AR model.
The parameter matrices may be estimated by multivariate
least squares, but there are other methods such as maximium
likelihood estimation.
Interesting
properties
of
parameter
matrices
There are a few interesting properties associated with the phi
or AR parameter matrices. Consider the following example
for a bivariate series with n =2, p = 2, and q = 0. The
ARMAV(2,0) model is:
Without loss of generality, assume that the X series is input and the Y series
are output and that the mean vector = (0,0).
Therefore, tranform the observation by subtracting their respective averages.
Diagonal
terms of
Phi matrix
The diagonal terms of each Phi matrix are the scalar estimates for each
series, in this case:
1.11
,
2.11
for the input series X,
1.22
, .
2.22
for the output series Y.
Transfer
mechanism
The lower off-diagonal elements represent the influence of the input on the
output.
This is called the "transfer" mechanism or transfer-function model as
discussed by Box and Jenkins in Chapter 11. The terms here correspond to
their terms.
The upper off-diagonal terms represent the influence of the output on the
input.
Feedback This is called "feedback". The presence of feedback can also be seen as a
high value for a coefficient in the correlation matrix of the residuals. A "true"
transfer model exists when there is no feedback.
This can be seen by expressing the matrix form into scalar form:
Delay Finally, delay or "dead' time can be measured by studying the lower off-
diagonal elements again.
If, for example,
1.21
is non-significant, the delay is 1 time period.
6.4.5.1. Example of Multivariate Time Series Analysis

Bivariate
Gas
Furance
Example
The gas furnace data from Box, Jenkins, and Reinsel, 1994 is used to
illustrate the analysis of a bivariate time series. Inside the gas furnace, air and
methane were combined in order to obtain a mixture of gases containing
CO
2
(carbon dioxide). The input series is the methane gas feedrate described
by
Methane Gas Input Feed = 0.60 - 0.04 X(t)
the CO
2
concentration was the output series, Y(t). In this experiment 296
successive pairs of observations (X
t,
Y
t
) were collected from continuous
records at 9-second intervals. For the analysis described here, only the first
60 pairs were used. We fit an ARV(2) model as described in 6.4.5.
Plots of
input and
output
series
The plots of the input and output series are displayed below.
Model
Fitting
The scalar form of the ARV(2) model is the following.
The equation for x
t
corresponds to gas rate while the equation for y
t
corresponds to CO
2
concentration.
The parameter estimates for the equation associated with gas rate are the
following.
Estimate Std. Err. t value Pr(>|t|)
a
1t
0.003063 0.035769 0.086 0.932
1.11
1.683225 0.123128 13.671 < 2e-16
2.11
-0.860205 0.165886 -5.186 3.44e-06
1.12
-0.076224 0.096947 -0.786 0.435
2.12
0.044774 0.082285 0.544 0.589
Multiple R-Squared: 0.9387
p-value: < 2.2e-16
The parameter estimates for the equation associated with CO
2
concentration
are the following.
Estimate Std. Err. t value Pr(>|t|)
a
2t
-0.03372 0.01615 -2.088 0.041641
1.22
1.22630 0.04378 28.013 < 2e-16
2.22
-0.40927 0.03716 -11.015 2.57e-15
0.22898 0.05560 4.118 0.000134
1.21
2.21
-0.80532 0.07491 -10.751 6.29e-15
Multiple R-Squared: 0.9985
F-statistic: 8978 based on 4 and 53 degrees of freedom
p-value: < 2.2e-16
Box-Ljung tests performed for each series to test the randomness of the first
24 residuals were not significant. The p-values for the tests using CO
2
concentration residuals and gas rate residuals were 0.4 and 0.6, respectively.
Forecasting The forecasting method is an extension of the model and follows the theory
outlined in the previous section. The forecasted values of the next six
observations (61-66) and the associated 90 % confidence limits are shown
below for each series.
90% Lower Concentration 90% Upper
Observation Limit Forecast Limit
----------- --------- -------- ---------
61 51.0 51.2 51.4
62 51.0 51.3 51.6
63 50.6 51.0 51.4
64 49.8 50.5 51.1
65 48.7 50.0 51.3
66 47.6 49.7 51.8
90% Lower Rate 90% Upper
Observation Limit Forecast Limit
----------- --------- -------- ---------
61 0.795 1.231 1.668
62 0.439 1.295 2.150
63 0.032 1.242 2.452
64 -0.332 1.128 2.588
65 -0.605 1.005 2.614
66 -0.776 0.908 2.593
6.5. Tutorials

6.5. Tutorials
Tutorial
contents
1. What do we mean by "Normal" data?
2. What do we do when data are "Non-normal"?
3. Elements of Matrix Algebra
1. Numerical Examples
2. Determinant and Eigenstructure
4. Elements of Multivariate Analysis
1. Mean vector and Covariance Matrix
2. The Multivariate Normal Distribution
3. Hotelling's T
2
1. Example of Hotelling's T
2
Test
2. Example 1 (continued)
3. Example 2 (multiple groups)
4. Hotelling's T
2
Chart
5. Principal Components
1. Properties of Principal Components
2. Numerical Example
6.5.1. What do we mean by "Normal" data?

6.5. Tutorials
The Normal
distribution
model
"Normal" data are data that are drawn (come from) a
population that has a normal distribution. This distribution is
inarguably the most important and the most frequently used
distribution in both the theory and application of statistics. If
X is a normal random variable, then the probability
distribution of X is
Normal
probability
distribution
Parameters
of normal
distribution
The parameters of the normal distribution are the mean and
the standard deviation (or the variance
2
). A special
notation is employed to indicate that X is normally distributed
with these parameters, namely
X ~ N( , ) or X ~ N( ,
2
).
Shape is
symmetric
and unimodal
The shape of the normal distribution is symmetric and
unimodal. It is called the bell-shaped or Gaussian
distribution after its inventor, Gauss (although De Moivre
also deserves credit).
The visual appearance is given below.
Property of
probability
distributions
is that area
under curve
equals one
A property of a special class of non-negative functions,
called probability distributions, is that the area under the
curve equals unity. One finds the area under any portion of
the curve by integrating the distribution between the specified
limits. The area under the bell-shaped curve of the normal
distribution can be shown to be equal to 1, and therefore the
normal distribution is a probability distribution.
Interpretation
of
There is a simple interpretation of
68.27% of the population fall between +/- 1
The
cumulative
normal
distribution
The cumulative normal distribution is defined as the
probability that the normal variate is less than or equal to
some value v, or
Unfortunately this integral cannot be evaluated in closed
form and one has to resort to numerical methods. But even
so, tables for all possible values of and would be
required. A change of variables rescues the situation. We let
Now the evaluation can be made independently of and ;
that is,
where (.) is the cumulative distribution function of the
standard normal distribution ( = 0, = 1).
Tables for the
cumulative
standard
normal
distribution
Tables of the cumulative standard normal distribution are
given in every statistics textbook and in the handbook. A rich
variety of approximations can be found in the literature on
numerical methods.
For example, if = 0 and = 1 then the area under the curve
from - 1 to + 1 is the area from 0 - 1 to 0 + 1, which
is 0.6827. Since most standard normal tables give area to the
left of the lookup value, they will have for z = 1 an area of
.8413 and for z = -1 an area of .1587. By subtraction we
obtain the area between -1 and +1 to be .8413 - .1587 =
.6826.
6.5.2. What do we do when data are non-normal

6.5. Tutorials
6.5.2. What to do when data are non-normal
Often it is
possible to
transform non-
normal data
into
approximately
normal data
Non-normality is a way of life, since no characteristic (height,
weight, etc.) will have exactly a normal distribution. One
strategy to make non-normal data resemble normal data is by
using a transformation. There is no dearth of transformations in
statistics; the issue is which one to select for the situation at
hand. Unfortunately, the choice of the "best" transformation is
generally not obvious.
This was recognized in 1964 by G.E.P. Box and D.R. Cox. They
wrote a paper in which a useful family of power transformations
was suggested. These transformations are defined only for
positive data values. This should not pose any problem because
a constant can always be added if the set of observations
contains one or more negative values.
The Box-Cox power transformations are given by
The Box-Cox
Transformation
Given the vector of data observations x = x
1
, x
2
, ...x
n
, one way
to select the power is to use the that maximizes the
logarithm of the likelihood function
The logarithm
of the
likelihood
function
where
is the arithmetic mean of the transformed data.
Confidence
bound for
In addition, a confidence bound (based on the likelihood ratio
statistic) can be constructed for as follows: A set of values
that represent an approximate 100(1- )% confidence bound for
is formed from those that satisfy
where denotes the maximum likelihood estimator for and
2
1-, 1
is the 100(1- ) percentile of the chi-square distribution
with 1 degree of freedom.
Example of the
Box-Cox
scheme
To illustrate the procedure, we used the data from Johnson and
Wichern's textbook (Prentice Hall 1988), Example 4.14. The
observations are microwave radiation measurements.
Sample data
.15 .09 .18 .10 .05 .12 .08
.05 .08 .10 .07 .02 .01 .10
.10 .10 .02 .10 .01 .40 .10
.05 .03 .05 .15 .10 .15 .09
.08 .18 .10 .20 .11 .30 .02
.20 .20 .30 .30 .40 .30 .05
Table of log-
likelihood
values for
various values
of
The values of the log-likelihood function obtained by varying
from -2.0 to 2.0 are given below.
LLF LLF LLF
-2.0 7.1146 -0.6 89.0587 0.7 103.0322
-1.9 14.1877 -0.5 92.7855 0.8 101.3254
-1.8 21.1356 -0.4 96.0974 0.9 99.3403
-1.7 27.9468 -0.3 98.9722 1.0 97.1030
-1.6 34.6082 -0.2 101.3923 1.1 94.6372
-1.5 41.1054 -0.1 103.3457 1.2 91.9643
-1.4 47.4229 0.0 104.8276 1.3 89.1034
-1.3 53.5432 0.1 105.8406 1.4 86.0714
1.2 59.4474 0.2 106.3947 1.5 82.8832
-1.1 65.1147 0.3 106.5069 1.6 79.5521
-0.9 75.6471 0.4 106.1994 1.7 76.0896
-0.8 80.4625 0.5 105.4985 1.8 72.5061
-0.7 84.9421 0.6 104.4330 1.9 68.8106
This table shows that = .3 maximizes the log-likelihood
function (LLF). This becomes 0.28 if a second digit of accuracy
is calculated.
The Box-Cox transform is also discussed in Chapter 1 under the
Box Cox Linearity Plot and the Box Cox Normality Plot. The
Box-Cox normality plot discussion provides a graphical method
for choosing to transform a data set to normality. The criterion
used to choose for the Box-Cox linearity plot is the value of
that maximizes the correlation between the transformed x-values
and the y-values when making a normal probability plot of the
(transformed) data.
6.5.3. Elements of Matrix Algebra

6.5. Tutorials
Elementary Matrix Algebra
Basic
definitions
and
operations of
matrix
algebra -
needed for
multivariate
analysis
Vectors and matrices are arrays of numbers. The algebra
for symbolic operations on them is different from the
algebra for operations on scalars, or single numbers. For
example there is no division in matrix algebra, although
there is an operation called "multiplying by an inverse". It
is possible to express the exact equivalent of matrix algebra
equations in terms of scalar algebra expressions, but the
results look rather messy.
It can be said that the matrix algebra notation is shorthand
for the corresponding scalar longhand.
Vectors A vector is a column of numbers
The scalars a
i
are the elements of vector a.
Transpose The transpose of a, denoted by a', is the row arrangement
of the elements of a.
Sum of two
vectors
The sum of two vectors (say, a and b) is the vector of sums
of corresponding elements.
The difference of two vectors is the vector of differences of
corresponding elements.
Product of
a'b
The product a'b is a scalar formed by
which may be written in shortcut notation as
where a
i
and b
i
are the ith elements of vector a and b,
respectively.
Product of
ab'
The product ab' is a square matrix
Product of
scalar times a
vector
The product of a scalar k, times a vector a is k times each
element of a
A matrix is a
rectangular
table of
numbers
A matrix is a rectangular table of numbers, with p rows and
n columns. It is also referred to as an array of n column
vectors of length p. Thus
is a p by n matrix. The typical element of A is a
ij
, denoting
the element of row i and column j.
Matrix
addition and
Matrices are added and subtracted on an element-by-
element basis. Thus
subtraction
Matrix
multiplication
Matrix multiplication involves the computation of the sum
of the products of elements from a row of the first matrix
(the premultiplier on the left) and a column of the second
matrix (the postmultiplier on the right). This sum of
products is computed for every combination of rows and
columns. For example, if A is a 2 x 3 matrix and B is a 3 x
2 matrix, the product AB is
Thus, the product is a 2 x 2 matrix. This came about as
follows: The number of columns of A must be equal to the
number of rows of B. In this case this is 3. If they are not
equal, multiplication is impossible. If they are equal, then
the number of rows of the product AB is equal to the
number of rows of A and the number of columns is equal to
the number of columns of B.
Example of
3x2 matrix
multiplied by
a 2x3
It follows that the result of the product BA is a 3 x 3 matrix
General case
for matrix
multiplication
In general, if A is a k x p matrix and B is a p x n matrix, the
product AB is a k x n matrix. If k = n, then the product BA
can also be formed. We say that matrices conform for the
operations of addition, subtraction or multiplication when
their respective orders (numbers of row and columns) are
such as to permit the operations. Matrices that do not
conform for addition or subtraction cannot be added or
subtracted. Matrices that do not conform for multiplication
cannot be multiplied.
6.5.3.1. Numerical Examples

6.5. Tutorials
Numerical
examples of
matrix
operations
Numerical examples of the matrix operations described on
the previous page are given here to clarify these operations.
Sample
matrices
If
then
Matrix
addition,
subtraction,
and
multipication
and
Multiply
matrix by a
scalar
To multiply a a matrix by a given scalar, each element of
the matrix is multiplied by that scalar
Pre-
multiplying
matrix by
transpose of
a vector
Pre-multiplying a p x n matrix by the transpose of a p-
element vector yields a n-element transpose
Post-
multiplying
matrix by
vector
Post-multiplying a p x n matrix by an n-element vector
yields an n-element vector
Quadratic
form
It is not possible to pre-multiply a matrix by a column
vector, nor to post-multiply a matrix by a row vector. The
matrix product a'Ba yields a scalar and is called a quadratic
form. Note that B must be a square matrix if a'Ba is to
conform to multiplication. Here is an example of a quadratic
form
Inverting a
matrix
The matrix analog of division involves an operation called
inverting a matrix. Only square matrices can be inverted.
Inversion is a tedious numerical procedure and it is best
performed by computers. There are many ways to invert a
matrix, but ultimately whichever method is selected by a
program is immaterial. If you wish to try one method by
hand, a very popular numerical method is the Gauss-Jordan
method.
Identity
matrix
To augment the notion of the inverse of a matrix, A
-1
(A
inverse) we notice the following relation
A
-1
A = A A
-1
= I
I is a matrix of form
I is called the identity matrix and is a special case of a
diagonal matrix. Any matrix that has zeros in all of the off-
diagonal positions is a diagonal matrix.
6.5.3.2. Determinant and Eigenstructure

6.5. Tutorials
A matrix
determinant is
difficult to
define but a
very useful
number
Unfortunately, not every square matrix has an inverse
(although most do). Associated with any square matrix is a
single number that represents a unique function of the
numbers in the matrix. This scalar function of a square
matrix is called the determinant. The determinant of a
matrix A is denoted by |A|. A formal definition for the
deteterminant of a square matrix A = (a
ij
) is somewhat
beyond the scope of this Handbook. Consult any good
linear algebra textbook if you are interested in the
mathematical details.
Singular
matrix
As is the case of inversion of a square matrix, calculation
of the determinant is tedious and computer assistance is
needed for practical calculations. If the determinant of the
(square) matrix is exactly zero, the matrix is said to be
singular and it has no inverse.
Determinant
of variance-
covariance
matrix
Of great interest in statistics is the determinant of a square
symmetric matrix D whose diagonal elements are sample
variances and whose off-diagonal elements are sample
covariances. Symmetry means that the matrix and its
transpose are identical (i.e., A = A'). An example is
where s
1
and s
2
are sample standard deviations and r
ij
is
the sample correlation.
D is the sample variance-covariance matrix for
observations of a multivariate vector of p elements. The
determinant of D, in this case, is sometimes called the
generalized variance.
Characteristic
equation
In addition to a determinant and possibly an inverse, every
square matrix has associated with it a characteristic
equation. The characteristic equation of a matrix is formed
by subtracting some particular value, usually denoted by
the greek letter (lambda), from each diagonal element of
the matrix, such that the determinant of the resulting
matrix is equal to zero. For example, the characteristic
equation of a second order (2 x 2) matrix A may be
written as
Definition of
the
characteristic
equation for
2x2 matrix
Eigenvalues of
a matrix
For a matrix of order p, there may be as many as p
different values for that will satisfy the equation. These
different values are called the eigenvalues of the matrix.
Eigenvectors
of a matrix
Associated with each eigenvalue is a vector, v, called the
eigenvector. The eigenvector satisfies the equation
Av = v
Eigenstructure
of a matrix
If the complete set of eigenvalues is arranged in the
diagonal positions of a diagonal matrix V, the following
relationship holds
AV = VL
This equation specifies the complete eigenstructure of A.
Eigenstructures and the associated theory figure heavily in
multivariate procedures and the numerical evaluation of L
and V is a central computing problem.
6.5.4. Elements of Multivariate Analysis

6.5. Tutorials
Multivariate
analysis
Multivariate analysis is a branch of statistics concerned
with the analysis of multiple measurements, made on one or
several samples of individuals. For example, we may wish
to measure length, width and weight of a product.
Multiple
measurement,
or
observation,
as row or
column
vector
A multiple measurement or observation may be expressed
as
x = [4 2 0.6]
referring to the physical properties of length, width and
weight, respectively. It is customary to denote multivariate
quantities with bold letters. The collection of measurements
on x is called a vector. In this case it is a row vector. We
could have written x as a column vector.
Matrix to
represent
more than
one multiple
measurement
If we take several such measurements, we record them in a
rectangular array of numbers. For example, the X matrix
below represents 5 observations, on each of three variables.
By
convention,
rows
typically
represent
In this case the number of rows, (n = 5), is the number of
observations, and the number of columns, (p = 3), is the
number of variables that are measured. The rectangular
array is an assembly of n row vectors of length p. This array
is called a matrix, or, more specifically, a n by p matrix. Its
observations
and columns
represent
variables
name is X. The names of matrices are usually written in
bold, uppercase letters, as in Section 6.5.3. We could just as
well have written X as a p (variables) by n (measurements)
matrix as follows:
Definition of
Transpose
A matrix with rows and columns exchanged in this manner
is called the transpose of the original matrix.
6.5.4.1. Mean Vector and Covariance Matrix

6.5. Tutorials
The first step in analyzing multivariate data is computing the
mean vector and the variance-covariance matrix.
Sample
data
matrix
Consider the following matrix:
The set of 5 observations, measuring 3 variables, can be
described by its mean vector and variance-covariance matrix.
The three variables, from left to right are length, width, and
height of a certain object, for example. Each row vector X
i
is
another observation of the three variables (or components).
Definition
of mean
vector and
variance-
covariance
matrix
The mean vector consists of the means of each variable and
the variance-covariance matrix consists of the variances of the
variables along the main diagonal and the covariances between
each pair of variables in the other matrix positions.
The formula for computing the covariance of the variables X
and Y is
with and denoting the means of X and Y, respectively.
Mean
vector and
variance-
covariance
matrix for
sample
data
matrix
The results are:

where the mean vector contains the arithmetic averages of the
three variables and the (unbiased) variance-covariance matrix
S is calculated by
where n = 5 for this example.
Thus, 0.025 is the variance of the length variable, 0.0075 is the
covariance between the length and the width variables,
0.00175 is the covariance between the length and the height
variables, 0.007 is the variance of the width variable, 0.00135
is the covariance between the width and height variables and
.00043 is the variance of the height variable.
Centroid,
dispersion
matix
The mean vector is often referred to as the centroid and the
variance-covariance matrix as the dispersion or dispersion
matrix. Also, the terms variance-covariance matrix and
covariance matrix are used interchangeably.

6.5.4.2. The Multivariate Normal Distribution

6.5. Tutorials
Multivariate
normal
model
When multivariate data are analyzed, the multivariate normal model is
the most commonly used model.
The multivariate normal distribution model extends the univariate normal
distribution model to fit vector observations.
Definition
of
multivariate
normal
distribution
A p-dimensional vector of random variables
is said to have a multivariate normal distribution if its density function
f(X) is of the form
where m = (m
1
, ..., m
p
) is the vector of means and is the variance-
covariance matrix of the multivariate normal distribution. The shortcut
notation for this density is
Univariate
normal
distribution
When p = 1, the one-dimensional vector X = X
1
has the normal
distribution with mean m and variance
2
Bivariate
normal
distribution
When p = 2, X = (X
1
,X
2
) has the bivariate normal distribution with a
two-dimensional vector of means, m = (m
1
,m
2
) and covariance matrix
The correlation between the two random variables is given by
6.5.4.3. Hotelling's T squared

6.5. Tutorials
6.5.4.3. Hotelling's T squared
Hotelling's
T
2
distribution
A multivariate method that is the multivariate counterpart of
Student's-t and which also forms the basis for certain
multivariate control charts is based on Hotelling's T
2
distribution, which was introduced by Hotelling (1947).
Univariate
t-test for
mean
Recall, from Section 1.3.5.2,
has a t distribution provided that X is normally distributed,
and can be used as long as X doesn't differ greatly from a
normal distribution. If we wanted to test the hypothesis that
=
0
, we would then have
so that
Generalize
to p
variables
When t
2
is generalized to p variables it becomes
with

S
-1
is the inverse of the sample variance-covariance matrix,
S, and n is the sample size upon which each
i
, i = 1, 2, ..., p,
is based. (The diagonal elements of S are the variances and
the off-diagonal elements are the covariances for the p
6.5.4.3. Hotelling's T squared
variables. This is discussed further in Section 6.5.4.3.1.)
Distribution
of T
2
It is well known that when =
0
with F
(p,n-p)
representing the F distribution with p degrees of
freedom for the numerator and n - p for the denominator.
Thus, if were specified to be
0
, this could be tested by
taking a single p-variate sample of size n, then computing T
2
and comparing it with
for a suitably chosen .
Result does
not apply
directly to
multivariate
Shewhart-
type charts
Although this result applies to hypothesis testing, it does not
apply directly to multivariate Shewhart-type charts (for
which there is no
0
), although the result might be used as an
approximation when a large sample is used and data are in
subgroups, with the upper control limit (UCL) of a chart
based on the approximation.
Three-
sigma limits
from
univariate
control
chart
When a univariate control chart is used for Phase I (analysis
of historical data), and subsequently for Phase II (real-time
process monitoring), the general form of the control limits is
the same for each phase, although this need not be the case.
Specifically, three-sigma limits are used in the univariate
case, which skirts the relevant distribution theory for each
Phase.
Selection of
different
control
limit forms
for each
Phase
Three-sigma units are generally not used with multivariate
charts, however, which makes the selection of different
control limit forms for each Phase (based on the relevant
distribution theory), a natural choice.
6.5.4.3.1. T2 Chart for Subgroup Averages -- Phase I

6.5. Tutorials
6.5.4.3.1.
T
2
Chart for Subgroup Averages --
Phase I
Estimate
with
Since is generally unknown, it is necessary to estimate
analogous to the way that is estimated when an chart is
used. Specifically, when there are rational subgroups, is
estimated by , with
Obtaining
the
i
Each
i
, i = 1, 2, ..., p, is obtained the same way as with an
chart, namely, by taking k subgroups of size n and computing
.
Here is used to denote the average for the lth subgroup of
the ith variable. That is,
with x
ilr
denoting the rth observation (out of n) for the ith
variable in the lth subgroup.
Estimating
the
variances
and
covariances
The variances and covariances are similarly averaged over the
subgroups. Specifically, the s
ij
elements of the variance-
covariance matrix S are obtained as
with s
ijl
for i j denoting the sample covariance between
variables X
i
and X
j
for the lth subgroup, and s
ij
for i = j
denotes the sample variance of X
i
. The variances (= s
iil
)
for subgroup l and for variables i = 1, 2, ..., p are computed as
.
Similarly, the covariances s
ijl
between variables X
i
and X
j
for
subgroup l are computed as
.
Compare
T
2
against
control
values
As with an chart (or any other chart), the k subgroups
would be tested for control by computing k values of T
2
and
comparing each against the UCL. If any value falls above the
UCL (there is no lower control limit), the corresponding
subgroup would be investigated.
Formula
for plotted
T
2
values
Thus, one would plot
for the jth subgroup (j = 1, 2, ..., k), with denoting a vector
with p elements that contains the subgroup averages for each
of the p characteristics for the jth subgroup. ( is the
inverse matrix of the "pooled" variance-covariance matrix,
, which is obtained by averaging the subgroup variance-
covariance matrices over the k subgroups.)
Formula
for the
upper
control
limit
Each of the k values of given in the equation above would
be compared with
Lower
control
limits
A lower control limit is generally not used in multivariate
control chart applications, although some control chart
methods do utilize a LCL. Although a small value for
might seem desirable, a value that is very small would likely
indicate a problem of some type as we would not expect
every element of to be virtually equal to every element in
.
Delete out-
of-control
points once
cause
discovered
and
As with any Phase I control chart procedure, if there are any
points that plot above the UCL and can be identified as
corresponding to out-of-control conditions that have been
corrected, the point(s) should be deleted and the UCL
recomputed. The remaining points would then be compared
with the new UCL and the process continued as long as
corrected necessary, remembering that points should be deleted only if
their correspondence with out-of-control conditions can be
identified and the cause(s) of the condition(s) were removed.
6.5.4.3.2. T2 Chart for Subgroup Averages -- Phase II

6.5. Tutorials
6.5.4.3.2.
T
2
Chart for Subgroup Averages -- Phase II
Phase II
requires
recomputing
S
p
and
and
different
control
limits
Determining the UCL that is to be subsequently applied to future
subgroups entails recomputing, if necessary, S
p
and , and using a
constant and an F-value that are different from the form given for the
Phase I control limits. The form is different because different
distribution theory is involved since future subgroups are assumed to be
independent of the "current" set of subgroups that is used in calculating
S
p
and . (The same thing happens with charts; the problem is simply
ignored through the use of 3-sigma limits, although a different approach
should be used when there is a small number of subgroups -- and the
necessary theory has been worked out.)
Illustration To illustrate, assume that a subgroups had been discarded (with possibly
a = 0) so that k - a subgroups are used in obtaining and . We shall
let these two values be represented by and to distinguish them
from the original values, and , before any subgroups are deleted.
Future values to be plotted on the multivariate chart would then be
obtained from
with denoting an arbitrary vector containing the averages for
the p characteristics for a single subgroup obtained in the future. Each
of these future values would be plotted on the multivariate chart and
compared with
Phase II
control
limits
with a denoting the number of the original subgroups that are deleted
before computing and . Notice that the equation for the control
limits for Phase II given here does not reduce to the equation for the
control limits for Phase I when a = 0, nor should we expect it to since
the Phase I UCL is used when testing for control of the entire set of
subgroups that is used in computing and .
6.5.4.3.3. Chart for Individual Observations -- Phase I

6.5. Tutorials
6.5.4.3.3. Chart for Individual Observations --
Phase I
Multivariate
individual
control
charts
Control charts for multivariate individual observations can
be constructed, just as charts can be constructed for
univariate individual observations.
Constructing
the control
chart
Assume there are m historical multivariate observations to be
tested for control, so that Q
j
, j = 1, 2, ...., m are computed,
with
Control
limits
Each value of Q
j
is compared against control limits of
with B( ) denoting the beta distribution with parameters p/2
and (m-p-1)/2. These limits are due to Tracy, Young and
Mason (1992). Note that a LCL is stated, unlike the other
multivariate control chart procedures given in this section.
Although interest will generally be centered at the UCL, a
value of Q below the LCL should also be investigated, as
this could signal problems in data recording.
Delete
points if
special
cause(s) are
identified
and
corrected
As in the case when subgroups are used, if any points plot
outside these control limits and special cause(s) that were
subsequently removed can be identified, the point(s) would
be deleted and the control limits recomputed, making the
appropriate adjustments on the degrees of freedom, and re-
testing the remaining points against the new limits.
6.5.4.3.3. Chart for Individual Observations -- Phase I
6.5.4.3.4. Chart for Individual Observations -- Phase II

6.5. Tutorials
6.5.4.3.4. Chart for Individual Observations --
Phase II
Control
limits
In Phase II, each value of Q
j
would be plotted against the
UCL of
with, as before, p denoting the number of characteristics.
Further
Information
The control limit expressions given in this section and the
immediately preceding sections are given in Ryan (2000,
Chapter 9).
6.5.4.3.5. Charts for Controlling Multivariate Variability

6.5. Tutorials
6.5.4.3.5. Charts for Controlling Multivariate
Variability
No
satisfactory
charts for
multivariate
variability
Unfortunately, there are no charts for controlling multivariate
variability, with either subgroups or individual observations,
that are simple, easy-to-understand and implement, and
statistically defensible. Methods based on the generalized
variance have been proposed for subgroup data, but such
methods have been criticized by Ryan (2000, Section 9.4)
and some references cited therein. For individual
observations, the multivariate analogue of a univariate
moving range chart might be considered as an estimator of
the variance-covariance matrix for Phase I, although the
distribution of the estimator is unknown.
6.5.4.3.6. Constructing Multivariate Charts

6.5. Tutorials
6.5.4.3.6. Constructing Multivariate Charts
Multivariate
control
charts not
commonly
available in
statistical
software
Although control charts were originally constructed and
maintained by hand, it would be extremely impractical to try
to do that with the chart procedures that were presented in
Sections 6.5.4.3.1-6.5.4.3.4. Unfortunately, the well-known
statistical software packages do not have capability for the
four procedures just outlined. However, Dataplot, which is
used for case studies and tutorials throughout this e-
Handbook, does have that capability.
6.5.5. Principal Components

6.5. Tutorials
Dimension
reduction tool
A Multivariate Analysis problem could start out with a
substantial number of correlated variables. Principal
Component Analysis is a dimension-reduction tool that
can be used advantageously in such situations. Principal
component analysis aims at reducing a large set of
variables to a small set that still contains most of the
information in the large set.
Principal
factors
The technique of principal component analysis enables us
to create and use a reduced set of variables, which are
called principal factors. A reduced set is much easier to
analyze and interpret. To study a data set that results in the
estimation of roughly 500 parameters may be difficult, but
if we could reduce these to 5 it would certainly make our
day. We will show in what follows how to achieve
substantial dimension reduction.
Inverse
transformaion
not possible
While these principal factors represent or replace one or
more of the original variables, it should be noted that they
are not just a one-to-one transformation, so inverse
transformations are not possible.
Original data
matrix
To shed a light on the structure of principal components
analysis, let us consider a multivariate data matrix X, with
n rows and p columns. The p elements of each row are
scores or measurements on a subject such as height, weight
and age.
Linear
function that
maximizes
variance
Next, standardize the X matrix so that each column mean is
0 and each column variance is 1. Call this matrix Z. Each
column is a vector variable, z
i
, i = 1, . . . , p. The main idea
behind principal component analysis is to derive a linear
function y for each of the vector variables z
i
. This linear
function possesses an extremely important property;
namely, its variance is maximized.
Linear
function is
component of
z
This linear function is referred to as a component of z. To
illustrate the computation of a single element for the jth y
vector, consider the product y = z v' where v' is a column
vector of V and V is a p x p coefficient matrix that carries
the p-element variable z into the derived n-element variable
y. V is known as the eigen vector matrix. The dimension of
z is 1 x p, the dimension of v' is p x 1. The scalar algebra
for the component score for the ith individual of y
j
, j = 1,
...p is:
y
ji
= v'
1
z
1i
+ v'
2
z
2i
+ ... + v'
p
z
pi
This becomes in matrix notation for all of the y:
Y = ZV
Mean and
dispersion
matrix of y
The mean of y is m
y
= V'm
z
= 0, because m
z
= 0.
The dispersion matrix of y is
D
y
= V'D
z
V = V'RV
R is
correlation
matrix
Now, it can be shown that the dispersion matrix D
z
of a
standardized variable is a correlation matrix. Thus R is the
correlation matrix for z.
Number of
parameters to
estimate
increases
rapidly as p
increases
At this juncture you may be tempted to say: "so what?". To
answer this let us look at the intercorrelations among the
elements of a vector variable. The number of parameters to
be estimated for a p-element variable is
p means
p variances
(p
2
- p)/2 covariances
for a total of 2p + (p
2
-p)/2 parameters.
So
If p = 2, there are 5 parameters
Uncorrelated
variables
require no
covariance
estimation
All these parameters must be estimated and interpreted.
That is a herculean task, to say the least. Now, if we could
transform the data so that we obtain a vector of
uncorrelated variables, life becomes much more bearable,
since there are no covariances.
6.5.5.1. Properties of Principal Components

6.5. Tutorials
Orthogonalizing Transformations
Transformation
from z to y
The equation y = V'z represents a transformation, where y
is the transformed variable, z is the original standardized
variable and V is the premultiplier to go from z to y.
Orthogonal
transformations
simplify things
To produce a transformation vector for y for which the
elements are uncorrelated is the same as saying that we
want V such that D
y
is a diagonal matrix. That is, all the
off-diagonal elements of D
y
must be zero. This is called
an orthogonalizing transformation.
Infinite number
of values for V
There are an infinite number of values for V that will
produce a diagonal D
y
for any correlation matrix R. Thus
the mathematical problem "find a unique V such that D
y
is diagonal" cannot be solved as it stands. A number of
famous statisticians such as Karl Pearson and Harold
Hotelling pondered this problem and suggested a
"variance maximizing" solution.
Principal
components
maximize
variance of the
transformed
elements, one
by one
Hotelling (1933) derived the "principal components"
solution. It proceeds as follows: for the first principal
component, which will be the first element of y and be
defined by the coefficients in the first column of V,
(denoted by v
1
), we want a solution such that the variance
of y
1
will be maximized.
Constrain v to
generate a
unique solution
The constraint on the numbers in v
1
is that the sum of the
squares of the coefficients equals 1. Expressed
mathematically, we wish to maximize
where
y
1i
= v
1
'

z
i
and v
1
'v
1
= 1 ( this is called "normalizing " v
1
).
Computation of
first principal
component
from R and v
1
Substituting the middle equation in the first yields
where R is the correlation matrix of Z, which, in turn, is
the standardized matrix of X, the original data matrix.
Therefore, we want to maximize v
1
'Rv
1
subject to v
1
'v
1
= 1.
The eigenstructure
Lagrange
multiplier
approach
Let
>
introducing the restriction on v
1
via the Lagrange
multiplier approach. It can be shown (T.W. Anderson,
1958, page 347, theorem 8) that the vector of partial
derivatives is
and setting this equal to zero, dividing out 2 and factoring
gives
This is known as "the problem of the eigenstructure of
R".
Set of p
homogeneous
equations
The partial differentiation resulted in a set of p
homogeneous equations, which may be written in matrix
form as follows
The characteristic equation
Characterstic
equation of R is
a polynomial of
The characteristic equation of R is a polynomial of
degree p, which is obtained by expanding the determinant
of
degree p
and solving for the roots
j
, j = 1, 2, ..., p.
Largest
eigenvalue
Specifically, the largest eigenvalue,
1
, and its associated
vector, v
1
, are required. Solving for this eigenvalue and
vector is another mammoth numerical task that can
realistically only be performed by a computer. In general,
software is involved and the algorithms are complex.
Remainig p
eigenvalues
After obtaining the first eigenvalue, the process is
repeated until all p eigenvalues are computed.
Full
eigenstructure
of R
To succinctly define the full eigenstructure of R, we
introduce another matrix L, which is a diagonal matrix
with
j
in the jth position on the diagonal. Then the full
eigenstructure of R is given as
RV = VL
where
V'V = VV' = I
and
V'RV = L = D
y
Principal Factors
Scale to zero
means and unit
variances
It was mentioned before that it is helpful to scale any
transformation y of a vector variable z so that its elements
have zero means and unit variances. Such a standardized
transformation is called a factoring of z, or of R, and
each linear component of the transformation is called a
factor.
Deriving unit
variances for
principal
components
Now, the principal components already have zero means,
but their variances are not 1; in fact, they are the
eigenvalues, comprising the diagonal elements of L. It is
possible to derive the principal factor with unit variance
from the principal component as follows
or for all factors:
substituting V'z for y we have
where
B = VL
-1/2
B matrix The matrix B is then the matrix of factor score
coefficients for principal factors.
How many Eigenvalues?
Dimensionality
of the set of
factor scores
The number of eigenvalues, N, used in the final set
determines the dimensionality of the set of factor scores.
For example, if the original test consisted of 8
measurements on 100 subjects, and we extract 2
eigenvalues, the set of factor scores is a matrix of 100
rows by 2 columns.
Eigenvalues
greater than
unity
Each column or principal factor should represent a
number of original variables. Kaiser (1966) suggested a
rule-of-thumb that takes as a value for N, the number of
eigenvalues larger than unity.
Factor Structure
Factor
structure
matrix S
The primary interpretative device in principal components
is the factor structure, computed as
S = VL
1/2
S is a matrix whose elements are the correlations between
the principal components and the variables. If we retain,
for example, two eigenvalues, meaning that there are two
principal components, then the S matrix consists of two
columns and p (number of variables) rows.
Table showing
relation
between
variables and
principal
components
Principal Component
Variable 1 2
1 r
11
r
12
2 r
21
r
22
3 r
31
r
32
4 r
41
r
42
The r
ij
are the correlation coefficients between variable i
and principal component j, where i ranges from 1 to 4
and j from 1 to 2.
The
communality
SS' is the source of the "explained" correlations among
the variables. Its diagonal is called "the communality".
Rotation
Factor analysis If this correlation matrix, i.e., the factor structure matrix,
does not help much in the interpretation, it is possible to
rotate the axis of the principal components. This may
result in the polarization of the correlation coefficients.
Some practitioners refer to rotation after generating the
factor structure as factor analysis.
Varimax
rotation
A popular scheme for rotation was suggested by Henry
Kaiser in 1958. He produced a method for orthogonal
rotation of factors, called the varimax rotation, which
cleans up the factors as follows:
for each factor, high loadings (correlations) will
result for a few variables; the rest will be near
zero.
Example The following computer output from a principal
component analysis on a 4-variable data set, followed by
varimax rotation of the factor structure, will illustrate his
point.
Before Rotation After Rotation
Variable Factor
1
Factor
2
Factor
1
Factor
2
1 .853 -.989 .997 .058
2 .634 .762 .089 .987
3 .858 -.498 .989 .076
4 .633 .736 .103 .965
Communality
Formula for
communality
statistic
A measure of how well the selected factors (principal
components) "explain" the variance of each of the
variables is given by a statistic called communality. This
is defined by
Explanation of
communality
statistic
That is: the square of the correlation of variable k with
factor i gives the part of the variance accounted for by
that factor. The sum of these squares for n factors is the
communality, or explained variable for that variable
(row).
Roadmap to solve the V matrix
Main steps to
obtaining
eigenstructure
for a
correlation
matrix
In summary, here are the main steps to obtain the
eigenstructure for a correlation matrix.
1. Compute R, the correlation matrix of the original
data. R is also the correlation matrix of the
standardized data.
2. Obtain the characteristic equation of R which is a
polynomial of degree p (the number of variables),
obtained from expanding the determinant of |R- I|
= 0 and solving for the roots
i
, that is:
1
,
2
, ...
,
p
.
3. Then solve for the columns of the V matrix, (v
1
, v
2
,
..v
p
). The roots, ,
i
, are called the eigenvalues (or
latent values). The columns of V are called the
eigenvectors.
6.5.5.2. Numerical Example

6.5. Tutorials
Calculation
of principal
components
example
A numerical example may clarify the mechanics of principal
component analysis.
Sample data
set
Let us analyze the following 3-variate dataset with 10 observations. Each
observation consists of 3 measurements on a wafer: thickness, horizontal
displacement and vertical displacement.
Compute the
correlation
matrix
First compute the correlation matrix
Solve for the
roots of R
Next solve for the roots of R, using software
value proportion
1 1.769 .590
2 .927 .899
3 .304 1.000
Notice that
Each eigenvalue satisfies |R- I| = 0.
The sum of the eigenvalues = 3 = p, which is equal to the trace of
R (i.e., the sum of the main diagonal elements).
The determinant of R is the product of the eigenvalues.
The product is
1
x
2
x
3
= .499.
Compute the
first column
of the V
matrix
Substituting the first eigenvalue of 1.769 and R in the appropriate
equation we obtain
This is the matrix expression for 3 homogeneous equations with 3
unknowns and yields the first column of V: .64 .69 -.34 (again, a
computerized solution is indispensable).
Compute the
remaining
columns of
the V matrix
Repeating this procedure for the other 2 eigenvalues yields the matrix V
Notice that if you multiply V by its transpose, the result is an identity
matrix, V'V=I.
Compute the
L
1/2
matrix
Now form the matrix L
1/2
, which is a diagonal matrix whose elements
are the square roots of the eigenvalues of R. Then obtain S, the factor
structure, using S = V L
1/2
So, for example, .91 is the correlation between variable 2 and the first
principal component.
Compute the
communality
Next compute the communality, using the first two eigenvalues only
Diagonal
elements
report how
much of the
variability is
explained
Communality consists of the diagonal elements.
var
1 .8662
2 .8420
3 .9876
This means that the first two principal components "explain" 86.62% of
the first variable, 84.20 % of the second variable, and 98.76% of the
third.
Compute the
coefficient
matrix
The coefficient matrix, B, is formed using the reciprocals of the
diagonals of L
1/2
Compute the
principal
factors
Finally, we can compute the factor scores from ZB, where Z is X
converted to standard score form. These columns are the principal
factors.
Principal
factors
control
chart
These factors can be plotted against the indices, which could be times. If
time is used, the resulting plot is an example of a principal factors
control chart.
6.6. Case Studies in Process Monitoring

Detailed
Examples
The general points of the first five sections are illustrated in
this section using data from physical science and engineering
applications. Each example is presented step-by-step in the
text, and is often cross-linked with the relevant sections of the
chapter describing the analysis in general.
Contents:
Section 6
1. Lithography Process Example
2. Aerosol Particle Size Example
6.6.1. Lithography Process

Lithography
Process
This case study illustrates the use of control charts in
analyzing a lithography process.
2. Graphical Representation of the Data
3. Subgroup Analysis
4. Shewhart Control Chart

Case Study for SPC in Batch Processing Environment
Semiconductor
processing
creates
multiple
sources of
variability to
monitor
One of the assumptions in using classical Shewhart SPC charts
is that the only source of variation is from part to part (or
within subgroup variation). This is the case for most continuous
processing situations. However, many of today's processing
situations have different sources of variation. The
semiconductor industry is one of the areas where the
processing creates multiple sources of variation.
In semiconductor processing, the basic experimental unit is a
silicon wafer. Operations are performed on the wafer, but
individual wafers can be grouped multiple ways. In the
diffusion area, up to 150 wafers are processed in one time in a
diffusion tube. In the etch area, single wafers are processed
individually. In the lithography area, the light exposure is done
on sub-areas of the wafer. There are many times during the
production of a computer chip where the experimental unit
varies and thus there are different sources of variation in this
batch processing environment.
The following is a case study of a lithography process. Five
sites are measured on each wafer, three wafers are measured in
a cassette (typically a grouping of 24 - 25 wafers) and thirty
cassettes of wafers are used in the study. The width of a line is
the measurement under study. There are two line width
variables. The first is the original data and the second has been
cleaned up somewhat. This case study uses the raw data. The
entire data table is 450 rows long with six columns.
Case study
data: wafer
line width
measurements
Raw
Cleaned
Line
Line
Cassette Wafer Site Width Sequence
Width
=====================================================
1 1 Top 3.199275 1
3.197275
1 1 Lef 2.253081 2
2.249081
1 1 Cen 2.074308 3
2.068308
1 1 Rgt 2.418206 4
2.410206
1 1 Bot 2.393732 5
2.383732
1 2 Top 2.654947 6
2.642947
1 2 Lef 2.003234 7
1.989234
1 2 Cen 1.861268 8
1.845268
1 2 Rgt 2.136102 9
2.118102
1 2 Bot 1.976495 10
1.956495
1 3 Top 2.887053 11
2.865053
1 3 Lef 2.061239 12
2.037239
1 3 Cen 1.625191 13
1.599191
1 3 Rgt 2.304313 14
2.276313
1 3 Bot 2.233187 15
2.203187
2 1 Top 3.160233 16
3.128233
2 1 Lef 2.518913 17
2.484913
2 1 Cen 2.072211 18
2.036211
2 1 Rgt 2.287210 19
2.249210
2 1 Bot 2.120452 20
2.080452
2 2 Top 2.063058 21
2.021058
2 2 Lef 2.217220 22
2.173220
2 2 Cen 1.472945 23
1.426945
2 2 Rgt 1.684581 24
1.636581
2 2 Bot 1.900688 25
1.850688
2 3 Top 2.346254 26
2.294254
2 3 Lef 2.172825 27
2.118825
2 3 Cen 1.536538 28
1.480538
2 3 Rgt 1.966630 29
1.908630
2 3 Bot 2.251576 30
2.191576
3 1 Top 2.198141 31
2.136141
3 1 Lef 1.728784 32
1.664784
3 1 Cen 1.357348 33
1.291348
3 1 Rgt 1.673159 34
1.605159
3 1 Bot 1.429586 35
1.359586
3 2 Top 2.231291 36
2.159291
3 2 Lef 1.561993 37
1.487993
3 2 Cen 1.520104 38
1.444104
3 2 Rgt 2.066068 39
1.988068
3 2 Bot 1.777603 40
1.697603
3 3 Top 2.244736 41
2.162736
3 3 Lef 1.745877 42
1.661877
3 3 Cen 1.366895 43
1.280895
3 3 Rgt 1.615229 44
1.527229
3 3 Bot 1.540863 45
1.450863
4 1 Top 2.929037 46
2.837037
4 1 Lef 2.035900 47
1.941900
4 1 Cen 1.786147 48
1.690147
4 1 Rgt 1.980323 49
1.882323
4 1 Bot 2.162919 50
2.062919
4 2 Top 2.855798 51
2.753798
4 2 Lef 2.104193 52
2.000193
4 2 Cen 1.919507 53
1.813507
4 2 Rgt 2.019415 54
1.911415
4 2 Bot 2.228705 55
2.118705
4 3 Top 3.219292 56
3.107292
4 3 Lef 2.900430 57
2.786430
4 3 Cen 2.171262 58
2.055262
4 3 Rgt 3.041250 59
2.923250
4 3 Bot 3.188804 60
3.068804
5 1 Top 3.051234 61
2.929234
5 1 Lef 2.506230 62
2.382230
5 1 Cen 1.950486 63
1.824486
5 1 Rgt 2.467719 64
2.339719
5 1 Bot 2.581881 65
2.451881
5 2 Top 3.857221 66
3.725221
5 2 Lef 3.347343 67
3.213343
5 2 Cen 2.533870 68
2.397870
5 2 Rgt 3.190375 69
3.052375
5 2 Bot 3.362746 70
3.222746
5 3 Top 3.690306 71
3.548306
5 3 Lef 3.401584 72
3.257584
5 3 Cen 2.963117 73
2.817117
5 3 Rgt 2.945828 74
2.797828
5 3 Bot 3.466115 75
3.316115
6 1 Top 2.938241 76
2.786241
6 1 Lef 2.526568 77
2.372568
6 1 Cen 1.941370 78
1.785370
6 1 Rgt 2.765849 79
2.607849
6 1 Bot 2.382781 80
2.222781
6 2 Top 3.219665 81
3.057665
6 2 Lef 2.296011 82
2.132011
6 2 Cen 2.256196 83
2.090196
6 2 Rgt 2.645933 84
2.477933
6 2 Bot 2.422187 85
2.252187
6 3 Top 3.180348 86
3.008348
6 3 Lef 2.849264 87
2.675264
6 3 Cen 1.601288 88
1.425288
6 3 Rgt 2.810051 89
2.632051
6 3 Bot 2.902980 90
2.722980
7 1 Top 2.169679 91
1.987679
7 1 Lef 2.026506 92
1.842506
7 1 Cen 1.671804 93
1.485804
7 1 Rgt 1.660760 94
1.472760
7 1 Bot 2.314734 95
2.124734
7 2 Top 2.912838 96
2.720838
7 2 Lef 2.323665 97
2.129665
7 2 Cen 1.854223 98
1.658223
7 2 Rgt 2.391240 99 2.19324
7 2 Bot 2.196071 100
1.996071
7 3 Top 3.318517 101
3.116517
7 3 Lef 2.702735 102
2.498735
7 3 Cen 1.959008 103
1.753008
7 3 Rgt 2.512517 104
2.304517
7 3 Bot 2.827469 105
2.617469
8 1 Top 1.958022 106
1.746022
8 1 Lef 1.360106 107
1.146106
8 1 Cen 0.971193 108
0.755193
8 1 Rgt 1.947857 109
1.729857
8 1 Bot 1.643580 110 1.42358
8 2 Top 2.357633 111
2.135633
8 2 Lef 1.757725 112
1.533725
8 2 Cen 1.165886 113
0.939886
8 2 Rgt 2.231143 114
2.003143
8 2 Bot 1.311626 115
1.081626
8 3 Top 2.421686 116
2.189686
8 3 Lef 1.993855 117
1.759855
8 3 Cen 1.402543 118
1.166543
8 3 Rgt 2.008543 119
1.770543
8 3 Bot 2.139370 120
1.899370
9 1 Top 2.190676 121
1.948676
9 1 Lef 2.287483 122
2.043483
9 1 Cen 1.698943 123
1.452943
9 1 Rgt 1.925731 124
1.677731
9 1 Bot 2.057440 125
1.807440
9 2 Top 2.353597 126
2.101597
9 2 Lef 1.796236 127
1.542236
9 2 Cen 1.241040 128
0.985040
9 2 Rgt 1.677429 129
1.419429
9 2 Bot 1.845041 130
1.585041
9 3 Top 2.012669 131
1.750669
9 3 Lef 1.523769 132
1.259769
9 3 Cen 0.790789 133
0.524789
9 3 Rgt 2.001942 134
1.733942
9 3 Bot 1.350051 135
1.080051
10 1 Top 2.825749 136
2.553749
10 1 Lef 2.502445 137
2.228445
10 1 Cen 1.938239 138
1.662239
10 1 Rgt 2.349497 139
2.071497
10 1 Bot 2.310817 140
2.030817
10 2 Top 3.074576 141
2.792576
10 2 Lef 2.057821 142
1.773821
10 2 Cen 1.793617 143
1.507617
10 2 Rgt 1.862251 144
1.574251
10 2 Bot 1.956753 145
1.666753
10 3 Top 3.072840 146
2.780840
10 3 Lef 2.291035 147
1.997035
10 3 Cen 1.873878 148
1.577878
10 3 Rgt 2.475640 149
2.177640
10 3 Bot 2.021472 150
1.721472
11 1 Top 3.228835 151
2.926835
11 1 Lef 2.719495 152
2.415495
11 1 Cen 2.207198 153
1.901198
11 1 Rgt 2.391608 154
2.083608
11 1 Bot 2.525587 155
2.215587
11 2 Top 2.891103 156
2.579103
11 2 Lef 2.738007 157
2.424007
11 2 Cen 1.668337 158
1.352337
11 2 Rgt 2.496426 159
2.178426
11 2 Bot 2.417926 160
2.097926
11 3 Top 3.541799 161
3.219799
11 3 Lef 3.058768 162
2.734768
11 3 Cen 2.187061 163
1.861061
11 3 Rgt 2.790261 164
2.462261
11 3 Bot 3.279238 165
2.949238
12 1 Top 2.347662 166
2.015662
12 1 Lef 1.383336 167
1.049336
12 1 Cen 1.187168 168
0.851168
12 1 Rgt 1.693292 169
1.355292
12 1 Bot 1.664072 170
1.324072
12 2 Top 2.385320 171
2.043320
12 2 Lef 1.607784 172
1.263784
12 2 Cen 1.230307 173
0.884307
12 2 Rgt 1.945423 174
1.597423
12 2 Bot 1.907580 175
1.557580
12 3 Top 2.691576 176
2.339576
12 3 Lef 1.938755 177
1.584755
12 3 Cen 1.275409 178
0.919409
12 3 Rgt 1.777315 179
1.419315
12 3 Bot 2.146161 180
1.786161
13 1 Top 3.218655 181
2.856655
13 1 Lef 2.912180 182
2.548180
13 1 Cen 2.336436 183
1.970436
13 1 Rgt 2.956036 184
2.588036
13 1 Bot 2.423235 185
2.053235
13 2 Top 3.302224 186
2.930224
13 2 Lef 2.808816 187
2.434816
13 2 Cen 2.340386 188
1.964386
13 2 Rgt 2.795120 189
2.417120
13 2 Bot 2.865800 190
2.485800
13 3 Top 2.992217 191
2.610217
13 3 Lef 2.952106 192
2.568106
13 3 Cen 2.149299 193
1.763299
13 3 Rgt 2.448046 194
2.060046
13 3 Bot 2.507733 195
2.117733
14 1 Top 3.530112 196
3.138112
14 1 Lef 2.940489 197
2.546489
14 1 Cen 2.598357 198
2.202357
14 1 Rgt 2.905165 199
2.507165
14 1 Bot 2.692078 200
2.292078
14 2 Top 3.764270 201
3.362270
14 2 Lef 3.465960 202
3.061960
14 2 Cen 2.458628 203
2.052628
14 2 Rgt 3.141132 204
2.733132
14 2 Bot 2.816526 205
2.406526
14 3 Top 3.217614 206
2.805614
14 3 Lef 2.758171 207
2.344171
14 3 Cen 2.345921 208
1.929921
14 3 Rgt 2.773653 209
2.355653
14 3 Bot 3.109704 210
2.689704
15 1 Top 2.177593 211
1.755593
15 1 Lef 1.511781 212
1.087781
15 1 Cen 0.746546 213
0.320546
15 1 Rgt 1.491730 214
1.063730
15 1 Bot 1.268580 215
0.838580
15 2 Top 2.433994 216
2.001994
15 2 Lef 2.045667 217
1.611667
15 2 Cen 1.612699 218
1.176699
15 2 Rgt 2.082860 219
1.644860
15 2 Bot 1.887341 220
1.447341
15 3 Top 1.923003 221
1.481003
15 3 Lef 2.124461 222
1.680461
15 3 Cen 1.945048 223
1.499048
15 3 Rgt 2.210698 224
1.762698
15 3 Bot 1.985225 225
1.535225
16 1 Top 3.131536 226
2.679536
16 1 Lef 2.405975 227
1.951975
16 1 Cen 2.206320 228
1.750320
16 1 Rgt 3.012211 229
2.554211
16 1 Bot 2.628723 230
2.168723
16 2 Top 2.802486 231
2.340486
16 2 Lef 2.185010 232
1.721010
16 2 Cen 2.161802 233
1.695802
16 2 Rgt 2.102560 234
1.634560
16 2 Bot 1.961968 235
1.491968
16 3 Top 3.330183 236
2.858183
16 3 Lef 2.464046 237
1.990046
16 3 Cen 1.687408 238
1.211408
16 3 Rgt 2.043322 239
1.565322
16 3 Bot 2.570657 240
2.090657
17 1 Top 3.352633 241
2.870633
17 1 Lef 2.691645 242
2.207645
17 1 Cen 1.942410 243
1.456410
17 1 Rgt 2.366055 244
1.878055
17 1 Bot 2.500987 245
2.010987
17 2 Top 2.886284 246
2.394284
17 2 Lef 2.292503 247
1.798503
17 2 Cen 1.627562 248
1.131562
17 2 Rgt 2.415076 249
1.917076
17 2 Bot 2.086134 250
1.586134
17 3 Top 2.554848 251
2.052848
17 3 Lef 1.755843 252
1.251843
17 3 Cen 1.510124 253
1.004124
17 3 Rgt 2.257347 254
1.749347
17 3 Bot 1.958592 255
1.448592
18 1 Top 2.622733 256
2.110733
18 1 Lef 2.321079 257
1.807079
18 1 Cen 1.169269 258
0.653269
18 1 Rgt 1.921457 259
1.403457
18 1 Bot 2.176377 260
1.656377
18 2 Top 3.313367 261
2.791367
18 2 Lef 2.559725 262
2.035725
18 2 Cen 2.404662 263
1.878662
18 2 Rgt 2.405249 264
1.877249
18 2 Bot 2.535618 265
2.005618
18 3 Top 3.067851 266
2.535851
18 3 Lef 2.490359 267
1.956359
18 3 Cen 2.079477 268
1.543477
18 3 Rgt 2.669512 269
2.131512
18 3 Bot 2.105103 270
1.565103
19 1 Top 4.293889 271
3.751889
19 1 Lef 3.888826 272
3.344826
19 1 Cen 2.960655 273
2.414655
19 1 Rgt 3.618864 274
3.070864
19 1 Bot 3.562480 275
3.012480
19 2 Top 3.451872 276
2.899872
19 2 Lef 3.285934 277
2.731934
19 2 Cen 2.638294 278
2.082294
19 2 Rgt 2.918810 279
2.360810
19 2 Bot 3.076231 280
2.516231
19 3 Top 3.879683 281
3.317683
19 3 Lef 3.342026 282
2.778026
19 3 Cen 3.382833 283
2.816833
19 3 Rgt 3.491666 284
2.923666
19 3 Bot 3.617621 285
3.047621
20 1 Top 2.329987 286
1.757987
20 1 Lef 2.400277 287
1.826277
20 1 Cen 2.033941 288
1.457941
20 1 Rgt 2.544367 289
1.966367
20 1 Bot 2.493079 290
1.913079
20 2 Top 2.862084 291
2.280084
20 2 Lef 2.404703 292
1.820703
20 2 Cen 1.648662 293
1.062662
20 2 Rgt 2.115465 294
1.527465
20 2 Bot 2.633930 295
2.043930
20 3 Top 3.305211 296
2.713211
20 3 Lef 2.194991 297
1.600991
20 3 Cen 1.620963 298
1.024963
20 3 Rgt 2.322678 299
1.724678
20 3 Bot 2.818449 300
2.218449
21 1 Top 2.712915 301
2.110915
21 1 Lef 2.389121 302
1.785121
21 1 Cen 1.575833 303
0.969833
21 1 Rgt 1.870484 304
1.262484
21 1 Bot 2.203262 305
1.593262
21 2 Top 2.607972 306
1.995972
21 2 Lef 2.177747 307
1.563747
21 2 Cen 1.246016 308
0.630016
21 2 Rgt 1.663096 309
1.045096
21 2 Bot 1.843187 310
1.223187
21 3 Top 2.277813 311
1.655813
21 3 Lef 1.764940 312
1.140940
21 3 Cen 1.358137 313
0.732137
21 3 Rgt 2.065713 314
1.437713
21 3 Bot 1.885897 315
1.255897
22 1 Top 3.126184 316
2.494184
22 1 Lef 2.843505 317
2.209505
22 1 Cen 2.041466 318
1.405466
22 1 Rgt 2.816967 319
2.178967
22 1 Bot 2.635127 320
1.995127
22 2 Top 3.049442 321
2.407442
22 2 Lef 2.446904 322
1.802904
22 2 Cen 1.793442 323
1.147442
22 2 Rgt 2.676519 324
2.028519
22 2 Bot 2.187865 325
1.537865
22 3 Top 2.758416 326
2.106416
22 3 Lef 2.405744 327
1.751744
22 3 Cen 1.580387 328
0.924387
22 3 Rgt 2.508542 329
1.850542
22 3 Bot 2.574564 330
1.914564
23 1 Top 3.294288 331
2.632288
23 1 Lef 2.641762 332
1.977762
23 1 Cen 2.105774 333
1.439774
23 1 Rgt 2.655097 334
1.987097
23 1 Bot 2.622482 335
1.952482
23 2 Top 4.066631 336
3.394631
23 2 Lef 3.389733 337
2.715733
23 2 Cen 2.993666 338
2.317666
23 2 Rgt 3.613128 339
2.935128
23 2 Bot 3.213809 340
2.533809
23 3 Top 3.369665 341
2.687665
23 3 Lef 2.566891 342
1.882891
23 3 Cen 2.289899 343
1.603899
23 3 Rgt 2.517418 344
1.829418
23 3 Bot 2.862723 345
2.172723
24 1 Top 4.212664 346
3.520664
24 1 Lef 3.068342 347
2.374342
24 1 Cen 2.872188 348
2.176188
24 1 Rgt 3.040890 349
2.342890
24 1 Bot 3.376318 350
2.676318
24 2 Top 3.223384 351
2.521384
24 2 Lef 2.552726 352
1.848726
24 2 Cen 2.447344 353
1.741344
24 2 Rgt 3.011574 354
2.303574
24 2 Bot 2.711774 355
2.001774
24 3 Top 3.359505 356
2.647505
24 3 Lef 2.800742 357
2.086742
24 3 Cen 2.043396 358
1.327396
24 3 Rgt 2.929792 359
2.211792
24 3 Bot 2.935356 360
2.215356
25 1 Top 2.724871 361
2.002871
25 1 Lef 2.239013 362
1.515013
25 1 Cen 2.341512 363
1.615512
25 1 Rgt 2.263617 364
1.535617
25 1 Bot 2.062748 365
1.332748
25 2 Top 3.658082 366
2.926082
25 2 Lef 3.093268 367
2.359268
25 2 Cen 2.429341 368
1.693341
25 2 Rgt 2.538365 369
1.800365
25 2 Bot 3.161795 370
2.421795
25 3 Top 3.178246 371
2.436246
25 3 Lef 2.498102 372
1.754102
25 3 Cen 2.445810 373
1.699810
25 3 Rgt 2.231248 374
1.483248
25 3 Bot 2.302298 375
1.552298
26 1 Top 3.320688 376
2.568688
26 1 Lef 2.861800 377
2.107800
26 1 Cen 2.238258 378
1.482258
26 1 Rgt 3.122050 379
2.364050
26 1 Bot 3.160876 380
2.400876
26 2 Top 3.873888 381
3.111888
26 2 Lef 3.166345 382
2.402345
26 2 Cen 2.645267 383
1.879267
26 2 Rgt 3.309867 384
2.541867
26 2 Bot 3.542882 385
2.772882
26 3 Top 2.586453 386
1.814453
26 3 Lef 2.120604 387
1.346604
26 3 Cen 2.180847 388
1.404847
26 3 Rgt 2.480888 389
1.702888
26 3 Bot 1.938037 390
1.158037
27 1 Top 4.710718 391
3.928718
27 1 Lef 4.082083 392
3.298083
27 1 Cen 3.533026 393
2.747026
27 1 Rgt 4.269929 394
3.481929
27 1 Bot 4.038166 395
3.248166
27 2 Top 4.237233 396
3.445233
27 2 Lef 4.171702 397
3.377702
27 2 Cen 3.04394 398
2.247940
27 2 Rgt 3.91296 399
3.114960
27 2 Bot 3.714229 400
2.914229
27 3 Top 5.168668 401
4.366668
27 3 Lef 4.823275 402
4.019275
27 3 Cen 3.764272 403
2.958272
27 3 Rgt 4.396897 404
3.588897
27 3 Bot 4.442094 405
3.632094
28 1 Top 3.972279 406
3.160279
28 1 Lef 3.883295 407
3.069295
28 1 Cen 3.045145 408
2.229145
28 1 Rgt 3.51459 409
2.696590
28 1 Bot 3.575446 410
2.755446
28 2 Top 3.024903 411
2.202903
28 2 Lef 3.099192 412
2.275192
28 2 Cen 2.048139 413
1.222139
28 2 Rgt 2.927978 414
2.099978
28 2 Bot 3.15257 415
2.322570
28 3 Top 3.55806 416
2.726060
28 3 Lef 3.176292 417
2.342292
28 3 Cen 2.852873 418
2.016873
28 3 Rgt 3.026064 419
2.188064
28 3 Bot 3.071975 420
2.231975
29 1 Top 3.496634 421
2.654634
29 1 Lef 3.087091 422
2.243091
29 1 Cen 2.517673 423
1.671673
29 1 Rgt 2.547344 424
1.699344
29 1 Bot 2.971948 425
2.121948
29 2 Top 3.371306 426
2.519306
29 2 Lef 2.175046 427
1.321046
29 2 Cen 1.940111 428
1.084111
29 2 Rgt 2.932408 429
2.074408
29 2 Bot 2.428069 430
1.568069
29 3 Top 2.941041 431
2.079041
29 3 Lef 2.294009 432
1.430009
29 3 Cen 2.025674 433
1.159674
29 3 Rgt 2.21154 434
1.343540
29 3 Bot 2.459684 435
1.589684
30 1 Top 2.86467 436
1.992670
30 1 Lef 2.695163 437
1.821163
30 1 Cen 2.229518 438
1.353518
30 1 Rgt 1.940917 439
1.062917
30 1 Bot 2.547318 440
1.667318
30 2 Top 3.537562 441
2.655562
30 2 Lef 3.311361 442
2.427361
30 2 Cen 2.767771 443
1.881771
30 2 Rgt 3.388622 444
2.500622
30 2 Bot 3.542701 445
2.652701
30 3 Top 3.184652 446
2.292652
30 3 Lef 2.620947 447
1.726947
30 3 Cen 2.697619 448
1.801619
30 3 Rgt 2.860684 449
1.962684
30 3 Bot 2.758571 450
1.858571
6.6.1.2. Graphical Representation of the Data

The first step in analyzing the data is to generate some
simple plots of the response and then of the response versus
the various factors.
4-Plot of
Data
Interpretation This 4-plot shows the following.
location and scale are not constant over time. This
indicates that the three factors do in fact have an
effect of some kind.
2. The lag plot (upper right) indicates that there is some
mild autocorrelation in the data. This is not
unexpected as the data are grouped in a logical order
of the three factors (i.e., not randomly) and the run
sequence plot indicates that there are factor effects.
3. The histogram (lower left) shows that most of the
data fall between 1 and 5, with the center of the data
at about 2.2.
4. Due to the non-constant location and scale and
autocorrelation in the data, distributional inferences
from the normal probability plot (lower right) are not
meaningful.
The run sequence plot is shown at full size to show greater
detail. In addition, a numerical summary of the data is
generated.
Run
Sequence
Plot of Data
Numerical
Summary
Sample size = 450
Mean = 2.53228
Median = 2.45334
Minimum = 0.74655
Maximum = 5.16867
Range = 4.42212
Stan. Dev. = 0.69376
Autocorrelation = 0.60726
We are primarily interested in the mean and standard
deviation. From the summary, we see that the mean is 2.53
and the standard deviation is 0.69.
Plot response
against
individual
factors
The next step is to plot the response against each individual
factor. For comparison, we generate both a scatter plot and
a box plot of the data. The scatter plot shows more detail.
However, comparisons are usually easier to see with the
box plot, particularly as the number of data points and
groups become larger.
Scatter plot
of width
versus
cassette
Box plot of
width versus
cassette
Interpretation We can make the following conclusions based on the above
scatter and box plots.
1. There is considerable variation in the location for the
various cassettes. The medians vary from about 1.7 to
4.
2. There is also some variation in the scale.
3. There are a number of outliers.
Scatter plot
of width
versus wafer
Box plot of
width versus
wafer
1. The locations for the three wafers are relatively
constant.
2. The scales for the three wafers are relatively constant.
3. There are a few outliers on the high side.
4. It is reasonable to treat the wafer factor as
homogeneous.
Scatter plot
of width
versus site
Box plot of
width versus
site
1. There is some variation in location based on site. The
center site in particular has a lower median.
2. The scales are relatively constant across sites.
3. There are a few outliers.
DOE mean
and sd plots
We can use the DOE mean plot and the DOE standard
deviation plot to show the factor means and standard
deviations together for better comparison.
DOE mean
plot
DOE sd plot
Summary The above graphs show that there are differences between
the lots and the sites.
There are various ways we can create subgroups of this
dataset: each lot could be a subgroup, each wafer could be
a subgroup, or each site measured could be a subgroup
(with only one data value in each subgroup).
Recall that for a classical Shewhart means chart, the
average within subgroup standard deviation is used to
calculate the control limits for the means chart. However,
with a means chart you are monitoring the subgroup mean-
to-mean variation. There is no problem if you are in a
continuous processing situation - this becomes an issue if
you are operating in a batch processing environment.
We will look at various control charts based on different
subgroupings in 6.6.1.3.
6.6.1.3. Subgroup Analysis

Control
charts for
subgroups
The resulting classical Shewhart control charts for each
possible subgroup are shown below.
Site as
subgroup
The first pair of control charts use the site as the subgroup.
However, since site has a subgroup size of one we use the
control charts for individual measurements. A moving
average and a moving range chart are shown.
Moving
average
control chart
Moving
range control
chart
Wafer as
subgroup
The next pair of control charts use the wafer as the
subgroup. In this case, the subgroup size is five. A mean
and a standard deviation control chart are shown.
Mean control
chart
SD control
chart
There is no LCL for the standard deviation chart because of
the small subgroup size.
Cassette as
subgroup
The next pair of control charts use the cassette as the
subgroup. In this case, the subgroup size is 15. A mean and
a standard deviation control chart are shown.
Mean control
chart
SD control
chart
Interpretation Which of these subgroupings of the data is correct? As you
can see, each sugrouping produces a different chart. Part of
the answer lies in the manufacturing requirements for this
process. Another aspect that can be statistically determined
is the magnitude of each of the sources of variation. In
order to understand our data structure and how much
variation each of our sources contribute, we need to
perform a variance component analysis. The variance
component analysis for this data set is shown below.
Component
Variance
Component
Estimate
Cassette 0.2645
Wafer 0.0500
Site 0.1755
Variance
Component
Estimation
If your software does not generate the variance components
directly, they can be computed from a standard analysis of
variance output by equating mean squares (MS) to expected
mean squares (EMS).
The sum of squares and mean squares for a nested, random
effects model are shown below.
Degrees of Sum of
Source Freedom Squares
Mean Squares
-------------------- ---------- --------- --
----------
Cassette 29 127.40293
4.3932
Wafer(Cassette) 60 25.52089
0.4253
Site(Cassette, Wafer) 360 63.17865
0.1755
The expected mean squares for cassette, wafer within
cassette, and site within cassette and wafer, along with their
associated mean squares, are the following.
4.3932 = (3*5)*Var(cassettes) + 5*Var(wafer) +
Var(site)
0.4253 = 5*Var(wafer) + Var(site)
0.1755 = Var(site)
Solving these equations, we obtain the variance component
estimates 0.2645, 0.04997, and 0.1755 for cassettes, wafers,
and sites, respectively.
All of the analyses in this section can be completed using R
code.
6.6.1.4. Shewhart Control Chart

Choosing
the right
control
charts to
monitor
the
process
The largest source of variation in this data is the lot-to-lot
variation. So, using classical Shewhart methods, if we specify
our subgroup to be anything other than lot, we will be ignoring
the known lot-to-lot variation and could get out-of-control
points that already have a known, assignable cause - the data
comes from different lots. However, in the lithography
processing area the measurements of most interest are the site
level measurements, not the lot means. How can we get
around this seeming contradiction?
Chart
sources of
variation
separately
One solution is to chart the important sources of variation
separately. We would then be able to monitor the variation of
our process and truly understand where the variation is coming
from and if it changes. For this dataset, this approach would
require having two sets of control charts, one for the
individual site measurements and the other for the lot means.
This would double the number of charts necessary for this
process (we would have 4 charts for line width instead of 2).
Chart only
most
important
source of
variation
Another solution would be to have one chart on the largest
source of variation. This would mean we would have one set
of charts that monitor the lot-to-lot variation. From a
manufacturing standpoint, this would be unacceptable.
Use
boxplot
type chart
We could create a non-standard chart that would plot all the
individual data values and group them together in a boxplot
type format by lot. The control limits could be generated to
monitor the individual data values while the lot-to-lot variation
would be monitored by the patterns of the groupings. This
would take special programming and management intervention
to implement non-standard charts in most floor shop control
systems.
Alternate
form for
mean
control
chart
A commonly applied solution is the first option; have multiple
charts on this process. When creating the control limits for the
lot means, care must be taken to use the lot-to-lot variation
instead of the within lot variation. The resulting control charts
are: the standard individuals/moving range charts (as seen
previously), and a control chart on the lot means that is
different from the previous lot means chart. This new chart
uses the lot-to-lot variation to calculate control limits instead
of the average within-lot standard deviation. The
accompanying standard deviation chart is the same as seen
previously.
Mean
control
chart
using lot-
to-lot
variation
The control limits labeled with "UCL" and "LCL" are the
standard control limits. The control limits labeled with "UCL:
LL" and "LCL: LL" are based on the lot-to-lot variation.

View
Dataplot
Macro for
this Case
Study
Data Analysis Steps
Results and
Conclusions
more detailed
columns of numbers
into Dataplot,
variables CASSETTE,
WAFER, SITE,
WIDTH, and RUNSEQ.
2. Plot of the response variable
1. Numerical summary of WIDTH.
2. 4-Plot of WIDTH.
3. Run sequence plot of WIDTH.
1. The summary shows
the mean line width
is 2.53 and the
standard deviation
of the line
width is 0.69.
2. The 4-plot shows
non-constant
location and
scale and moderate
autocorrelation.
3. The run sequence
plot shows
non-constant
location and scale.
3. Generate scatter and box plots
against
individual factors.
1. Scatter plot of WIDTH versus
CASSETTE.
2. Box plot of WIDTH versus
CASSETTE.
WAFER.
WAFER.
SITE.
SITE.
7. DOE mean plot of WIDTH versus
CASSETTE, WAFER, and SITE.
8. DOE sd plot of WIDTH versus
CASSETTE, WAFER, and SITE.
1. The scatter plot
shows considerable
variation in
location.
2. The box plot
shows considerable
variation in
location and scale
and the prescence
of some outliers.
3. The scatter plot
shows minimal
variation in
location and scale.
4. The box plot
shows minimal
variation in
location and scale.
It also show
some outliers.
5. The scatter plot
shows some
variation in
location.
6. The box plot
shows some
variation in
location. Scale
seems relatively
constant.
Some outliers.
7. The DOE mean
plot shows effects
for CASSETTE and
SITE, no effect
for WAFER.
8. The DOE sd plot
shows effects
for CASSETTE and
SITE, no effect
for WAFER.
4. Subgroup analysis.
1. Generate a moving mean control
chart.
2. Generate a moving range control
chart.
3. Generate a mean control chart
for WAFER.
1. The moving mean
plot shows
a large number of
out-of-
control points.
2. The moving range
plot shows
a large number of
out-of-
4. Generate a sd control chart
for WAFER.
for CASSETTE.
6. Generate a sd control chart
for CASSETTE.
7. Generate an analysis of
variance. This is not
currently implemented in
DATAPLOT for nested
datasets.
using lot-to-lot variation.
control points.
3. The mean control
chart shows
a large number of
out-of-
control points.
4. The sd control
chart shows
no out-of-control
points.
5. The mean control
chart shows
a large number of
out-of-
control points.
6. The sd control
chart shows
no out-of-control
points.
7. The analysis of
variance and
components of
variance
calculations show
that
cassette to
cassette
variation is 54%
of the total
and site to site
variation
is 36% of the
total.
8. The mean control
chart shows one
point that is on
the boundary of
being out of
control.
6.6.2. Aerosol Particle Size

Box-
Jenkins
Modeling
of Aerosol
Particle
Size
This case study illustrates the use of Box-Jenkins modeling
with aerosol particle size data.
2. Model Identification
3. Model Estimation
4. Model Validation

Data Source The source of the data for this case study is Antuan Negiz
who analyzed these data while he was a post-doc in the
NIST Statistical Engineering Division from the Illinois
Institute of Technology.
Data
Collection
These data were collected from an aerosol mini-spray dryer
device. The purpose of this device is to convert a slurry
stream into deposited particles in a drying chamber. The
device injects the slurry at high speed. The slurry is
pulverized as it enters the drying chamber when it comes into
contact with a hot gas stream at low humidity. The liquid
contained in the pulverized slurry particles is vaporized, then
transferred to the hot gas stream leaving behind dried small-
sized particles.
The response variable is particle size, which is collected
equidistant in time. There are a variety of associated
variables that may affect the injection process itself and
hence the size and quality of the deposited particles. For this
case study, we restrict our analysis to the response variable.
Applications Such deposition process operations have many applications
from powdered laundry detergents at one extreme to ceramic
molding at an important other extreme. In ceramic molding,
the distribution and homogeneity of the particle sizes are
particularly important because after the molds are baked and
cured, the properties of the final molded ceramic product is
strongly affected by the intermediate uniformity of the base
ceramic particles, which in turn is directly reflective of the
quality of the initial atomization process in the aerosol
injection device.
Aerosol
Particle
Size
Dynamic
Modeling
and Control
The data set consists of particle sizes collected over time.
The basic distributional properties of this process are of
interest in terms of distributional shape, constancy of size,
and variation in size. In addition, this time series may be
examined for autocorrelation structure to determine a
prediction model of particle size as a function of time--such
a model is frequently autoregressive in nature. Such a high-
quality prediction equation would be essential as a first step
in developing a predictor-corrective recursive feedback
mechanism which would serve as the core in developing and
implementing real-time dynamic corrective algorithms. The
net effect of such algorthms is, of course, a particle size
distribution that is much less variable, much more stable in
nature, and of much higher quality. All of this results in final
ceramic mold products that are more uniform and predictable
across a wide range of important performance
characteristics.
For the purposes of this case study, we restrict the analysis to
determining an appropriate Box-Jenkins model of the particle
size.
Case study
data
115.36539
114.63150
114.63150
116.09940
116.34400
116.09940
116.34400
116.83331
116.34400
116.83331
117.32260
117.07800
117.32260
117.32260
117.81200
117.56730
118.30130
117.81200
118.30130
117.81200
118.30130
118.30130
118.54590
118.30130
117.07800
116.09940
118.30130
118.79060
118.05661
118.30130
118.54590
118.30130
118.54590
118.05661
118.30130
118.54590
118.30130
118.30130
118.30130
118.30130
118.05661
118.30130
117.81200
118.30130
117.32260
117.32260
117.56730
117.81200
117.56730
117.81200
117.81200
117.32260
116.34400
116.58870
116.83331
116.58870
116.83331
116.83331
117.32260
116.34400
116.09940
115.61010
115.61010
115.61010
115.36539
115.12080
115.61010
115.85471
115.36539
115.36539
115.36539
115.12080
114.87611
114.87611
115.12080
114.87611
114.87611
114.63150
114.63150
114.14220
114.38680
114.14220
114.63150
114.87611
114.38680
114.87611
114.63150
114.14220
114.14220
113.89750
114.14220
113.89750
113.65289
113.65289
113.40820
113.40820
112.91890
113.40820
112.91890
113.40820
113.89750
113.40820
113.65289
113.89750
113.65289
113.65289
113.89750
113.65289
113.16360
114.14220
114.38680
113.65289
113.89750
113.89750
113.40820
113.65289
113.89750
113.65289
113.65289
114.14220
114.38680
114.63150
115.61010
115.12080
114.63150
114.38680
113.65289
113.40820
113.40820
113.16360
113.16360
113.16360
113.16360
113.16360
112.42960
113.40820
113.40820
113.16360
113.16360
113.16360
113.16360
111.20631
112.67420
112.91890
112.67420
112.91890
113.16360
112.91890
112.67420
112.91890
112.67420
112.91890
113.16360
112.67420
112.67420
112.91890
113.16360
112.67420
112.91890
111.20631
113.40820
112.91890
112.67420
113.16360
113.65289
113.40820
114.14220
114.87611
114.87611
116.09940
116.34400
116.58870
116.09940
116.34400
116.83331
117.07800
117.07800
116.58870
116.83331
116.58870
116.34400
116.83331
116.83331
117.07800
116.58870
116.58870
117.32260
116.83331
118.79060
116.83331
117.07800
116.58870
116.83331
116.34400
116.58870
116.34400
116.34400
116.34400
116.09940
116.09940
116.34400
115.85471
115.85471
115.85471
115.61010
115.61010
115.61010
115.36539
115.12080
115.61010
115.85471
115.12080
115.12080
114.87611
114.87611
114.38680
114.14220
114.14220
114.38680
114.14220
114.38680
114.38680
114.38680
114.38680
114.38680
114.14220
113.89750
114.14220
113.65289
113.16360
112.91890
112.67420
112.42960
112.42960
112.42960
112.18491
112.18491
112.42960
112.18491
112.42960
111.69560
112.42960
112.42960
111.69560
111.94030
112.18491
112.18491
112.18491
111.94030
111.69560
111.94030
111.94030
112.42960
112.18491
112.18491
111.94030
112.18491
112.18491
111.20631
111.69560
111.69560
111.69560
111.94030
111.94030
112.18491
111.69560
112.18491
111.94030
111.69560
112.18491
110.96170
111.69560
111.20631
111.20631
111.45100
110.22771
109.98310
110.22771
110.71700
110.22771
111.20631
111.45100
111.69560
112.18491
112.18491
112.18491
112.42960
112.67420
112.18491
112.42960
112.18491
112.91890
112.18491
112.42960
111.20631
112.42960
112.42960
112.42960
112.42960
113.16360
112.18491
112.91890
112.91890
112.67420
112.42960
112.42960
112.42960
112.91890
113.16360
112.67420
113.16360
112.91890
112.42960
112.67420
112.91890
112.18491
112.91890
113.16360
112.91890
112.91890
112.91890
112.67420
112.42960
112.42960
113.16360
112.91890
112.67420
113.16360
112.91890
113.16360
112.91890
112.67420
112.91890
112.67420
112.91890
112.91890
112.91890
113.16360
112.91890
112.91890
112.18491
112.42960
112.42960
112.18491
112.91890
112.67420
112.42960
112.42960
112.18491
112.42960
112.67420
112.42960
112.42960
112.18491
112.67420
112.42960
112.42960
112.67420
112.42960
112.42960
112.42960
112.67420
112.91890
113.40820
113.40820
113.40820
112.91890
112.67420
112.67420
112.91890
113.65289
113.89750
114.38680
114.87611
114.87611
115.12080
115.61010
115.36539
115.61010
115.85471
116.09940
116.83331
116.34400
116.58870
116.58870
116.34400
116.83331
116.83331
116.83331
117.32260
116.83331
117.32260
117.56730
117.32260
117.07800
117.32260
117.81200
117.81200
117.81200
118.54590
118.05661
118.05661
117.56730
117.32260
117.81200
118.30130
118.05661
118.54590
118.05661
118.30130
118.05661
118.30130
118.30130
118.30130
118.05661
117.81200
117.32260
118.30130
118.30130
117.81200
117.07800
118.05661
117.81200
117.56730
117.32260
117.32260
117.81200
117.32260
117.81200
117.07800
117.32260
116.83331
117.07800
116.83331
116.83331
117.07800
115.12080
116.58870
116.58870
116.34400
115.85471
116.34400
116.34400
115.85471
116.58870
116.34400
115.61010
115.85471
115.61010
115.85471
115.12080
115.61010
115.61010
115.85471
115.61010
115.36539
114.87611
114.87611
114.63150
114.87611
115.12080
114.63150
114.87611
115.12080
114.63150
114.38680
114.38680
114.87611
114.63150
114.63150
114.63150
114.63150
114.63150
114.14220
113.65289
113.65289
113.89750
113.65289
113.40820
113.40820
113.89750
113.89750
113.89750
113.65289
113.65289
113.89750
113.40820
113.40820
113.65289
113.89750
113.89750
114.14220
113.65289
113.40820
113.40820
113.65289
113.40820
114.14220
113.89750
114.14220
113.65289
113.65289
113.65289
113.89750
113.16360
113.16360
113.89750
113.65289
113.16360
113.65289
113.40820
112.91890
113.16360
113.16360
113.40820
113.40820
113.65289
113.16360
113.40820
113.16360
113.16360
112.91890
112.91890
112.91890
113.65289
113.65289
113.16360
112.91890
112.67420
113.16360
112.91890
112.67420
112.91890
112.91890
112.91890
111.20631
112.91890
113.16360
112.42960
112.67420
113.16360
112.42960
112.67420
112.91890
112.67420
111.20631
112.42960
112.67420
112.42960
113.16360
112.91890
112.67420
112.91890
112.42960
112.67420
112.18491
112.91890
112.42960
112.18491
6.6.2.2. Model Identification

Check for
Stationarity,
Outliers,
Seasonality
The first step in the analysis is to generate a run sequence
plot of the response variable. A run sequence plot can
indicate stationarity (i.e., constant location and scale), the
presence of outliers, and seasonal patterns.
Non-stationarity can often be removed by differencing the
data or fitting some type of trend curve. We would then
attempt to fit a Box-Jenkins model to the differenced data or
to the residuals after fitting a trend curve.
Although Box-Jenkins models can estimate seasonal
components, the analyst needs to specify the seasonal period
(for example, 12 for monthly data). Seasonal components are
common for economic time series. They are less common for
engineering and scientific data.
Run Sequence
Plot
Interpretation
of the Run
Sequence Plot
We can make the following conclusions from the run
sequence plot.
1. The data show strong and positive autocorrelation.
2. There does not seem to be a significant trend or any
obvious seasonal pattern in the data.
The next step is to examine the sample autocorrelations using
the autocorrelation plot.
Autocorrelation
Plot
Interpretation
of the
Autocorrelation
Plot
The autocorrelation plot has a 95% confidence band, which
is constructed based on the assumption that the process is a
moving average process. The autocorrelation plot shows that
the sample autocorrelations are very strong and positive and
decay very slowly.
The autocorrelation plot indicates that the process is non-
stationary and suggests an ARIMA model. The next step is to
difference the data.
Run Sequence
Plot of
Differenced
Data
Interpretation
of the Run
Sequence Plot
The run sequence plot of the differenced data shows that the
mean of the differenced data is around zero, with the
differenced data less autocorrelated than the original data.
The next step is to examine the sample autocorrelations of
the differenced data.
Autocorrelation
Plot of the
Differenced
Data
Interpretation
of the
Autocorrelation
Plot of the
Differenced
Data
The autocorrelation plot of the differenced data with a 95%
confidence band shows that only the autocorrelation at lag 1
is significant. The autocorrelation plot together with run
sequence of the differenced data suggest that the differenced
data are stationary. Based on the autocorrelation plot, an
MA(1) model is suggested for the differenced data.
To examine other possible models, we produce the partial
autocorrelation plot of the differenced data.
Partial
Autocorrelation
Plot of the
Differenced
Data
Interpretation
of the Partial
Autocorrelation
Plot of the
Differenced
Data
The partial autocorrelation plot of the differenced data with
95% confidence bands shows that only the partial
autocorrelations of the first and second lag are significant.
This suggests an AR(2) model for the differenced data.
Akaike
Information
Criterion (AIC
and AICC)
Information-based criteria, such as the AIC or AICC (see
Brockwell and Davis (2002), pp. 171-174), can be used to
automate the choice of an appropriate model. Many software
programs for time series analysis will generate the AIC or
AICC for a broad range of models.
Whatever method is used for model identification, model
diagnostics should be performed on the selected model.
Based on the plots in this section, we will examine the
ARIMA(2,1,0) and ARIMA(0,1,1) models in detail.
6.6.2.3. Model Estimation

AR(2)
Model
Parameter
Estimates
The following parameter estimates were computed for the AR(2) model based on the
differenced data.

Parameter Standard 95 % Confidence
Source Estimate Error Interval
------ --------- -------- ----------------
Intercept -0.0050 0.0119
AR1 -0.4064 0.0419 (-0.4884, -0.3243)
AR2 -0.1649 0.0419 (-0.2469, -0.0829)

Number of Observations: 558
Degrees of Freedom: 558 - 3 = 555
Residual Standard Deviation: 0.4423
Both AR parameters are significant since the confidence intervals do not contain zero.
The model for the differenced data, Y
t
, is an AR(2) model:
with = 0.4423.
It is often more convenient to express the model in terms of the original data, X
t
, rather
than the differenced data. From the definition of the difference, Y
t
= X
t
- X
t-1
, we can
make the appropriate substitutions into the above equation:
to arrive at the model in terms of the original series:
MA(1)
Model
Parameter
Estimates
Alternatively, the parameter estimates for an MA(1) model based on the differenced data
are the following.
Parameter Standard 95 % Confidence
Source Estimate Error Interval
------ --------- -------- ----------------
Intercept -0.0051 0.0114
MA1 -0.3921 0.0366 (-0.4638, -0.3205)

Number of Observations: 558
Degrees of Freedom: 558 - 2 = 556
Residual Standard Deviation: 0.4434
The model for the differenced data, Y
t
, is an ARIMA(0,1,1) model:
with = 0.4434.
It is often more convenient to express the model in terms of the original data, X
t
, rather
than the differenced data. Making the appropriate substitutions into the above equation:
we arrive at the model in terms of the original series:
6.6.2.4. Model Validation

Residuals After fitting the model, we should check whether the model
is appropriate.
As with standard non-linear least squares fitting, the primary
tool for model diagnostic checking is residual analysis.
4-Plot of
Residuals from
ARIMA(2,1,0)
Model
The 4-plot is a convenient graphical technique for model
validation in that it tests the assumptions for the residuals on
a single graph.
Interpretation
of the 4-Plot
We can make the following conclusions based on the above
4-plot.
1. The run sequence plot shows that the residuals do not
violate the assumption of constant location and scale. It
also shows that most of the residuals are in the range (-
1, 1).
2. The lag plot indicates that the residuals are not
autocorrelated at lag 1.
3. The histogram and normal probability plot indicate that
the normal distribution provides an adequate fit for this
model.
Autocorrelation
Plot of
Residuals from
ARIMA(2,1,0)
Model
In addition, the autocorrelation plot of the residuals from the
ARIMA(2,1,0) model was generated.
Interpretation
of the
Autocorrelation
Plot
The autocorrelation plot shows that for the first 25 lags, all
sample autocorrelations except those at lags 7 and 18 fall
inside the 95 % confidence bounds indicating the residuals
appear to be random.
Test the
Randomness of
Residuals From
the
ARIMA(2,1,0)
Model Fit
We apply the Box-Ljung test to the residuals from the
ARIMA(2,1,0) model fit to determine whether residuals are
random. In this example, the Box-Ljung test shows that the
first 24 lag autocorrelations among the residuals are zero (p-
value = 0.080), indicating that the residuals are random and
that the model provides an adequate fit to the data.
4-Plot of
Residuals from
ARIMA(0,1,1)
Model
The 4-plot is a convenient graphical technique for model
validation in that it tests the assumptions for the residuals on
a single graph.
Interpretation
of the 4-Plot
from the
ARIMA(0,1,1)
Model
We can make the following conclusions based on the above
4-plot.
1. The run sequence plot shows that the residuals do not
violate the assumption of constant location and scale. It
also shows that most of the residuals are in the range (-
1, 1).
2. The lag plot indicates that the residuals are not
autocorrelated at lag 1.
3. The histogram and normal probability plot indicate that
the normal distribution provides an adequate fit for this
model.
This 4-plot of the residuals indicates that the fitted model is
adequate for the data.
Autocorrelation
Plot of
Residuals from
ARIMA(0,1,1)
Model
The autocorrelation plot of the residuals from ARIMA(0,1,1)
was generated.
Interpretation
of the
Autocorrelation
Plot
Similar to the result for the ARIMA(2,1,0) model, it shows
that for the first 25 lags, all sample autocorrelations expect
those at lags 7 and 18 fall inside the 95% confidence bounds
indicating the residuals appear to be random.
Test the
Randomness of
Residuals From
the
ARIMA(0,1,1)
Model Fit
The Box-Ljung test is also applied to the residuals from the
ARIMA(0,1,1) model. The test indicates that there is at least
one non-zero autocorrelation amont the first 24 lags. We
conclude that there is not enough evidence to claim that the
residuals are random (p-value = 0.026).
Summary Overall, the ARIMA(0,1,1) is an adequate model. However,
the ARIMA(2,1,0) is a little better than the ARIMA(0,1,1).

View
Dataplot
Macro for
this Case
Study
Data Analysis Steps
Results and
Conclusions
more detailed
1. Read in the data. 1. You have read
one column of numbers
into Dataplot,
variable Y.
2. Model identification plots
1. Run sequence plot of Y.
2. Autocorrelation plot of Y.
3. Run sequence plot of the
differenced data of Y.
1. The run sequence
plot shows that the
data show strong
and positive
autocorrelation.
2. The
indicates
significant
autocorrelation
and that the
data are not
stationary.
4. Autocorrelation plot of the
differenced data of Y.
5. Partial autocorrelation plot
of the differenced data of Y.
3. The run sequence
plot shows that the
differenced data
appear to be
stationary
and do not
exhibit seasonality.
4. The
of the
differenced data
suggests an
ARIMA(0,1,1)
model may be
appropriate.
5. The partial
suggests an
ARIMA(2,1,0) model
may
be appropriate.
3. Estimate the model.
1. ARIMA(2,1,0) fit of Y.
2. ARIMA(0,1,1) fit of Y.
1. The ARMA fit
generates parameter
estimates for the
ARIMA(2,1,0)
model.
2. The ARMA fit
generates parameter
estimates for the
ARIMA(0,1,1)
model.
4. Model validation.
1. Generate a 4-plot of the
residuals from the ARIMA(2,1,0)
model.
2. Generate an autocorrelation plot
of the residuals from the
ARIMA(2,1,0) model.
3. Perform a Ljung-Box test of
randomness for the residuals from
the ARIMA(2,1,0) model.
4. Generate a 4-plot of the
residuals from the ARIMA(0,1,1)
model.
5. Generate an autocorrelation plot
of the residuals from the
ARIMA(0,1,1) model.
6. Perform a Ljung-Box test of
randomness for the residuals from
the ARIMA(0,1,1) model.
1. The 4-plot shows
that the
assumptions for
the residuals
are satisfied.
2. The
of the
residuals
indicates that the
residuals are
random.
3. The Ljung-Box
test indicates
that the
residuals are
random.
4. The 4-plot shows
that the
assumptions for
the residuals
are satisfied.
5. The
of the
residuals
indicates that the
residuals are
random.
6. The Ljung-Box
test indicates
that the
residuals are not
random at the 95%
level, but
are random at the
99% level.
6.7. References

6.7. References
Selected References
Time Series Analysis
Abraham, B. and Ledolter, J. (1983). Statistical Methods for Forecasting,
Wiley, New York, NY.
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series
Analysis, Forecasting and Control, 3rd ed. Prentice Hall, Englewood Clifs,
NJ.
Box, G. E. P. and McGregor, J. F. (1974). "The Analysis of Closed-Loop
Dynamic Stochastic Systems", Technometrics, Vol. 16-3.
Brockwell, Peter J. and Davis, Richard A. (1987). Time Series: Theory and
Methods, Springer-Verlang.
Brockwell, Peter J. and Davis, Richard A. (2002). Introduction to Time
Series and Forecasting, 2nd. ed., Springer-Verlang.
Chatfield, C. (1996). The Analysis of Time Series, 5th ed., Chapman & Hall,
New York, NY.
DeLurgio, S. A. (1998). Forecasting Principles and Applications, Irwin
McGraw-Hill, Boston, MA.
Ljung, G. and Box, G. (1978). "On a Measure of Lack of Fit in Time Series
Models", Biometrika, 65, 297-303.
Nelson, C. R. (1973). Applied Time Series Analysis for Managerial
Forecasting, Holden-Day, Boca-Raton, FL.
Makradakis, S., Wheelwright, S. C. and McGhee, V. E. (1983). Forecasting:
Methods and Applications, 2nd ed., Wiley, New York, NY.
Statistical Process and Quality Control
Army Chemical Corps (1953). Master Sampling Plans for Single, Duplicate,
Double and Multiple Sampling, Manual No. 2.
Bissell, A. F. (1990). "How Reliable is Your Capability Index?", Applied
Statistics, 39, 331-340.
Champ, C.W., and Woodall, W.H. (1987). "Exact Results for Shewhart
6.7. References
Control Charts with Supplementary Runs Rules", Technometrics, 29, 393-
399.
Duncan, A. J. (1986). Quality Control and Industrial Statistics, 5th ed.,
Irwin, Homewood, IL.
Hotelling, H. (1947). Multivariate Quality Control. In C. Eisenhart, M. W.
Hastay, and W. A. Wallis, eds. Techniques of Statistical Analysis. New
York: McGraw-Hill.
Juran, J. M. (1997). "Early SQC: A Historical Supplement", Quality
Progress, 30(9) 73-81.
Montgomery, D. C. (2000). Introduction to Statistical Quality Control, 4th
ed., Wiley, New York, NY.
Kotz, S. and Johnson, N. L. (1992). Process Capability Indices, Chapman &
Hall, London.
Lowry, C. A., Woodall, W. H., Champ, C. W., and Rigdon, S. E. (1992). "A
Multivariate Exponentially Weighted Moving Average Chart",
Technometrics, 34, 46-53.
Lucas, J. M. and Saccucci, M. S. (1990). "Exponentially weighted moving
average control schemes: Properties and enhancements", Technometrics 32,
1-29.
Ott, E. R. and Schilling, E. G. (1990). Process Quality Control, 2nd ed.,
McGraw-Hill, New York, NY.
Quesenberry, C. P. (1993). "The effect of sample size on estimated limits for
and X control charts", Journal of Quality Technology, 25(4) 237-247.
Ryan, T.P. (2000). Statistical Methods for Quality Improvement, 2nd ed.,
Wiley, New York, NY.
Ryan, T. P. and Schwertman, N. C. (1997). "Optimal limits for attributes
control charts", Journal of Quality Technology, 29 (1), 86-98.
Schilling, E. G. (1982). Acceptance Sampling in Quality Control, Marcel
Dekker, New York, NY.
Tracy, N. D., Young, J. C. and Mason, R. L. (1992). "Multivariate Control
Charts for Individual Observations", Journal of Quality Technology, 24(2),
88-95.
Woodall, W. H. (1997). "Control Charting Based on Attribute Data:
Bibliography and Review", Journal of Quality Technology, 29, 172-183.
Woodall, W. H., and Adams, B. M. (1993); "The Statistical Design of
CUSUM Charts", Quality Engineering, 5(4), 559-570.
Zhang, Stenback, and Wardrop (1990). "Interval Estimation of the Process
Capability Index", Communications in Statistics: Theory and Methods,
19(21), 4455-4470.
6.7. References
Statistical Analysis
Anderson, T. W. (1984). Introduction to Multivariate Statistical Analysis,
2nd ed., Wiley New York, NY.
Johnson, R. A. and Wichern, D. W. (1998). Applied Multivariate Statistical
Analysis, Fourth Ed., Prentice Hall, Upper Saddle River, NJ.
7. Product and Process Comparisons
http://www.itl.nist.gov/div898/handbook/prc/prc.htm[6/27/2012 2:43:09 PM]

7. Product and Process
Comparisons
This chapter presents the background and specific analysis techniques
needed to compare the performance of one or more processes against known
standards or one another.
1. Introduction
1. Scope
2. Assumptions
3. Statistical Tests
4. Confidence Intervals
5. Equivalence of Tests and
Intervals
6. Outliers
7. Trends
2. Comparisons: One Process
1. Comparing to a Distribution
2. Comparing to a Nominal
Mean
3. Comparing to Nominal
Variability
4. Fraction Defective
5. Defect Density
6. Location of Population
Values
3. Comparisons: Two Processes
1. Means: Normal Data
2. Variability: Normal Data
3. Fraction Defective
4. Failure Rates
5. Means: General Case
4. Comparisons: Three +
Processes
1. Comparing Populations
2. Comparing Variances
3. Comparing Means
4. Variance Components
5. Comparing Categorical
Datasets
6. Comparing Fraction
Defectives
7. Multiple Comparisons
Detailed table of contents
http://www.itl.nist.gov/div898/handbook/prc/prc_d.htm[6/27/2012 2:42:08 PM]

7. Product and Process Comparisons - Detailed Table of Contents [7.]
1. What is the scope? [7.1.1.]
2. What assumptions are typically made? [7.1.2.]
3. What are statistical tests? [7.1.3.]
1. Critical values and p values [7.1.3.1.]
4. What are confidence intervals? [7.1.4.]
5. What is the relationship between a test and a confidence interval? [7.1.5.]
6. What are outliers in the data? [7.1.6.]
7. What are trends in sequential process or product data? [7.1.7.]
2. Comparisons based on data from one process [7.2.]
1. Do the observations come from a particular distribution? [7.2.1.]
1. Chi-square goodness-of-fit test [7.2.1.1.]
2. Kolmogorov- Smirnov test [7.2.1.2.]
3. Anderson-Darling and Shapiro-Wilk tests [7.2.1.3.]
2. Are the data consistent with the assumed process mean? [7.2.2.]
1. Confidence interval approach [7.2.2.1.]
2. Sample sizes required [7.2.2.2.]
3. Are the data consistent with a nominal standard deviation? [7.2.3.]
1. Confidence interval approach [7.2.3.1.]
4. Does the proportion of defectives meet requirements? [7.2.4.]
1. Confidence intervals [7.2.4.1.]
5. Does the defect density meet requirements? [7.2.5.]
6. What intervals contain a fixed percentage of the population values? [7.2.6.]
1. Approximate intervals that contain most of the population values [7.2.6.1.]
2. Percentiles [7.2.6.2.]
3. Tolerance intervals for a normal distribution [7.2.6.3.]
4. Tolerance intervals based on the largest and smallest observations [7.2.6.4.]
3. Comparisons based on data from two processes [7.3.]
1. Do two processes have the same mean? [7.3.1.]
1. Analysis of paired observations [7.3.1.1.]
2. Confidence intervals for differences between means [7.3.1.2.]
2. Do two processes have the same standard deviation? [7.3.2.]
3. How can we determine whether two processes produce the same proportion of defectives? [7.3.3.]
4. Assuming the observations are failure times, are the failure rates (or Mean Times To Failure) for two
distributions the same? [7.3.4.]
5. Do two arbitrary processes have the same central tendency? [7.3.5.]
4. Comparisons based on data from more than two processes [7.4.]
http://www.itl.nist.gov/div898/handbook/prc/prc_d.htm[6/27/2012 2:42:08 PM]
1. How can we compare several populations with unknown distributions (the Kruskal-Wallis
test)? [7.4.1.]
2. Assuming the observations are normal, do the processes have the same variance? [7.4.2.]
3. Are the means equal? [7.4.3.]
1. 1-Way ANOVA overview [7.4.3.1.]
2. The 1-way ANOVA model and assumptions [7.4.3.2.]
3. The ANOVA table and tests of hypotheses about means [7.4.3.3.]
4. 1-Way ANOVA calculations [7.4.3.4.]
5. Confidence intervals for the difference of treatment means [7.4.3.5.]
6. Assessing the response from any factor combination [7.4.3.6.]
7. The two-way ANOVA [7.4.3.7.]
8. Models and calculations for the two-way ANOVA [7.4.3.8.]
4. What are variance components? [7.4.4.]
5. How can we compare the results of classifying according to several categories? [7.4.5.]
6. Do all the processes have the same proportion of defects? [7.4.6.]
7. How can we make multiple comparisons? [7.4.7.]
1. Tukey's method [7.4.7.1.]
2. Scheffe's method [7.4.7.2.]
3. Bonferroni's method [7.4.7.3.]
4. Comparing multiple proportions: The Marascuillo procedure [7.4.7.4.]
7.1. Introduction
http://www.itl.nist.gov/div898/handbook/prc/section1/prc1.htm[6/27/2012 2:42:13 PM]

7.1. Introduction
Goals of
this
section
The primary goal of this section is to lay a foundation for
understanding statistical tests and confidence intervals that are
useful for making decisions about processes and comparisons
among processes. The materials covered are:
Scope
Assumptions
Introduction to hypothesis testing
Introduction to confidence intervals
Relationship between hypothesis testing and confidence
intervals
Outlier detection
Detection of sequential trends in data or processes
Hypothesis
testing and
confidence
intervals
This chapter explores the types of comparisons which can be
made from data and explains hypothesis testing, confidence
intervals, and the interpretation of each.
7.1.1. What is the scope?

7.1. Introduction
7.1.1. What is the scope?
Data from
one
process
This section deals with introductory material related to
comparisons that can be made on data from one process for
cases where the process standard deviation may be known or
unknown.
7.1.2. What assumptions are typically made?

7.1. Introduction
7.1.2. What assumptions are typically made?
Validity of
tests
The validity of the tests described in this chapter depend
on the following assumptions:
1. The data come from a single process that can be
represented by a single statistical distribution.
2. The distribution is a normal distribution.
3. The data are uncorrelated over time.
An easy method for checking the assumption of a single
normal distribution is to construct a histogram of the data.
Clarification The tests described in this chapter depend on the
assumption of normality, and the data should be examined
for departures from normality before the tests are applied.
However, the tests are robust to small departures from
normality; i.e., they work fairly well as long as the data are
bell-shaped and the tails are not heavy. Quantitative
methods for checking the normality assumption are
discussed in the next section.
Another graphical method for testing the normality
assumption is the normal probability plot.
A graphical method for testing for correlation among
measurements is a time-lag plot. Correlation may not be a
problem if measurements are properly structured over time.
Correlation problems often occur when measurements are
made close together in time.
7.1.3. What are statistical tests?

7.1. Introduction
What is
meant by a
statistical
test?
A statistical test provides a mechanism for making
quantitative decisions about a process or processes. The
intent is to determine whether there is enough evidence to
"reject" a conjecture or hypothesis about the process. The
conjecture is called the null hypothesis. Not rejecting may be
a good result if we want to continue to act as if we "believe"
the null hypothesis is true. Or it may be a disappointing
result, possibly indicating we may not yet have enough data
to "prove" something by rejecting the null hypothesis.
For more discussion about the meaning of a statistical
hypothesis test, see Chapter 1.
Concept of
null
hypothesis
A classic use of a statistical test occurs in process control
studies. For example, suppose that we are interested in
ensuring that photomasks in a production process have mean
linewidths of 500 micrometers. The null hypothesis, in this
case, is that the mean linewidth is 500 micrometers. Implicit
in this statement is the need to flag photomasks which have
mean linewidths that are either much greater or much less
than 500 micrometers. This translates into the alternative
hypothesis that the mean linewidths are not equal to 500
micrometers. This is a two-sided alternative because it guards
against alternatives in opposite directions; namely, that the
linewidths are too small or too large.
The testing procedure works this way. Linewidths at random
positions on the photomask are measured using a scanning
electron microscope. A test statistic is computed from the
data and tested against pre-determined upper and lower
critical values. If the test statistic is greater than the upper
critical value or less than the lower critical value, the null
hypothesis is rejected because there is evidence that the mean
linewidth is not 500 micrometers.
One-sided
tests of
hypothesis
Null and alternative hypotheses can also be one-sided. For
example, to ensure that a lot of light bulbs has a mean
lifetime of at least 500 hours, a testing program is
implemented. The null hypothesis, in this case, is that the
mean lifetime is greater than or equal to 500 hours. The
complement or alternative hypothesis that is being guarded
against is that the mean lifetime is less than 500 hours. The
test statistic is compared with a lower critical value, and if it
is less than this limit, the null hypothesis is rejected.
Thus, a statistical test requires a pair of hypotheses; namely,
H
0
: a null hypothesis
H
a
: an alternative hypothesis.
Significance
levels
The null hypothesis is a statement about a belief. We may
doubt that the null hypothesis is true, which might be why we
are "testing" it. The alternative hypothesis might, in fact, be
what we believe to be true. The test procedure is constructed
so that the risk of rejecting the null hypothesis, when it is in
fact true, is small. This risk, , is often referred to as the
significance level of the test. By having a test with a small
value of , we feel that we have actually "proved" something
when we reject the null hypothesis.
Errors of
the second
kind
The risk of failing to reject the null hypothesis when it is in
fact false is not chosen by the user but is determined, as one
might expect, by the magnitude of the real discrepancy. This
risk, , is usually referred to as the error of the second kind.
Large discrepancies between reality and the null hypothesis
are easier to detect and lead to small errors of the second
kind; while small discrepancies are more difficult to detect
and lead to large errors of the second kind. Also the risk
increases as the risk decreases. The risks of errors of the
second kind are usually summarized by an operating
characteristic curve (OC) for the test. OC curves for several
types of tests are shown in (Natrella, 1962).
Guidance in
this chapter
This chapter gives methods for constructing test statistics and
their corresponding critical values for both one-sided and
two-sided tests for the specific situations outlined under the
scope. It also provides guidance on the sample sizes required
for these tests.
Further guidance on statistical hypothesis testing,
significance levels and critical regions, is given in Chapter 1.
7.1.3.1. Critical values and p values

7.1. Introduction
7.1.3.1. Critical values and p values
Determination
of critical
values
Critical values for a test of hypothesis depend upon a test
statistic, which is specific to the type of test, and the
significance level, , which defines the sensitivity of the
test. A value of = 0.05 implies that the null hypothesis is
rejected 5% of the time when it is in fact true. The choice
of is somewhat arbitrary, although in practice values of
0.1, 0.05, and 0.01 are common. Critical values are
essentially cut-off values that define regions where the test
statistic is unlikely to lie; for example, a region where the
critical value is exceeded with probability if the null
hypothesis is true. The null hypothesis is rejected if the test
statistic lies within this region which is often referred to as
the rejection region(s). Critical values for specific tests of
hypothesis are tabled in chapter 1.
Information in
this chapter
This chapter gives formulas for the test statistics and points
to the appropriate tables of critical values for tests of
hypothesis regarding means, standard deviations, and
proportion defectives.
P values Another quantitative measure for reporting the result of a
test of hypothesis is the p-value. The p-value is the
probability of the test statistic being at least as extreme as
the one observed given that the null hypothesis is true. A
small p-value is an indication that the null hypothesis is
false.
Good practice It is good practice to decide in advance of the test how
small a p-value is required to reject the test. This is exactly
analagous to choosing a significance level, for test. For
example, we decide either to reject the null hypothesis if
the test statistic exceeds the critical value (for = 0.05) or
analagously to reject the null hypothesis if the p-value is
smaller than 0.05. It is important to understand the
relationship between the two concepts because some
statistical software packages report p-values rather than
critical values.
7.1.4. What are confidence intervals?

7.1. Introduction
How do we
form a
confidence
interval?
The purpose of taking a random sample from a lot or
population and computing a statistic, such as the mean from
the data, is to approximate the mean of the population. How
well the sample statistic estimates the underlying population
value is always an issue. A confidence interval addresses this
issue because it provides a range of values which is likely to
contain the population parameter of interest.
Confidence
levels
Confidence intervals are constructed at a confidence level,
such as 95%, selected by the user. What does this mean? It
means that if the same population is sampled on numerous
occasions and interval estimates are made on each occasion,
the resulting intervals would bracket the true population
parameter in approximately 95% of the cases. A confidence
stated at a level can be thought of as the inverse of a
significance level, .
One and
two-sided
confidence
intervals
In the same way that statistical tests can be one or two-sided,
confidence intervals can be one or two-sided. A two-sided
confidence interval brackets the population parameter from
above and below. A one-sided confidence interval brackets
the population parameter either from above or below and
furnishes an upper or lower bound to its magnitude.
Example of
a two-
sided
confidence
interval
For example, a 100( )% confidence interval for the mean
of a normal population is;
where is the sample mean, z
1-/2
is the 1-/2 critical value
of the standard normal distribution which is found in the table
of the standard normal distribution, is the known population
standard deviation, and N is the sample size.
Guidance
in this
chapter
This chapter provides methods for estimating the population
parameters and confidence intervals for the situations
described under the scope.
Problem
with
unknown
standard
deviation
In the normal course of events, population standard deviations
are not known, and must be estimated from the data.
Confidence intervals, given the same confidence level, are by
necessity wider if the standard deviation is estimated from
limited data because of the uncertainty in this estimate.
Procedures for creating confidence intervals in this situation
are described fully in this chapter.
More information on confidence intervals can also be found in
Chapter 1.
7.1.5. What is the relationship between a test and a confidence interval?

7.1. Introduction
7.1.5. What is the relationship between a test
and a confidence interval?
There is a
correspondence
between
hypothesis
testing and
confidence
intervals
In general, for every test of hypothesis there is an
equivalent statement about whether the hypothesized
parameter value is included in a confidence interval. For
example, consider the previous example of linewidths
where photomasks are tested to ensure that their
linewidths have a mean of 500 micrometers. The null and
alternative hypotheses are:
H
0
: mean linewidth = 500 micrometers
H
a
: mean linewidth 500 micrometers
Hypothesis test
for the mean For the test, the sample mean, , is calculated from N
linewidths chosen at random positions on each
photomask. For the purpose of the test, it is assumed that
the standard deviation, , is known from a long history
of this process. A test statistic is calculated from these
sample statistics, and the null hypothesis is rejected if:
where z
/2
and z
1-/2
are tabled values from the normal
distribution.
Equivalent
confidence
interval
With some algebra, it can be seen that the null hypothesis
is rejected if and only if the value 500 micrometers is not
in the confidence interval
Equivalent
confidence
interval
In fact, all values bracketed by this interval would be
accepted as null values for a given set of test data.
7.1.5. What is the relationship between a test and a confidence interval?
7.1.6. What are outliers in the data?

7.1. Introduction
Definition
of outliers
An outlier is an observation that lies an abnormal distance from other values in a
random sample from a population. In a sense, this definition leaves it up to the
analyst (or a consensus process) to decide what will be considered abnormal. Before
abnormal observations can be singled out, it is necessary to characterize normal
observations.
Ways to
describe
data
Two activities are essential for characterizing a set of data:
1. Examination of the overall shape of the graphed data for important features,
including symmetry and departures from assumptions. The chapter on
Exploratory Data Analysis (EDA) discusses assumptions and summarization
of data in detail.
2. Examination of the data for unusual observations that are far removed from
the mass of data. These points are often referred to as outliers. Two graphical
techniques for identifying outliers, scatter plots and box plots, along with an
analytic procedure for detecting outliers when the distribution is normal
(Grubbs' Test), are also discussed in detail in the EDA chapter.
Box plot
construction
The box plot is a useful graphical display for describing the behavior of the data in
the middle as well as at the ends of the distributions. The box plot uses the median
and the lower and upper quartiles (defined as the 25th and 75th percentiles). If the
lower quartile is Q1 and the upper quartile is Q2, then the difference (Q2 - Q1) is
called the interquartile range or IQ.
Box plots
with fences
A box plot is constructed by drawing a box between the upper and lower quartiles
with a solid line drawn across the box to locate the median. The following quantities
(called fences) are needed for identifying extreme values in the tails of the
distribution:
1. lower inner fence: Q1 - 1.5*IQ
2. upper inner fence: Q2 + 1.5*IQ
3. lower outer fence: Q1 - 3*IQ
4. upper outer fence: Q2 + 3*IQ
Outlier
detection
criteria
A point beyond an inner fence on either side is considered a mild outlier. A point
beyond an outer fence is considered an extreme outlier.
Example of
an outlier
The data set of N = 90 ordered observations as shown below is examined for
outliers:
box plot
30, 171, 184, 201, 212, 250, 265, 270, 272, 289, 305, 306, 322, 322, 336, 346, 351,
370, 390, 404, 409, 411, 436, 437, 439, 441, 444, 448, 451, 453, 470, 480, 482, 487,
494, 495, 499, 503, 514, 521, 522, 527, 548, 550, 559, 560, 570, 572, 574, 578, 585,
592, 592, 607, 616, 618, 621, 629, 637, 638, 640, 656, 668, 707, 709, 719, 737, 739,
752, 758, 766, 792, 792, 794, 802, 818, 830, 832, 843, 858, 860, 869, 918, 925, 953,
991, 1000, 1005, 1068, 1441
The computations are as follows:
Median = (n+1)/2 largest data point = the average of the 45th and 46th
ordered points = (559 + 560)/2 = 559.5
Lower quartile = .25(N+1)th ordered point = 22.75th ordered point = 411 +
.75(436-411) = 429.75
Upper quartile = .75(N+1)th ordered point = 68.25th ordered point = 739
+.25(752-739) = 742.25
Interquartile range = 742.25 - 429.75 = 312.5
Lower inner fence = 429.75 - 1.5 (312.5) = -39.0
Upper inner fence = 742.25 + 1.5 (312.5) = 1211.0
Lower outer fence = 429.75 - 3.0 (312.5) = -507.75
Upper outer fence = 742.25 + 3.0 (312.5) = 1679.75
From an examination of the fence points and the data, one point (1441) exceeds the
upper inner fence and stands out as a mild outlier; there are no extreme outliers.
Histogram
with box
plot
A histogram with an overlaid box plot are shown below.
The outlier is identified as the largest value in the data set, 1441, and appears as the
circle to the right of the box plot.
Outliers
may contain
important
information
Outliers should be investigated carefully. Often they contain valuable information
about the process under investigation or the data gathering and recording process.
Before considering the possible elimination of these points from the data, one should
try to understand why they appeared and whether it is likely similar values will
continue to appear. Of course, outliers are often bad data points.
7.1.7. What are trends in sequential process or product data?

7.1. Introduction
7.1.7. What are trends in sequential process or
product data?
Detecting
trends by
plotting
the data
points to
see if a
line with
an
obviously
non-zero
slope fits
the points
Detecting trends is equivalent to comparing the process values
to what we would expect a series of numbers to look like if
there were no trends. If we see a significant departure from a
model where the next observation is equally likely to go up or
down, then we would reject the hypothesis of "no trend".
A common way of investigating for trends is to fit a straight
line to the data and observe the line's direction (or slope). If
the line looks horizontal, then there is no evidence of a trend;
otherwise there is. Formally, this is done by testing whether
the slope of the line is significantly different from zero. The
methodology for this is covered in Chapter 4.
Other
trend tests
A non-parametric approach for detecting significant trends
known as the Reverse Arrangement Test is described in
Chapter 8.
7.2. Comparisons based on data from one process

7.2. Comparisons based on data from one
process
Questions
answered in
this section
For a single process, the current state of the process can be
compared with a nominal or hypothesized state. This
section outlines techniques for answering the following
questions from data gathered from a single process:
1. Do the observations come from a particular
distribution?
1. Chi-Square Goodness-of-Fit test for a
continuous or discrete distribution
2. Kolmogorov- Smirnov test for a continuous
distribution
3. Anderson-Darling and Shapiro-Wilk tests for
a continuous distribution
2. Are the data consistent with the assumed process
mean?
1. Confidence interval approach
2. Sample sizes required
3. Are the data consistent with a nominal standard
deviation?
1. Confidence interval approach
4. Does the proportion of defectives meet
requirements?
1. Confidence intervals
5. Does the defect density meet requirements?
6. What intervals contain a fixed percentage of the
data?
1. Approximate intervals that contain most of the
population values
2. Percentiles
3. Tolerance intervals
4. Tolerance intervals based on the smallest and
largest observations
General forms
of testing
These questions are addressed either by an hypothesis test
or by a confidence interval.
Parametric vs.
non-
parametric
All hypothesis-testing procedures can be broadly described
as either parametric or non-parametric/distribution-free.
Parametric test procedures are those that:
testing
1. Involve hypothesis testing of specified parameters
(such as "the population mean=50 grams"...).
2. Require a stringent set of assumptions about the
underlying sampling distributions.
When to use
nonparametric
methods?
When do we require non-parametric or distribution-free
methods? Here are a few circumstances that may be
candidates:
1. The measurements are only categorical; i.e., they are
nominally scaled, or ordinally (in ranks) scaled.
2. The assumptions underlying the use of parametric
methods cannot be met.
3. The situation at hand requires an investigation of
such features as randomness, independence,
symmetry, or goodness of fit rather than the testing
of hypotheses about specific values of particular
population parameters.
Difference
between non-
parametric
and
distribution-
free
Some authors distinguish between non-parametric and
distribution-free procedures.
Distribution-free test procedures are broadly defined as:
1. Those whose test statistic does not depend on the
form of the underlying population distribution from
which the sample data were drawn, or
2. Those for which the data are nominally or ordinally
scaled.
Nonparametric test procedures are defined as those that
are not concerned with the parameters of a distribution.
Advantages of
nonparametric
methods.
Distribution-free or nonparametric methods have several
advantages, or benefits:
1. They may be used on all types of data-categorical
data, which are nominally scaled or are in rank form,
called ordinally scaled, as well as interval or ratio-
scaled data.
2. For small sample sizes they are easy to apply.
3. They make fewer and less stringent assumptions
than their parametric counterparts.
4. Depending on the particular procedure they may be
almost as powerful as the corresponding parametric
procedure when the assumptions of the latter are
met, and when this is not the case, they are generally
more powerful.
Disadvantages
of
nonparametric
methods
Of course there are also disadvantages:
1. If the assumptions of the parametric methods can be
met, it is generally more efficient to use them.
2. For large sample sizes, data manipulations tend to
become more laborious, unless computer software is
available.
3. Often special tables of critical values are needed for
the test statistic, and these values cannot always be
generated by computer software. On the other hand,
the critical values for the parametric tests are readily
available and generally easy to incorporate in
computer programs.
7.2.1. Do the observations come from a particular distribution?

7.2.1. Do the observations come from a
particular distribution?
Data are
often
assumed to
come from
a particular
distribution.
Goodness-of-fit tests indicate whether or not it is reasonable
to assume that a random sample comes from a specific
distribution. Statistical techniques often rely on observations
having come from a population that has a distribution of a
specific form (e.g., normal, lognormal, Poisson, etc.).
Standard control charts for continuous measurements, for
instance, require that the data come from a normal
distribution. Accurate lifetime modeling requires specifying
the correct distributional model. There may be historical or
theoretical reasons to assume that a sample comes from a
particular population, as well. Past data may have
consistently fit a known distribution, for example, or theory
may predict that the underlying population should be of a
specific form.
Hypothesis
Test model
for
Goodness-
of-fit
Goodness-of-fit tests are a form of hypothesis testing where
the null and alternative hypotheses are
H
0
: Sample data come from the stated distribution.
H
A
: Sample data do not come from the stated distribution.
Parameters
may be
assumed or
estimated
from the
data
One needs to consider whether a simple or composite
hypothesis is being tested. For a simple hypothesis, values of
the distribution's parameters are specified prior to drawing
the sample. For a composite hypothesis, one or more of the
parameters is unknown. Often, these parameters are estimated
using the sample observations.
A simple hypothesis would be:
H
0
: Data are from a normal distribution, = 0 and = 1.
A composite hypothesis would be:
H
0
: Data are from a normal distribution, unknown and .
Composite hypotheses are more common because they allow
us to decide whether a sample comes from any distribution of
a specific type. In this situation, the form of the distribution
is of interest, regardless of the values of the parameters.
Unfortunately, composite hypotheses are more difficult to
work with because the critical values are often hard to
compute.
Problems
with
censored
data
A second issue that affects a test is whether the data are
censored. When data are censored, sample values are in some
way restricted. Censoring occurs if the range of potential
values are limited such that values from one or both tails of
the distribution are unavailable (e.g., right and/or left
censoring - where high and/or low values are missing).
Censoring frequently occurs in reliability testing, when either
the testing time or the number of failures to be observed is
fixed in advance. A thorough treatment of goodness-of-fit
testing under censoring is beyond the scope of this document.
See D'Agostino & Stephens (1986) for more details.
Three types
of tests will
be covered
Three goodness-of-fit tests are examined in detail:
1. Chi-square test for continuous and discrete
distributions;
2. Kolmogorov-Smirnov test for continuous distributions
based on the empirical distribution function (EDF);
3. Anderson-Darling test for continuous distributions.
A more extensive treatment of goodness-of-fit techniques is
presented in D'Agostino & Stephens (1986). Along with the
tests mentioned above, other general and specific tests are
examined, including tests based on regression and graphical
techniques.
7.2.1.1. Chi-square goodness-of-fit test

7.2.1.1. Chi-square goodness-of-fit test
Choice of
number of
groups for
"Goodness
of Fit" tests
is important
- but only
useful rules
of thumb
can be given
The test requires that the data first be grouped. The actual
number of observations in each group is compared to the
expected number of observations and the test statistic is
calculated as a function of this difference. The number of
groups and how group membership is defined will affect the
power of the test (i.e., how sensitive it is to detecting
departures from the null hypothesis). Power will not only be
affected by the number of groups and how they are defined,
but by the sample size and shape of the null and underlying
(true) distributions. Despite the lack of a clear "best
method", some useful rules of thumb can be given.
Group
Membership
When data are discrete, group membership is unambiguous.
Tabulation or cross tabulation can be used to categorize the
data. Continuous data present a more difficult challenge.
One defines groups by segmenting the range of possible
values into non-overlapping intervals. Group membership
can then be defined by the endpoints of the intervals. In
general, power is maximized by choosing endpoints such
that group membership is equiprobable (i.e., the probabilities
associated with an observation falling into a given group are
divided as evenly as possible across the intervals). Many
commercial software packages follow this procedure.
Rule-of-
thumb for
number of
groups
One rule-of-thumb suggests using the value 2n
2/5
as a good
starting point for choosing the number of groups. Another
well known rule-of-thumb requires every group to have at
least 5 data points.
Computation
of the chi-
square
goodness-
of-fit test
The formulas for the computation of the chi-square goodnes-
of-fit test are given in the EDA chapter.
7.2.1.2. Kolmogorov- Smirnov test

7.2.1.2. Kolmogorov- Smirnov test
The K-S
test is a
good
alternative
to the chi-
square
test.
The Kolmogorov-Smirnov (K-S) test was originally proposed
in the 1930's in papers by Kolmogorov (1933) and Smirnov
(1936). Unlike the Chi-Square test, which can be used for
testing against both continuous and discrete distributions, the
K-S test is only appropriate for testing data against a
continuous distribution, such as the normal or Weibull
distribution. It is one of a number of tests that are based on the
empirical cumulative distribution function (ECDF).
K-S
procedure
Details on the construction and interpretation of the K-S test
statistic, D, and examples for several distributions are outlined
in Chapter 1.
The
probability
associated
with the
test
statistic is
difficult to
compute.
Critical values associated with the test statistic, D, are difficult
to compute for finite sample sizes, often requiring Monte
Carlo simulation. However, some general purpose statistical
software programs support the Kolmogorov-Smirnov test at
least for some of the more common distributions. Tabled
values can be found in Birnbaum (1952). A correction factor
can be applied if the parameters of the distribution are
estimated with the same data that are being tested. See
D'Agostino and Stephens (1986) for details.
7.2.1.3. Anderson-Darling and Shapiro-Wilk tests

7.2.1.3. Anderson-Darling and Shapiro-Wilk
tests
Purpose:
Test for
distributional
adequacy
The Anderson-Darling Test
The Anderson-Darling test (Stephens, 1974) is used to test
if a sample of data comes from a specific distribution. It is a
modification of the Kolmogorov-Smirnov (K-S) test and
gives more weight to the tails of the distribution than does
the K-S test. The K-S test is distribution free in the sense
that the critical values do not depend on the specific
distribution being tested.
Requires
critical
values for
each
distribution
The Anderson-Darling test makes use of the specific
distribution in calculating critical values. This has the
advantage of allowing a more sensitive test and the
disadvantage that critical values must be calculated for each
distribution. Tables of critical values are not given in this
handbook (see Stephens 1974, 1976, 1977, and 1979)
because this test is usually applied with a statistical
software program that produces the relevant critical values.
Currently, Dataplot computes critical values for the
Anderson-Darling test for the following distributions:
normal
lognormal
Weibull
extreme value type I.
Anderson-
Darling
procedure
Details on the construction and interpretation of the
Anderson-Darling test statistic, A
2
, and examples for
several distributions are outlined in Chapter 1.
Shapiro-Wilk
test for
normality
The Shapiro-Wilk Test For Normality
The Shapiro-Wilk test, proposed in 1965, calculates a W
statistic that tests whether a random sample, x
1
, x
2
, ..., x
n
comes from (specifically) a normal distribution . Small
values of W are evidence of departure from normality and
percentage points for the W statistic, obtained via Monte
Carlo simulations, were reproduced by Pearson and Hartley
(1972, Table 16). This test has done very well in
7.2.1.3. Anderson-Darling and Shapiro-Wilk tests
comparison studies with other goodness of fit tests.
The W statistic is calculated as follows:
where the x
(i)
are the ordered sample values (x
(1)
is the
smallest) and the a
i
are constants generated from the means,
variances and covariances of the order statistics of a sample
of size n from a normal distribution (see Pearson and
Hartley (1972, Table 15).
For more information about the Shapiro-Wilk test the reader
is referred to the original Shapiro and Wilk (1965) paper
and the tables in Pearson and Hartley (1972).
7.2.2. Are the data consistent with the assumed process mean?

7.2.2. Are the data consistent with the assumed
process mean?
The testing
of H
0
for a
single
population
mean
Given a random sample of measurements, Y
1
, ..., Y
N
, there
are three types of questions regarding the true mean of the
population that can be addressed with the sample data. They
are:
1. Does the true mean agree with a known standard or
assumed mean?
2. Is the true mean of the population less than a given
standard?
3. Is the true mean of the population at least as large as a
given standard?
Typical null
hypotheses
The corresponding null hypotheses that test the true mean, ,
against the standard or assumed mean, are:
1.
2.
3.
Test
statistic
where the
standard
deviation is
not known
The basic statistics for the test are the sample mean and the
standard deviation. The form of the test statistic depends on
whether the poulation standard deviation, , is known or is
estimated from the data at hand. The more typical case is
where the standard deviation must be estimated from the
data, and the test statistic is
where the sample mean is
and the sample standard deviation is
with N - 1 degrees of freedom.
Comparison
with critical
values
For a test at significance level , where is chosen to be
small, typically 0.01, 0.05 or 0.10, the hypothesis associated
with each case enumerated above is rejected if:
1. | t | t
1-/2, N-1
2. t t
1-, N-1
3. t t
, N-1
where t
1-/2, N-1
is the 1-/2 critical value from the t
distribution with N - 1 degrees of freedom and similarly for
cases (2) and (3). Critical values can be found in the t-table
in Chapter 1.
Test
statistic
where the
standard
deviation is
known
If the standard deviation is known, the form of the test
statistic is
For case (1), the test statistic is compared with z
1-/2
, which
is the 1-/2 critical value from the standard normal
distribution, and similarly for cases (2) and (3).
Caution If the standard deviation is assumed known for the purpose
of this test, this assumption should be checked by a test of
hypothesis for the standard deviation.
An
illustrative
example of
the t-test
The following numbers are particle (contamination) counts
for a sample of 10 semiconductor silicon wafers:
50 48 44 56 61 52 53 55 67 51
The mean = 53.7 counts and the standard deviation = 6.567
counts.
The test is
two-sided
Over a long run the process average for wafer particle counts
has been 50 counts per wafer, and on the basis of the sample,
we want to test whether a change has occurred. The null
hypothesis that the process mean is 50 counts is tested against
the alternative hypothesis that the process mean is not equal
to 50 counts. The purpose of the two-sided alternative is to
rule out a possible process change in either direction.
Critical
values
For a significance level of = 0.05, the chances of
erroneously rejecting the null hypothesis when it is true are 5
% or less. (For a review of hypothesis testing basics, see
Chapter 1).
Even though there is a history on this process, it has not been
stable enough to justify the assumption that the standard
deviation is known. Therefore, the appropriate test statistic is
the t-statistic. Substituting the sample mean, sample standard
deviation, and sample size into the formula for the test
statistic gives a value of
t = 1.782
with degrees of freedom N - 1 = 9. This value is tested
against the critical value
t
1-0.025;9
= 2.262
from the t-table where the critical value is found under the
column labeled 0.975 for the probability of exceeding the
critical value and in the row for 9 degrees of freedom. The
critical value is based on instead of because of the
two-sided alternative (two-tailed test) which requires equal
probabilities in each tail of the distribution that add to .
Conclusion Because the value of the test statistic falls in the interval (-
2.262, 2.262), we cannot reject the null hypothesis and,
therefore, we may continue to assume the process mean is 50
counts.
7.2.2.1. Confidence interval approach

Testing using
a confidence
interval
The hypothesis test results in a "yes" or "no" answer. The null
hypothesis is either rejected or not rejected. There is another way of
testing a mean and that is by constructing a confidence interval about
the true but unknown mean.
General form
of confidence
intervals
where the
standard
deviation is
unknown
Tests of hypotheses that can be made from a single sample of data
were discussed on the foregoing page. As with null hypotheses,
confidence intervals can be two-sided or one-sided, depending on the
question at hand. The general form of confidence intervals, for the
three cases discussed earlier, where the standard deviation is unknown
are:
1. Two-sided confidence interval for :
2. Lower one-sided confidence interval for :
3. Upper one-sided confidence interval for :
where t
/2, N-1
is the /2 critical value from the t distribution with N - 1
degrees of freedom and similarly for cases (2) and (3). Critical values
can be found in the t table in Chapter 1.
Confidence
level
The confidence intervals are constructed so that the probability of the
interval containing the mean is 1 - . Such intervals are referred to as
100(1- )% confidence intervals.
A 95%
confidence
interval for
The corresponding confidence interval for the test of hypothesis
example on the foregoing page is shown below. A 95 % confidence
interval for the population mean of particle counts per wafer is given by
the example
Interpretation The 95 % confidence interval includes the null hypothesis if, and only
if, it would be accepted at the 5 % level. This interval includes the null
hypothesis of 50 counts so we cannot reject the hypothesis that the
process mean for particle counts is 50. The confidence interval includes
all null hypothesis values for the population mean that would be
accepted by an hypothesis test at the 5 % significance level. This
assumes, of course, a two-sided alternative.
7.2.2.2. Sample sizes required

The
computation
of sample
sizes depends
on many
things, some
of which
have to be
assumed in
advance
Perhaps one of the most frequent questions asked of a statistician is,
"How many measurements should be included in the sample?
"
Unfortunately, there is no correct answer without additional
information (or assumptions). The sample size required for an
experiment designed to investigate the behavior of an unknown
population mean will be influenced by the following:
value selected for , the risk of rejecting a true hypothesis
value of , the risk of accepting a false null hypothesis when
a particular value of the alternative hypothesis is true.
value of the population standard deviation.
Application -
estimating a
minimum
sample size,
N, for
limiting the
error in the
estimate of
the mean
For example, suppose that we wish to estimate the average daily
yield, , of a chemical process by the mean of a sample, Y
1
, ..., Y
N
,
such that the error of estimation is less than with a probability of
95%. This means that a 95% confidence interval centered at the
sample mean should be
and if the standard deviation is known,
The critical value from the normal distribution for 1-/2 = 0.975 is
1.96. Therefore,
Limitation
and
interpretation
A restriction is that the standard deviation must be known. Lacking
an exact value for the standard deviation requires some
accommodation, perhaps the best estimate available from a previous
experiment.
Controlling
the risk of
accepting a
false
hypothesis
To control the risk of accepting a false hypothesis, we set not only
, the probability of rejecting the null hypothesis when it is true,
but also , the probability of accepting the null hypothesis when in
fact the population mean is where is the difference or shift
we want to detect.
Standard
deviation
assumed to
be known
The minimum sample size, N, is shown below for two- and one-
sided tests of hypotheses with assumed to be known.
The quantities z
1-/2
and z
1-
are critical values from the normal
distribution.
Note that it is usual to state the shift, , in units of the standard
deviation, thereby simplifying the calculation.
Example
where the
shift is stated
in terms of
the standard
deviation
For a one-sided hypothesis test where we wish to detect an increase
in the population mean of one standard deviation, the following
information is required: , the significance level of the test, and ,
the probability of failing to detect a shift of one standard deviation.
For a test with = 0.05 and = 0.10, the minimum sample size
required for the test is
N = (1.645 + 1.282)
2
= 8.567 ~ 9.
More often
we must
compute the
sample size
with the
population
standard
deviation
being
unknown
The procedures for computing sample sizes when the standard
deviation is not known are similar to, but more complex, than when
the standard deviation is known. The formulation depends on the t
distribution where the minimum sample size is given by
The drawback is that critical values of the t distribution depend on
known degrees of freedom, which in turn depend upon the sample
size which we are trying to estimate.
Iterate on the
initial
estimate
using critical
values from
Therefore, the best procedure is to start with an intial estimate based
on a sample standard deviation and iterate. Take the example
discussed above where the the minimum sample size is computed to
be N = 9. This estimate is low. Now use the formula above with
degrees of freedom N - 1 = 8 which gives a second estimate of
the t table
N = (1.860 + 1.397)
2
= 10.6 ~11.
It is possible to apply another iteration using degrees of freedom 10,
but in practice one iteration is usually sufficient. For the purpose of
this example, results have been rounded to the closest integer;
however, computer programs for finding critical values from the t
distribution allow non-integer degrees of freedom.
Table
showing
minimum
sample sizes
for a two-
sided test
The table below gives sample sizes for a two-sided test of
hypothesis that the mean is a given value, with the shift to be
detected a multiple of the standard deviation. For a one-sided test at
significance level , look under the value of 2 in column 1. Note
that this table is based on the normal approximation (i.e., the
standard deviation is known).
Sample Size Table for Two-Sided Tests
.01 .01 98 25 11
.01 .05 73 18 8
.01 .10 61 15 7
.01 .20 47 12 6
.01 .50 27 7 3
.05 .01 75 19 9
.05 .05 53 13 6
.05 .10 43 11 5
.05 .20 33 8 4
.05 .50 16 4 3
.10 .01 65 16 8
.10 .05 45 11 5
.10 .10 35 9 4
.10 .20 25 7 3
.10 .50 11 3 3
.20 .01 53 14 6
.20 .05 35 9 4
.20 .10 27 7 3
.20 .20 19 5 3
.20 .50 7 3 3
7.2.3. Are the data consistent with a nominal standard deviation?

7.2.3. Are the data consistent with a nominal
standard deviation?
The testing of
H
0
for a single
population
mean
Given a random sample of measurements, Y
1
, ..., Y
N
, there
are three types of questions regarding the true standard
deviation of the population that can be addressed with the
sample data. They are:
1. Does the true standard deviation agree with a
nominal value?
2. Is the true standard deviation of the population less
than or equal to a nominal value?
3. Is the true stanard deviation of the population at
least as large as a nominal value?
Corresponding
null
hypotheses
The corresponding null hypotheses that test the true
standard deviation, , against the nominal value, are:
1. H
0
: =
2. H
0
: <=
3. H
0
: >=
Test statistic The basic test statistic is the chi-square statistic
with N - 1 degrees of freedom where s is the sample
standard deviation; i.e.,
.
Comparison
with critical
values
For a test at significance level , where is chosen to be
small, typically 0.01, 0.05 or 0.10, the hypothesis
associated with each case enumerated above is rejected if:
where
2
/2
is the critical value from the chi-square
distribution with N - 1 degrees of freedom and similarly
for cases (2) and (3). Critical values can be found in the
chi-square table in Chapter 1.
Warning Because the chi-square distribution is a non-negative,
asymmetrical distribution, care must be taken in looking
up critical values from tables. For two-sided tests, critical
values are required for both tails of the distribution.
Example
A supplier of 100 ohm
.
cm silicon wafers claims that his
fabrication process can produce wafers with sufficient
consistency so that the standard deviation of resistivity for
the lot does not exceed 10 ohm
.
cm. A sample of N = 10
wafers taken from the lot has a standard deviation of 13.97
ohm.cm. Is the suppliers claim reasonable? This question
falls under null hypothesis (2) above. For a test at
significance level, = 0.05, the test statistic,
is compared with the critical value,
2
0.95, 9
= 16.92.
Since the test statistic (17.56) exceeds the critical value
(16.92) of the chi-square distribution with 9 degrees of
freedom, the manufacturer's claim is rejected.

Confidence
intervals
for the
standard
deviation
Confidence intervals for the true standard deviation can be
constructed using the chi-square distribution. The 100(1- )%
confidence intervals that correspond to the tests of hypothesis
on the previous page are given by
1. Two-sided confidence interval for
2. Lower one-sided confidence interval for
3. Upper one-sided confidence interval for
where for case (1),
2
/2
is the critical value from the
chi-square distribution with N - 1 degrees of freedom and
similarly for cases (2) and (3). Critical values can be found in
the chi-square table in Chapter 1.
Choice of
risk level
can
change the
conclusion
Confidence interval (1) is equivalent to a two-sided test for the
standard deviation. That is, if the hypothesized or nominal
value, , is not contained within these limits, then the
hypothesis that the standard deviation is equal to the nominal
value is rejected.
A dilemma
of
hypothesis
testing
A change in can lead to a change in the conclusion. This
poses a dilemma. What should be? Unfortunately, there is
no clear-cut answer that will work in all situations. The usual
strategy is to set small so as to guarantee that the null
hypothesis is wrongly rejected in only a small number of
cases. The risk, , of failing to reject the null hypothesis when
it is false depends on the size of the discrepancy, and also
depends on . The discussion on the next page shows how to
choose the sample size so that this risk is kept small for
specific discrepancies.

Sample sizes
to minimize
risk of false
acceptance
The following procedure for computing sample sizes for
tests involving standard deviations follows W. Diamond
(1989). The idea is to find a sample size that is large
enough to guarantee that the risk, , of accepting a false
hypothesis is small.
Alternatives
are specific
departures
from the null
hypothesis
This procedure is stated in terms of changes in the variance,
not the standard deviation, which makes it somewhat
difficult to interpret. Tests that are generally of interest are
stated in terms of , a discrepancy from the hypothesized
variance. For example:
1. Is the true variance larger than its hypothesized value
by ?
2. Is the true variance smaller than its hypothesized
value by ?
That is, the tests of interest are:
1. H
0
:
2. H
0
:
Interpretation The experimenter wants to assure that the probability of
erroneously accepting the null hypothesis of unchanged
variance is at most . The sample size, N, required for this
type of detection depends on the factor, ; the significance
level, ; and the risk, .
First choose
the level of
significance
and beta risk
The sample size is determined by first choosing appropriate
values of and and then following the directions below
to find the degrees of freedom, , from the chi-square
distribution.
The
calculations
should be
done by
creating a
table or
First compute
Then generate a table of degrees of freedom, , say
spreadsheet
between 1 and 200. For case (1) or (2) above, calculate
and the corresponding value of C
for each value of degrees

of freedom in the table where
The value of where C
is closest to is the correct

degrees of freedom and
N = + 1
Hints on
using
software
packages to
do the
calculations
The quantity
2
1-,
is the critical value from the chi-
square distribution with degrees of freedom which is
exceeded with probability . It is sometimes referred to as
the percent point function (PPF) or the inverse chi-square
function. The probability that is evaluated to get C
is
called the cumulative density function (CDF).
Example Consider the case where the variance for resistivity
measurements on a lot of silicon wafers is claimed to be
100 (ohm
.
cm)
2
. A buyer is unwilling to accept a shipment
if is greater than 55 ohm
.
cm for a particular lot. This
problem falls under case (1) above. How many samples are
needed to assure risks of = 0.05 and = 0.01?
Calculations If software is available to compute the roots (or zero
values) of a univariate function, then we can determine the
sample size by finding the roots of a function that calculates
C
for a given value of . The procedure is:

1. Define constants.
= 0.05
= 0.01
= 55

0
2
= 100
R = 1 + /
0
2
2. Create a function, Cnu.
Cnu = F( F
-1
(, )/R, ) -

F(x, ) returns the probability of a chi-
square random
variable with degrees of freedom that is
less than
or equal to x and
F
-1
(, ) returns x such that F(x, ) = .
3. Find the value of for which the function,
Cnu, is zero.
Using this procedure, Cnu is zero when is 169.3.
Therefore, the minimum sample size needed to guarantee
the risk level is N = 170.
Alternatively, we can determine the sample size by simply
printing computed values of Cnu for various values of .
1. Define constants.
= 0.05
= 55

0
2
= 100
R = 1 + /
0
2
2. Generate Cnu for values of from 1 to 200.
Bnu = F
-1
(, ) / R
Cnu = F(Bnu, )
The values of Cnu generated for between 165 and 175
degrees of freedom are shown below.
Bnu Cnu
165 126.4344 0.0114
166 127.1380 0.0110
167 127.8414 0.0107
168 128.5446 0.0104
169 129.2477 0.0101
170 129.9506 0.0098
171 130.6533 0.0095
172 131.3558 0.0092
173 132.0582 0.0090
174 132.7604 0.0087
175 133.4625 0.0085
The value of Cnu closest to 0.01 is 0.0101, which is
associated with = 169 degrees of freedom. Therefore, the
minimum sample size needed to guarantee the risk level is
N = 170.
The calculations used in this section can be performed
using both Dataplot code and R code.
7.2.4. Does the proportion of defectives meet requirements?

7.2.4. Does the proportion of defectives meet
requirements?
Testing
proportion
defective is
based on the
binomial
distribution
The proportion of defective items in a manufacturing
process can be monitored using statistics based on the
observed number of defectives in a random sample of size
N from a continuous manufacturing process, or from a
large population or lot. The proportion defective in a
sample follows the binomial distribution where p is the
probability of an individual item being found defective.
Questions of interest for quality control are:
1. Is the proportion of defective items within prescribed
limits?
2. Is the proportion of defective items less than a
prescribed limit?
3. Is the proportion of defective items greater than a
prescribed limit?
Hypotheses
regarding
proportion
defective
The corresponding hypotheses that can be tested are:
1. p p
0
2. p p
0
3. p p
0
where p
0
is the prescribed proportion defective.
Test statistic
based on a
normal
approximation
Given a random sample of measurements Y
1
, ..., Y
N
from a
population, the proportion of items that are judged
defective from these N measurements is denoted . The
test statistic
depends on a normal approximation to the binomial
distribution that is valid for large N, (N > 30). This
approximation simplifies the calculations using critical
values from the table of the normal distribution as shown
below.
Restriction on
sample size
Because the test is approximate, N needs to be large for the
test to be valid. One criterion is that N should be chosen so
that
min{Np
0
, N(1 - p
0
)} 5
For example, if p
0
= 0.1, then N should be at least 50 and
if p
0
= 0.01, then N should be at least 500. Criteria for
choosing a sample size in order to guarantee detecting a
change of size are discussed on another page.
One and two-
sided tests for
proportion
defective
Tests at the 1 - confidence level corresponding to
hypotheses (1), (2), and (3) are shown below. For
hypothesis (1), the test statistic, z, is compared with z
1-/2
,
the critical value from the normal distribution that is
exceeded with probability and similarly for (2) and
(3). If
1. | z | z
1-/2
2. z z
3. z z
1-
the null hypothesis is rejected.
Example of a
one-sided test
for proportion
defective
After a new method of processing wafers was introduced
into a fabrication process, two hundred wafers were tested,
and twenty-six showed some type of defect. Thus, for N=
200, the proportion defective is estimated to be = 26/200
= 0.13. In the past, the fabrication process was capable of
producing wafers with a proportion defective of at most
0.10. The issue is whether the new process has degraded
the quality of the wafers. The relevant test is the one-sided
test (3) which guards against an increase in proportion
defective from its historical level.
Calculations
for a one-
sided test of
proportion
defective
For a test at significance level = 0.05, the hypothesis of
no degradation is validated if the test statistic z is less than
the critical value, z
0.95
= 1.645. The test statistic is
computed to be
Interpretation Because the test statistic is less than the critical value
(1.645), we cannot reject hypothesis (3) and, therefore, we
cannot conclude that the new fabrication method is
degrading the quality of the wafers. The new process may,
indeed, be worse, but more evidence would be needed to
reach that conclusion at the 95% confidence level.
7.2.4.1. Confidence intervals

Confidence
intervals
using the
method of
Agresti and
Coull
The method recommended by Agresti and Coull (1998) and also by Brown, Cai
and DasGupta (2001) (the methodology was originally developed by Wilson in
1927) is to use the form of the confidence interval that corresponds to the
hypothesis test given in Section 7.2.4. That is, solve for the two values of p
0
(say,
p
upper
and p
lower
) that result from setting z = z
1-/2
and solving for p
0
= p
upper
,
and then setting z = z
/2
and solving for p
0
= p
lower
. (Here, as in Section 7.2.4,
z
/2
denotes the variate value from the standard normal distribution such that the
area to the left of the value is /2.) Although solving for the two values of p
0
might sound complicated, the appropriate expressions can be obtained by
straightforward but slightly tedious algebra. Such algebraic manipulation isn't
necessary, however, as the appropriate expressions are given in various sources.
Specifically, we have
Formulas
for the
confidence
intervals
Procedure
does not
strongly
depend on
values of p
and n
This approach can be substantiated on the grounds that it is the exact algebraic
counterpart to the (large-sample) hypothesis test given in section 7.2.4 and is also
supported by the research of Agresti and Coull. One advantage of this procedure
is that its worth does not strongly depend upon the value of n and/or p, and
indeed was recommended by Agresti and Coull for virtually all combinations of
n and p.
Another
advantage
is that the
lower limit
cannot be
negative
Another advantage is that the lower limit cannot be negative. That is not true for
the confidence expression most frequently used:
A confidence limit approach that produces a lower limit which is an impossible
value for the parameter for which the interval is constructed is an inferior
approach. This also applies to limits for the control charts that are discussed in
Chapter 6.
One-sided
confidence
intervals
A one-sided confidence interval can also be constructed simply by replacing each
by in the expression for the lower or upper limit, whichever is desired.
The 95% one-sided interval for p for the example in the preceding section is:
Example
Conclusion
from the
example
Since the lower bound does not exceed 0.10, in which case it would exceed the
hypothesized value, the null hypothesis that the proportion defective is at most
0.10, which was given in the preceding section, would not be rejected if we used
the confidence interval to test the hypothesis. Of course a confidence interval has
value in its own right and does not have to be used for hypothesis testing.
Exact Intervals for Small Numbers of Failures and/or Small Sample Sizes
Constrution
of exact
two-sided
confidence
intervals
based on
the
binomial
distribution
If the number of failures is very small or if the sample size N is very small,
symmetical confidence limits that are approximated using the normal distribution
may not be accurate enough for some applications. An exact method based on the
binomial distribution is shown next. To construct a two-sided confidence interval
at the 100(1-)% confidence level for the true proportion defective p where N
d
defects are found in a sample of size N follow the steps below.
1. Solve the equation
for p
U
to obtain the upper 100(1-)% limit for p.
2. Next solve the equation
for p
L
to obtain the lower 100(1-)% limit for p.
Note The interval (p
L
, p
U
) is an exact 100(1-)% confidence interval for p. However,
it is not symmetric about the observed proportion defective, .
Binomial
confidence
interval
example
The equations above that determine p
L
and p
U
can be solved using readily
available functions. Take as an example the situation where twenty units are
sampled from a continuous production line and four items are found to be
defective. The proportion defective is estimated to be = 4/20 = 0.20. The steps
for calculating a 90 % confidence interval for the true proportion defective, p
follow.
1. Initalize constants.
alpha = 0.10
Nd = 4
N = 20
2. Define a function for upper limit (fu) and a function
for the lower limit (fl).
fu = F(Nd,pu,20) - alpha/2
fl = F(Nd-1,pl,20) - (1-alpha/2)
F is the cumulative density function for the
binominal distribution.

3. Find the value of pu that corresponds to fu = 0 and
the value of pl that corresponds to fl = 0 using software
to find the roots of a function.
The values of pu and pl for our example are:
pu = 0.401029
pl = 0.071354
Thus, a 90 % confidence interval for the proportion defective, p, is (0.071,
0.400). Whether or not the interval is truly "exact" depends on the software.
The calculations used in this example can be performed using both Dataplot code
and R code.

Derivation of
formula for
required
sample size
when testing
proportions
The method of determining sample sizes for testing proportions is similar
to the method for determining sample sizes for testing the mean. Although
the sampling distribution for proportions actually follows a binomial
distribution, the normal approximation is used for this derivation.
Minimum
sample size
If we are interested in detecting a change in the proportion defective of
size in either direction, the minimum sample size is
1. For a two-sided test
2. For a one-sided test
Interpretation
and sample
size for high
probability of
detecting a
change
This requirement on the sample size only guarantees that a change of size
is detected with 50% probability. The derivation of the sample size
when we are interested in protecting against a change with probability 1
- (where is small) is
1. For a two-sided test
2. For a one-sided test
where z
1-
is the critical value from the normal distribution that is
exceeded with probability .
Value for the
true
proportion
defective
The equations above require that p be known. Usually, this is not the case.
If we are interested in detecting a change relative to an historical or
hypothesized value, this value is taken as the value of p for this purpose.
Note that taking the value of the proportion defective to be 0.5 leads to the
largest possible sample size.
Example of
calculating
sample size
for testing
proportion
defective
Suppose that a department manager needs to be able to detect any change
above 0.10 in the current proportion defective of his product line, which is
running at approximately 10% defective. He is interested in a one-sided
test and does not want to stop the line except when the process has clearly
degraded and, therefore, he chooses a significance level for the test of 5%.
Suppose, also, that he is willing to take a risk of 10% of failing to detect a
change of this magnitude. With these criteria:
1. z
0.95
= 1.645; z
0.90
=1.282
2. = 0.10
3. p = 0.10
and the minimum sample size for a one-sided test procedure is
7.2.5. Does the defect density meet requirements?

Testing defect
densities is
based on the
Poisson
distribution
The number of defects observed in an area of size A units is often
assumed to have a Poisson distribution with parameter A x D,
where D is the actual process defect density (D is defects per
unit area). In other words:
The questions of primary interest for quality control are:
1. Is the defect density within prescribed limits?
2. Is the defect density less than a prescribed limit?
3. Is the defect density greater than a prescribed limit?
Normal
approximation
to the Poisson
We assume that AD is large enough so that the normal
approximation to the Poisson applies (in other words, AD > 10
for a reasonable approximation and AD > 20 for a good one).
That translates to
where is the standard normal distribution function.
Test statistic
based on a
normal
approximation
If, for a sample of area A with a defect density target of D
0
, a
defect count of C is observed, then the test statistic
can be used exactly as shown in the discussion of the test
statistic for fraction defectives in the preceding section.
Testing the
hypothesis
that the
process defect
density is less
than or equal
to D
0
For example, after choosing a sample size of area A (see below
for sample size calculation) we can reject that the process defect
density is less than or equal to the target D
0
if the number of
defects C in the sample is greater than C
A
, where
and z
1-
is the 100(1-) percentile of the standard normal
distribution. The test significance level is 100(1-). For a 90%
significance level use z
0.90
= 1.282 and for a 95% test use z
0.95
= 1.645. is the maximum risk that an acceptable process with a
defect density at least as low as D
0
"fails" the test.
Choice of
sample size
(or area) to
examine for
defects
In order to determine a suitable area A to examine for defects,
you first need to choose an unacceptable defect density level.
Call this unacceptable defect density D
1
= kD
0
, where k > 1.
We want to have a probability of less than or equal to is of
"passing" the test (and not rejecting the hypothesis that the true
level is D
0
or better) when, in fact, the true defect level is D
1
or
worse. Typically will be 0.2, 0.1 or 0.05. Then we need to
count defects in a sample size of area A, where A is equal to
Example Suppose the target is D
0
= 4 defects per wafer and we want to
verify a new process meets that target. We choose = 0.1 to be
the chance of failing the test if the new process is as good as D
0
( = the Type I error probability or the "producer's risk") and we
choose = 0.1 for the chance of passing the test if the new
process is as bad as 6 defects per wafer ( = the Type II error
probability or the "consumer's risk"). That means z
1-
= 1.282
and z
= -1.282.
The sample size needed is A wafers, where
which we round up to 9.
The test criteria is to "accept" that the new process meets target
unless the number of defects in the sample of 9 wafers exceeds
In other words, the reject criteria for the test of the new process
is 44 or more defects in the sample of 9 wafers.
Note: Technically, all we can say if we run this test and end up
not rejecting is that we do not have statistically significant
evidence that the new process exceeds target. However, the way
we chose the sample size for this test assures us we most likely
would have had statistically significant evidence for rejection if
the process had been as bad as 1.5 times the target.
7.2.6. What intervals contain a fixed percentage of the population values?

7.2.6. What intervals contain a fixed percentage
of the population values?
Observations
tend to
cluster
around the
median or
mean
Empirical studies have demonstrated that it is typical for a
large number of the observations in any study to cluster near
the median. In right-skewed data this clustering takes place
to the left of (i.e., below) the median and in left-skewed
data the observations tend to cluster to the right (i.e., above)
the median. In symmetrical data, where the median and the
mean are the same, the observations tend to distribute
equally around these measures of central tendency.
Various
methods
Several types of intervals about the mean that contain a
large percentage of the population values are discussed in
this section.
Approximate intervals that contain most of the
population values
Percentiles
Tolerance intervals for a normal distribution
Tolerance intervals based on the smallest and largest
observations
7.2.6.1. Approximate intervals that contain most of the population values

7.2.6.1. Approximate intervals that contain
most of the population values
Empirical
intervals
A rule of thumb is that where there is no evidence of
significant skewness or clustering, two out of every three
observations (67%) should be contained within a distance of
one standard deviation of the mean; 90% to 95% of the
observations should be contained within a distance of two
standard deviations of the mean; 99-100% should be
contained within a distance of three standard deviations. This
rule can help identify outliers in the data.
Intervals
that apply
to any
distribution
The Bienayme-Chebyshev rule states that regardless of how
the data are distributed, the percentage of observations that are
contained within a distance of k tandard deviations of the
mean is at least (1 - 1/k
2
)100%.
Exact
intervals
for the
normal
distribution
The Bienayme-Chebyshev rule is conservative because it
applies to any distribution. For a normal distribution, a higher
percentage of the observations are contained within k standard
deviations of the mean as shown in the following table.
Percentage of observations contained between the mean
and k standard deviations
k, No. of
Standard
Deviations
Empircal Rule
Bienayme-
Chebychev
Normal
Distribution
1 67% N/A 68.26%
2 90-95% at least 75% 95.44%
3 99-100%
at least
88.89%
99.73%
4 N/A
at least
93.75%
99.99%
7.2.6.2. Percentiles

Definitions of
order
statistics and
ranks
For a series of measurements Y
1
, ..., Y
N
, denote the data
ordered in increasing order of magnitude by Y
[1]
, ..., Y
[N]
.
These ordered data are called order statistics. If Y
[j]
is the
order statistic that corresponds to the measurement Y
i
, then
the rank for Y
i
is j; i.e.,
Definition of
percentiles
Order statistics provide a way of estimating proportions of
the data that should fall above and below a given value,
called a percentile. The pth percentile is a value, Y
(p)
, such
that at most (100p) % of the measurements are less than
this value and at most 100(1- p) % are greater. The 50th
percentile is called the median.
Percentiles split a set of ordered data into hundredths.
(Deciles split ordered data into tenths). For example, 70 %
of the data should fall below the 70th percentile.
Estimation of
percentiles
Percentiles can be estimated from N measurements as
follows: for the pth percentile, set p(N+1) equal to k + d for
k an integer, and d, a fraction greater than or equal to 0 and
less than 1.
1. For 0 < k < N,
2. For k = 0, Y(p) = Y
[1]
3. For k = N, Y(p) = Y
[N]
Example and
interpretation
For the purpose of illustration, twelve measurements from a
gage study are shown below. The measurements are
resistivities of silicon wafers measured in ohm
.
cm.
i Measurements Order stats Ranks
1 95.1772 95.0610 9
2 95.1567 95.0925 6
3 95.1937 95.1065 10
4 95.1959 95.1195 11
5 95.1442 95.1442 5
6 95.0610 95.1567 1
7 95.1591 95.1591 7
8 95.1195 95.1682 4
9 95.1065 95.1772 3
10 95.0925 95.1937 2
11 95.1990 95.1959 12
12 95.1682 95.1990 8
To find the 90th percentile, p(N+1) = 0.9(13) =11.7; k = 11,
and d = 0.7. From condition (1) above, Y(0.90) is estimated
to be 95.1981 ohm
.
cm. This percentile, although it is an
estimate from a small sample of resistivities measurements,
gives an indication of the percentile for a population of
resistivity measurements.
Note that
there are
other ways of
calculating
percentiles in
common use
Some software packages set 1+p(N-1) equal to k + d, then
proceed as above. The two methods give fairly similar
results.
A third way of calculating percentiles (given in some
elementary textbooks) starts by calculating pN. If that is not
an integer, round up to the next highest integer k and use
Y
[k]
as the percentile estimate. If pN is an integer k, use
0.5(Y
[k]
+Y
[k+1]
).
Definition of
Tolerance
Interval
An interval covering population percentiles can be
interpreted as "covering a proportion p of the population
with a level of confidence, say, 90 %." This is known as a
tolerance interval.
7.2.6.3. Tolerance intervals for a normal distribution

Definition of
a tolerance
interval
A confidence interval covers a population parameter with a stated confidence, that
is, a certain proportion of the time. There is also a way to cover a fixed proportion
of the population with a stated confidence. Such an interval is called a tolerance
interval. The endpoints of a tolerance interval are called tolerance limits. An
application of tolerance intervals to manufacturing involves comparing specification
limits prescribed by the client with tolerance limits that cover a specified proportion
of the population.
Difference
between
confidence
and tolerance
intervals
Confidence limits are limits within which we expect a given population parameter,
such as the mean, to lie. Statistical tolerance limits are limits within which we
expect a stated proportion of the population to lie.
Not related to
engineering
tolerances
Statistical tolerance intervals have a probabilistic interpretation. Engineering
tolerances are specified outer limits of acceptability which are usually prescribed by
a design engineer and do not necessarily reflect a characteristic of the actual
measurements.
Three types of
tolerance
intervals
Three types of questions can be addressed by tolerance intervals. Question (1) leads
to a two-sided interval; questions (2) and (3) lead to one-sided intervals.
1. What interval will contain p percent of the population measurements?
2. What interval guarantees that p percent of population measurements will not
fall below a lower limit?
3. What interval guarantees that p percent of population measurements will not
exceed an upper limit?
Tolerance
intervals for
measurements
from a
normal
distribution
For the questions above, the corresponding tolerance intervals are defined by lower
(L) and upper (U) tolerance limits which are computed from a series of
measurements Y
1
, ..., Y
N
:
1.
2.
3.
where the k factors are determined so that the intervals cover at least a proportion p
of the population with confidence, .
Calculation If the data are from a normally distributed population, an approximate value for the
of k factor for
a two-sided
tolerance
limit for a
normal
distribution
factor as a function of p and for a two-sided tolerance interval (Howe, 1969) is
where
2
1-, N-1
, is the critical value of the chi-square distribution with degrees of
freedom, N - 1, that is exceeded with probability and z
1-(1-p)/2
is the critical value
of the normal distribution which is exceeded with probability (1-p)/2.
Example of
calculation
For example, suppose that we take a sample of N = 43 silicon wafers from a lot and
measure their thicknesses in order to find tolerance limits within which a proportion
p = 0.90 of the wafers in the lot fall with probability = 0.99.
Use of tables
in calculating
two-sided
tolerance
intervals
Values of the k factor as a function of p and are tabulated in some textbooks, such
as Dixon and Massey (1969). To use the tables in this handbook, follow the steps
outlined below:
1. Calculate = (1 - p)/2 = 0.05
2. Go to the page describing critical values of the normal distribution and in the
summary table under the column labeled 0.95 find z
1-(1-p)/2
= z
0.95
= 1.645.
3. Go to the table of lower critical values of the chi-square distribution and
under the column labeled 0.01 in the row labeled degrees of freedom = 42,
find
2
1-, N-1
=
2
0.01, 42
= 23.650.
4. Calculate
The tolerance limits are then computed from the sample mean, , and standard
deviation, s, according to case(1).
Important
notes
The notation for the critical value of the chi-square distribution can be confusing.
Values as tabulated are, in a sense, already squared; whereas the critical value for
the normal distribution must be squared in the formula above.
Some software is capable of computing a tolerance intervals for a given set of data
so that the user does not need to perform all the calculations. All the tolerance
intervals shown in this section can be computed using both Dataplot code and R
code. R and Dataplot examples include the case where a tolerance interval is
computed automatically from a data set.
Calculation
of a one-
sided
tolerance
interval for a
normal
The calculation of an approximate k factor for one-sided tolerance intervals comes
directly from the following set of formulas (Natrella, 1963):
distribution
A one-sided
tolerance
interval
example
For the example above, it may also be of interest to guarantee with 0.99 probability
(or 99% confidence) that 90% of the wafers have thicknesses less than an upper
tolerance limit. This problem falls under case (3). The calculations for the k
1
factor
for a one-sided tolerance interval are:
Tolerance
factor based
on the non-
central t
distribution
The value of k
1
can also be computed using the inverse cumulative distribution
function for the non-central t distribution. This method may give more accurate
results for small values of N. The value of k
1
using the non-central t distribution
(using the same example as above) is:
where is the non-centrality parameter.
In this case, the difference between the two computations is negligble (1.8752
versus 1.8740). However, the difference becomes more pronounced as the value of
N gets smaller (in particular, for N 10). For example, if N = 43 is replaced with N
= 6, the non-central t method returns a value of 4.4111 for k
1
while the method
based on the Natrella formuals returns a value of 5.2808.
The disadvantage of the non-central t method is that it depends on the inverse
cumulative distribution function for the non-central t distribution. This function is
not available in many statistical and spreadsheet software programs, but it is
available in Dataplot and R (see Dataplot code and R code). The Natrella formulas
only depend on the inverse cumulative distribution function for the normal
distribution (which is available in just about all statistical and spreadsheet software
programs). Unless you have small samples (say N 10), the difference in the
methods should not have much practical effect.
7.2.6.4. Tolerance intervals based on the largest and smallest observations

7.2.6.4. Tolerance intervals based on the largest and smallest
observations
Tolerance
intervals can
be constructed
for a
distribution of
any form
The methods on the previous pages for computing tolerance limits are based on the
assumption that the measurements come from a normal distribution. If the distribution is
not normal, tolerance intervals based on this assumption will not provide coverage for the
intended proportion p of the population. However, there are methods for achieving the
intended coverage if the form of the distribution is not known, but these methods may
produce substantially wider tolerance intervals.
Risks
associated
with making
assumptions
about the
distribution
There are situations where it would be particularly dangerous to make unwarranted
assumptions about the exact shape of the distribution, for example, when testing the
strength of glass for airplane windshields where it is imperative that a very large
proportion of the population fall within acceptable limits.
Tolerance
intervals
based on
largest and
smallest
observations
One obvious choice for a two-sided tolerance interval for an unknown distribution is the
interval between the smallest and largest observations from a sample of Y
1
, ..., Y
N
measurements. Given the sample size N and coverage p, an equation from Hahn and
Meeker (p. 91),
allows us to calculate the confidence of the tolerance interval. For example, the
confidence levels for selected coverages between 0.5 and 0.9999 are shown below for N
= 25.
Confidence Coverage
1.000 0.5000
0.993 0.7500
0.729 0.9000
0.358 0.9500
0.129 0.9750
0.026 0.9900
0.007 0.9950
0.0 0.9990
0.0 0.9995
0.0 0.9999
Note that if 99 % confidence is required, the interval that covers the entire sample data
set is guaranteed to achieve a coverage of only 75 % of the population values.
What is the
optimal
sample size?
Another question of interest is, "How large should a sample be so that one can be
assured with probability that the tolerance interval will contain at least a proportion p of
the population?"
7.2.6.4. Tolerance intervals based on the largest and smallest observations
Approximation
for N
A rather good approximation for the required sample size is given by
where
2
, 4
is the critical value of the chi-square distribution with 4 degrees of freedom
that is exceeded with probability .
Example of
the effect of p
on the sample
size
Suppose we want to know how many measurements to make in order to guarantee that
the interval between the smallest and largest observations covers a proportion p of the
population with probability = 0.95. From the table for the upper critical value of the
chi-square distribution, look under the column labeled 0.95 in the row for 4 degrees of
freedom. The value is found to be
2
0.95, 4
= 9.488 and calculations are shown below for
p equal to 0.90 and 0.99.
For p = 0.90, = 0.95:
For p = 0.99, = 0.95:
These calculations demonstrate that requiring the tolerance interval to cover a very large
proportion of the population may lead to an unacceptably large sample size.
7.3. Comparisons based on data from two processes

7.3. Comparisons based on data from two
processes
Outline for
this section
In many manufacturing environments it is common to have
two or more processes performing the same task or generating
similar products. The following pages describe tests covering
several of the most common and useful cases for two
processes.
1. Do two processes have the same mean?
1. Tests when the standard deviations are equal
2. Tests when the standard deviations are unequal
3. Tests for paired data
2. Do two processes have the same standard deviation?
3. Do two processes produce the same proportion of
defectives?
4. If the observations are failure times, are the failure rates
(or mean times to failure) the same?
5. Do two arbitrary processes have the same central
tendency?
Example of
a dual
track
process
For example, in an automobile manufacturing plant, there may
exist several assembly lines producing the same part. If one
line goes down for some reason, parts can still be produced
and production will not be stopped. For example, if the parts
are piston rings for a particular model car, the rings produced
by either line should conform to a given set of specifications.
How does one confirm that the two processes are in fact
producing rings that are similar? That is, how does one
determine if the two processes are similar?
The goal is
to
determine
if the two
processes
are similar
In order to answer this question, data on piston rings are
collected for each process. For example, on a particular day,
data on the diameters of ten piston rings from each process
are measured over a one-hour time frame.
To determine if the two processes are similar, we are
interested in answering the following questions:
1. Do the two processes produce piston rings with the
same diameter?
2. Do the two processes have similar variability in the
diameters of the rings produced?
Unknown
standard
deviation
The second question assumes that one does not know the
standard deviation of either process and therefore it must be
estimated from the data. This is usually the case, and the tests
in this section assume that the population standard deviations
are unknown.
Assumption
of a
normal
distribution
The statistical methodology used (i.e., the specific test to be
used) to answer these two questions depends on the
underlying distribution of the measurements. The tests in this
section assume that the data are normally distributed.
7.3.1. Do two processes have the same mean?

Testing
hypotheses
related to
the means of
two
processes
Given two random samples of measurements,
Y
1
, ..., Y
N
and Z
1
, ..., Z
N
from two independent processes (the Y's are sampled from process 1
and the Z's are sampled from process 2), there are three types of
questions regarding the true means of the processes that are often
asked. They are:
1. Are the means from the two processes the same?
2. Is the mean of process 1 less than or equal to the mean of
process 2?
3. Is the mean of process 1 greater than or equal to the mean of
process 2?
Typical null
hypotheses
The corresponding null hypotheses that test the true mean of the first
process, , against the true mean of the second process, are:
1. H
0
: =
2. H
0
: < or equal to
3. H
0
: > or equal to
Note that as previously discussed, our choice of which null hypothesis
to use is typically made based on one of the following considerations:
1. When we are hoping to prove something new with the sample
data, we make that the alternative hypothesis, whenever
possible.
2. When we want to continue to assume a reasonable or traditional
hypothesis still applies, unless very strong contradictory
evidence is present, we make that the null hypothesis, whenever
possible.
Basic
statistics
from the two
processes
The basic statistics for the test are the sample means
;
and the sample standard deviations
with degrees of freedom and respectively.
Form of the
test statistic
where the
two
processes
have
equivalent
standard
deviations
If the standard deviations from the two processes are equivalent, and
this should be tested before this assumption is made, the test statistic
is
where the pooled standard deviation is estimated as
with degrees of freedom .
Form of the
test statistic
where the
two
processes do
NOT have
equivalent
standard
deviations
If it cannot be assumed that the standard deviations from the two
processes are equivalent, the test statistic is
The degrees of freedom are not known exactly but can be estimated
using the Welch-Satterthwaite approximation
Test
strategies
The strategy for testing the hypotheses under (1), (2) or (3) above is to
calculate the appropriate t statistic from one of the formulas above,
and then perform a test at significance level , where is chosen to
be small, typically .01, .05 or .10. The hypothesis associated with each
case enumerated above is rejected if:
1. | t | t
1-/2,
2. t t
1-,
3. t t
,
Explanation
of critical
values
The critical values from the t table depend on the significance level
and the degrees of freedom in the standard deviation. For hypothesis
(1) t
1-/2,
is the 1-/2 critical value from the t table with degrees
of freedom and similarly for hypotheses (2) and (3).
Example of
unequal
number of
data points
A new procedure (process 2) to assemble a device is introduced and
tested for possible improvement in time of assembly. The question
being addressed is whether the mean, , of the new assembly process
is smaller than the mean, , for the old assembly process (process 1).
We choose to test hypothesis (2) in the hope that we will reject this
null hypothesis and thereby feel we have a strong degree of
confidence that the new process is an improvement worth
implementing. Data (in minutes required to assemble a device) for
both the new and old processes are listed below along with their
relevant statistics.
Device Process 1 (Old) Process 2 (New)
1 32 36
2 37 31
3 35 30
4 28 31
5 41 34
6 44 36
7 35 29
8 31 32
9 34 31
10 38
11 42
Mean 36.0909 32.2222
Standard deviation 4.9082 2.5386
No. measurements 11 9
Degrees freedom 10 8
Computation
of the test
statistic
From this table we generate the test statistic
with the degrees of freedom approximated by
Decision
process
For a one-sided test at the 5% significance level, go to the t table for
0.95 signficance level, and look up the critical value for degrees of
freedom = 16. The critical value is 1.746. Thus, hypothesis (2) is
rejected because the test statistic (t = 2.269) is greater than 1.746 and,
therefore, we conclude that process 2 has improved assembly time
(smaller mean) over process 1.
7.3.1.1. Analysis of paired observations

Definition of
paired
comparisons
Given two random samples,
Y
1
, ..., Y
N
and Z
1
, ..., Z
N
from two populations, the data are said to be paired if the ith
measurement on the first sample is naturally paired with the
ith measurement on the second sample. For example, if N
supposedly identical products are chosen from a production
line, and each one, in turn, is tested with first one measuring
device and then with a second measuring device, it is
possible to decide whether the measuring devices are
compatible; i.e., whether there is a difference between the
two measurement systems. Similarly, if "before" and "after"
measurements are made with the same device on N objects, it
is possible to decide if there is a difference between "before"
and "after"; for example, whether a cleaning process changes
an important characteristic of an object. Each "before"
measurement is paired with the corresponding "after"
measurement, and the differences
are calculated.
Basic
statistics for
the test
The mean and standard deviation for the differences are
calculated as
and
with = N - 1 degrees of freedom.
Test statistic
based on the
t
The paired-sample t test is used to test for the difference of
two means before and after a treatment. The test statistic is:
distribution
The hypotheses described on the foregoing page are rejected
if:
1. | t | t
1-/2,
2. t t
1-,
3. t t
,
where for hypothesis (1) t
1-/2,
is the 1-/2 critical value
from the t distribution with degrees of freedom and
similarly for cases (2) and (3). Critical values can be found
in the t table in Chapter 1.
7.3.1.2. Confidence intervals for differences between means

7.3.1.2. Confidence intervals for differences between means
Definition of
confidence
interval for
difference
between
population
means
Given two random samples,
Y
1
, ..., Y
N
and Z
1
, ..., Z
N
from two populations, two-sided confidence intervals with 100 (1- )% coverage for the
difference between the unknown population means, and , are shown in the table
below. Relevant statistics for paired observations and for unpaired observations are
shown elsewhere.
Two-sided confidence intervals with 100(1- )% coverage for - :
Paired observations
Unpaired observations
Interpretation
of confidence
interval
One interpretation of the confidence interval for means is that if zero is contained within
the confidence interval, the two population means are equivalent.
7.3.2. Do two processes have the same standard deviation?

7.3.2. Do two processes have the same standard
deviation?
Testing
hypotheses
related to
standard
deviations
from two
processes
Given two random samples of measurements,
Y
1
, ..., Y
N
and Z
1
, ..., Z
N
from two independent processes, there are three types of
questions regarding the true standard deviations of the
processes that can be addressed with the sample data. They
are:
1. Are the standard deviations from the two processes the
same?
2. Is the standard deviation of one process less than the
standard deviation of the other process?
3. Is the standard deviation of one process greater than
the standard deviation of the other process?
Typical null
hypotheses
The corresponding null hypotheses that test the true standard
deviation of the first process, , against the true standard
deviation of the second process, are:
1. H
0
: =
2. H
0
:
3. H
0
:
Basic
statistics
from the two
processes
The basic statistics for the test are the sample variances
and degrees of freedom and , respectively.
Form of the
test statistic
The test statistic is
Test
strategies
The strategy for testing the hypotheses under (1), (2) or (3)
above is to calculate the F statistic from the formula above,
and then perform a test at significance level , where is
chosen to be small, typically 0.01, 0.05 or 0.10. The
hypothesis associated with each case enumerated above is
rejected if:
1. or
2.
3.
Explanation
of critical
values
The critical values from the F table depend on the
significance level and the degrees of freedom in the standard
deviations from the two processes. For hypothesis (1):
is the upper critical value from the F table
with
degrees of freedom for the numerator and
degrees of freedom for the denominator
and
is the upper critical value from the F table
with
degrees of freedom for the numerator and
degrees of freedom for the denominator.
Caution on
looking up
critical
values
The F distribution has the property that
which means that only upper critical values are required for
two-sided tests. However, note that the degrees of freedom
are interchanged in the ratio. For example, for a two-sided
test at significance level 0.05, go to the F table labeled "2.5%
significance level".
For , reverse the order of the degrees of
freedom; i.e., look across the top of the table for
and down the table for .
For , look across the top of the table for
and down the table for .
Critical values for cases (2) and (3) are defined similarly,
except that the critical values for the one-sided tests are
based on rather than on .
Two-sided
confidence
interval
The two-sided confidence interval for the ratio of the two
unknown variances (squares of the standard deviations) is
shown below.
Two-sided confidence interval with 100(1- )% coverage
for:
One interpretation of the confidence interval is that if the
quantity "one" is contained within the interval, the standard
deviations are equivalent.
Example of
unequal
number of
data points
A new procedure to assemble a device is introduced and
tested for possible improvement in time of assembly. The
question being addressed is whether the standard deviation,
, of the new assembly process is better (i.e., smaller) than
the standard deviation, , for the old assembly process.
Therefore, we test the null hypothesis that . We form
the hypothesis in this way because we hope to reject it, and
therefore accept the alternative that is less than . This is
hypothesis (2). Data (in minutes required to assemble a
device) for both the old and new processes are listed on an
earlier page. Relevant statistics are shown below:
Process 1 Process 2
Mean 36.0909 32.2222
Standard deviation 4.9082 2.5874
No. measurements 11 9
Degrees freedom 10 8
Computation
of the test
statistic
From this table we generate the test statistic
Decision
process
For a test at the 5% significance level, go to the F table for
5% signficance level, and look up the critical value for
numerator degrees of freedom = 10 and
denominator degrees of freedom = 8. The critical
value is 3.35. Thus, hypothesis (2) can be rejected because
the test statistic (F = 3.60) is greater than 3.35. Therefore, we
accept the alternative hypothesis that process 2 has better
precision (smaller standard deviation) than process 1.
7.3.3. How can we determine whether two processes produce the same proportion of defectives?

7.3.3. How can we determine whether two
processes produce the same proportion of
defectives?
Case 1: Large Samples (Normal Approximation to
Binomial)
The
hypothesis of
equal
proportions
can be tested
using a z
statistic
If the samples are reasonably large we can use the normal
approximation to the binomial to develop a test similar to
testing whether two normal means are equal.
Let sample 1 have x
1
defects out of n
1
and sample 2 have
x
2
defects out of n
2
. Calculate the proportion of defects for
each sample and the z statistic below:
where
Compare |z| to the normal z
1-/2
table value for a two-
sided test. For a one-sided test, assuming the alternative
hypothesis is p
1
> p
2
, compare z to the normal z
1-
table
value. If the alternative hypothesis is p
1
< p
2
, compare z to
z
.
Case 2: An Exact Test for Small Samples
The Fisher
Exact
Probability
test is an
excellent
choice for
small samples
The Fisher Exact Probability Test is an excellent
nonparametric technique for analyzing discrete data (either
nominal or ordinal), when the two independent samples are
small in size. It is used when the results from two
independent random samples fall into one or the other of
two mutually exclusive classes (i.e., defect versus good, or
successes vs failures).
Example of a
2x2
In other words, every subject in each group has one of two
possible scores. These scores are represented by frequencies
contingency
table
in a 2x2 contingency table. The following discussion, using
a 2x2 contingency table, illustrates how the test operates.
We are working with two independent groups, such as
experiments and controls, males and females, the Chicago
Bulls and the New York Knicks, etc.
- + Total
Group
I
A B A+B
Group
II
C D C+D
Total A+C B+D N
The column headings, here arbitrarily indicated as plus and
minus, may be of any two classifications, such as: above
and below the median, passed and failed, Democrat and
Republican, agree and disagree, etc.
Determine
whether two
groups differ
in the
proportion
with which
they fall into
two
classifications
Fisher's test determines whether the two groups differ in
the proportion with which they fall into the two
classifications. For the table above, the test would
determine whether Group I and Group II differ significantly
in the proportion of plusses and minuses attributed to them.
The method proceeds as follows:
The exact probability of observing a particular set of
frequencies in a 2 2 table, when the marginal totals are
regarded as fixed, is given by the hypergeometric
distribution
But the test does not just look at the observed case. If
needed, it also computes the probability of more extreme
outcomes, with the same marginal totals. By "more
extreme", we mean relative to the null hypothesis of equal
proportions.
Example of
Fisher's test
This will become clear in the next illustrative example.
Consider the following set of 2 x 2 contingency tables:
Observed Data
More extreme outcomes with same
marginals
(a) (b) (c)
2 5 7
3 2 5
5 7 12
1 6 7
4 1 5
5 7 12
0 7 7
5 0 5
5 7 12
Table (a) shows the observed frequencies and tables (b)
and (c) show the two more extreme distributions of
frequencies that could occur with the same marginal totals
7, 5. Given the observed data in table (a) , we wish to test
the null hypothesis at, say, = 0.05.
Applying the previous formula to tables (a), (b), and (c),
we obtain
The probability associated with the occurrence of values as
extreme as the observed results under H
0
is given by
adding these three p's:
.26515 + .04419 + .00126 = .31060
So p = 0.31060 is the probability that we get from Fisher's
test. Since 0.31060 is larger than , we cannot reject the
null hypothesis.
Tocher's Modification
Tocher's
modification
makes
Fisher's test
less
conservative
Tocher (1950) showed that a slight modification of the
Fisher test makes it a more useful test. Tocher starts by
isolating the probability of all cases more extreme than the
observed one. In this example that is
p
b
+ p
c
= .04419 + .00126 = .04545
Now, if this probability is larger than , we cannot reject
H
0
. But if this probability is less than , while the
probability that we got from Fisher's test is greater than
(as is the case in our example) then Tocher advises to
compute the following ratio:
For the data in the example, that would be
Now we go to a table of random numbers and at random
draw a number between 0 and 1. If this random number is
smaller than the ratio above of 0.0172, we reject H
0
. If it is
larger we cannot reject H
0
. This added small probability of
rejecting H
0
brings the test procedure Type I error (i.e.,
value) to exactly 0.05 and makes the Fisher test less
conservative.
The test is a one-tailed test. For a two-tailed test, the value
of p obtained from the formula must be doubled.
A difficulty with the Tocher procedure is that someone else
analyzing the same data would draw a different random
number and possibly make a different decision about the
validity of H
0
.
7.3.4. Assuming the observations are failure times, are the failure rates (or Mean Times To Failure) for two distributions the same?

7.3.4. Assuming the observations are failure
times, are the failure rates (or Mean
Times To Failure) for two distributions
the same?
Comparing
two
exponential
distributions
is to
compare the
means or
hazard rates
The comparison of two (or more) life distributions is a
common objective when performing statistical analyses of
lifetime data. Here we look at the one-parameter exponential
distribution case.
In this case, comparing two exponential distributions is
equivalent to comparing their means (or the reciprocal of
their means, known as their hazard rates).
Type II Censored data
Definition
of Type II
censored
data
Definition: Type II censored data occur when a life test is
terminated exactly when a pre-specified number of failures
have occurred. The remaining units have not yet failed. If n
units were on test, and the pre-specified number of failures is
r (where r is less than or equal to n), then the test ends at t
r
= the time of the r-th failure.
Two
exponential
samples
oredered by
time
Suppose we have Type II censored data from two
exponential distributions with means
1
and
2
. We have two
samples from these distributions, of sizes n
1
on test with r
1
failures and n
2
on test with r
2
failures, respectively. The
observations are time to failure and are therefore ordered by
time.
Test of
equality of
1
and
2
and
confidence
Letting
interval for
1
/
2
Then
and
with T
1
and T
2
independent. Thus
where
and
has an F distribution with (2r
1
, 2r
2
) degrees of freedom.
Tests of equality of
1
and
2
can be performed using tables
of the F distribution or computer programs. Confidence
intervals for
1
/
2
, which is the ratio of the means or the
hazard rates for the two distributions, are also readily
obtained.
Numerical
example
A numerical application will illustrate the concepts outlined
above.
For this example,
H
0
:
1
/
2
= 1
H
a
:
1
/
2
1
Two samples of size 10 from exponential distributions were
put on life test. The first sample was censored after 7 failures
and the second sample was censored after 5 failures. The
times to failure were:
Sample 1: 125 189 210 356 468 550 610
Sample 2: 170 234 280 350 467
So r
1
= 7, r
2
= 5 and t
1,(r1)
= 610, t
2,(r2)
=467.
Then T
1
= 4338 and T
2
= 3836.
The estimator for
1
is 4338 / 7 = 619.71 and the estimator
for
2
is 3836 / 5 = 767.20.
The ratio of the estimators = U = 619.71 / 767.20 = .808.
If the means are the same, the ratio of the estimators, U,
follows an F distribution with 2r
1
, 2r
2
degrees of freedom.
The P(F < .808) = .348. The associated p-value is 2(.348) =
.696. Based on this p-value, we find no evidence to reject the
null hypothesis (that the true but unknown ratio = 1). Note
that this is a two-sided test, and we would reject the null
hyposthesis if the p-value is either too small (i.e., less or
equal to .025) or too large (i.e., greater than or equal to .975)
for a 95% significance level test.
We can also put a 95% confidence interval around the ratio
of the two means. Since the .025 and .975 quantiles of
F
(14,10)
are 0.3178 and 3.5504, respectively, we have
Pr(U/3.5504 <
1
/
2
< U/.3178) = .95
and (.228, 2.542) is a 95% confidence interval for the ratio
of the unknown means. The value of 1 is within this range,
which is another way of showing that we cannot reject the
null hypothesis at the 95% significance level.
7.3.5. Do two arbitrary processes have the same central tendency?

7.3.5. Do two arbitrary processes have the same central
tendency?
The
nonparametric
equivalent of
the t test is
due to Mann
and Whitney,
called the U
test
By "arbitrary" we mean that we make no underlying assumptions about
normality or any other distribution. The test is called the Mann-Whitney U
Test, which is the nonparametric equivalent of the t test for means.
The U-test (as the majority of nonparametric tests) uses the rank sums of the
two samples.
Procedure The test is implemented as follows.
1. Rank all (n
1
+ n
2
) observations in ascending order. Ties receive the
average of their observations.
2. Calculate the sum of the ranks, call these T
a
and T
b
3. Calculate the U statistic,
U
a
= n
1
(n
2
) + 0.5(n
1
)(n
1
+ 1) - T
a
or
U
b
= n
1
(n
2
) + 0.5(n
2
)(n
2
+ 1) - T
b
where U
a
+ U
b
= n
1
(n
2
).
Null
Hypothesis
The null hypothesis is: the two populations have the same central tendency.
The alternative hypothesis is: The central tendencies are NOT the same.
Test statistic The test statistic, U, is the smaller of U
a
and U
b
. For sample sizes larger than
20, we can use the normal z as follows:
z = [ U - E(U)] /
where
The critical value is the normal tabled z for /2 for a two-tailed test or z at
level, for a one-tail test.
For small samples, tables are readily available in most textbooks on
nonparametric statistics.
Example
An illustrative
example of the
U test
Two processing systems were used to clean wafers. The following data
represent the (coded) particle counts. The null hypothesis is that there is no
difference between the central tendencies of the particle counts; the alternative
hypothesis is that there is a difference. The solution shows the typical kind of
output software for this procedure would generate, based on the large sample
approximation.
Group A Rank Group B Rank
.55 8 .49 5
.67 15.5 .68 17
.43 1 .59 9.5
.51 6 .72 19
.48 3.5 .67 15.5
.60 11 .75 20.5
.71 18 .65 13.5
.53 7 .77 22
.44 2 .62 12
.65 13.5 .48 3.5
.75 20.5 .59 9.5
N Sum of Ranks U Std. Dev of U Median
A 11 106.000 81.000 15.229 0.540
B 11 147.000 40.000 15.229 0.635
For U = 40.0 and E[U] = 0.5(n
1
)(n
2
) = 60.5, the test statistic is
where
For a two-sided test with significance level = 0.05, the critical value is z
1-/2
= 1.96. Since |z| is less than the critical value, we do not reject the null
hypothesis and conclude that there is not enough evidence to claim that two
groups have different central tendencies.
7.4. Comparisons based on data from more than two processes

7.4. Comparisons based on data from more
than two processes
Introduction This section begins with a nonparametric procedure for
comparing several populations with unknown distributions.
Then the following topics are discussed:
Comparing variances
Comparing means (ANOVA technique)
Estimating variance components
Comparing categorical data
Comparing population proportion defectives
Making multiple comparisons
7.4.1. How can we compare several populations with unknown distributions (the Kruskal-Wallis test)?

7.4.1. How can we compare several populations
with unknown distributions (the Kruskal-
Wallis test)?
The Kruskal-Wallis (KW) Test for Comparing
Populations with Unknown Distributions
A
nonparametric
test for
comparing
population
medians by
Kruskal and
Wallis
The KW procedure tests the null hypothesis that k samples
from possibly different populations actually originate from
similar populations, at least as far as their central
tendencies, or medians, are concerned. The test assumes
that the variables under consideration have underlying
continuous distributions.
In what follows assume we have k samples, and the
sample size of the i-th sample is n
i
, i = 1, 2, . . ., k.
Test based on
ranks of
combined data
In the computation of the KW statistic, each observation is
replaced by its rank in an ordered combination of all the k
samples. By this we mean that the data from the k samples
combined are ranked in a single series. The minimum
observation is replaced by a rank of 1, the next-to-the-
smallest by a rank of 2, and the largest or maximum
observation is replaced by the rank of N, where N is the
total number of observations in all the samples (N is the
sum of the n
i
).
Compute the
sum of the
ranks for each
sample
The next step is to compute the sum of the ranks for each
of the original samples. The KW test determines whether
these sums of ranks are so different by sample that they are
not likely to have all come from the same population.
Test statistic
follows a
2
distribution
It can be shown that if the k samples come from the same
population, that is, if the null hypothesis is true, then the
test statistic, H, used in the KW procedure is distributed
approximately as a chi-square statistic with df = k - 1,
provided that the sample sizes of the k samples are not too
small (say, n
i
>4, for all i). H is defined as follows:
where
k = number of samples (groups)
n
i
= number of observations for the i-th sample or
group
N = total number of observations (sum of all the n
i
)
R
i
= sum of ranks for group i
Example
An illustrative
example
The following data are from a comparison of four
investment firms. The observations represent percentage of
growth during a three month period.for recommended
funds.

A B C D
4.2 3.3 1.9 3.5
4.6 2.4 2.4 3.1
3.9 2.6 2.1 3.7
4.0 3.8 2.7 4.1
2.8 1.8 4.4
Step 1: Express the data in terms of their ranks
A B C D
17 10 2 11
19 4.5 4.5 9
14 6 3 12
15 13 7 16
8 1 18
SUM 65 41.5 17.5 66
Compute the
test statistic
The corresponding H test statistic is
From the chi-square table in Chapter 1, the critical value
for 1- = 0.95 with df = k-1 = 3 is 7.812. Since 13.678 >
7.812, we reject the null hypothesis.
Note that the rejection region for the KW procedure is one-
sided, since we only reject the null hypothesis when the H
statistic is too large.
7.4.2. Assuming the observations are normal, do the processes have the same variance?

7.4.2. Assuming the observations are normal, do
the processes have the same variance?
Before
comparing
means, test
whether the
variances
are equal
Techniques for comparing means of normal populations
generally assume the populations have the same variance.
Before using these ANOVA techniques, it is advisable to test
whether this assumption of homogeneity of variance is
reasonable. The following procedure is widely used for this
purpose.
Bartlett's Test for Homogeneity of Variances
Null
hypothesis
Bartlett's test is a commonly used test for equal variances.
Let's examine the null and alternative hypotheses.
against
Test
statistic
Assume we have samples of size n
i
from the i-th population,
i = 1, 2, . . . , k, and the usual variance estimates from each
sample:
where
Now introduce the following notation:
j
= n
j
- 1 (the
j
are
the degrees of freedom) and
The Bartlett's test statistic M is defined by
Distribution
of the test
statistic
When none of the degrees of freedom is small, Bartlett
showed that M is distributed approximately as . The chi-
square approximation is generally acceptable if all the n
i
are
at least 5.
Bias
correction
This is a slightly biased test, according to Bartlett. It can be
improved by dividing M by the factor
Instead of M, it is suggested to use M/C for the test statistic.
Bartlett's
test is not
robust
This test is not robust, it is very sensitive to departures from
normality.
An alternative description of Bartlett's test appears in Chapter
1.
Gear Data Example (from Chapter 1):
An
illustrative
example of
Bartlett's
test
Gear diameter measurements were made on 10 batches of
product. The complete set of measurements appears in
Chapter 1. Bartlett's test was applied to this dataset leading to
a rejection of the assumption of equal batch variances at the
.05 critical value level. applied to this dataset
The Levene Test for Homogeneity of Variances
The Levene
test for
equality of
variances
Levene's test offers a more robust alternative to Bartlett's
procedure. That means it will be less likely to reject a true
hypothesis of equality of variances just because the
distributions of the sampled populations are not normal.
When non-normality is suspected, Levene's procedure is a
better choice than Bartlett's.
Levene's test is described in Chapter 1. This description also
includes an example where the test is applied to the gear
data. Levene's test does not reject the assumption of equality
of batch variances for these data. This differs from the
conclusion drawn from Bartlett's test and is a better answer
if, indeed, the batch population distributions are non-normal.
7.4.3. Are the means equal?

Test
equality of
means
The procedure known as the Analysis of Variance or ANOVA
is used to test hypotheses concerning means when we have
several populations.
The Analysis of Variance (ANOVA)
The ANOVA
procedure
is one of the
most
powerful
statistical
techniques
ANOVA is a general technique that can be used to test the
hypothesis that the means among two or more groups are
equal, under the assumption that the sampled populations are
normally distributed.
A couple of questions come immediately to mind: what
means? and why analyze variances in order to derive
conclusions about the means?
Both questions will be answered as we delve further into the
subject.
Introduction
to ANOVA
To begin, let us study the effect of temperature on a passive
component such as a resistor. We select three different
temperatures and observe their effect on the resistors. This
experiment can be conducted by measuring all the
participating resistors before placing n resistors each in three
different ovens.
Each oven is heated to a selected temperature. Then we
measure the resistors again after, say, 24 hours and analyze
the responses, which are the differences between before and
after being subjected to the temperatures. The temperature is
called a factor. The different temperature settings are called
levels. In this example there are three levels or settings of the
factor Temperature.
What is a
factor?
A factor is an independent treatment variable whose
settings (values) are controlled and varied by the
experimenter. The intensity setting of a factor is the level.
Levels may be quantitative numbers or, in many
cases, simply "present" or "not present" ("0" or
"1").
The 1-way In the experiment above, there is only one factor,
ANOVA temperature, and the analysis of variance that we will be
using to analyze the effect of temperature is called a one-way
or one-factor ANOVA.
The 2-way
or 3-way
ANOVA
We could have opted to also study the effect of positions in
the oven. In this case there would be two factors, temperature
and oven position. Here we speak of a two-way or two-
factor ANOVA. Furthermore, we may be interested in a third
factor, the effect of time. Now we deal with a three-way or
three-factorANOVA. In each of these ANOVA's we test a
variety of hypotheses of equality of means (or average
responses when the factors are varied).
Hypotheses
that can be
tested in an
ANOVA
First consider the one-way ANOVA. The null hypothesis is:
there is no difference in the population means of the different
levels of factor A (the only factor).
The alternative hypothesis is: the means are not the same.
For the 2-way ANOVA, the possible null hypotheses are:
1. There is no difference in the means of factor A
2. There is no difference in means of factor B
3. There is no interaction between factors A and B
The alternative hypothesis for cases 1 and 2 is: the means are
not equal.
The alternative hypothesis for case 3 is: there is an
interaction between A and B.
For the 3-way ANOVA: The main effects are factors A, B
and C. The 2-factor interactions are: AB, AC, and BC. There
is also a three-factor interaction: ABC.
For each of the seven cases the null hypothesis is the same:
there is no difference in means, and the alternative hypothesis
is the means are not equal.
The n-way
ANOVA
In general, the number of main effects and interactions can
be found by the following expression:
The first term is for the overall mean, and is always 1. The
second term is for the number of main effects. The third term
is for the number of 2-factor interactions, and so on. The last
term is for the n-factor interaction and is always 1.
In what follows, we will discuss only the 1-way and 2-way
ANOVA.
7.4.3.1. 1-Way ANOVA overview

Overview and
principles
This section gives an overview of the one-way ANOVA.
First we explain the principles involved in the 1-way
ANOVA.
Partition
response into
components
In an analysis of variance the variation in the response
measurements is partitoned into components that
correspond to different sources of variation.
The goal in this procedure is to split the total variation in
the data into a portion due to random error and portions
due to changes in the values of the independent
variable(s).
Variance of n
measurements
The variance of n measurements is given by
where is the mean of the n measurements.
Sums of
squares and
degrees of
freedom
The numerator part is called the sum of squares of
deviations from the mean, and the denominator is called
the degrees of freedom.
The variance, after some algebra, can be rewritten as:
The first term in the numerator is called the "raw sum of
squares" and the second term is called the "correction term
for the mean". Another name for the numerator is the
"corrected sum of squares", and this is usually abbreviated
by Total SS or SS(Total).
The SS in a 1-way ANOVA can be split into two
components, called the "sum of squares of treatments" and
"sum of squares of error", abbreviated as SST and SSE,
respectively.
The guiding
principle
behind
ANOVA is the
decomposition
of the sums of
squares, or
Total SS
Algebraically, this is expressed by
where k is the number of treatments and the bar over the
y.. denotes the "grand" or "overall" mean. Each n
i
is the
number of observations for treatment i. The total number of
observations is N (the sum of the n
i
).
Note on
subscripting
Don't be alarmed by the double subscripting. The total SS
can be written single or double subscripted. The double
subscript stems from the way the data are arranged in the
data table. The table is usually a rectangular array with k
columns and each column consists of n
i
rows (however, the
lengths of the rows, or the n
i
, may be unequal).
Definition of
"Treatment"
We introduced the concept of treatment. The definition is:
A treatment is a specific combination of factor levels
whose effect is to be compared with other treatments.
7.4.3.2. The 1-way ANOVA model and assumptions

7.4.3.2. The 1-way ANOVA model and
assumptions
A model
that
describes
the
relationship
between the
response
and the
treatment
(between
the
dependent
and
independent
variables)
The mathematical model that describes the relationship
between the response and treatment for the one-way ANOVA
is given by
where Y
ij
represents the j-th observation (j = 1, 2, ...n
i
) on the
i-th treatment (i = 1, 2, ..., k levels). So, Y
23
represents the
third observation using level 2 of the factor. is the common
effect for the whole experiment,
i
represents the i-th
treatment effect and
ij
represents the random error present in
the j-th observation on the i-th treatment.
Fixed
effects
model
The errors
ij
are assumed to be normally and independently
(NID) distributed, with mean zero and variance . is
always a fixed parameter and are considered to
be fixed parameters if the levels of the treatment are fixed,
and not a random sample from a population of possible
levels. It is also assumed that is chosen so that
holds. This is the fixed effects model.
Random
effects
model
If the k levels of treatment are chosen at random, the model
equation remains the same. However, now the
i
's are
random variables assumed to be NID(0, ). This is the
random effects model.
Whether the levels are fixed or random depends on how these
levels are chosen in a given experiment.
7.4.3.3. The ANOVA table and tests of hypotheses about means

7.4.3.3. The ANOVA table and tests of
hypotheses about means
Sums of
Squares help
us compute
the variance
estimates
displayed in
ANOVA
Tables
The sums of squares SST and SSE previously computed for
the one-way ANOVA are used to form two mean squares,
one for treatments and the second for error. These mean
squares are denoted by MST and MSE, respectively. These
are typically displayed in a tabular form, known as an
ANOVA Table. The ANOVA table also shows the statistics
used to test hypotheses about the population means.
Ratio of MST
and MSE
When the null hypothesis of equal means is true, the two
mean squares estimate the same quantity (error variance),
and should be of approximately equal magnitude. In other
words, their ratio should be close to 1. If the null hypothesis
is false, MST should be larger than MSE.
Divide sum of
squares by
degrees of
freedom to
obtain mean
squares
The mean squares are formed by dividing the sum of
squares by the associated degrees of freedom.
Let N = n
i
. Then, the degrees of freedom for treatment,
DFT = k - 1, and the degrees of freedom for error, DFE =
N

- k.
The corresponding mean squares are:
MST = SST / DFT
MSE = SSE / DFE
The F-test The test statistic, used in testing the equality of treatment
means is: F = MST / MSE.
The critical value is the tabular value of the F distribution,
based on the chosen level and the degrees of freedom
DFT and DFE.
The calculations are displayed in an ANOVA table, as
follows:
ANOVA table
Source SS DF MS F
Treatments SST k-1 SST / (k-1) MST/MSE
Error SSE N-k SSE / (N-k)
Total
(corrected)
SS N-1
The word "source" stands for source of variation. Some
authors prefer to use "between" and "within" instead of
"treatments" and "error", respectively.
ANOVA Table Example
A numerical
example
The data below resulted from measuring the difference in
resistance resulting from subjecting identical resistors to
three different temperatures for a period of 24 hours. The
sample size of each group was 5. In the language of Design
of Experiments, we have an experiment in which each of
three treatments was replicated 5 times.
Level 1 Level 2 Level 3
6.9 8.3 8.0
5.4 6.8 10.5
5.8 7.8 8.1
4.6 9.2 6.9
4.0 6.5 9.3
means 5.34 7.72 8.56
The resulting ANOVA table is
Example
ANOVA table
Source SS DF MS F
Treatments 27.897 2 13.949 9.59
Error 17.452 12 1.454
Total (corrected) 45.349 14
Correction Factor 779.041 1
Interpretation
of the
ANOVA table
The test statistic is the F value of 9.59. Using an of .05,
we have that F
.05; 2, 12
= 3.89 (see the F distribution table in
Chapter 1). Since the test statistic is much larger than the
critical value, we reject the null hypothesis of equal
population means and conclude that there is a (statistically)
significant difference among the population means. The p-
value for 9.59 is .00325, so the test statistic is significant at
that level.
Techniques
for further
analysis
The populations here are resistor readings while operating
under the three different temperatures. What we do not
know at this point is whether the three means are all
different or which of the three means is different from the
other two, and by how much.
There are several techniques we might use to further
analyze the differences. These are:
constructing confidence intervals around the
difference of two means,
estimating combinations of factor levels with
confidence bounds
multiple comparisons of combinations of factor levels
tested simultaneously.
7.4.3.4. 1-Way ANOVA calculations

Formulas
for 1-way
ANOVA
hand
calculations
Although computer programs that do ANOVA calculations
now are common, for reference purposes this page describes
how to calculate the various entries in an ANOVA table.
Remember, the goal is to produce two variances (of
treatments and error) and their ratio. The various
computational formulas will be shown and applied to the data
from the previous example.
Step 1:
compute
CM
STEP 1 Compute CM, the correction for the mean.
Step 2:
compute
total SS
STEP 2 Compute the total SS.
The total SS = sum of squares of all observations - CM
The 829.390 SS is called the "raw" or "uncorrected " sum of
squares.
Step 3:
compute
SST
STEP 3 Compute SST, the treatment sum of squares.
First we compute the total (sum) for each treatment.
T
1
= (6.9) + (5.4) + ... + (4.0) = 26.7
T
2
= (8.3) + (6.8) + ... + (6.5) = 38.6
T
1
= (8.0) + (10.5) + ... + (9.3) = 42.8
Then
Step 4:
compute
SSE
STEP 4 Compute SSE, the error sum of squares.
Here we utilize the property that the treatment sum of squares
plus the error sum of squares equals the total sum of squares.
Hence, SSE = SS Total - SST = 45.349 - 27.897 = 17.45.
Step 5:
Compute
MST, MSE,
and F
STEP 5 Compute MST, MSE and their ratio, F.
MST is the mean square of treatments, MSE is the mean
square of error (MSE is also frequently denoted by ).
MST = SST / (k-1) = 27.897 / 2 = 13.949
MSE = SSE / (N-k) = 17.452/ 12 = 1.454
where N is the total number of observations and k is the
number of treatments. Finally, compute F as
F = MST / MSE = 9.59
That is it. These numbers are the quantities that are
assembled in the ANOVA table that was shown previously.
7.4.3.5. Confidence intervals for the difference of treatment means

7.4.3.5. Confidence intervals for the difference
of treatment means
Confidence
intervals for
the
difference
between two
means
This page shows how to construct a confidence interval
around (
i
-
j
) for the one-way ANOVA by continuing the
example shown on a previous page.
Formula for
the
confidence
interval
The formula for a (1- ) 100% confidence interval for the
difference between two treatment means is:
where = MSE.
Computation
of the
confidence
interval for
3
-
1
For the example, we have the following quantities for the
formula:
3
= 8.56
1
= 5.34
t
0.975, 12
= 2.179
Substituting these values yields (8.56 - 5.34) 2.179(0.763)
or 3.22 1.616.
That is, the confidence interval is from 1.604 to 4.836.
Additional
95%
confidence
intervals
A 95% confidence interval for
3
-
2
is: from -1.787 to
3.467.
A 95% confidence interval for
2
-
1
is: from -0.247 to
5.007.
Contrasts Later on the topic of estimating more general linear
7.4.3.5. Confidence intervals for the difference of treatment means
discussed
later
combinations of means (primarily contrasts) will be
discussed, including how to put confidence bounds around
contrasts.
7.4.3.6. Assessing the response from any factor combination

7.4.3.6. Assessing the response from any factor
combination
Contrasts This page treats how to estimate and put confidence bounds
around the response to different combinations of factors.
Primary focus is on the combinations that are known as
contrasts. We begin, however, with the simple case of a
single factor-level mean.
Estimation of a Factor Level Mean With Confidence
Bounds
Estimating
factor level
means
An unbiased estimator of the factor level mean
i
in the 1-
way ANOVA model is given by:
where
Variance of
the factor
level means
The variance of this sample mean estimator is
Confidence
intervals for
the factor
level means
It can be shown that:
has a t distribution with (N - k) degrees of freedom for the
ANOVA model under consideration, where N is the total
number of observations and k is the number of factor levels
or groups. The degrees of freedom are the same as were
used to calculate the MSE in the ANOVA table. That is: dfe
(degrees of freedom for error) = N - k. From this we can
calculate (1- )100% confidence limits for each
i
. These
are given by:
Example 1
Example for
a 4-level
treatment (or
4 different
treatments)
The data in the accompanying table resulted from an
experiment run in a completely randomized design in which
each of four treatments was replicated five times.
Total Mean
Group 1 6.9 5.4 5.8 4.6 4.0 26.70 5.34
Group 2 8.3 6.8 7.8 9.2 6.5 38.60 7.72
Group 3 8.0 10.5 8.1 6.9 9.3 42.80 8.56
Group 4 5.8 3.8 6.1 5.6 6.2 27.50 5.50
All Groups 135.60 6.78
1-Way
ANOVA
table layout
This experiment can be illustrated by the table layout for
this 1-way ANOVA experiment shown below:
Level Sample j
i 1 2 ... 5 Sum Mean N
1 Y
11
Y
12
... Y
15
Y
1. 1.
n
1
2 Y
21
Y
22
... Y
25
Y
2. 2.
n
2
3 Y
31
Y
32
... Y
35
Y
3. 3.
n
3
4 Y
41
Y
42
... Y
45
Y
4. 4.
n
4
All Y
. ..
n
t
ANOVA
table
The resulting ANOVA table is
Source SS DF MS F
Treatments 38.820 3 12.940 9.724
Error 21.292 16 1.331
Total (Corrected) 60.112 19
Mean 919.368 1
Total (Raw) 979.480 20
The estimate for the mean of group 1 is 5.34, and the
sample size is n
1
= 5.
Computing
the
confidence
interval
Since the confidence interval is two-sided, the entry (1 -
/2) value for the t table is (1 - 0.05/2) = 0.975, and the
associated degrees of freedom is N - 4, or 20 - 4 = 16.
From the t table in Chapter 1, we obtain t
0.975;16
= 2.120.
Next we need the standard error of the mean for group 1:
Hence, we obtain confidence limits 5.34 2.120 (0.5159)
and the confidence interval is
Definition and Estimation of Contrasts
Definition of
contrasts and
orthogonal
contrasts
Definitions
A contrast is a linear combination of 2 or more factor level
means with coefficients that sum to zero.
Two contrasts are orthogonal if the sum of the products of
corresponding coefficients (i.e., coefficients for the same
means) adds to zero.
Formally, the definition of a contrast is expressed below,
using the notation
i
for the i-th treatment mean:
C = c
1 1
+ c
2 2
+ ... + c
j j
+ ... + c
k k
where
c
1
+ c
2
+ ... + c
j
+ ... + c
k
= = 0
Simple contrasts include the case of the difference between
two factor means, such as
1
-
2
. If one wishes to compare
treatments 1 and 2 with treatment 3, one way of expressing
this is by:
1
+
2
- 2
3
. Note that
1
-
2
has coefficients +1, -1
1
+
2
- 2
3
has coefficients +1, +1, -2.
These coefficients sum to zero.
An example
of
orthogonal
contrasts
As an example of orthogonal contrasts, note the three
contrasts defined by the table below, where the rows denote
coefficients for the column treatment means.

1 2 3 4
c
1
+1 0 0 -1
c
2
0 +1 -1 0
c
3
+1 -1 -1 +1
Some
properties of
orthogonal
contrasts
The following is true:
1. The sum of the coefficients for each contrast is zero.
2. The sum of the products of coefficients of each pair
of contrasts is also 0 (orthogonality property).
3. The first two contrasts are simply pairwise
comparisons, the third one involves all the treatments.
Estimation of
contrasts
As might be expected, contrasts are estimated by taking the
same linear combination of treatment mean estimators. In
other words:
and
Note: These formulas hold for any linear combination of
treatment means, not just for contrasts.
Confidence Interval for a Contrast
Confidence
intervals for
contrasts
An unbiased estimator for a contrast C is given by
The estimator of is
The estimator is normally distributed because it is a
linear combination of independent normal random variables.
It can be shown that:
is distributed as t
N-r
for the one-way ANOVA model under
discussion.
Therefore, the 1- confidence limits for C are:
Example 2 (estimating contrast)
Contrast to
estimate
We wish to estimate, in our previous example, the
following contrast:
and construct a 95 % confidence interval for C.
Computing
the point
estimate and
standard
error
The point estimate is:
Applying the formulas above we obtain
and
and the standard error is = 0.5159.
Confidence
interval
For a confidence coefficient of 95 % and df = 20 - 4 = 16,
t
0.975,16
= 2.12. Therefore, the desired 95 % confidence
interval is -0.5 2.12(0.5159) or
(-1.594, 0.594).
Estimation of Linear Combinations
Estimating
linear
combinations
Sometimes we are interested in a linear combination of the
factor-level means that is not a contrast. Assume that in our
sample experiment certain costs are associated with each
group. For example, there might be costs associated with
each factor as follows:
Factor Cost in $
1 3
2 5
3 2
4 1
The following linear combination might then be of interest:
Coefficients
do not have
to sum to
zero for
linear
combinations
This resembles a contrast, but the coefficients c
i
do not
sum to zero. A linear combination is given by the
definition:
with no restrictions on the coefficients c
i
.
Confidence
interval
identical to
contrast
Confidence limits for a linear combination C are obtained in
precisely the same way as those for a contrast, using the
same calculation for the point estimator and estimated
variance.
7.4.3.7. The two-way ANOVA

Definition
of a
factorial
experiment
The 2-way ANOVA is probably the most popular layout in
the Design of Experiments. To begin with, let us define a
factorial experiment:
An experiment that utilizes every combination of factor
levels as treatments is called a factorial experiment.
Model for
the two-
way
factorial
experiment
In a factorial experiment with factor A at a levels and factor
B at b levels, the model for the general layout can be written
as
where is the overall mean response,
i
is the effect due to
the i-th level of factor A,
j
is the effect due to the j-th level
of factor B and
ij
is the effect due to any interaction
between the i-th level of A and the j-th level of B.
Fixed
factors and
fixed effects
models
At this point, consider the levels of factor A and of factor B
chosen for the experiment to be the only levels of interest to
the experimenter such as predetermined levels for
temperature settings or the length of time for process step.
The factors A and B are said to be fixed factors and the
model is a fixed-effects model. Random actors will be
discussed later.
When an a x b factorial experiment is conducted with an
equal number of observations per treatment combination, the
total (corrected) sum of squares is partitioned as:
SS(total) = SS(A) + SS(B) + SS(AB) + SSE
where AB represents the interaction between A and B.
For reference, the formulas for the sums of squares are:
The
breakdown
of the total
(corrected
for the
mean) sums
of squares
The resulting ANOVA table for an a x b factorial experiment
is
Source SS df MS
Factor A SS(A) (a - 1) MS(A) = SS(A)/(a-
1)
Factor B SS(B) (b - 1) MS(B) = SS(B)/(b-
1)
Interaction AB SS(AB) (a-1)(b-
1)
MS(AB)=
SS(AB)/(a-1)(b-1)
Error SSE (N - ab) SSE/(N - ab)
Total
(Corrected)
SS(Total) (N - 1)
The
ANOVA
table can
be used to
test
hypotheses
about the
effects and
interactions
The various hypotheses that can be tested using this ANOVA
table concern whether the different levels of Factor A, or
Factor B, really make a difference in the response, and
whether the AB interaction is significant (see previous
discussion of ANOVA hypotheses).
7.4.3.8. Models and calculations for the two-way ANOVA

7.4.3.8. Models and calculations for the two-way
ANOVA
Basic Layout
The
balanced
2-way
factorial
layout
Factor A has 1, 2, ..., a levels. Factor B has 1, 2, ..., b levels. There are
ab treatment combinations (or cells) in a complete factorial layout.
Assume that each treatment cell has r independent obsevations (known
as replications). When each cell has the same number of replications,
the design is a balanced factorial. In this case, the abrdata points
{y
ijk
} can be shown pictorially as follows:
Factor B
1 2 ... b
1 y
111
, y
112
, ..., y
11r
y
121
, y
122
, ..., y
12r
... y
1b1
, y
1b2
, ..., y
1br
2 y
211
, y
212
, ..., y
21r
y
221
, y
222
, ..., y
22r
... y
2b1
, y
2b2
, ..., y
2br
Factor
A
.
.
... .... ...
a y
a11
, y
a12
, ..., y
a1r
y
a21
, y
a22
, ..., y
a2r
... y
ab1
, y
ab2
, ..., y
abr
How to
obtain
sums of
squares
for the
balanced
factorial
layout
Next, we will calculate the sums of squares needed for the ANOVA
table.
Let A
i
be the sum of all observations of level i of factor A, i = 1,
... ,a. The A
i
are the row sums.
Let B
j
be the sum of all observations of level j of factor B, j = 1,
...,b. The B
j
are the column sums.
Let (AB)
ij
be the sum of all observations of level i of A and
level j of B. These are cell sums.
Let r be the number of replicates in the experiment; that is: the
number of times each factorial treatment combination appears in
the experiment.
Then the total number of observations for each level of factor A is rb
and the total number of observations for each level of factor B is raand
the total number of observations for each interaction is r.
Finally, the total number of observations n in the experiment is abr.
With the help of these expressions we arrive (omitting derivations) at
These expressions are used to calculate the ANOVA table entries for
the (fixed effects) 2-way ANOVA.
Two-Way ANOVA Example:
Data An evaluation of a new coating applied to 3 different materials was
conducted at 2 different laboratories. Each laboratory tested 3 samples
from each of the treated materials. The results are given in the next
table:
Materials (B)
LABS (A) 1 2 3
4.1 3.1 3.5
1 3.9 2.8 3.2
4.3 3.3 3.6
2.7 1.9 2.7
2 3.1 2.2 2.3
2.6 2.3 2.5
Row and
column
sums
The preliminary part of the analysis yields a table of row and column
sums.
Material (B)
Lab (A) 1 2 3 Total (A
i
)
1 12.3 9.2 10.3 31.8
2 8.4 6.4 7.5 22.3
Total (B
j
) 20.7 15.6 17.8 54.1
ANOVA
table
From this table we generate the ANOVA table.
Source SS df MS F p-value
A 5.0139 1 5.0139 100.28 0
B 2.1811 2 1.0906 21.81 .0001
AB 0.1344 2 0.0672 1.34 .298
Error 0.6000 12 0.0500
Total (Corr) 7.9294 17
7.4.4. What are variance components?

Fixed and Random Factors and Components of
Variance
A fixed level
of a factor or
variable
means that
the levels in
the
experiment
are the only
ones we are
interested in
In the previous example, the levels of the factor
temperature were considered as fixed; that is, the three
temperatures were the only ones that we were interested in
(this may sound somewhat unlikely, but let us accept it
without opposition). The model employed for fixed levels is
called a fixed model. When the levels of a factor are
random, such as operators, days, lots or batches, where the
levels in the experiment might have been chosen at random
from a large number of possible levels, the model is called
a random model, and inferences are to be extended to all
levels of the population.
Random
levels are
chosen at
random from
a large or
infinite set of
levels
In a random model the experimenter is often interested in
estimating components of variance. Let us run an example
that analyzes and interprets a component of variance or
random model.
Components of Variance Example for Random Factors
Data for the
example
A company supplies a customer with a larger number of
batches of raw materials. The customer makes three sample
determinations from each of 5 randomly selected batches to
control the quality of the incoming material. The model is
and the k levels (e.g., the batches) are chosen at random
from a population with variance . The data are shown
below
Batch
1 2 3 4 5
74 68 75 72 79
76 71 77 74 81
75 72 77 73 79
ANOVA table
for example
A 1-way ANOVA is performed on the data with the
following results:
ANOVA
Source SS df MS EMS
Treatment (batches) 147.74 4 36.935 + 3
Error 17.99 10 1.799
Total (corrected) 165.73 14
Interpretation
of the
ANOVA table
The computations that produce the SS are the same for both
the fixed and the random effects model. For the random
model, however, the treatment sum of squares, SST, is an
estimate of { + 3 }. This is shown in the EMS
(Expected Mean Squares) column of the ANOVA table.
The test statistic from the ANOVA table is F = 36.94 / 1.80
= 20.5.
If we had chosen an value of .01, then the F value from
the table in Chapter 1 for a df of 4 in the numerator and 10
in the denominator is 5.99.
Method of
moments
Since the test statistic is larger than the critical value, we
reject the hypothesis of equal means. Since these batches
were chosen via a random selection process, it may be of
interest to find out how much of the variance in the
experiment might be attributed to batch diferences and how
much to random error. In order to answer these questions,
we can use the EMS column. The estimate of is 1.80 and
the computed treatment mean square of 36.94 is an estimate
of + 3 . Setting the MS values equal to the EMS values
(this is called the Method of Moments), we obtain
where we use s
2
since these are estimators of the
corresponding
2
's.
Computation
of the
Solving these expressions
components
of variance
The total variance can be estimated as
Interpretation In terms of percentages, we see that 11.71/13.51 = 86.7
percent of the total variance is attributable to batch
differences and 13.3 percent to error variability within the
batches.
7.4.5. How can we compare the results of classifying according to several categories?

7.4.5. How can we compare the results of classifying
according to several categories?
Contingency
Table
approach
When items are classified according to two or more criteria, it is often of
interest to decide whether these criteria act independently of one another.
For example, suppose we wish to classify defects found in wafers produced
in a manufacturing plant, first according to the type of defect and, second,
according to the production shift during which the wafers were produced. If
the proportions of the various types of defects are constant from shift to
shift, then classification by defects is independent of the classification by
production shift. On the other hand, if the proportions of the various defects
vary from shift to shift, then the classification by defects depends upon or is
contingent upon the shift classification and the classifications are dependent.
In the process of investigating whether one method of classification is
contingent upon another, it is customary to display the data by using a cross
classification in an array consisting of r rows and c columns called a
contingency table. A contingency table consists of r x c cells representing
the r x c possible outcomes in the classification process. Let us construct an
industrial case:
Industrial
example
A total of 309 wafer defects were recorded and the defects were classified as
being one of four types, A, B, C, or D. At the same time each wafer was
identified according to the production shift in which it was manufactured, 1,
2, or 3.
Contingency
table
classifying
defects in
wafers
according to
type and
production
shift
These counts are presented in the following table.
Type of Defects
Shift A B C D Total
1 15(22.51) 21(20.99) 45(38.94) 13(11.56) 94
2 26(22.9) 31(21.44) 34(39.77) 5(11.81) 96
3 33(28.50) 17(26.57) 49(49.29) 20(14.63) 119
Total 74 69 128 38 309
(Note: the numbers in parentheses are the expected cell frequencies).
Column
probabilities
Let p
A
be the probability that a defect will be of type A. Likewise, define p
B
,
p
C
, and p
D
as the probabilities of observing the other three types of defects.
These probabilities, which are called the column probabilities, will satisfy
the requirement
p
A
+ p
B
+ p
C
+ p
D
= 1
Row
probabilities
By the same token, let p
i
(i=1, 2, or 3) be the row probability that a defect
will have occurred during shift i, where
p
1
+ p
2
+ p
3
= 1
Multiplicative
Law of
Probability
Then if the two classifications are independent of each other, a cell
probability will equal the product of its respective row and column
probabilities in accordance with the Multiplicative Law of Probability.
Example of
obtaining
column and
row
probabilities
For example, the probability that a particular defect will occur in shift 1 and
is of type A is (p
1
) (p
A
). While the numerical values of the cell probabilities
are unspecified, the null hypothesis states that each cell probability will equal
the product of its respective row and column probabilities. This condition
implies independence of the two classifications. The alternative hypothesis is
that this equality does not hold for at least one cell.
In other words, we state the null hypothesis as H
0
: the two classifications are
independent, while the alternative hypothesis is H
a
: the classifications are
dependent.
To obtain the observed column probability, divide the column total by the
grand total, n. Denoting the total of column j as c
j
, we get
Similarly, the row probabilities p
1
, p
2
, and p
3
are estimated by dividing the
row totals r
1
, r
2
, and r
3
by the grand total n, respectively
Expected cell
frequencies
Denote the observed frequency of the cell in row i and column jof the
contingency table by n
ij
. Then we have
Estimated
expected cell
frequency
when H
0
is
true.
In other words, when the row and column classifications are independent, the
estimated expected value of the observed cell frequency n
ij
in an r x c
contingency table is equal to its respective row and column totals divided by
the total frequency.
The estimated cell frequencies are shown in parentheses in the contingency
table above.
Test statistic From here we use the expected and observed frequencies shown in the table
to calculate the value of the test statistic
df = (r-1)(c-
1)
The next step is to find the appropriate number of degrees of freedom
associated with the test statistic. Leaving out the details of the derivation, we
state the result:
The number of degrees of freedom associated with a contingency
table consisting of r rows and c columns is (r-1) (c-1).
So for our example we have (3-1) (4-1) = 6 d.f.
Testing the
null
hypothesis
In order to test the null hypothesis, we compare the test statistic with the
critical value of
2
1-/2
at a selected value of . Let us use = 0.05. Then
the critical value is
2
0.95,6
= 12.5916 (see the chi square table in Chapter
1). Since the test statistic of 19.18 exceeds the critical value, we reject the
null hypothesis and conclude that there is significant evidence that the
proportions of the different defect types vary from shift to shift. In this case,
the p-value of the test statistic is 0.00387.
7.4.6. Do all the processes have the same proportion of defects?

7.4.6. Do all the processes have the same
proportion of defects?
The contingency table approach
Testing for
homogeneity
of proportions
using the chi-
square
distribution
via
contingency
tables
When we have samples from n populations (i.e., lots,
vendors, production runs, etc.), we can test whether there
are significant differences in the proportion defectives for
these populations using a contingency table approach. The
contingency table we construct has two rows and n
columns.
To test the null hypothesis of no difference in the
proportions among the n populations
H
0
: p
1
= p
2
= ... = p
n
against the alternative that not all n population proportions
are equal
H
1
: Not all p
i
are equal (i = 1, 2, ..., n)
The chi-square
test statistic
we use the following test statistic:
where f
o
is the observed frequency in a given cell of a 2 x
n contingency table, and f
c
is the theoretical count or
expected frequency in a given cell if the null hypothesis
were true.
The critical
value
The critical value is obtained from the
2
distribution
table with degrees of freedom (2-1)(n-1) = n-1, at a given
level of significance.
An illustrative example
Data for the
example
Diodes used on a printed circuit board are produced in lots
of size 4000. To study the homogeneity of lots with
respect to a demanding specification, we take random
samples of size 300 from 5 consecutive lots and test the
diodes. The results are:
Lot
Results 1 2 3 4 5 Totals
Nonconforming 36 46 42 63 38 225
Conforming 264 254 258 237 262 1275
Totals 300 300 300 300 300 1500
Computation
of the overall
proportion of
nonconforming
units
Assuming the null hypothesis is true, we can estimate the
single overall proportion of nonconforming diodes by
pooling the results of all the samples as
Computation
of the overall
proportion of
conforming
units
We estimate the proportion of conforming ("good") diodes
by the complement 1 - 0.15 = 0.85. Multiplying these two
proportions by the sample sizes used for each lot results in
the expected frequencies of nonconforming and
conforming diodes. These are presented below:
Table of
expected
frequencies
Lot
Results 1 2 3 4 5 Totals
Nonconforming 45 45 45 45 45 225
Conforming 255 255 255 255 255 1275
Totals 300 300 300 300 300 1500
Null and
alternate
hypotheses
To test the null hypothesis of homogeneity or equality of
proportions
H
0
: p
1
= p
2
= ... = p
5
against the alternative that not all 5 population proportions
are equal
H
1
: Not all p
i
are equal (i = 1, 2, ...,5)
Table for
computing the
test statistic
we use the observed and expected values from the tables
above to compute the
2
test statistic. The calculations are
presented below:
2 2
f
o
f
c
(f
o
- f
c
)
(f
o
- f
c
) (f
o
- f
c
) / f
c
36 45 -9 81 1.800
46 45 1 1 0.022
42 45 -3 9 0.200
63 45 18 324 7.200
38 45 -7 49 1.089
264 225 9 81 0.318
254 255 -1 1 0.004
258 255 3 9 0.035
237 255 -18 324 1.271
262 255 7 49 0.192
12.131
Conclusions If we choose a .05 level of significance, the critical value
of
2
with 4 degrees of freedom is 9.488 (see the chi
square distribution table in Chapter 1). Since the test
statistic (12.131) exceeds this critical value, we reject the
null hypothesis.
7.4.7. How can we make multiple comparisons?

What to do
after
equality of
means is
rejected
When processes are compared and the null hypothesis of
equality (or homogeneity) is rejected, all we know at that
point is that there is no equality amongst them. But we do
not know the form of the inequality.
Typical
questions
Questions concerning the reason for the rejection of the null
hypothesis arise in the form of:
"Which mean(s) or proportion (s) differ from a
standard or from each other?"
"Does the mean of treatment 1 differ from that of
treatment 2?"
"Does the average of treatments 1 and 2 differ from
the average of treatments 3 and 4?"
Multiple
Comparison
test
procedures
are needed
One popular way to investigate the cause of rejection of the
null hypothesis is a Multiple Comparison Procedure. These
are methods which examine or compare more than one pair
of means or proportions at the same time.
Note: Doing pairwise comparison procedures over and over
again for all possible pairs will not, in general, work. This is
because the overall significance level is not as specified for
a single pair comparison.
ANOVA F
test is a
preliminary
test
The ANOVA uses the F test to determine whether there
exists a significant difference among treatment means or
interactions. In this sense it is a preliminary test that informs
us if we should continue the investigation of the data at
hand.
If the null hypothesis (no difference among treatments or
interactions) is accepted, there is an implication that no
relation exists between the factor levels and the response.
There is not much we can learn, and we are finished with the
analysis.
When the F test rejects the null hypothesis, we usually want
to undertake a thorough analysis of the nature of the factor-
level effects.
Procedures
for
examining
factor-level
effects
Previously, we discussed several procedures for examining
particular factor-level effects. These were
Estimation of the Difference Between Two Factor
Means
Estimation of Factor Level Effects
Confidence Intervals For A Contrast
Determine
contrasts in
advance of
observing
the
experimental
results
These types of investigations should be done on
combinations of factors that were determined in advance of
observing the experimental results, or else the confidence
levels are not as specified by the procedure. Also, doing
several comparisons might change the overall confidence
level (see note above). This can be avoided by carefully
selecting contrasts to investigate in advance and making sure
that:
the number of such contrasts does not exceed the
number of degrees of freedom between the treatments
only orthogonal contrasts are chosen.
However, there are also several powerful multiple
comparison procedures we can use after observing the
experimental results.
Tests on Means after Experimentation
Procedures
for
performing
multiple
comparisons
If the decision on what comparisons to make is withheld
until after the data are examined, the following procedures
can be used:
Tukey's Method to test all possible pairwise
differences of means to determine if at least one
difference is significantly different from 0.
Scheff's Method to test all possible contrasts at the
same time, to see if at least one is significantly
different from 0.
Bonferroni Method to test, or put simultaneous
confidence intervals around, a pre-selected group of
contrasts
Multiple Comparisons Between Proportions
Procedure
for
proportion
defective
data
When we are dealing with population proportion defective
data, the Marascuilo procedure can be used to
simultaneously examine comparisons between all groups
after the data have been collected.
7.4.7.1. Tukey's method

Tukey's
method
considers
all possible
pairwise
differences
of means at
the same
time
The Tukey method applies simultaneously to the set of all
pairwise comparisons
{
i
-
j
}
The confidence coefficient for the set, when all sample sizes
are equal, is exactly 1- . For unequal sample sizes, the
confidence coefficient is greater than 1- . In other words,
the Tukey method is conservative when there are unequal
sample sizes.
Studentized Range Distribution
The
studentized
range q
The Tukey method uses the studentized range distribution.
Suppose we have r independent observations y
1
, ..., y
r
from
a normal distribution with mean and variance
2
. Let w be
the range for this set , i.e., the maximum minus the
minimum. Now suppose that we have an estimate s
2
of the
variance
2
which is based on degrees of freedom and is
independent of the y
i
. The studentized range is defined as
The
distribution
of q is
tabulated in
many
textbooks
and can be
calculated
using
Dataplot
The distribution of q has been tabulated and appears in many
textbooks on statistics. In addition, Dataplot has a CDF
function (SRACDF) and a percentile function (SRAPPF) for
q.
As an example, let r = 5 and = 10. The 95th percentile is
q
.05;5,10
= 4.65. This means:
So, if we have five observations from a normal distribution,
the probability is .95 that their range is not more than 4.65
times as great as an independent sample standard deviation
estimate for which the estimator has 10 degrees of freedom.

Tukey's Method
Confidence
limits for
Tukey's
method
The Tukey confidence limits for all pairwise comparisons
with confidence coefficient of at least 1- are:
Notice that the point estimator and the estimated variance are
the same as those for a single pairwise comparison that was
illustrated previously. The only difference between the
confidence limits for simultaneous comparisons and those for
a single comparison is the multiple of the estimated standard
deviation.
Also note that the sample sizes must be equal when using the
studentized range approach.
Example
Data We use the data from a previous example.
Set of all
pairwise
comparisons
The set of all pairwise comparisons consists of:
2
-
1
,
3
-
1
,
1
-
4
,
2
-
3
,
2
-
4
,
3
-
4
Confidence
intervals for
each pair
Assume we want a confidence coefficient of 95 percent, or
.95. Since r = 4 and n
t
= 20, the required percentile of the
studentized range distribution is q
.05; 4,16
. Using the Tukey
method for each of the six comparisons yields:
Conclusions The simultaneous pairwise comparisons indicate that the
differences
1
-
4
and
2
-
3
are not significantly different
from 0 (their confidence intervals include 0), and all the
other pairs are significantly different.
Unequal
sample sizes
It is possible to work with unequal sample sizes. In this case,
one has to calculate the estimated standard deviation for each
pairwise comparison. The Tukey procedure for unequal
sample sizes is sometimes referred to as the Tukey-Kramer
Method.
7.4.7.2. Scheffe's method

Scheffe's
method tests
all possible
contrasts at
the same
time
Scheff's method applies to the set of estimates of all
possible contrasts among the factor level means, not just the
pairwise differences considered by Tukey's method.
Definition of
contrast
An arbitrary contrast is defined by
where
Infinite
number of
contrasts
Technically there is an infinite number of contrasts. The
simultaneous confidence coefficient is exactly 1- , whether
the factor level sample sizes are equal or unequal.
Estimate and
variance for
C
As was described earlier, we estimate C by:
for which the estimated variance is:
Simultaneous
confidence
interval
It can be shown that the probability is 1 - that all
confidence limits of the type
are correct simultaneously.
Scheffe method example
Contrasts to
estimate
We wish to estimate, in our previous experiment, the
following contrasts
and construct 95 percent confidence intervals for them.
Compute the
point
estimates of
the
individual
contrasts
The point estimates are:
Compute the
point
estimate and
variance of
C
Applying the formulas above we obtain in both cases:
and
where = 1.331 was computed in our previous example.
The standard error = .5158 (square root of .2661).
Scheffe
confidence
interval
For a confidence coefficient of 95 percent and degrees of
freedom in the numerator of r - 1 = 4 - 1 = 3, and in the
denominator of 20 - 4 = 16, we have:
The confidence limits for C
1
are -.5 3.12(.5158) = -.5
1.608, and for C
2
they are .34 1.608.
The desired simultaneous 95 percent confidence intervals
are
-2.108 C
1
1.108
-1.268 C
2
1.948
Comparison
to confidence
interval for a
single
contrast
Recall that when we constructed a confidence interval for a
single contrast, we found the 95 percent confidence interval:
-1.594 C 0.594
As expected, the Scheff confidence interval procedure that
generates simultaneous intervals for all contrasts is
considerabley wider.
Comparison of Scheff's Method with Tukey's Method
Tukey
preferred
when only
pairwise
comparisons
are of
interest
If only pairwise comparisons are to be made, the Tukey
method will result in a narrower confidence limit, which is
preferable.
Consider for example the comparison between
3
and
1
.
Tukey: 1.13 <
3
-
1
< 5.31
Scheff: 0.95 <
3
-
1
< 5.49
which gives Tukey's method the edge.
The normalized contrast, using sums, for the Scheff method
is 4.413, which is close to the maximum contrast.
Scheffe
preferred
when many
contrasts are
of interest
In the general case when many or all contrasts might be of
interest, the Scheff method tends to give narrower
confidence limits and is therefore the preferred method.
7.4.7.3. Bonferroni's method

Simple
method
The Bonferroni method is a simple method that allows
many comparison statements to be made (or confidence
intervals to be constructed) while still assuring an overall
confidence coefficient is maintained.
Applies for a
finite number
of contrasts
This method applies to an ANOVA situation when the
analyst has picked out a particular set of pairwise
comparisons or contrasts or linear combinations in advance.
This set is not infinite, as in the Scheff case, but may
exceed the set of pairwise comparisons specified in the
Tukey procedure.
Valid for
both equal
and unequal
sample sizes
The Bonferroni method is valid for equal and unequal
sample sizes. We restrict ourselves to only linear
combinations or comparisons of treatment level means
(pairwise comparisons and contrasts are special cases of
linear combinations). We denote the number of statements
or comparisons in the finite set by g.
Bonferroni
general
inequality
Formally, the Bonferroni general inequality is presented by:
where A
i
and its complement are any events.
Interpretation
of Bonferroni
inequality
In particular, if each A
i
is the event that a calculated
confidence interval for a particular linear combination of
treatments includes the true value of that combination, then
the left-hand side of the inequality is the probability that all
the confidence intervals simultaneously cover their
respective true values. The right-hand side is one minus the
sum of the probabilities of each of the intervals missing
their true values. Therefore, if simultaneous multiple
interval estimates are desired with an overall confidence
coefficient 1- , one can construct each interval with
confidence coefficient (1- /g), and the Bonferroni
inequality insures that the overall confidence coefficient is
at least 1- .
Formula for
Bonferroni
confidence
interval
In summary, the Bonferroni method states that the
confidence coefficient is at least 1- that simultaneously all
the following confidence limits for the g linear
combinations C
i
are "correct" (or capture their respective
true values):
where
Example using Bonferroni method
Contrasts to
estimate
We wish to estimate, as we did using the Scheffe method,
the following linear combinations (contrasts):
and construct 95 % confidence intervals around the
estimates.
Compute the
point
estimates of
the individual
contrasts
The point estimates are:
Compute the
point
estimate and
variance of C
As before, for both contrasts, we have
and
where = 1.331 was computed in our previous example.
The standard error is .5158 (the square root of .2661).
Compute the
Bonferroni
simultaneous
confidence
interval
For a 95 % overall confidence coefficient using the
Bonferroni method, the t value is t
1-0.05/(2*2),16
= t
0.9875,16
= 2.473 (from the t table in Chapter 1). Now we can
calculate the confidence intervals for the two contrasts. For
C
1
we have confidence limits -0.5 2.473 (.5158) and for
C
2
we have confidence limits 0.34 2.473 (0.5158).
Thus, the confidence intervals are:
-1.776 C
1
0.776
-0.936 C
2
1.616
Comparison
to Scheffe
interval
Notice that the Scheff interval for C
1
is:
-2.108 C
1
1.108
which is wider and therefore less attractive.
Comparison of Bonferroni Method with Scheff and
Tukey Methods
No one
comparison
method is
uniformly
best - each
has its uses
1. If all pairwise comparisons are of interest, Tukey has
the edge. If only a subset of pairwise comparisons are
required, Bonferroni may sometimes be better.
2. When the number of contrasts to be estimated is
small, (about as many as there are factors) Bonferroni
is better than Scheff. Actually, unless the number of
desired contrasts is at least twice the number of
factors, Scheff will always show wider confidence
bands than Bonferroni.
3. Many computer packages include all three methods.
So, study the output and select the method with the
smallest confidence band.
4. No single method of multiple comparisons is
uniformly best among all the methods.
7.4.7.4. Comparing multiple proportions: The Marascuillo procedure

7.4.7.4. Comparing multiple proportions: The
Marascuillo procedure
Testing for
equal
proportions of
defects
Earlier, we discussed how to test whether several
populations have the same proportion of defects. The
example given there led to rejection of the null hypothesis
of equality.
Marascuilo
procedure
allows
comparison of
all possible
pairs of
proportions
Rejecting the null hypothesis only allows us to conclude
that not (in this case) all lots are equal with respect to the
proportion of defectives. However, it does not tell us which
lot or lots caused the rejection.
The Marascuilo procedure enables us to simultaneously test
the differences of all pairs of proportions when there are
several populations under investigation.
The Marascuillo Procedure
Step 1:
compute
differences p
i
- p
j
Assume we have samples of size n
i
(i = 1, 2, ..., k) from k
populations. The first step of this procedure is to compute
the differences p
i
- p
j
, (where i is not equal to j) among all
k(k-1)/2 pairs of proportions.
The absolute values of these differences are the test-
statistics.
Step 2:
compute test
statistics
Step 2 is to pick a significance level and compute the
corresponding critical values for the Marascuilo procedure
from
Step 3:
compare test
statistics
against
corresponding
critical values
The third and last step is to compare each of the k(k-1)/2
test statistics against its corresponding critical r
ij
value.
Those pairs that have a test statistic that exceeds the
critical value are significant at the level.
Example
Sample
proportions
To illustrate the Marascuillo procedure, we use the data
from the previous example. Since there were 5 lots, there
are (5 x 4)/2 = 10 possible pairwise comparisons to be
made and ten critical ranges to compute. The five sample
proportions are:
p
1
= 36/300 = .120
p
2
= 46/300 = .153
p
3
= 42/300 = .140
p
4
= 63/300 = .210
p
5
= 38/300 = .127
Table of
critical values
For an overall level of significance of 0.05, the critical
value of the chi-square distribution having four degrees of
freedom is
2
0.95,4
= 9.488 and the square root of 9.488 is
3.080. Calculating the 10 absolute differences and the 10
critical values leads to the following summary table.
contrast value critical range significant
|p
1
- p
2
| .033 0.086 no
|p
1
- p
3
| .020 0.085 no
|p
1
- p
4
| .090 0.093 no
|p
1
- p
5
| .007 0.083 no
|p
2
- p
3
| .013 0.089 no
|p
2
- p
4
| .057 0.097 no
|p
2
- p
5
| .026 0.087 no
|p
3
- p
4
| .070 0.095 no
|p
3
- p
5
| .013 0.086 no
|p
4
- p
5
| .083 0.094 no
The table of critical values can be generated using both
No individual
contrast is
statistically
significant
A difference is statistically significant if its value exceeds
the critical range value. In this example, even though the
null hypothesis of equality was rejected earlier, there is not
enough data to conclude any particular difference is
significant. Note, however, that all the comparisons
involving population 4 come the closest to significance -
leading us to suspect that more data might actually show
that population 4 does have a significantly higher
proportion of defects.
7.5. References

7.5. References
Primary
References
Agresti, A. and Coull, B. A. (1998). Approximate is better
than "exact" for interval estimation of binomial proportions",
The American Statistician, 52(2), 119-126.
Berenson M.L. and Levine D.M. (1996) Basic Business
Statistics, Prentice-Hall, Englewood Cliffs, New Jersey.
Bhattacharyya, G. K., and R. A. Johnson, (1997). Statistical
Concepts and Methods, John Wiley and Sons, New York.
Birnbaum, Z. W. (1952). "Numerical tabulation of the
distribution of Kolmogorov's statistic for finite sample size",
Journal of the American Statistical Association, 47, page 425.
Brown, L. D. Cai, T. T. and DasGupta, A. (2001). Interval
estimation for a binomial proportion", Statistical Science,
16(2), 101-133.
Diamond, W. J. (1989). Practical Experiment Designs, Van-
Nostrand Reinhold, New York.
Dixon, W. J. and Massey, F.J. (1969). Introduction to
Statistical Analysis, McGraw-Hill, New York.
Draper, N. and Smith, H., (1981). Applied Regression
Analysis, John Wiley & Sons, New York.
Fliess, J. L., Levin, B. and Paik, M. C. (2003). Statistical
Methods for Rates and Proportions, Third Edition, John Wiley
& Sons, New York.
Hahn, G. J. and Meeker, W. Q. (1991). Statistical Intervals: A
Guide for Practitioners, John Wiley & Sons, New York.
Hicks, C. R. (1973). Fundamental Concepts in the Design of
Experiments, Holt, Rinehart and Winston, New York.
Hollander, M. and Wolfe, D. A. (1973). Nonparametric
Statistical Methods, John Wiley & Sons, New York.
Howe, W. G. (1969). "Two-sided Tolerance Limits for Normal
Populations - Some Improvements", Journal of the Americal
Statistical Association, 64 , pages 610-620.
Kendall, M. and Stuart, A. (1979). The Advanced Theory of
7.5. References
Statistics, Volume 2: Inference and Relationship. Charles
Griffin & Co. Limited, London.
Mendenhall, W., Reinmuth, J. E. and Beaver, R. J. Statistics
for Management and Economics, Duxbury Press, Belmont,
CA.
Montgomery, D. C. (1991). Design and Analysis of
Experiments, John Wiley & Sons, New York.
Moore, D. S. (1986). "Tests of Chi-Square Type". From
Goodness-of-Fit Techniques (D'Agostino & Stephens eds.).
Myers, R. H., (1990). Classical and Modern Regression with
Applications, PWS-Kent, Boston, MA.
Neter, J., Wasserman, W. and Kutner, M. H. (1990). Applied
Linear Statistical Models, 3rd Edition, Irwin, Boston, MA.
Lawless, J. F., (1982). Statistical Models and Methods for
Lifetime Data, John Wiley & Sons, New York.
Pearson, A. V., and Hartley, H. O. (1972). Biometrica Tables
for Statisticians, Vol 2, Cambridge, England, Cambridge
University Press.
Sarhan, A. E. and Greenberg, B. G. (1956). "Estimation of
location and scale parameters by order statistics from singly
and double censored samples," Part I, Annals of Mathematical
Statistics, 27, 427-451.
Searle, S. S., Casella, G. and McCulloch, C. E. (1992).
Variance Components, John Wiley & Sons, New York.
Siegel, S. (1956). Nonparametric Statistics, McGraw-Hill,
New York.
Shapiro, S. S. and Wilk, M. B. (1965). "An analysis of
variance test for normality (complete samples)", Biometrika,
52, 3 and 4, pages 591-611.
Some Additional References and Bibliography
Books D'Agostino, R. B. and Stephens, M. A. (1986). Goodness-of-
FitTechniques, Marcel Dekker, Inc., New York.
Hicks, C. R. 1973. Fundamental Concepts in the Design of
Experiments. Holt, Rhinehart and Winston,New-York
Miller, R. G., Jr. (1981). Simultaneous Statistical Inference,
Springer-Verlag, New York.
Neter, Wasserman, and Whitmore (1993). Applied Statistics,
4th Edition, Allyn and Bacon, Boston, MA.
7.5. References
Neter, J., Wasserman, W. and Kutner, M. H. (1990). Applied
Linear Statistical Models, 3rd Edition, Irwin, Boston, MA.
Scheffe, H. (1959). The Analysis of Variance, John Wiley,
New-York.
Articles Begun, J. M. and Gabriel, K. R. (1981). "Closure of the
Newman-Keuls Multiple Comparisons Procedure", Journal of
the American Statistical Association, 76, page 374.
Carmer, S. G. and Swanson, M. R. (1973. "Evaluation of Ten
Pairwise Multiple Comparison Procedures by Monte-Carlo
Methods", Journal of the American Statistical Association, 68,
pages 66-74.
Duncan, D. B. (1975). "t-Tests and Intervals for Comparisons
suggested by the Data" Biometrics, 31, pages 339-359.
Dunnett, C. W. (1980). "Pairwise Multiple Comparisons in the
Homogeneous Variance for Unequal Sample Size Case",
Journal of the American Statistical Association, 75, page 789.
Einot, I. and Gabriel, K. R. (1975). "A Study of the Powers of
Several Methods of Multiple Comparison", Journal of the
American Statistical Association, 70, page 351.
Gabriel, K. R. (1978). "A Simple Method of Multiple
Comparisons of Means", Journal of the American Statistical
Association, 73, page 364.
Hochburg, Y. (1974). "Some Conservative Generalizations of
the T-Method in Simultaneous Inference", Journal of
Multivariate Analysis, 4, pages 224-234.
Kramer, C. Y. (1956). "Extension of Multiple Range Tests to
Group Means with Unequal Sample Sizes", Biometrics, 12,
pages 307-310.
Marcus, R., Peritz, E. and Gabriel, K. R. (1976). "On Closed
Testing Procedures with Special Reference to Ordered
Analysis of Variance", Biometrics, 63, pages 655-660.
Ryan, T. A. (1959). "Multiple Comparisons in Psychological
Research", Psychological Bulletin, 56, pages 26-47.
Ryan, T. A. (1960). "Significance Tests for Multiple
Comparisons of Proportions, Variances, and Other Statistics",
Psychological Bulletin, 57, pages 318-328.
Scheffe, H. (1953). "A Method for Judging All Contrasts in the
Analysis of Variance", Biometrika,40, pages 87-104.
Sidak, Z., (1967). "Rectangular Confidence Regions for the
Means of Multivariate Normal Distributions", Journal of the
American Statistical Association, 62, pages 626-633.
7.5. References
Tukey, J. W. (1953). The Problem of Multiple Comparisons,
Unpublished Manuscript.
Waller, R. A. and Duncan, D. B. (1969). "A Bayes Rule for the
Symmetric Multiple Comparison Problem", Journal of the
American Statistical Association 64, pages 1484-1504.
Waller, R. A. and Kemp, K. E. (1976). "Computations of
Bayesian t-Values for Multiple Comparisons", Journal of
Statistical Computation and Simulation, 75, pages 169-172.
Welsch, R. E. (1977). "Stepwise Multiple Comparison
Procedure", Journal of the American Statistical Association,
72, page 359.
8. Assessing Product Reliability
http://www.itl.nist.gov/div898/handbook/apr/apr.htm[6/27/2012 2:50:13 PM]

This chapter describes the terms, models and techniques used to evaluate
and predict product reliability.
1. Introduction
1. Why important?
2. Basic terms and models
3. Common difficulties
4. Modeling "physical
acceleration"
5. Common acceleration models
6. Basic non-repairable lifetime
distributions
7. Basic models for repairable
systems
8. Evaluate reliability "bottom-
up"
9. Modeling reliability growth
10. Bayesian methodology
2. Assumptions/Prerequisites
1. Choosing appropriate life
distribution
2. Plotting reliability data
3. Testing assumptions
4. Choosing a physical
acceleration model
5. Models and assumptions for
Bayesian methods
3. Reliability Data Collection
1. Planning reliability assessment
tests
4. Reliability Data Analysis
1. Estimating parameters from
censored data
2. Fitting an acceleration model
3. Projecting reliability at use
conditions
4. Comparing reliability between
two or more populations
5. Fitting system repair rate
models
6. Estimating reliability using a
Bayesian gamma prior
Click here for a detailed table of contents
http://www.itl.nist.gov/div898/handbook/apr/apr_d.htm[6/27/2012 2:48:49 PM]

8. Assessing Product Reliability - Detailed Table of Contents [8.]
1. Why is the assessment and control of product reliability important? [8.1.1.]
1. Quality versus reliability [8.1.1.1.]
2. Competitive driving factors [8.1.1.2.]
3. Safety and health considerations [8.1.1.3.]
2. What are the basic terms and models used for reliability evaluation? [8.1.2.]
1. Repairable systems, non-repairable populations and lifetime distribution models [8.1.2.1.]
2. Reliability or survival function [8.1.2.2.]
3. Failure (or hazard) rate [8.1.2.3.]
4. "Bathtub" curve [8.1.2.4.]
5. Repair rate or ROCOF [8.1.2.5.]
3. What are some common difficulties with reliability data and how are they overcome? [8.1.3.]
1. Censoring [8.1.3.1.]
2. Lack of failures [8.1.3.2.]
4. What is "physical acceleration" and how do we model it? [8.1.4.]
5. What are some common acceleration models? [8.1.5.]
1. Arrhenius [8.1.5.1.]
2. Eyring [8.1.5.2.]
3. Other models [8.1.5.3.]
6. What are the basic lifetime distribution models used for non-repairable populations? [8.1.6.]
1. Exponential [8.1.6.1.]
2. Weibull [8.1.6.2.]
3. Extreme value distributions [8.1.6.3.]
4. Lognormal [8.1.6.4.]
5. Gamma [8.1.6.5.]
6. Fatigue life (Birnbaum-Saunders) [8.1.6.6.]
7. Proportional hazards model [8.1.6.7.]
7. What are some basic repair rate models used for repairable systems? [8.1.7.]
1. Homogeneous Poisson Process (HPP) [8.1.7.1.]
2. Non-Homogeneous Poisson Process (NHPP) - power law [8.1.7.2.]
3. Exponential law [8.1.7.3.]
8. How can you evaluate reliability from the "bottom-up" (component failure mode to system failure
rate)? [8.1.8.]
1. Competing risk model [8.1.8.1.]
2. Series model [8.1.8.2.]
3. Parallel or redundant model [8.1.8.3.]
4. R out of N model [8.1.8.4.]
5. Standby model [8.1.8.5.]
6. Complex systems [8.1.8.6.]
9. How can you model reliability growth? [8.1.9.]
1. NHPP power law [8.1.9.1.]
2. Duane plots [8.1.9.2.]
http://www.itl.nist.gov/div898/handbook/apr/apr_d.htm[6/27/2012 2:48:49 PM]
3. NHPP exponential law [8.1.9.3.]
10. How can Bayesian methodology be used for reliability evaluation? [8.1.10.]
2. Assumptions/Prerequisites [8.2.]
1. How do you choose an appropriate life distribution model? [8.2.1.]
1. Based on failure mode [8.2.1.1.]
2. Extreme value argument [8.2.1.2.]
3. Multiplicative degradation argument [8.2.1.3.]
4. Fatigue life (Birnbaum-Saunders) model [8.2.1.4.]
5. Empirical model fitting - distribution free (Kaplan-Meier) approach [8.2.1.5.]
2. How do you plot reliability data? [8.2.2.]
1. Probability plotting [8.2.2.1.]
2. Hazard and cum hazard plotting [8.2.2.2.]
3. Trend and growth plotting (Duane plots) [8.2.2.3.]
3. How can you test reliability model assumptions? [8.2.3.]
1. Visual tests [8.2.3.1.]
2. Goodness of fit tests [8.2.3.2.]
3. Likelihood ratio tests [8.2.3.3.]
4. Trend tests [8.2.3.4.]
4. How do you choose an appropriate physical acceleration model? [8.2.4.]
5. What models and assumptions are typically made when Bayesian methods are used for reliability
evaluation? [8.2.5.]
3. Reliability Data Collection [8.3.]
1. How do you plan a reliability assessment test? [8.3.1.]
1. Exponential life distribution (or HPP model) tests [8.3.1.1.]
2. Lognormal or Weibull tests [8.3.1.2.]
3. Reliability growth (Duane model) [8.3.1.3.]
4. Accelerated life tests [8.3.1.4.]
5. Bayesian gamma prior model [8.3.1.5.]
4. Reliability Data Analysis [8.4.]
1. How do you estimate life distribution parameters from censored data? [8.4.1.]
1. Graphical estimation [8.4.1.1.]
2. Maximum likelihood estimation [8.4.1.2.]
3. A Weibull maximum likelihood estimation example [8.4.1.3.]
2. How do you fit an acceleration model? [8.4.2.]
1. Graphical estimation [8.4.2.1.]
2. Maximum likelihood [8.4.2.2.]
3. Fitting models using degradation data instead of failures [8.4.2.3.]
3. How do you project reliability at use conditions? [8.4.3.]
4. How do you compare reliability between two or more populations? [8.4.4.]
5. How do you fit system repair rate models? [8.4.5.]
1. Constant repair rate (HPP/exponential) model [8.4.5.1.]
2. Power law (Duane) model [8.4.5.2.]
3. Exponential law model [8.4.5.3.]
6. How do you estimate reliability using the Bayesian gamma prior model? [8.4.6.]
7. References For Chapter 8: Assessing Product Reliability [8.4.7.]
8.1. Introduction
http://www.itl.nist.gov/div898/handbook/apr/section1/apr1.htm[6/27/2012 2:48:55 PM]

8.1. Introduction
This section introduces the terminology and models that will
be used to describe and quantify product reliability. The
terminology, probability distributions and models used for
reliability analysis differ in many cases from those used in
other statistical applications.
Detailed
contents of
Section 1
1. Introduction
1. Why is the assessment and control of product
reliability important?
1. Quality versus reliability
2. Competitive driving factors
3. Safety and health considerations
2. What are the basic terms and models used for
reliability evaluation?
1. Repairable systems, non-repairable
populations and lifetime distribution
models
2. Reliability or survival function
3. Failure (or hazard) rate
4. "Bathtub" curve
5. Repair rate or ROCOF
3. What are some common difficulties with
reliability data and how are they overcome?
1. Censoring
2. Lack of failures
4. What is "physical acceleration" and how do we
model it?
5. What are some common acceleration models?
1. Arrhenius
2. Eyring
3. Other models
6. What are the basic lifetime distribution models
used for non-repairable populations?
1. Exponential
2. Weibull
3. Extreme value distributions
4. Lognormal
5. Gamma
6. Fatigue life (Birnbaum-Saunders)
7. Proportional hazards model
7. What are some basic repair rate models used for
repairable systems?
1. Homogeneous Poisson Process (HPP)
8.1. Introduction
2. Non-Homogeneous Poisson Process
(NHPP) with power law
3. Exponential law
8. How can you evaluate reliability from the
"bottom- up" (component failure mode to system
failure rates)?
1. Competing risk model
2. Series model
3. Parallel or redundant model
4. R out of N model
5. Standby model
6. Complex systems
9. How can you model reliability growth?
1. NHPP power law
2. Duane plots
3. NHPP exponential law
10. How can Bayesian methodology be used for
8.1.1. Why is the assessment and control of product reliability important?

8.1. Introduction
8.1.1. Why is the assessment and control of
product reliability important?
We depend
on,
demand,
and expect
reliable
products
In today's technological world nearly everyone depends upon
the continued functioning of a wide array of complex
machinery and equipment for their everyday health, safety,
mobility and economic welfare. We expect our cars,
computers, electrical appliances, lights, televisions, etc. to
function whenever we need them - day after day, year after
year. When they fail the results can be catastrophic: injury,
loss of life and/or costly lawsuits can occur. More often,
repeated failure leads to annoyance, inconvenience and a
lasting customer dissatisfaction that can play havoc with the
responsible company's marketplace position.
Shipping
unreliable
products
can
destroy a
company's
reputation
It takes a long time for a company to build up a reputation for
reliability, and only a short time to be branded as "unreliable"
after shipping a flawed product. Continual assessment of new
product reliability and ongoing control of the reliability of
everything shipped are critical necessities in today's
competitive business arena.
8.1.1.1. Quality versus reliability

8.1. Introduction
8.1.1.1. Quality versus reliability
Reliability
is "quality
changing
over time"
The everyday usage term "quality of a product" is loosely
taken to mean its inherent degree of excellence. In industry,
this is made more precise by defining quality to be
"conformance to requirements at the start of use". Assuming
the product specifications adequately capture customer
requirements, the quality level can now be precisely
measured by the fraction of units shipped that meet
specifications.
A motion
picture
instead of a
snapshot
But how many of these units still meet specifications after a
week of operation? Or after a month, or at the end of a one
year warranty period? That is where "reliability" comes in.
Quality is a snapshot at the start of life and reliability is a
motion picture of the day-by-day operation. Time zero
defects are manufacturing mistakes that escaped final test.
The additional defects that appear over time are "reliability
defects" or reliability fallout.
Life
distributions
model
fraction
fallout over
time
The quality level might be described by a single fraction
defective. To describe reliability fallout a probability model
that describes the fraction fallout over time is needed. This is
known as the life distribution model.
8.1.1.2. Competitive driving factors

8.1. Introduction
8.1.1.2. Competitive driving factors
Reliability
is a major
economic
factor in
determining
a product's
success
Accurate prediction and control of reliability plays an
important role in the profitability of a product. Service costs
for products within the warranty period or under a service
contract are a major expense and a significant pricing factor.
Proper spare part stocking and support personnel hiring and
training also depend upon good reliability fallout predictions.
On the other hand, missing reliability targets may invoke
contractual penalties and cost future business.
Companies that can economically design and market products
that meet their customers' reliability expectations have a
strong competitive advantage in today's marketplace.
8.1.1.3. Safety and health considerations

8.1. Introduction
8.1.1.3. Safety and health considerations
Some failures
have serious
social
consequences
and this
should be
taken into
account
when
planning
reliability
studies
Sometimes equipment failure can have a major impact on
human safety and/or health. Automobiles, planes, life
support equipment, and power generating plants are a few
examples.
From the point of view of "assessing product reliability", we
treat these kinds of catastrophic failures no differently from
the failure that occurs when a key parameter measured on a
manufacturing tool drifts slightly out of specification,
calling for an unscheduled maintenance action.
It is up to the reliability engineer (and the relevant
customer) to define what constitutes a failure in any
reliability study. More resource (test time and test units)
should be planned for when an incorrect reliability
assessment could negatively impact safety and/or health.
8.1.2. What are the basic terms and models used for reliability evaluation?

8.1. Introduction
8.1.2. What are the basic terms and models used
for reliability evaluation?
Reliability
methods
and
terminology
began with
19th
century
insurance
companies
Reliability theory developed apart from the mainstream of
probability and statistics, and was used primarily as a tool to
help nineteenth century maritime and life insurance
companies compute profitable rates to charge their customers.
Even today, the terms "failure rate" and "hazard rate" are
often used interchangeably.
The following sections will define some of the concepts,
terms, and models we need to describe, estimate and predict
reliability.
8.1.2.1. Repairable systems, non-repairable populations and lifetime distribution models

8.1. Introduction
8.1.2.1. Repairable systems, non-repairable populations
and lifetime distribution models
Life
distribution
models
describe
how non-
repairable
populations
fail over
time
A repairable system is one which can be restored to satisfactory operation by
any action, including parts replacements or changes to adjustable settings.
When discussing the rate at which failures occur during system operation
time (and are then repaired) we will define a Rate Of Occurrence Of Failure
(ROCF) or "repair rate". It would be incorrect to talk about failure rates or
hazard rates for repairable systems, as these terms apply only to the first
failure times for a population of non repairable components.
A non-repairable population is one for which individual items that fail are
removed permanently from the population. While the system may be
repaired by replacing failed units from either a similar or a different
population, the members of the original population dwindle over time until
all have eventually failed.
We begin with models and definitions for non-repairable populations. Repair
rates for repairable populations will be defined in a later section.
The theoretical population models used to describe unit lifetimes are known
as Lifetime Distribution Models. The population is generally considered to
be all of the possible unit lifetimes for all of the units that could be
manufactured based on a particular design and choice of materials and
manufacturing process. A random sample of size n from this population is
the collection of failure times observed for a randomly selected group of n
units.
Any
continuous
PDF
defined
only for
non-
negative
values can
be a
lifetime
distribution
model
A lifetime distribution model can be any probability density function (or
PDF) f(t) defined over the range of time from t = 0 to t = infinity. The
corresponding cumulative distribution function (or CDF) F(t) is a very
useful function, as it gives the probability that a randomly selected unit will
fail by time t. The figure below shows the relationship between f(t) and F(t)
and gives three descriptions of F(t).
1. F(t) = the area under the PDF f(t) to the left of
t.
2. F(t) = the probability that a single randomly
chosen new unit will fail by time t.
3. F(t) = the proportion of the entire population
that fails by time t.
The figure above also shows a shaded area under f(t) between the two times
t
1
and t
2
. This area is [F(t
2
) - F(t
1
)] and represents the proportion of the
population that fails between times t
1
and t
2
(or the probability that a brand
new randomly chosen unit will survive to time t
1
but fail before time t
2
).
Note that the PDF f(t) has only non-negative values and eventually either
becomes 0 as t increases, or decreases towards 0. The CDF F(t) is
monotonically increasing and goes from 0 to 1 as t approaches infinity. In
other words, the total area under the curve is always 1.
The
Weibull
model is a
good
example of
a life
distribution
The 2-parameter Weibull distribution is an example of a popular F(t). It has
the CDF and PDF equations given by:
where is the "shape" parameter and is a scale parameter called the
characteristic life.
Example: A company produces automotive fuel pumps that fail according to
a Weibull life distribution model with shape parameter = 1.5 and scale
parameter 8,000 (time measured in use hours). If a typical pump is used 800
hours a year, what proportion are likely to fail within 5 years?
Solution: The probability associated with the 800*5 quantile of a Weibull
distribution with = 1.5 and = 8000 is 0.298. Thus about 30% of the
pumps will fail in the first 5 years.
Functions for computing PDF values and CDF values, are available in both
8.1.2.2. Reliability or survival function

8.1. Introduction
8.1.2.2. Reliability or survival function
Survival is the
complementary
event to failure
The Reliability FunctionR(t), also known as the Survival
Function S(t), is defined by:
R(t) = S(t) = the probability a unit survives beyond time t.
Since a unit either fails, or survives, and one of these two
mutually exclusive alternatives must occur, we have
R(t) = 1 - F(t), F(t) = 1 - R(t)
Calculations using R(t) often occur when building up from
single components to subsystems with many components.
For example, if one microprocessor comes from a
population with reliability function R
m
(t) and two of them
are used for the CPU in a system, then the system CPU
has a reliability function given by
R
cpu
(t) = R
m
2
(t)
The reliability
of the system is
the product of
the reliability
functions of
the
components
since both must survive in order for the system to survive.
This building up to the system from the individual
components will be discussed in detail when we look at
the "Bottom-Up" method. The general rule is: to calculate
the reliability of a system of independent components,
multiply the reliability functions of all the components
together.
8.1.2.3. Failure (or hazard) rate

8.1. Introduction
The
failure
rate is the
rate at
which the
population
survivors
at any
given
instant are
"falling
over the
cliff"
The failure rate is defined for non repairable populations as the
(instantaneous) rate of failure for the survivors to time t during
the next instant of time. It is a rate per unit of time similar in
meaning to reading a car speedometer at a particular instant and
seeing 45 mph. The next instant the failure rate may change and
the units that have already failed play no further role since only
the survivors count.
The failure rate (or hazard rate) is denoted by h(t) and calculated
from
The failure rate is sometimes called a "conditional failure rate"
since the denominator 1 - F(t) (i.e., the population survivors)
converts the expression into a conditional rate, given survival
past time t.
Since h(t) is also equal to the negative of the derivative of
ln{R(t)}, we have the useful identity:
If we let
be the Cumulative Hazard Function, we then have F(t) = 1 - e
-
H(t)
. Two other useful identities that follow from these formulas
are:
It is also sometimes useful to define an average failure rate over
any interval (T
1
, T
2
) that "averages" the failure rate over that
interval. This rate, denoted by AFR(T
1
,T
2
), is a single number
that can be used as a specification or target for the population
failure rate over that interval. If T
1
is 0, it is dropped from the
expression. Thus, for example, AFR(40,000) would be the
average failure rate for the population over the first 40,000 hours
of operation.
The formulas for calculating AFR's are:
8.1.2.4. "Bathtub" curve

8.1. Introduction
A plot of
the
failure
rate
over
time for
most
products
yields a
curve
that
looks
like a
drawing
of a
bathtub
If enough units from a given population are observed operating and failing over
time, it is relatively easy to compute week-by-week (or month-by-month)
estimates of the failure rate h(t). For example, if N
12
units survive to start the
13th month of life and r
13
of them fail during the next month (or 720 hours) of
life, then a simple empirical estimate of h(t) averaged across the 13th month of
life (or between 8640 hours and 9360 hours of age), is given by (r
13
/ N
12
*
720). Similar estimates are discussed in detail in the section on Empirical Model
Fitting.
Over many years, and across a wide variety of mechanical and electronic
components and systems, people have calculated empirical population failure
rates as units age over time and repeatedly obtained a graph such as shown
below. Because of the shape of this failure rate curve, it has become widely
known as the "Bathtub" curve.
The initial region that begins at time zero when a customer first begins to use the
product is characterized by a high but rapidly decreasing failure rate. This region
is known as the Early Failure Period (also referred to as Infant Mortality
Period, from the actuarial origins of the first bathtub curve plots). This
decreasing failure rate typically lasts several weeks to a few months.
Next, the failure rate levels off and remains roughly constant for (hopefully) the
majority of the useful life of the product. This long period of a level failure rate
is known as the Intrinsic Failure Period (also called the Stable Failure
Period) and the constant failure rate level is called the Intrinsic Failure Rate.
Note that most systems spend most of their lifetimes operating in this flat
portion of the bathtub curve
Finally, if units from the population remain in use long enough, the failure rate
begins to increase as materials wear out and degradation failures occur at an ever
increasing rate. This is the Wearout Failure Period.
NOTE: The Bathtub Curve also applies (based on much empirical evidence) to
Repairable Systems. In this case, the vertical axis is the Repair Rate or the Rate
of Occurrence of Failures (ROCOF).
8.1.2.5. Repair rate or ROCOF

8.1. Introduction
8.1.2.5. Repair rate or ROCOF
Repair
Rate
models are
based on
counting
the
cumulative
number of
failures
over time
A different approach is used for modeling the rate of
occurrence of failure incidences for a repairable system. In
this chapter, these rates are called repair rates (not to be
confused with the length of time for a repair, which is not
discussed in this chapter). Time is measured by system power-
on-hours from initial turn-on at time zero, to the end of
system life. Failures occur as given system ages and the
system is repaired to a state that may be the same as new, or
better, or worse. The frequency of repairs may be increasing,
decreasing, or staying at a roughly constant rate.
Let N(t) be a counting function that keeps track of the
cumulative number of failures a given system has had from
time zero to time t. N(t) is a step function that jumps up one
every time a failure occurs and stays at the new level until the
next failure.
Every system will have its own observed N(t) function over
time. If we observed the N(t) curves for a large number of
similar systems and "averaged" these curves, we would have
an estimate of M(t) = the expected number (average number)
of cumulative failures by time t for these systems.
The Repair
Rate (or
ROCOF)
is the
mean rate
of failures
per unit
time
The derivative of M(t), denoted m(t), is defined to be the
Repair Rate or the Rate Of Occurrence Of Failures at Time
t or ROCOF.
Models for N(t), M(t) and m(t) will be described in the section
on Repair Rate Models.
8.1.3. What are some common difficulties with reliability data and how are they overcome?

8.1. Introduction
8.1.3. What are some common difficulties with
reliability data and how are they
overcome?
The
Paradox
of
Reliability
Analysis:
The more
reliable a
product is,
the harder
it is to get
the failure
data
needed to
"prove" it
is reliable!
There are two closely related problems that are typical with
reliability data and not common with most other forms of
statistical data. These are:
Censoring (when the observation period ends, not all
units have failed - some are survivors)
Lack of Failures (if there is too much censoring, even
though a large number of units may be under
observation, the information in the data is limited due to
the lack of actual failures)
These problems cause considerable practical difficulty when
planning reliability assessment tests and analyzing failure data.
Some solutions are discussed in the next two sections.
Typically, the solutions involve making additional assumptions
and using complicated models.
8.1.3.1. Censoring

8.1. Introduction
8.1.3.1. Censoring
When not
all units
on test fail
we have
censored
data
Consider a situation in which we are reliability testing n (non repairable) units
taken randomly from a population. We are investigating the population to
determine if its failure rate is acceptable. In the typical test scenario, we have a
fixed time T to run the units to see if they survive or fail. The data obtained are
called Censored Type I data.
Censored Type I Data
During the T hours of test we observe r failures (where r can be any number
from 0 to n). The (exact) failure times are t
1
, t
2
, ..., t
r
and there are (n - r) units
that survived the entire T-hour test without failing. Note that T is fixed in
advance and r is random, since we don't know how many failures will occur until
the test is run. Note also that we assume the exact times of failure are recorded
when there are failures.
This type of censoring is also called "right censored" data since the times of
failure to the right (i.e., larger than T) are missing.
Another (much less common) way to test is to decide in advance that you want
to see exactly r failure times and then test until they occur. For example, you
might put 100 units on test and decide you want to see at least half of them fail.
Then r = 50, but T is unknown until the 50th fail occurs. This is called Censored
Type II data.
Censored Type II Data
We observe t
1
, t
2
, ..., t
r
, where r is specified in advance. The test ends at time T
= t
r
, and (n-r) units have survived. Again we assume it is possible to observe
the exact time of failure for failed units.
Type II censoring has the significant advantage that you know in advance how
many failure times your test will yield - this helps enormously when planning
adequate tests. However, an open-ended random test time is generally
impractical from a management point of view and this type of testing is rarely
seen.
Sometimes
we don't
even know
the exact
time of
failure
Readout or Interval Data
Sometimes exact times of failure are not known; only an interval of time in
which the failure occurred is recorded. This kind of data is called Readout or
Interval data and the situation is shown in the figure below:
8.1.3.1. Censoring
.
Multicensored Data
In the most general case, every unit observed yields exactly one of the following
three types of information:
a run-time if the unit did not fail while under observation
an exact failure time
an interval of time during which the unit failed.
The units may all have different run-times and/or readout intervals.
Many
special
methods
have been
developed
to handle
censored
data
How do we handle censored data?
Many statistical methods can be used to fit models and estimate failure rates,
even with censored data. In later sections we will discuss the Kaplan-Meier
approach, Probability Plotting, Hazard Plotting, Graphical Estimation, and
Maximum Likelihood Estimation.
Separating out Failure Modes
Note that when a data set consists of failure times that can be sorted into several
different failure modes, it is possible (and often necessary) to analyze and model
each mode separately. Consider all failures due to modes other than the one
being analyzed as censoring times, with the censored run-time equal to the time
it failed due to the different (independent) failure mode. This is discussed further
in the competing risk section and later analysis sections.
8.1.3.2. Lack of failures

8.1. Introduction
Failure
data is
needed to
accurately
assess and
improve
reliability
- this
poses
problems
when
testing
highly
reliable
parts
When fitting models and estimating failure rates from
reliability data, the precision of the estimates (as measured by
the width of the confidence intervals) tends to vary inversely
with the square root of the number of failures observed - not
the number of units on test or the length of the test. In other
words, a test where 5 fail out of a total of 10 on test gives
more information than a test with 1000 units but only 2
failures.
Since the number of failures r is critical, and not the sample
size n on test, it becomes increasingly difficult to assess the
failure rates of highly reliable components. Parts like memory
chips, that in typical use have failure rates measured in parts
per million per thousand hours, will have few or no failures
when tested for reasonable time periods with affordable
sample sizes. This gives little or no information for
accomplishing the two primary purposes of reliability testing,
namely:
accurately assessing population failure rates
obtaining failure mode information to feedback for
product improvement.
Testing at
much
higher
than
typical
stresses
can yield
failures
but models
are then
needed to
relate
these back
to use
stress
How can tests be designed to overcome an expected lack of
failures?
The answer is to make failures occur by testing at much higher
stresses than the units would normally see in their intended
application. This creates a new problem: how can these
failures at higher-than-normal stresses be related to what
would be expected to happen over the course of many years at
normal use stresses? The models that relate high stress
reliability to normal use reliability are called acceleration
models.
8.1.4. What is "physical acceleration" and how do we model it?

8.1. Introduction
8.1.4. What is "physical acceleration" and how
do we model it?
When
changing
stress is
equivalent to
multiplying
time to fail
by a
constant, we
have true
(physical)
acceleration
Physical Acceleration (sometimes called True
Acceleration or just Acceleration) means that operating a
unit at high stress (i.e., higher temperature or voltage or
humidity or duty cycle, etc.) produces the same failures that
would occur at typical-use stresses, except that they happen
much quicker.
Failure may be due to mechanical fatigue, corrosion,
chemical reaction, diffusion, migration, etc. These are the
same causes of failure under normal stress; the time scale is
simply different.
An
Acceleration
Factor is the
constant
multiplier
between the
two stress
levels
When there is true acceleration, changing stress is equivalent
to transforming the time scale used to record when failures
occur. The transformations commonly used are linear,
which means that time-to-fail at high stress just has to be
multiplied by a constant (the acceleration factor) to obtain
the equivalent time-to-fail at use stress.
We use the following notation:
t
s
= time-to-fail at stress
t
u
= corresponding time-to-fail at
use
F
s
(t) = CDF at stress F
u
(t) = CDF at use
f
s
(t) = PDF at stress f
u
(t) = PDF at use
h
s
(t) = failure rate at
stress
h
u
(t) = failure rate at use
Then, an acceleration factor AF between stress and use
means the following relationships hold:
Linear Acceleration Relationships
Time-to-Fail
t
u
= AF t
s
Failure Probability
F
u
(t) = F
s
(t/AF)
Reliability
R
u
(t) = R
s
(t/AF)
PDF or Density Function
f
u
(t) = (1/AF)f
s
(t/AF)
Failure Rate
h
u
(t) = (1/AF) h
s
(t/AF)
8.1.4. What is "physical acceleration" and how do we model it?
Each failure
mode has its
own
acceleration
factor
Failure data
should be
separated by
failure mode
when
analyzed, if
acceleration
is relevant
Probability
plots of data
from
different
stress cells
have the
same slope
(if there is
acceleration)
Note: Acceleration requires that there be a stress dependent
physical process causing change or degradation that leads to
failure. In general, different failure modes will be affected
differently by stress and have different acceleration factors.
Therefore, it is unlikely that a single acceleration factor will
apply to more than one failure mechanism. In general,
different failure modes will be affected differently by stress
and have different acceleration factors. Separate out
different types of failure when analyzing failure data.
Also, a consequence of the linear acceleration relationships
shown above (which follows directly from "true
acceleration") is the following:
The Shape Parameter for the key life
distribution models (Weibull, Lognormal) does
not change for units operating under different
stresses. Probability plots of data from different
stress cells will line up roughly parallel.
These distributions and probability plotting will be
discussed in later sections.
8.1.5. What are some common acceleration models?

8.1. Introduction
8.1.5. What are some common acceleration
models?
Acceleration
models
predict time
to fail as a
function of
stress
Acceleration factors show how time-to-fail at a particular
operating stress level (for one failure mode or mechanism)
can be used to predict the equivalent time to fail at a
different operating stress level.
A model that predicts time-to-fail as a function of stress
would be even better than a collection of acceleration
factors. If we write t
f
= G(S), with G(S) denoting the model
equation for an arbitrary stress level S, then the acceleration
factor between two stress levels S
1
and S
2
can be evaluated
simply by AF = G(S
1
)/G(S
2
). Now we can test at the higher
stress S
2
, obtain a sufficient number of failures to fit life
distribution models and evaluate failure rates, and use the
Linear Acceleration Relationships Table to predict what will
occur at the lower use stress S
1
.
A model that predicts time-to-fail as a function of operating
stresses is known as an acceleration model.
Acceleration
models are
often
derived
from
physics or
kinetics
models
related to
the failure
mechanism
Acceleration models are usually based on the physics or
chemistry underlying a particular failure mechanism.
Successful empirical models often turn out to be
approximations of complicated physics or kinetics models,
when the theory of the failure mechanism is better
understood. The following sections will consider a variety of
powerful and useful models:
Arrhenius
Eyring
Other Models
8.1.5.1. Arrhenius

8.1. Introduction
8.1.5.1. Arrhenius
The
Arrhenius
model
predicts
failure
acceleration
due to
temperature
increase
One of the earliest and most successful acceleration models
predicts how time-to-fail varies with temperature. This
empirically based model is known as the Arrhenius equation.
It takes the form
with T denoting temperature measured in degrees Kelvin
(273.16 + degrees Celsius) at the point when the failure
process takes place and k is Boltzmann's constant (8.617 x
10
-5
in ev/K). The constant A is a scaling factor that drops
out when calculating acceleration factors, with H
(pronounced "Delta H") denoting the activation energy,
which is the critical parameter in the model.
The
Arrhenius
activation
energy,
H, is all you
need to
know to
calculate
temperature
acceleration
The value of H depends on the failure mechanism and the
materials involved, and typically ranges from .3 or .4 up to
1.5, or even higher. Acceleration factors between two
temperatures increase exponentially as H increases.
The acceleration factor between a higher temperature T
2
and
a lower temperature T
1
is given by
Using the value of k given above, this can be written in
terms of T in degrees Celsius as
Note that the only unknown parameter in this formula is
H.
Example: The acceleration factor between 25C and 125C
is 133 if H = .5 and 17,597 if H = 1.0.
The Arrhenius model has been used successfully for failure
8.1.5.1. Arrhenius
mechanisms that depend on chemical reactions, diffusion
processes or migration processes. This covers many of the
non mechanical (or non material fatigue) failure modes that
cause electronic equipment failure.
8.1.5.2. Eyring

8.1. Introduction
8.1.5.2. Eyring
The Eyring
model has a
theoretical
basis in
chemistry
and quantum
mechanics
and can be
used to
model
acceleration
when many
stresses are
involved
Henry Eyring's contributions to chemical reaction rate theory
have led to a very general and powerful model for
acceleration known as the Eyring Model. This model has
several key features:
It has a theoretical basis from chemistry and quantum
mechanics.
If a chemical process (chemical reaction, diffusion,
corrosion, migration, etc.) is causing degradation
leading to failure, the Eyring model describes how the
rate of degradation varies with stress or, equivalently,
how time to failure varies with stress.
The model includes temperature and can be expanded
to include other relevant stresses.
The temperature term by itself is very similar to the
Arrhenius empirical model, explaining why that model
has been so successful in establishing the connection
between the H parameter and the quantum theory
concept of "activation energy needed to cross an
energy barrier and initiate a reaction".
The model for temperature and one additional stress takes
the general form:
for which S
1
could be some function of voltage or current or
any other relevant stress and the parameters , H, B, and
C determine acceleration between stress combinations. As
with the Arrhenius Model, k is Boltzmann's constant and
temperature is in degrees Kelvin.
If we want to add an additional non-thermal stress term, the
model becomes
and as many stresses as are relevant can be included by
adding similar terms.
8.1.5.2. Eyring
Models with
multiple
stresses
generally
have no
interaction
terms -
which means
you can
multiply
acceleration
factors due
to different
stresses
Note that the general Eyring model includes terms that have
stress and temperature interactions (in other words, the
effect of changing temperature varies, depending on the
levels of other stresses). Most models in actual use do not
include any interaction terms, so that the relative change in
acceleration factors when only one stress changes does not
depend on the level of the other stresses.
In models with no interaction, you can compute acceleration
factors for each stress and multiply them together. This
would not be true if the physical mechanism required
interaction terms - but, at least to first approximations, it
seems to work for most examples in the literature.
The Eyring
model can
also be used
to model
rate of
degradation
leading to
failure as a
function of
stress
Advantages of the Eyring Model
Can handle many stresses.
Can be used to model degradation data as well as
failure data.
The H parameter has a physical meaning and has
been studied and estimated for many well known
failure mechanisms and materials.
In practice,
the Eyring
Model is
usually too
complicated
to use in its
most general
form and
must be
"customized"
or simplified
for any
particular
failure
mechanism
Disadvantages of the Eyring Model
Even with just two stresses, there are 5 parameters to
estimate. Each additional stress adds 2 more unknown
parameters.
Many of the parameters may have only a second-
order effect. For example, setting = 0 works quite
well since the temperature term then becomes the
same as in the Arrhenius model. Also, the constants C
and E are only needed if there is a significant
temperature interaction effect with respect to the other
stresses.
The form in which the other stresses appear is not
specified by the general model and may vary
according to the particular failure mechanism. In other
words, S
1
may be voltage or ln (voltage) or some
other function of voltage.
Many well-known models are simplified versions of the
Eyring model with appropriate functions of relevant stresses
chosen for S
1
and S
2
. Some of these will be shown in the
Other Models section. The trick is to find the right
simplification to use for a particular failure mechanism.
8.1.5.3. Other models

8.1. Introduction
Many
useful 1, 2
and 3
stress
models are
simple
Eyring
models.
Six are
described
This section will discuss several acceleration models whose
successful use has been described in the literature.
The (Inverse) Power Rule for Voltage
The Exponential Voltage Model
Two Temperature/Voltage Models
The Electromigration Model
Three Stress Models (Temperature, Voltage and
Humidity)
The Coffin-Manson Mechanical Crack Growth Model
The (Inverse) Power Rule for Voltage
This model, used for capacitors, has only voltage dependency
and takes the form:
This is a very simplified Eyring model with , H, and C all
0, and S = lnV, and = -B.
The Exponential Voltage Model
In some cases, voltage dependence is modeled better with an
exponential model:
Two Temperature/Voltage Models
Temperature/Voltage models are common in the literature and
take one of the two forms given below:
Again, these are just simplified two stress Eyring models with
the appropriate choice of constants and functions of voltage.
The Electromigration Model
Electromigration is a semiconductor failure mechanism where
open failures occur in metal thin film conductors due to the
movement of ions toward the anode. This ionic movement is
accelerated high temperatures and high current density. The
(modified Eyring) model takes the form
with J denoting the current density. H is typically between
.5 and 1.2 electron volts, while an n around 2 is common.
Three-Stress Models (Temperature, Voltage and
Humidity)
Humidity plays an important role in many failure mechanisms
that depend on corrosion or ionic movement. A common 3-
stress model takes the form
Here RH is percent relative humidity. Other obvious variations
on this model would be to use an exponential voltage term
and/or an exponential RH term.
Even this simplified Eyring 3-stress model has 4 unknown
parameters and an extensive experimental setup would be
required to fit the model and calculate acceleration factors.
The
Coffin-
Manson
Model is a
useful
non-
Eyring
model for
crack
growth or
material
fatigue
The Coffin-Manson Mechanical Crack Growth Model
Models for mechanical failure, material fatigue or material
deformation are not forms of the Eyring model. These models
typically have terms relating to cycles of stress or frequency of
use or change in temperatures. A model of this type known as
the (modified) Coffin-Manson model has been used
successfully to model crack growth in solder and other metals
due to repeated temperature cycling as equipment is turned on
and off. This model takes the form
with
N
f
= the number of cycles to fail
f = the cycling frequency
T = the temperature range during a cycle
and G(T
max
) is an Arrhenius term evaluated at the maximum
temperature reached in each cycle.
Typical values for the cycling frequency exponent and the
temperature range exponent are around -1/3 and 2,
respectively (note that reducing the cycling frequency reduces
the number of cycles to failure). The H activation energy
term in G(T
max
) is around 1.25.
8.1.6. What are the basic lifetime distribution models used for non-repairable populations?

8.1. Introduction
8.1.6. What are the basic lifetime distribution
models used for non-repairable
populations?
A handful
of lifetime
distribution
models
have
enjoyed
great
practical
success
There are a handful of parametric models that have
successfully served as population models for failure times
arising from a wide range of products and failure
mechanisms. Sometimes there are probabilistic arguments
based on the physics of the failure mode that tend to justify
the choice of model. Other times the model is used solely
because of its empirical success in fitting actual failure data.
Seven models will be described in this section:
1. Exponential
2. Weibull
3. Extreme Value
4. Lognormal
5. Gamma
6. Birnbaum-Saunders
7. Proportional hazards
8.1.6.1. Exponential

8.1. Introduction
All the key
formulas
for using
the
exponential
model
Formulas and Plots
The exponential model, with only one unknown parameter, is the simplest of all
life distribution models. The key equations for the exponential are shown below:
Note that the failure rate reduces to the constant for any time. The exponential
distribution is the only distribution to have a constant failure rate. Also, another
name for the exponential mean is the Mean Time To Fail or MTTF and we
have MTTF = 1/.
The cumulative hazard function for the exponential is just the integral of the
failure rate or H(t) = t.
The PDF for the exponential has the familiar shape shown below.
The
Exponential
distribution
'shape'
The
Exponential
CDF
Below is an example of typical exponential lifetime data displayed in Histogram
form with corresponding exponential PDF drawn through the histogram.
Histogram
of
Exponential
Data
The
Exponential
models the
flat portion
of the
"bathtub"
curve -
where most
systems
spend most
of their
'lives'
Uses of the Exponential Distribution Model
1. Because of its constant failure rate property, the exponential distribution is
an excellent model for the long flat "intrinsic failure" portion of the
Bathtub Curve. Since most components and systems spend most of their
lifetimes in this portion of the Bathtub Curve, this justifies frequent use of
the exponential distribution (when early failures or wear out is not a
concern).
2. Just as it is often useful to approximate a curve by piecewise straight line
segments, we can approximate any failure rate curve by week-by-week or
month-by-month constant rates that are the average of the actual changing
rate during the respective time durations. That way we can approximate
any model by piecewise exponential distribution segments patched
together.
3. Some natural phenomena have a constant failure rate (or occurrence rate)
property; for example, the arrival rate of cosmic ray alpha particles or
Geiger counter tics. The exponential model works well for inter arrival
times (while the Poisson distribution describes the total number of events
in a given period). When these events trigger failures, the exponential life
distribution model will naturally apply.
Exponential
probability
plot
We can generate a probability plot of normalized exponential data, so that a
perfect exponential fit is a diagonal line with slope 1. The probability plot for
100 normalized random exponential observations ( = 0.01) is shown below.
We can calculate the exponential PDF and CDF at 100 hours for the case where
= 0.01. The PDF value is 0.0037 and the CDF value is 0.6321.
Functions for computing exponential PDF values, CDF values, and for producing
probability plots, are found in both Dataplot code and R code.
8.1.6.2. Weibull

8.1. Introduction
8.1.6.2. Weibull
Weibull
Formulas
Formulas and Plots
The Weibull is a very flexible life distribution model with two parameters. It has CDF and
PDF and other key formulas given by:
with the scale parameter (the Characteristic Life), (gamma) the Shape Parameter, and
is the Gamma function with (N) = (N-1)! for integer N.
The cumulative hazard function for the Weibull is the integral of the failure rate or
A more general three-parameter form of the Weibull includes an additional waiting time
parameter (sometimes called a shift or location parameter). The formulas for the 3-
parameter Weibull are easily obtained from the above formulas by replacing t by (t - )
wherever t appears. No failure can occur before hours, so the time scale starts at , and not
0. If a shift parameter is known (based, perhaps, on the physics of the failure mode), then all
you have to do is subtract from all the observed failure times and/or readout times and
8.1.6.2. Weibull
analyze the resulting shifted data with a two-parameter Weibull.
NOTE: Various texts and articles in the literature use a variety of different symbols for the
same Weibull parameters. For example, the characteristic life is sometimes called c ( = nu or
= eta) and the shape parameter is also called m (or = beta). To add to the confusion, some
software uses as the characteristic life parameter and as the shape parameter. Some
authors even parameterize the density function differently, using a scale parameter .
Special Case: When = 1, the Weibull reduces to the Exponential Model, with = 1/ = the
mean time to fail (MTTF).
Depending on the value of the shape parameter , the Weibull model can empirically fit a
wide range of data histogram shapes. This is shown by the PDF example curves below.
Weibull
data
'shapes'
From a failure rate model viewpoint, the Weibull is a natural extension of the constant failure
rate exponential model since the Weibull has a polynomial failure rate with exponent { - 1}.
This makes all the failure rate curves shown in the following plot possible.
Weibull
failure rate
'shapes'
8.1.6.2. Weibull
The Weibull
is very
flexible and
also has
theoretical
justification
in many
applications
Uses of the Weibull Distribution Model
1. Because of its flexible shape and ability to model a wide range of failure rates, the
Weibull has been used successfully in many applications as a purely empirical model.
2. The Weibull model can be derived theoretically as a form of Extreme Value
Distribution, governing the time to occurrence of the "weakest link" of many competing
failure processes. This may explain why it has been so successful in applications such
as capacitor, ball bearing, relay and material strength failures.
3. Another special case of the Weibull occurs when the shape parameter is 2. The
distribution is called the Rayleigh Distribution and it turns out to be the theoretical
probability model for the magnitude of radial error when the x and y coordinate errors
are independent normals with 0 mean and the same standard deviation.
Weibull
probability
plot
We generated 100 Weibull random variables using T = 1000, = 1.5 and = 5000. To see
how well these random Weibull data points are actually fit by a Weibull distribution, we
generated the probability plot shown below. Note the log scale used is base 10.
8.1.6.2. Weibull
If the data follow a Weibull distribution, the points should follow a straight line.
We can comput the PDF and CDF values for failure time T = 1000, using the example
Weibull distribution with = 1.5 and = 5000. The PDF value is 0.000123 and the CDF
value is 0.08556.
Functions for computing Weibull PDF values, CDF values, and for producing probability
plots, are found in both Dataplot code and R code.
8.1.6.3. Extreme value distributions

8.1. Introduction
The Extreme
Value
Distribution
usually
refers to the
distribution
of the
minimum of
a large
number of
unbounded
random
observations
Description, Formulas and Plots
We have already referred to Extreme Value Distributions when describing the uses of the
Weibull distribution. Extreme value distributions are the limiting distributions for the
minimum or the maximum of a very large collection of random observations from the same
arbitrary distribution. Gumbel (1958) showed that for any well-behaved initial distribution
(i.e., F(x) is continuous and has an inverse), only a few models are needed, depending on
whether you are interested in the maximum or the minimum, and also if the observations are
bounded above or below.
In the context of reliability modeling, extreme value distributions for the minimum are
frequently encountered. For example, if a system consists of n identical components in series,
and the system fails when the first of these components fails, then system failure times are the
minimum of n random component failure times. Extreme value theory says that, independent
of the choice of component model, the system model will approach a Weibull as n becomes
large. The same reasoning can also be applied at a component level, if the component failure
occurs when the first of many similar competing failure processes reaches a critical level.
The distribution often referred to as the Extreme Value Distribution (Type I) is the limiting
distribution of the minimum of a large number of unbounded identically distributed random
variables. The PDF and CDF are given by:
Extreme
Value
Distribution
formulas
and PDF
shapes
If the x values are bounded below (as is the case with times of failure) then the limiting
distribution is the Weibull. Formulas and uses of the Weibull have already been discussed.
PDF Shapes for the (minimum) Extreme Value Distribution (Type I) are shown in the
following figure.
The natural
log of
Weibull
data is
extreme
value data
Uses of the Extreme Value Distribution Model
1. In any modeling application for which the variable of interest is the minimum of many
random factors, all of which can take positive or negative values, try the extreme value
distribution as a likely candidate model. For lifetime distribution modeling, since failure
times are bounded below by zero, the Weibull distribution is a better choice.
2. The Weibull distribution and the extreme value distribution have a useful mathematical
relationship. If t
1
, t
2
, ...,t
n
are a sample of random times of fail from a Weibull
distribution, then ln t
1
, ln t
2
, ...,ln t
n
are random observations from the extreme value
distribution. In other words, the natural log of a Weibull random time is an extreme
value random observation.
Because of this relationship, computer programs designed for the extreme value
distribution can be used to analyze Weibull data. The situation exactly parallels using
normal distribution programs to analyze lognormal data, after first taking natural
logarithms of the data points.
Probability
plot for the
extreme
value
distribution
Assume = ln 200,000 = 12.206 and = 1/2 = 0.5. The extreme value distribution associated
with these parameters could be obtained by taking natural logarithms of data from a Weibull
population with characteristic life = 200,000 and shape = 2.
We generate 100 random numbers from this extreme value distribution and construct the
following probability plot.
Data from an extreme value distribution will line up approximately along a straight line when
this kind of plot is constructed. The slope of the line is an estimate of , "y-axis" value on the
line corresponding to the "x-axis" 0 point is an estimate of . For the graph above, these turn
out to be very close to the actual values of and .
For the example extreme value distribution with = ln 200,000 = 12.206 and = 1/2 = 0.5,
the PDF values corresponding to the points 5, 8, 10, 12, 12.8. are 0.110E-5, 0.444E-3, 0.024,
0.683 and 0.247. and the CDF values corresponding to the same points are 0.551E-6, 0.222E-
3, 0.012, 0.484 and 0.962.
Functions for computing extreme value distribution PDF values, CDF values, and for
producing probability plots, are found in both Dataplot code and R code.
8.1.6.4. Lognormal

8.1. Introduction
8.1.6.4. Lognormal
Lognormal
Formulas and
relationship
to the normal
distribution
Formulas and Plots
The lognormal life distribution, like the Weibull, is a very flexible model that can empirically
fit many types of failure data. The two-parameter form has parameters is the shape
parameter and T
50
is the median (a scale parameter).
Note: If time to failure, t
f
, has a lognormal distribution, then the (natural) logarithm of time to
failure has a normal distribution with mean = ln T
50
and standard deviation . This makes
lognormal data convenient to work with; just take natural logarithms of all the failure times
and censoring times and analyze the resulting normal data. Later on, convert back to real time
and lognormal parameters using as the lognormal shape and T
50
= e
as the (median) scale

parameter.
Below is a summary of the key formulas for the lognormal.
8.1.6.4. Lognormal
Note: A more general three-parameter form of the lognormal includes an additional waiting
time parameter (sometimes called a shift or location parameter). The formulas for the
three-parameter lognormal are easily obtained from the above formulas by replacing t by (t -
) wherever t appears. No failure can occur before hours, so the time scale starts at and
not 0. If a shift parameter is known (based, perhaps, on the physics of the failure mode),
then all you have to do is subtract from all the observed failure times and/or readout times
and analyze the resulting shifted data with a two-parameter lognormal.
Examples of lognormal PDF and failure rate plots are shown below. Note that lognormal
shapes for small sigmas are very similar to Weibull shapes when the shape parameter is
large and large sigmas give plots similar to small Weibull 's. Both distributions are very
flexible and it is often difficult to choose which to use based on empirical fits to small
samples of (possibly censored) data.
Lognormal
data 'shapes'
8.1.6.4. Lognormal
Lognormal
failure rate
'shapes'
A very
flexible model
that also can
apply
(theoretically)
to many
degradation
process
Uses of the Lognormal Distribution Model
1. As shown in the preceding plots, the lognormal PDF and failure rate shapes are flexible
enough to make the lognormal a very useful empirical model. In addition, the
relationship to the normal (just take natural logarithms of all the data and time points
and you have "normal" data) makes it easy to work with mathematically, with many
good software analysis programs available to treat normal data.
2. The lognormal model can be theoretically derived under assumptions matching many
8.1.6.4. Lognormal
failure modes
failure degradation processes common to electronic (semiconductor) failure
mechanisms. Some of these are: corrosion, diffusion, migration, crack growth,
electromigration, and, in general, failures resulting from chemical reactions or
processes. That does not mean that the lognormal is always the correct model for these
mechanisms, but it does perhaps explain why it has been empirically successful in so
many of these cases.
A brief sketch of the theoretical arguments leading to a lognormal model follows.
Applying the Central Limit Theorem to small additive errors in the log
domain and justifying a normal model is equivalent to justifying the
lognormal model in real time when a process moves towards failure based
on the cumulative effect of many small "multiplicative" shocks. More
precisely, if at any instant in time a degradation process undergoes a small
increase in the total amount of degradation that is proportional to the
current total amount of degradation, then it is reasonable to expect the time
to failure (i.e., reaching a critical amount of degradation) to follow a
lognormal distribution (Kolmogorov, 1941).
A more detailed description of the multiplicative degradation argument appears in a
later section.
Lognormal
probability
plot
We generated 100 random numbers from a lognormal distribution with shape 0.5 and median
life 20,000. To see how well these random lognormal data points are fit by a lognormal
distribution, we generate the lognormal probability plot shown below. Points that line up
approximately on a straight line indicates a good fit to a lognormal (with shape 0.5). The time
that corresponds to the (normalized) x-axis T
50
of 1 is the estimated T
50
according to the data.
In this case it is close to 20,000, as expected.
8.1.6.4. Lognormal
For a lognormal distribution at time T = 5000 with = 0.5 and T
50
= 20,000, the PDF value is
0.34175E-5, the CDF value is 0.002781, and the failure rate is 0.3427E-5.
Functions for computing lognormal distribution PDF values, CDF values, failure rates, and for
8.1.6.5. Gamma

8.1. Introduction
8.1.6.5. Gamma
Formulas
for the
gamma
model
Formulas and Plots
There are two ways of writing (parameterizing) the gamma distribution that are common in
the literature. In addition, different authors use different symbols for the shape and scale
parameters. Below we show two ways of writing the gamma, with "shape" parameter a = ,
and "scale" parameter b = 1/.
The
exponential
is a special
case of the
gamma
Note: When a = 1, the gamma reduces to an exponential distribution with b = .
Another well-known statistical distribution, the Chi-Square, is also a special case of the
gamma. A Chi-Square distribution with n degrees of freedom is the same as a gamma with a
= n/2 and b = 0.5 (or = 2).
The following plots give examples of gamma PDF, CDF and failure rate shapes.
Shapes for
gamma
data
8.1.6.5. Gamma
Gamma
CDF
shapes
Gamma
failure rate
shapes
8.1.6.5. Gamma
The
gamma is
used in
"Standby"
system
models and
also for
Bayesian
reliability
analysis
Uses of the Gamma Distribution Model
1. The gamma is a flexible life distribution model that may offer a good fit to some sets of
failure data. It is not, however, widely used as a life distribution model for common
failure mechanisms.
2. The gamma does arise naturally as the time-to-first fail distribution for a system with
standby exponentially distributed backups. If there are n-1 standby backup units and the
system and all backups have exponential lifetimes with parameter , then the total
lifetime has a gamma distribution with a = n and b = . Note: when a is a positive
integer, the gamma is sometimes called an Erlang distribution. The Erlang distribution
is used frequently in queuing theory applications.
3. A common use of the gamma model occurs in Bayesian reliability applications. When a
system follows an HPP (exponential) model with a constant repair rate , and it is
desired to make use of prior information about possible values of , a gamma Bayesian
prior for is a convenient and popular choice.
Gamma
probability
plot
We generated 100 random gamma data points using shape parameter = 2 and scale
parameter = 30. A gamma probability plot of the 100 data points is shown below.
8.1.6.5. Gamma
The value of the shape parameter can be estimated from data using
the squared ratio of mean failure time to the standard deviation of the failure times.
Using an example solved in the section on standby models, where = 2, = 30, and t = 24
months, the PDF, CDF, reliability, and failure rate are the following.
PDF = 0.01198
CDF = 0.19121
Reliability = 0.80879
Failure Rate = 0.01481
Functions for computing gamma distribution PDF values, CDF values, reliability values,
failure rates, and for producing probability plots, are found in both Dataplot code and R code.
8.1.6.5. Gamma
8.1.6.6. Fatigue life (Birnbaum-Saunders)

8.1. Introduction
A model
based on
cycles of
stress
causing
degradation
or crack
growth
In 1969, Birnbaum and Saunders described a life distribution model that could be derived
from a physical fatigue process where crack growth causes failure. Since one of the best ways
to choose a life distribution model is to derive it from a physical/statistical argument that is
consistent with the failure mechanism, the Birnbaum-Saunders fatigue life distribution is
worth considering.
Formulas
and shapes
for the
fatigue life
model
Formulas and Plots for the Birnbaum-Saunders Model
The PDF, CDF, mean and variance for the Birnbaum-Saunders distribution are shown below.
The parameters are: , a shape parameter; and , a scale parameter. These are the parameters
we will use in our discussion, but there are other choices also common in the literature (see the
parameters used for the derivation of the model).
PDF shapes for the model vary from highly skewed and long tailed (small gamma values) to
nearly symmetric and short tailed as gamma increases. This is shown in the figure below.
Corresponding failure rate curves are shown in the next figure.
If crack
growth in
each stress
cycle is a
random
Derivation and Use of the Birnbaum-Saunders Model:
Consider a material that continually undergoes cycles of stress loads. During each cycle, a
dominant crack grows towards a critical length that will cause failure. Under repeated
application of n cycles of loads, the total extension of the dominant crack can be written as
amount
independent
of past
cycles of
growth, the
Fatigue Life
mode model
may apply.
and we assume the Y
j
are independent and identically distributed non-negative random
variables with mean and variance
2
. Suppose failure occurs at the N-th cycle, when W
n
first exceeds a constant critical value w. If n is large, we can use a central limit theorem
argument to conclude that
Since there are many cycles, each lasting a very short time, we can replace the discrete
number of cycles N needed to reach failure by the continuous time t
f
needed to reach failure.
The CDF F(t) of t
f
is given by
Here denotes the standard normal CDF. Writing the model with parameters and is an
alternative way of writing the Birnbaum-Saunders distribution that is often used ( = and
= , as compared to the way the formulas were parameterized earlier in this section).
Note:
The critical assumption in the derivation, from a physical point of view, is that the crack
growth during any one cycle is independent of the growth during any other cycle. Also, the
growth has approximately the same random distribution, from cycle to cycle. This is a very
different situation from the proportional degradation argument used to derive a log normal
distribution model, with the rate of degradation at any point in time depending on the total
amount of degradation that has occurred up to that time.
This kind of
physical
degradation
is consistent
with
Miner's
Rule.
The Birnbaum-Saunders assumption, while physically restrictive, is consistent with a
deterministic model from materials physics known as Miner's Rule (Miner's Rule implies that
the damage that occurs after n cycles, at a stress level that produces a fatigue life of N cycles,
is proportional to n/N). So, when the physics of failure suggests Miner's Rule applies, the
Birnbaum-Saunders model is a reasonable choice for a life distribution model.
Birnbaum-
Saunders
probability
plot
We generated 100 random numbers from a Birnbaum-Saunders distribution where = 5000
and = 2, and created a fatigue life probability plot of the 100 data points.
If the points on the probability plot line up roughly on a straight line, as expected for data we
generated, then the data are correctly modeled by the Birnbaum-Saunders distribution.
The PDF value at time t = 4000 for a Birnbaum-Saunders (fatigue life) distribution with
parameters = 5000 and = 2 is 4.987e-05 and the CDF value is 0.455.
Functions for computing Birnbaum-Saunders distribution PDF values, CDF values, and for
8.1.6.7. Proportional hazards model

8.1. Introduction
The
proportional
hazards
model is
often used
in survival
analysis
(medical
testing)
studies. It is
not used
much with
engineering
data
The proportional hazards model, proposed by Cox (1972),
has been used primarily in medical testing analysis, to model
the effect of secondary variables on survival. It is more like
an acceleration model than a specific life distribution model,
and its strength lies in its ability to model and test many
inferences about survival without making any specific
assumptions about the form of the life distribution model.
This section will give only a brief description of the
proportional hazards model, since it has limited engineering
applications.
Proportional Hazards Model Assumption
Let z = {x, y, ...} be a vector of 1 or more explanatory
variables believed to affect lifetime. These variables may be
continuous (like temperature in engineering studies, or
dosage level of a particular drug in medical studies) or they
may be indicator variables with the value 1 if a given factor
or condition is present, and 0 otherwise.
Let the hazard rate for a nominal (or baseline) set z
0
=
(x
0
,y
0
, ...) of these variables be given by h
0
(t), with h
0
(t)
denoting legitimate hazard function (failure rate) for some
unspecified life distribution model.
The
proportional
hazard
model
assumes
changing a
stress
variable (or
explanatory
variable)
has the
effect of
multiplying
the hazard
rate by a
constant.
The proportional hazards model assumes we can write the
changed hazard function for a new value of z as
h
z
(t) = g(z)h
0
(t)
In other words, changing z, the explanatory variable vector,
results in a new hazard function that is proportional to the
nominal hazard function, and the proportionality constant is
a function of z, g(z), independent of the time variable t.
A common and useful form for f(z) is the Log Linear Model
which has the equation: g(x) = e
ax
for one variable, g(x,y) =
e
ax+by
for two variables, etc.
Properties and Applications of the Proportional Hazards
Model
1. The proportional hazards model is equivalent to the
acceleration factor concept if and only if the life
distribution model is a Weibull (which includes the
exponential model, as a special case). For a Weibull
with shape parameter , and an acceleration factor AF
between nominal use fail time t
0
and high stress fail
time t
s
(with t
0
= AFt
s
) we have g(s) = AF . In other
words, h
s
(t) = AF h
0
(t).
2. Under a log-linear model assumption for g(z), without
any further assumptions about the life distribution
model, it is possible to analyze experimental data and
compute maximum likelihood estimates and use
likelihood ratio tests to determine which explanatory
variables are highly significant. In order to do this kind
of analysis, however, special software is needed.
More details on the theory and applications of the
proportional hazards model may be found in Cox and Oakes
(1984).
8.1.7. What are some basic repair rate models used for repairable systems?

8.1. Introduction
8.1.7. What are some basic repair rate models
used for repairable systems?
Models for
repair
rates of
repairable
systems
N(t), M(t) and m(t) were defined in the section on Repair
Rates. Repair rate models are defined by first picking a
functional form for M(t), the expected number of cumulative
failures by time t. Taking the derivative of this gives the repair
rate model m(t).
In the next three sections we will describe three models, of
increasing complexity, for M(t). They are: the Homogeneous
Poisson Process, the Non-Homogeneous Poisson Process
following a Power law, and the Non-Homogeneous Poisson
Process following an Exponential law.
8.1.7.1. Homogeneous Poisson Process (HPP)

8.1. Introduction
8.1.7.1. Homogeneous Poisson Process (HPP)
Repair rate
(ROCOF)
models and
formulas
The simplest useful model for M(t) is M(t) = t and the repair rate (or ROCOF) is the
constant m(t) = . This model comes about when the interarrival times between failures
are independent and identically distributed according to the exponential distribution,
with parameter . This basic model is also known as a Homogeneous Poisson Process
(HPP). The following formulas apply:
HPP model
fits flat
portion of
"bathtub"
curve
Despite the simplicity of this model, it is widely used for repairable equipment and
systems throughout industry. Justification for this comes, in part, from the shape of the
empirical Bathtub Curve. Most systems (or complex tools or equipment) spend most of
their "lifetimes" operating in the long flat constant repair rate portion of the Bathtub
Curve. The HPP is the only model that applies to that portion of the curve, so it is the
most popular model for system reliability evaluation and reliability test planning.
Planning reliability assessment tests (under the HPP assumption) is covered in a later
section, as is estimating the MTBF from system failure data and calculating upper and
lower confidence limits.
Poisson
relationship
In the HPP model, the probability of having exactly k failures by time T is given by the
Poisson distribution with mean T (see formula for P{N(T) = k} above).
8.1.7.2. Non-Homogeneous Poisson Process (NHPP) - power law

8.1. Introduction
8.1.7.2. Non-Homogeneous Poisson Process
(NHPP) - power law
The repair
rate for a
NHPP
following the
Power law
A flexible model (that has been very successful in many
applications) for the expected number of failures in the first t
hours, M(t), is given by the polynomial
The repair rate (or ROCOF) for this model is
The Power
law model is
very flexible
and contains
the HPP
(exponential)
model as a
special case
The HPP model has a the constant repair rate m(t) = . If
we substitute an arbitrary function (t) for , we have a
Non Homogeneous Poisson Process (NHPP) with Intensity
Function . If
then we have an NHPP with a Power Law intensity
function (the "intensity function" is another name for the
repair rate m(t)).
Because of the polynomial nature of the ROCOF, this model
is very flexible and can model both increasing (b>1 or <
0) and decreasing (0 < b < 1 or 0 < < 1)) failure rates.
When b = 1 or = 0, the model reduces to the HPP
constant repair rate model.
Probabilities
of failure for
all NHPP
processes
can easily be
calculated
based on the
Poisson
formula
Probabilities of a given number of failures for the NHPP
model are calculated by a straightforward generalization of
the formulas for the HPP. Thus, for any NHPP
and for the Power Law model:
8.1.7.2. Non-Homogeneous Poisson Process (NHPP) - power law
The Power
Law model
is also called
the Duane
Model and
the AMSAA
model
Other names for the Power Law model are: the Duane
Model and the AMSAA model. AMSAA stands for the
United States Army Materials System Analysis Activity,
where much theoretical work describing the Power Law
model was performed in the 1970's.
It is also
called a
Weibull
Process - but
this name is
misleading
and should
be avoided
The time to the first fail for a Power Law process has a
Weibull distribution with shape parameter b and
characteristic life a. For this reason, the Power Law model is
sometimes called a Weibull Process. This name is confusing
and should be avoided, however, since it mixes a life
distribution model applicable to the lifetimes of a non-
repairable population with a model for the inter-arrival
times of failures of a repairable population.
For any NHPP process with intensity function m(t), the
distribution function (CDF) for the inter-arrival time to
the next failure, given a failure just occurred at time T, is
given by
Once a
failure
occurs, the
waiting time
to the next
failure for an
NHPP has a
simple CDF
formula
In particular, for the Power Law the waiting time to the next
failure, given a failure at time T, has distribution function
This inter arrival time CDF can be used to derive a simple
algorithm for simulating NHPP Power Law Data.
8.1.7.3. Exponential law

8.1. Introduction
8.1.7.3. Exponential law
The
Exponential
Law is
another
flexible
NHPP
model
An NHPP with ROCOF or intensity function given by
is said to follow an Exponential Law. This is also called the
log-linear model or the Cox-Lewis model.
A system whose repair rate follows this flexible model is
improving if < 0 and deteriorating if >0. When = 0,
the Exponential Law reduces to the HPP constant repair rate
model
8.1.8. How can you evaluate reliability from the "bottom-up" (component failure mode to system failure rate)?

8.1. Introduction
8.1.8. How can you evaluate reliability from the
"bottom-up" (component failure mode to
system failure rate)?
Several
simple
models can
be used to
calculate
system
failure
rates,
starting
with failure
rates for
failure
modes
within
individual
system
components
This section deals with models and methods that apply to
non-repairable components and systems. Models for failure
rates (and not repair rates) are described. The next section
covers models for (repairable) system reliability growth.
We use the Competing Risk Model to go from component
failure modes to component failure rates. Next we use the
Series Model to go from components to assemblies and
systems. These models assume independence and "first
failure mode to reach failure causes both the component and
the system to fail".
If some components are "in parallel", so that the system can
survive one (or possibly more) component failures, we have
the parallel or redundant model. If an assembly has n identical
components, at least r of which must be working for the
system to work, we have what is known as the r out of n
model.
The standby model uses redundancy like the parallel model,
except that the redundant unit is in an off-state (not exercised)
until called upon to replace a failed unit.
This section describes these various models. The last
subsection shows how complex systems can be evaluated
using the various models as building blocks.
8.1.8.1. Competing risk model

8.1. Introduction
8.1.8. How can you evaluate reliability from the "bottom-up" (component failure mode to
Use the
competing
risk model
when the
failure
mechanisms
are
independent
and the first
mechanism
failure
causes the
component
to fail
Assume a (replaceable) component or unit has k different
ways it can fail. These are called failure modes and
underlying each failure mode is a failure mechanism.
The Competing Risk Model evaluates component reliability
by "building up" from the reliability models for each failure
mode.
The following 3 assumptions are needed:
1. Each failure mechanism leading to a particular type of
failure (i.e., failure mode) proceeds independently of
every other one, at least until a failure occurs.
2. The component fails when the first of all the
competing failure mechanisms reaches a failure state.
3. Each of the k failure modes has a known life
distribution model F
i
(t).
The competing risk model can be used when all three
assumptions hold. If R
c
(t), F
c
(t), and h
c
(t) denote the
reliability, CDF and failure rate for the component,
respectively, and R
i
(t), F
i
(t) and h
i
(t) are the reliability, CDF
and failure rate for the i-th failure mode, respectively, then
the competing risk model formulas are:
Multiply
reliabilities
and add
failure rates
Think of the competing risk model in the following way:
All the failure mechanisms are having a race to
see which can reach failure first. They are not
allowed to "look over their shoulder or
sideways" at the progress the other ones are
making. They just go their own way as fast as
they can and the first to reach "failure" causes
the component to fail.
Under these conditions the component reliability
is the product of the failure mode reliabilities and
the component failure rate is just the sum of the
failure mode failure rates.
Note that the above holds for any arbitrary life distribution
model, as long as "independence" and "first mechanism
failure causes the component to fail" holds.
When we learn how to plot and analyze reliability data in
later sections, the methods will be applied separately to each
failure mode within the data set (considering failures due to
all other modes as "censored run times"). With this approach,
the competing risk model provides the glue to put the pieces
back together again.
8.1.8.2. Series model

8.1. Introduction
The series
model is used
to go from
individual
components
to the entire
system,
assuming the
system fails
when the first
component
fails and all
components
fail or
survive
independently
of one
another
The Series Model is used to build up from components to sub-assemblies
and systems. It only applies to non replaceable populations (or first failures
of populations of systems). The assumptions and formulas for the Series
Model are identical to those for the Competing Risk Model, with the k
failure modes within a component replaced by the n components within a
system.
The following 3 assumptions are needed:
1. Each component operates or fails independently of every other one, at
least until the first component failure occurs.
2. The system fails when the first component failure occurs.
3. Each of the n (possibly different) components in the system has a
known life distribution model F
i
(t).
Add failure
rates and
multiply
reliabilities
in the Series
Model
When the Series Model assumptions hold we have:
with the subscript S referring to the entire system and the subscript i
referring to the i-th component.
Note that the above holds for any arbitrary component life distribution
models, as long as "independence" and "first component failure causes the
system to fail" both hold.
The analogy to a series circuit is useful. The entire system has n components
in series. The system fails when current no longer flows and each component
operates or fails independently of all the others. The schematic below shows
a system with 5 components in series "replaced" by an "equivalent" (as far
as reliability is concerned) system with only one component.
8.1.8.3. Parallel or redundant model

8.1. Introduction
The parallel
model
assumes all n
components
that make up
a system
operate
independently
and the
system works
as long as at
least one
component
still works
The opposite of a series model, for which the first component failure causes
the system to fail, is a parallel model for which all the components have to
fail before the system fails. If there are n components, any (n-1) of them
may be considered redundant to the remaining one (even if the components
are all different). When the system is turned on, all the components operate
until they fail. The system reaches failure at the time of the last component
failure.
The assumptions for a parallel model are:
1. All components operate independently of one another, as far as
reliability is concerned.
2. The system operates as long as at least one component is still
operating. System failure occurs at the time of the last component
failure.
3. The CDF for each component is known.
Multiply
component
CDF's to get
the system
CDF for a
parallel
model
For a parallel model, the CDF F
s
(t) for the system is just the product of the
CDF's F
i
(t) for the components or
R
s
(t) and h
s
(t) can be evaluated using basic definitions, once we have F
s
(t).
The schematic below represents a parallel system with 5 components and the
(reliability) equivalent 1 component system with a CDF F
s
equal to the
product of the 5 component CDF's.
8.1.8.4. R out of N model

8.1. Introduction
An r out of
n model is
a system
that
survives
when at
least r of
its
components
are
working
(any r)
An "r out of n" system contains both the series system model
and the parallel system model as special cases. The system
has n components that operate or fail independently of one
another and as long as at least r of these components (any r)
survive, the system survives. System failure occurs when the
(n-r+1)th component failure occurs.
When r = n, the r out of n model reduces to the series model.
When r = 1, the r out of n model becomes the parallel model.
We treat here the simple case where all the components are
identical.
Formulas and assumptions for r out of n model (identical
components):
1. All components have the identical reliability function
R(t).
2. All components operate independently of one another
(as far as failure is concerned).
3. The system can survive any (n-r) of the components
failing. The system fails at the instant of the (n-r+1)th
component failure.
Formula
for an r out
of n system
where the
components
are
identical
System reliability is given by adding the probability of
exactly r components surviving to time t to the probability of
exactly (r+1) components surviving, and so on up to the
probability of all components surviving to time t. These are
binomial probabilities (with p = R(t)), so the system
reliability is given by:
Note: If we relax the assumption that all the components are
identical, then R
s
(t) would be the sum of probabilities
evaluated for all possible terms that could be formed by
picking at least r survivors and the corresponding failures.
The probability for each term is evaluated as a product of
R(t)'s and F(t)'s. For example, for n = 4 and r = 2, the system
reliability would be (abbreviating the notation for R(t) and
F(t) by using only R and F)
R
s
= R
1
R
2
F
3
F
4
+ R
1
R
3
F
2
F
4
+ R
1
R
4
F
2
F
3
+ R
2
R
3
F
1
F
4

+ R
2
R
4
F
1
F
3
+ R
3
R
4
F
1
F
2
+ R
1
R
2
R
3
F
4
+ R
1
R
3
R
4
F
2

+ R
1
R
2
R
4
F
3
+ R
2
R
3
R
4
F
1
+ R
1
R
2
R
3
R
4
8.1.8.5. Standby model

8.1. Introduction
The Standby
Model
evaluates
improved
reliability
when backup
replacements
are switched
on when
failures
occur.
A Standby Model refers to the case in which a key
component (or assembly) has an identical backup
component in an "off" state until needed. When the original
component fails, a switch turns on the "standby" backup
component and the system continues to operate.
In the simple case, assume the non-standby part of the
system has CDF F(t) and there are (n-1) identical backup
units that will operate in sequence until the last one fails. At
that point, the system finally fails.
The total system lifetime is the sum of n identically
distributed random lifetimes, each having CDF F(t).
Identical
backup
Standby
model leads
to
convolution
formulas
In other words, T
n
= t
1
+ t
2
+ ... + t
n
, where each t
i
has
CDF F(t) and T
n
has a CDF we denote by F
n
(t). This can be
evaluated using convolution formulas:
In general, convolutions are solved numerically. However,
for the special case when F(t) is the exponential model, the
above integrations can be solved in closed form.
Exponential
standby
systems lead
to a gamma
lifetime
model
Special Case: The Exponential (or Gamma) Standby Model
If F(t) has the exponential CDF (i.e., F(t) = 1 - e
-lt
), then
and the PDF f
n
(t) is the well-known gamma distribution.
Example: An unmanned space probe sent out to explore the
solar system has an onboard computer with reliability
characterized by the exponential distribution with a Mean
Time To Failure (MTTF) of 1/ = 30 months (a constant
failure rate of 1/30 = .033 fails per month). The probability
of surviving a two year mission is only e
-24/30
= .45. If,
however, a second computer is included in the probe in a
standby mode, the reliability at 24 months (using the above
formula for F
2
) becomes .8 .449 + .449 = .81. The failure
rate at 24 months (f
2
/[1-F
2
]) reduces to [(24/900)
.449]/.81 = .015 fails per month. At 12 months the failure
rate is only .0095 fails per month, which is less than 1/3 of
the failure rate calculated for the non-standby case.
Standby units (as the example shows) are an effective way
of increasing reliability and reducing failure rates, especially
during the early stages of product life. Their improvement
effect is similar to, but greater than, that of parallel
redundancy . The drawback, from a practical standpoint, is
the expense of extra components that are not needed for
functionality.
8.1.8.6. Complex systems

8.1. Introduction
Often the
reliability
of complex
systems can
be
evaluated
by
successive
applications
of Series
and/or
Parallel
model
formulas
Many complex systems can be diagrammed as combinations of Series
components, Parallel components, R out of N components and Standby
components. By using the formulas for these models, subsystems or sections
of the original system can be replaced by an "equivalent" single component
with a known CDF or Reliability function. Proceeding like this, it may be
possible to eventually reduce the entire system to one component with a
known CDF.
Below is an example of a complex system composed of both components in
parallel and components in series is reduced first to a series system and
finally to a one-component system.
Note: The reduction methods described above will work for many, but not
all, systems. Some systems with a complicated operational logic structure
will need a more formal structural analysis methodology. This methodology
deals with subjects such as event trees, Boolean representations, coherent
structures, cut sets and decompositions, and is beyond the present scope of
this Handbook.
8.1.9. How can you model reliability growth?

8.1. Introduction
A reliability
improvement
test is a
formal
procedure
aimed at
discovering
and fixing
system
reliability
flaws
During the early stages of developing and prototyping
complex systems, reliability often does not meet customer
requirements. A formal test procedure aimed at discovering
and fixing causes of unreliability is known as a Reliability
Improvement Test. This test focuses on system design,
system assembly and component selection weaknesses that
cause failures.
A typical reliability improvement test procedure would be to
run a prototype system, as the customer might for a period
of several weeks, while a multidisciplined team of engineers
and technicians (design, quality, reliability, manufacturing,
etc.) analyze every failure that occurs. This team comes up
with root causes for the failures and develops design and/or
assembly improvements to hopefully eliminate or reduce the
future occurrence of that type of failure. As the testing
continues, the improvements the team comes up with are
incorporated into the prototype, so it is expected that
reliability will improve during the course of the test.
Repair rates
should show
an
improvement
trend during
the course of
a reliability
improvement
test and this
can be
modeled
using a
NHPP
model
Another name for reliability improvement testing is TAAF
testing, standing for Test, Analyze And Fix.
While only one model applies when a repairable system has
no improvement or degradation trends (the constant repair
rate HPP model), there are infinitely many models that could
be used to describe a system with a decreasing repair rate
(reliability growth models).
Fortunately, one or two relatively simple models have been
very successful in a wide range of industrial applications.
Two models that have previously been described will be
used in this section. These models are the NHPP Power Law
Model and the NHPP Exponential Law Model. The Power
Law Model underlies the frequently used graphical
technique known as Duane Plotting.
8.1.9.1. NHPP power law

8.1. Introduction
If the Power
Law applies,
Repair Rates
improve
over time
according to
the formula
. The
exponent
lies between
0 and 1 and
is called the
reliability
growth slope
This repairable system model was described in Section 8.1.7.2.
The expected number of failures by time t has the form M(t) =
at
b
and the repair rate has the form m(t) = abt
b-1
. This will
model improvement when 0 < b < 1, with larger improvements
coming when b is smaller. As we will see in the next section on
Duane Plotting, it is convenient to define = 1 - b and =
ab, and write the repair rate as
m(t) =
Again we have improvement when 0 < < 1, with larger
improvement coming from larger values of . is known as
the Duane Plot slope or the reliability improvement Growth
Slope.
In terms of the original parameters for M(t), we have
Use of the Power Law model for reliability growth test data
generally assumes the following:
1. While the test is ongoing, system improvements are
introduced that produce continual improvements in the rate of
system repair.
2. Over a long enough period of time the effect of these
improvements can be modeled adequately by the continuous
polynomial repair rate improvement model .
When an
improvement
test ends,
the MTBF
stays
constant at
its last
achieved
value
3. When the improvement test ends at test time T and no
further improvement actions are ongoing, the repair rate has
been reduced to . The repair rate remains constant from
then on at this new (improved) level.
Assumption 3 means that when the test ends, the HPP constant
repair rate model takes over and the MTBF for the system from
then on is the reciprocal of the final repair rate or . If
we estimate the expected number of failures up to time T by the
actual number observed, the estimated MTBF at the end of a
reliability test (following the Power Law) is:
with T denoting the test time, r is the total number of test
failures and is the reliability growth slope. A formula for
estimating from system failure times is given in the Analysis
Section for the Power Law model.
Simulated
Data
Example
Simulating NHPP Power Law Data
Step 1: User inputs the positive constants a and b.
Step 2: Simulate a vector of n uniform (0,1) random numbers.
Call these U
1
, U
2
, U
3
, . . . U
n
.
Step 3: Calculate Y
1
= {-1/a * ln U
1
} ** 1/b
Step i: Calculate Y
i
= {(Y
i-1
** b) -1/a * ln U
i
}**1/b for i = 2,
. . ., n
The n numbers Y
1
, Y
2
, . . ., Y
n
are the desired repair times
simulated from an NHPP Power Law process with parameters
a, b (or = 1 - b and = ab).
Example
We generated n = 13 random repair times using the NHPP
Power Law process with a = 0.2 and b = 0.4. The resulting data
and a plot of failure number versus repair times are shown
below.
Failure Failure
Number Time
1 26
2 182
3 321
4 728
5 896
6 1268
7 1507
8 2325
9 3427
10 11871
11 11978
12 13562
13 15053
The NHPP power law process can be implemented using both
8.1.9.2. Duane plots

8.1. Introduction
A plot on log-
log paper of
successive
MTBF
estimates
versus system
time of fail
for reliability
improvement
test data is
called a
Duane Plot
The standard estimate of the MTBF for a system with a constant repair rate
(an HPP system) is T/r, with T denoting the total time the system was
observed and r is the total number of failures that occurred.
If we calculate successive MTBF estimates (called Cum MTBF Estimates),
every time a failure occurs for a system undergoing reliability improvement
testing, we typically see a sequence of mostly increasing numbers.
In 1964, J. T. Duane observed that when he plotted these cum MTBF
estimates versus the times of failure on log-log paper, the points tended to
line up following a straight line. This was true for many different sets of
reliability improvement data and many other engineers have seen similar
results over the last three decades. This type of plot is called a Duane Plot
and the slope of the best line through the points is called the reliability
growth slope or Duane plot slope.
Points on a
Duane plot
line up
approximately
on a straight
line if the
Power Law
model applies
Plotting a Duane Plot is simple. If the ith failure occurs at time t
i
, then plot t
i
divided by i (the "y"- axis value) versus the time t
i
(the "x"-axis value) on
log-log graph paper. Do this for all the test failures and draw the best
straight line you can following all these points.
Why does this "work"? Following the notation for repairable system models,
we are plotting estimates of {t/M(t)} versus the time of failure t. If M(t)
follows the Power Law (also described in the last section), then we are
plotting estimates of t/at
b
versus the time of fail t. This is the same as
plotting versus t, with = 1-b . For a log-log scale plot, this will
be a straight line with slope and intercept (when t = 1) of - log
10
a.
In other words, a straight line on a Duane plot is equivalent to the NHPP
Power Law Model with a reliability growth slope of = 1 - b and an "a"
parameter equal to 10
-intercept
.
Note: A useful empirical rule of thumb based on Duane plots made from
many reliability improvement tests across many industries is the following:
Duane plot
reliability
growth slopes
should lie
between 0.3
The reliability improvement slope for virtually all reliability
improvement tests will be between 0.3 and 0.6. The lower end
(0.3) describes a minimally effective test - perhaps the cross-
functional team is inexperienced or the system has many failure
mechanisms that are not well understood. The higher end (0.6)
and 0.6 approaches the empirical state of the art for reliability
improvement activities.
Examples of
Duane Plots
Duane Plot Example 1:
A reliability growth test lasted 1500 hours (approximately 10 weeks) and
recorded 8 failures at the following system hours: 33, 76, 145, 347, 555, 811,
1212, 1499. After calculating successive cum MTBF estimates, a Duane plot
shows these estimates versus system age on log vs log paper. The "best"
straight line through the data points corresponds to a NHPP Power Law
model with reliability growth slope equal to the slope of the line. This line
is an estimate of the theoretical model line (assuming the Power Law holds
during the course of the test) and the achieved MTBF at the end of the test is
given by
T / [r (1- )]
with T denoting the total test time and r the number of failures. Results for
this particular reliability growth test follow.

Failure # System Age of Failure Cum MTBF
1 33 33
2 76 38
3 145 48.3
4 347 86.8
5 555 111.0
6 811 135.2
7 1212 173.1
8 1499 187.3
The Duane plot indicates a reasonable fit to a Power Law NHPP model. The
reliability improvement slope (slope of line on Duane plot) is = 0.437
(using the formula given in the section on reliability data analysis for the
Power Law model) and the estimated MTBF achieved by the end of the
1500 hour test is 1500/(8 [1-0.437]) or 333 hours.
Duane Plot Example 2:
A Duane plot for the simulated Power Law data used in the Example in the
preceding section is shown below.
Duane plots can be produced using both Dataplot code and R code.
8.1.9.3. NHPP exponential law

8.1. Introduction
8.1.9.3. NHPP exponential law
The
Exponential
Law is
another
useful
reliability
growth
model to
try when
the Power
law is not
fitting well
When the data points in a Duane plot show obvious
curvature, a model that might fit better is the NHPP
Exponential Law.
For this model, if < 0, the repair rate improves over time
according to
The corresponding cumulative expected failures model is
This approaches the maximum value of A expected failures as
t goes to infinity, so the cumulative failures plot should
clearly be bending over and asymptotically approaching a
value .
Rule of thumb: First try a Duane plot and the Power law
model. If that shows obvious lack of fit, try the Exponential
Law model, estimating parameters using the formulas in the
Analysis Section for the Exponential law. A plot of cum fails
versus time, along with the estimated M(t) curve, can be used
to judge goodness of fit.

8.1.10. How can Bayesian methodology be used for reliability evaluation?
http://www.itl.nist.gov/div898/handbook/apr/section1/apr1a.htm[6/27/2012 2:49:30 PM]

8.1. Introduction
8.1.10. How can Bayesian methodology be used for
Several
Bayesian
Methods
overview
topics are
covered in
this section
This section gives an overview of the application of Bayesian techniques in
reliability investigations. The following topics are covered:
What is Bayesian Methodology ?
Bayes Formula, Prior and Posterior Distribution Models, and Conjugate
Priors
How Bayesian Methodology is used in System Reliability Evaluation
Advantages and Disadvantages of using Bayes Methodology
What is Bayesian Methodology?
Bayesian
analysis
considers
population
parameters
to be
random, not
fixed
Old
information,
or subjective
judgment, is
used to
determine a
prior
distribution
for these
population
parameters
It makes a great deal of practical sense to use all the information available, old
and/or new, objective or subjective, when making decisions under uncertainty.
This is especially true when the consequences of the decisions can have a
significant impact, financial or otherwise. Most of us make everyday personal
decisions this way, using an intuitive process based on our experience and
subjective judgments.
Mainstream statistical analysis, however, seeks objectivity by generally
restricting the information used in an analysis to that obtained from a current
set of clearly relevant data. Prior knowledge is not used except to suggest the
choice of a particular population model to "fit" to the data, and this choice is
later checked against the data for reasonableness.
Lifetime or repair models, as we saw earlier when we looked at repairable and
non repairable reliability population models, have one or more unknown
parameters. The classical statistical approach considers these parameters as
fixed but unknown constants to be estimated (i.e., "guessed at") using sample
data taken randomly from the population of interest. A confidence interval for
an unknown parameter is really a frequency statement about the likelihood that
numbers calculated from a sample capture the true parameter. Strictly speaking,
one cannot make probability statements about the true parameter since it is
fixed, not random.
The Bayesian approach, on the other hand, treats these population model
parameters as random, not fixed, quantities. Before looking at the current data,
we use old information, or even subjective judgments, to construct a prior
distribution model for these parameters. This model expresses our starting
assessment about how likely various values of the unknown parameters are. We
then make use of the current data (via Baye's formula) to revise this starting
assessment, deriving what is called the posterior distribution model for the
population model parameters. Parameter estimates, along with confidence
intervals (known as credibility intervals), are calculated directly from the
posterior distribution. Credibility intervals are legitimate probability statements
about the unknown parameters, since these parameters now are considered
random, not fixed.
It is unlikely in most applications that data will ever exist to validate a chosen
prior distribution model. Parametric Bayesian prior models are chosen because
of their flexibility and mathematical convenience. In particular, conjugate
priors (defined below) are a natural and popular choice of Bayesian prior
distribution models.
Bayes Formula, Prior and Posterior Distribution Models, and Conjugate
Priors
Bayes
formula
provides the
mathematical
tool that
combines
prior
knowledge
with current
data to
produce a
posterior
distribution
Bayes formula is a useful equation from probability theory that expresses the
conditional probability of an event A occurring, given that the event B has
occurred (written P(A|B)), in terms of unconditional probabilities and the
probability the event B has occurred, given that A has occurred. In other words,
Bayes formula inverts which of the events is the conditioning event. The
formula is
and P(B) in the denominator is further expanded by using the so-called "Law of
Total Probability" to write
with the events A
i
being mutually exclusive and exhausting all possibilities and
including the event A as one of the A
i
.
The same formula, written in terms of probability density function models,
takes the form:
where f(x| ) is the probability model, or likelihood function, for the observed
data x given the unknown parameter (or parameters) , g( ) is the prior
distribution model for and g( |x) is the posterior distribution model for
given that the data x have been observed.
When g( |x) and g( ) both belong to the same distribution family, g( ) and
f(x| ) are called conjugate distributions and g( ) is the conjugate prior for
f(x| ). For example, the Beta distribution model is a conjugate prior for the
proportion of successes p when samples have a binomial distribution. And the
Gamma model is a conjugate prior for the failure rate when sampling failure
times or repair times from an exponentially distributed population. This latter
conjugate pair (gamma, exponential) is used extensively in Bayesian system
reliability applications.
How Bayes Methodology is used in System Reliability Evaluation
Bayesian
system
reliability
evaluation
assumes the
system
MTBF is a
random
quantity
"chosen"
according to
a prior
distribution
model
Models and assumptions for using Bayes methodology will be described in a
later section. Here we compare the classical paradigm versus the Bayesian
paradigm when system reliability follows the HPP or exponential model (i.e.,
the flat portion of the Bathtub Curve).
Classical Paradigm For System Reliability Evaluation:
The MTBF is one fixed unknown value - there is no probability
associated with it
Failure data from a test or observation period allows you to make
inferences about the value of the true unknown MTBF
No other data are used and no judgment - the procedure is objective
and based solely on the test data and the assumed HPP model
Bayesian Paradigm For System Reliability Evaluation:
The MTBF is a random quantity with a probability distribution
The particular piece of equipment or system you are testing chooses an
MTBF from this distribution and you observe failure data that follow an
HPP model with that MTBF
Prior to running the test, you already have some idea of what the MTBF
probability distribution looks like based on prior test data or an consensus
engineering judgment
Advantages and Disadvantages of using Bayes Methodology
Pro's and
con's for
using
Bayesian
methods
While the primary motivation to use Bayesian reliability methods is typically a
desire to save on test time and materials cost, there are other factors that should
also be taken into account. The table below summarizes some of these "good
news" and "bad news" considerations.
Bayesian Paradigm: Advantages and Disadvantages
Pro's Con's
Uses prior information - this
"makes sense"
If the prior information is
encouraging, less new testing may
be needed to confirm a desired
MTBF at a given confidence
Confidence intervals are really
intervals for the (random) MTBF -
sometimes called "credibility
intervals"
Prior information may not be
accurate - generating
misleading conclusions
Way of inputting prior
information (choice of prior)
may not be correct
Customers may not accept
validity of prior data or
engineering judgements
There is no one "correct way"
of inputting prior information
and different approaches can
give different results
Results aren't objective and
don't stand by themselves
8.2. Assumptions/Prerequisites

This section describes how life distribution models and
acceleration models are typically chosen. Several graphical
and analytical methods for evaluating model fit are also
discussed.
Detailed
contents of
Section 2
2. Assumptions/Prerequisites
1. How do you choose an appropriate life distribution
model?
1. Based on failure mode
2. Extreme value argument
3. Multiplicative degradation argument
4. Fatigue life (Birnbaum-Saunders) argument
5. Empirical model fitting - distribution free
(Kaplan-Meier) approach
2. How do you plot reliability data?
1. Probability plotting
2. Hazard and cum hazard plotting
3. Trend and growth plotting (Duane plots)
3. How can you test reliability model assumptions?
1. Visual tests
2. Goodness of fit tests
3. Likelihood ratio tests
4. Trend tests
4. How do you choose an appropriate physical acceleration
model?
5. What models and assumptions are typically made when
Bayesian methods are used for reliability evaluation?

8.2.1. How do you choose an appropriate life distribution model?

8.2.1. How do you choose an appropriate life
distribution model?
Choose
models that
make sense,
fit the data
and,
hopefully,
have a
plausible
theoretical
justification
Life distribution models are chosen for one or more of the
following three reasons:
1. There is a physical/statistical argument that
theoretically matches a failure mechanism to a life
distribution model
2. A particular model has previously been used
successfully for the same or a similar failure
mechanism
3. A convenient model provides a good empirical fit to
all the failure data
Whatever method is used to choose a model, the model
should
"make sense" - for example, don't use an exponential
model with a constant failure rate to model a "wear
out" failure mechanism
pass visual and statistical tests for fitting the data.
Models like the lognormal and the Weibull are so flexible
that it is not uncommon for both to fit a small set of failure
data equally well. Yet, especially when projecting via
acceleration models to a use condition far removed from the
test data, these two models may predict failure rates that
differ by orders of magnitude. That is why it is more than an
academic exercise to try to find a theoretical justification for
using a particular distribution.
There are
several
useful
theoretical
arguments
to help
guide the
choice of a
model
We will consider three well-known arguments of this type:
Extreme value argument
Multiplicative degradation argument
Fatigue life (Birnbaum-Saunders) model
Note that physical/statistical arguments for choosing a life
distribution model are typically based on individual failure
modes.
For some
questions,
The Kaplan-Meier technique can be used when it is
appropriate to just "let the data points speak for themselves"
an
"empirical"
distribution-
free
approach
can be used
without making any model assumptions. However, you
generally need a considerable amount of data for this
approach to be useful, and acceleration modeling is much
more difficult.
8.2.1.1. Based on failure mode

8.2.1.1. Based on failure mode
Life
distribution
models and
physical
acceleration
models
typically
only apply
at the
individual
failure
mode level
Failure mode data are failure data sorted by types of failures.
Root cause analysis must be done on each failure incident in
order to characterize them by failure mode. While this may
be difficult and costly, it is a key part of any serious effort to
understand, model, project and improve component or
system reliability.
The natural place to apply both life distribution models and
physical acceleration models is at the failure mode level.
Each component failure mode will typically have its own life
distribution model. The same is true for acceleration models.
For the most part, these models only make sense at the
failure mode level, and not at the component or system level.
Once each mode (or mechanism) is modeled, the bottom-up
approach can be used to build up to the entire component or
system.
In particular, the arguments for choosing a life distribution
model described in the next 3 sections apply at the failure
mode level only. These are the Extreme value argument, the
Multiplicative degradation argument and the Fatigue life
(Birnbaum-Saunders) model.
The distribution-free (Kaplan - Meier) approach can be
applied at any level (mode, component, system, etc.).
8.2.1.2. Extreme value argument

If
component
or system
failure
occurs
when the
first of
many
competing
failure
processes
reaches a
critical
point, then
Extreme
Value
Theory
suggests
that the
Weibull
Distribution
will be a
good model
It is well known that the Central Limit Theorem suggests
that normal distributions will successfully model most
engineering data when the observed measurements arise from
the sum of many small random sources (such as measurement
errors). Practical experience validates this theory - the
normal distribution "works" for many engineering data sets.
Less known is the fact that Extreme Value Theory suggests
that the Weibull distribution will successfully model failure
times for mechanisms for which many competing similar
failure processes are "racing" to failure and the first to reach
it (i.e., the minimum of a large collection of roughly
comparable random failure times) produces the observed
failure time. Analogously, when a large number of roughly
equivalent runners are competing and the winning time is
recorded for many similar races, these times are likely to
follow a Weibull distribution.
Note that this does not mean that anytime there are several
failure mechanisms competing to cause a component or
system to fail, the Weibull model applies. One or a few of
these mechanisms may dominate the others and cause almost
all of the failures. Then the "minimum of a large number of
roughly comparable" random failure times does not apply and
the proper model should be derived from the distribution
models for the few dominating mechanisms using the
competing risk model.
On the other hand, there are many cases in which failure
occurs at the weakest link of a large number of similar
degradation processes or defect flaws. One example of this
occurs when modeling catastrophic failures of capacitors
caused by dielectric material breakdown. Typical dielectric
material has many "flaws" or microscopic sites where a
breakdown will eventually take place. These sites may be
thought of as competing with each other to reach failure
first. The Weibull model, as extreme value theory would
suggest, has been very successful as a life distribution model
for this failure mechanism.
8.2.1.3. Multiplicative degradation argument

The
lognormal
model can be
applied when
degradation
is caused by
random
shocks that
increase
degradation
at a rate
proportional
to the total
amount
already
present
A brief verbal description of the multiplicative degradation
argument (leading to a derivation of the lognormal model)
was given under Uses of the Lognormal Distribution Model.
Here a formal derivation will be outlined because it gives
insight into why the lognormal has been a successful model
for many failure mechanisms based on degradation
processes.
Let y
1
, y
2
, ...y
n
be measurements of the amount of
degradation for a particular failure process taken at
successive discrete instants of time as the process moves
towards failure. Assume the following relationships exist
between the y's:
where the are small, independent random perturbations
or "shocks" to the system that move the failure process
along. In other words, the increase in the amount of
degradation from one instant to the next is a small random
multiple of the total amount of degradation already present.
This is what is meant by multiplicative degradation. The
situation is analogous to a snowball rolling down a snow
covered hill; the larger it becomes, the faster it grows
because it is able to pick up even more snow.
We can express the total amount of degradation at the n-th
instant of time by
where x
0
is a constant and the are small random shocks.
Next we take natural logarithms of both sides and obtain:
Using a Central Limit Theorem argument we can conclude
that ln x
n
has approximately a normal distribution. But by
the properties of the lognormal distribution, this means that
x
n
(or the amount of degradation) will follow approximately
a lognormal model for any n (or at any time t). Since failure
occurs when the amount of degradation reaches a critical
point, time of failure will be modeled successfully by a
lognormal for this type of process.
Failure
mechanisms
that might be
successfully
modeled by
the
lognormal
distribution
based on the
multiplicative
degradation
model
What kinds of failure mechanisms might be expected to
follow a multiplicative degradation model? The processes
listed below are likely candidates:
1. Chemical reactions leading to the formation of new
compounds
2. Diffusion or migration of ions
3. Crack growth or propagation
Many semiconductor failure modes are caused by one of
these three degradation processes. Therefore, it is no
surprise that the lognormal model has been very successful
for the following semiconductor wear out failure
mechanisms:
1. Corrosion
2. Metal migration
3. Electromigration
4. Diffusion
5. Crack growth
8.2.1.4. Fatigue life (Birnbaum-Saunders) model

8.2.1.4. Fatigue life (Birnbaum-Saunders) model
A model
derived
from
random
crack
growth
occurring
during
many
independent
cycles of
stress
The derivation of the Fatigue Life model is based on repeated
cycles of stress causing degradation leading to eventual
failure. The typical example is crack growth. One key
assumption is that the amount of degradation during any
cycle is independent of the degradation in any other cycle,
with the same random distribution.
When this assumption matches well with a hypothesized
physical model describing the degradation process, one would
expect the Birnbaum-Saunders model to be a reasonable
distribution model candidate. (See the note in the derivation
for comments about the difference between the lognormal
model derivation and the Fatigue Life model assumptions.
Also see the comment on Miner's Rule).
8.2.1.5. Empirical model fitting - distribution free (Kaplan-Meier) approach

8.2.1.5. Empirical model fitting - distribution
free (Kaplan-Meier) approach
The
Kaplan-
Meier
procedure
gives CDF
estimates
for complete
or censored
sample data
without
assuming a
particular
distribution
model
The Kaplan-Meier (K-M) Product Limit procedure provides
quick, simple estimates of the Reliability function or the
CDF based on failure data that may even be multicensored.
No underlying model (such as Weibull or lognormal) is
assumed; K-M estimation is an empirical (non-parametric)
procedure. Exact times of failure are required, however.
Calculating Kaplan - Meier Estimates
The steps for calculating K-M estimates are the following:
1. Order the actual failure times from t
1
through t
r
, where
there are r failures
2. Corresponding to each t
i
, associate the number n
i
, with
n
i
= the number of operating units just before the the i-
th failure occurred at time t
i
3. Estimate R(t
1
) by (n
1
-1)/n
1
4. Estimate R(t
i
) by R(t
i-1
) (n
i
-1)/n
i
5. Estimate the CDF F(t
i
) by 1 - R(t
i
)
Note that unfailed units taken off test (i.e., censored) only
count up to the last actual failure time before they were
removed. They are included in the n
i
counts up to and
including that failure time, but not after.
Example of
K-M
estimate
calculations
A simple example will illustrate the K-M procedure. Assume
20 units are on life test and 6 failures occur at the following
times: 10, 32, 56, 98, 122, and 181 hours. There were 4
unfailed units removed from the test for other experiments at
the following times: 50 100 125 and 150 hours. The
remaining 10 unfailed units were removed from the test at
200 hours. The K-M estimates for this life test are:
R(10) = 19/20
R(32) = 19/20 x 18/19
R(56) = 19/20 x 18/19 x 16/17
R(98) = 19/20 x 18/19 x 16/17 x 15/16
8.2.1.5. Empirical model fitting - distribution free (Kaplan-Meier) approach
R(122) = 19/20 x 18/19 x 16/17 x 15/16 x 13/14
R(181) = 19/20 x 18/19 x 16/17 x 15/16 x 13/14 x 10/11
A General Expression for K-M Estimates
A general expression for the K-M estimates can be written.
Assume we have n units on test and order the observed times
for these n units from t
1
to t
n
. Some of these are actual
failure times and some are running times for units taken off
test before they fail. Keep track of all the indices
corresponding to actual failure times. Then the K-M
estimates are given by:
with the "hat" over R indicating it is an estimate and S is the
set of all subscripts j such that t
j
is an actual failure time. The
notation j S and t
j
less than or equal to t
i
means we only
form products for indices j that are in S and also correspond
to times of failure less than or equal to t
i
.
Once values for R(t
i
) are calculated, the CDF estimates are
F(t
i
) = 1 - R(t
i
)
A small
modification
of K-M
estimates
produces
better
results for
probability
plotting
Modified K-M Estimates
The K-M estimate at the time of the last failure is R(t
r
)= 0
and F(t
r
) = 1. This estimate has a pessimistic bias and
cannot be plotted (without modification) on a probability plot
since the CDF for standard reliability models asymptotically
approaches 1 as time approaches infinity. Better estimates for
graphical plotting can be obtained by modifying the K-M
estimates so that they reduce to the median rank estimates for
plotting Type I Censored life test data (described in the next
section). Modified K-M estimates are given by the formula
Once values for R(t
i
) are calculated, the CDF estimates are
F(t
i
) = 1 - R(t
i
)
8.2.2. How do you plot reliability data?

Create a
probability
plot and if the
points line up
approximately
on a straight
line, the
assumed
model is a
reasonable fit
Graphical plots of reliability data are quick, useful visual
tests of whether a particular model is consistent with the
observed data. The basic idea behind virtually all graphical
plotting techniques is the following:
Points calculated from the data are plotted on a
log-log scale and, as long as they line up
approximately on a straight line, the analyst
can conclude that the data are consistent with
the assumed model.
If the reliability data consist of (possibly multicensored)
failure data from a non repairable population (or a
repairable population for which only time to the first failure
is considered) then the models are life distribution models
such as the exponential, Weibull or lognormal. If the data
consist of repair times for a repairable system, then the
model might be the NHPP Power Law and the plot would
be a Duane Plot.
The kinds of plots we will consider for failure data from
non-repairable populations are:
Probability (CDF) plots
Hazard and Cum Hazard plots
For repairable populations we have
Trend plots (to check whether an HPP or exponential
model applies)
Duane plots (to check whether the NHPP Power Law
applies)
Later on (Section 8.4.2.1) we will also look at plots that
can be used to check acceleration model assumptions.
Note: Many of the plots discussed in this section can also
be used to obtain quick estimates of model parameters.
This will be covered in later sections. While there may be
other, more accurate ways of estimating parameters, simple
graphical estimates can be very handy, especially when
other techniques require software programs that are not
readily available.
8.2.2.1. Probability plotting

Use
probability
plots to see
your data
and visually
check
model
assumptions
Probability plots are simple visual ways of summarizing reliability data by plotting
CDF estimates versus time using a log-log scale.
The x axis is labeled "Time" and the y axis is labeled "cumulative percent" or
"percentile". There are rules, independent of the model, for calculating plotting
positions (points) from the reliability data. These only depend on the type of
censoring in the data and whether exact times of failure are recorded or only readout
times.
Plot each
failure
mode
separately
Remember that different failure modes can and should be separated out and
individually analyzed. When analyzing failure mode A, for example, treat failure
times from failure modes B, C, etc., as censored run times. Then repeat for failure
mode B, and so on.
Data points
line up
roughly on
a straight
line when
the model
chosen is
reasonable
When the points are plotted, the analyst fits a straight line to the data (either by eye,
or with the aid of a least squares fitting program). Every straight line on, say, a
Weibull probability plot uniquely corresponds to a particular Weibull life
distribution model and the same is true for lognormal or exponential plots. If the
points follow the line reasonably well, then the model is consistent with the data. If
it was your previously chosen model, there is no reason to question the choice. In
addition, there is a simple way to find the parameter estimates that correspond to the
fitted straight line.
Plotting
positions on
the x axis
depend on
the type of
data
censoring
Plotting Positions: Censored Data (Type I or Type II)
At the time t
i
of the i-th failure, we need an estimate of the CDF (or the cumulative
population percent failure). The simplest and most obvious estimate is just 100 i/n
(with a total of n units on test). This, however, is generally an overestimate (i.e.
biased). Various texts recommend corrections such as 100 (i-0.5)/n or 100
i/(n+1). Here, we recommend what are known as (approximate) median rank
estimates.
For each time t
i
of the i-th failure, calculate the CDF or percentile estimate using
100 (i - 0.3)/(n + 0.4).
Plotting Positions: Readout Data
Let the readout times be T
1
, T
2
, ..., T
k
and let the corresponding new failures
recorded at each readout be r
1
, r
2
, ..., r
k
. Again, there are n units on test.
For each readout time T
j
, calculate the CDF or percentile estimate using
Plotting Positions: Multicensored Data
The calculations are more complicated for multicensored data. K-M estimates
(described in a preceding section) can be used to obtain plotting positions at every
failure time. The more precise Modified K-M Estimates are recommended. They
reduce to the Censored Type I or the Censored Type II median rank estimates when
the data consist of only failures, without any removals except possibly at the end of
the test.
Reliability Models
Plotting
positions on
the y axis
depend on
the
reliability
model
The general idea is to take the model CDF equation and write it in such a way that a
function of F(t) is a linear equation of a function of t. This will be clear after a few
examples. In the formulas that follow, "ln" always means "natural logarithm", while
"log" always means "base 10 logarithm".
a) Exponential Model: Rewrite the exponential CDF as
If we let y = 1/{1 - F(t)} and x = t, then log(y) is linear in x with slope / ln10.
Thus, we can make an exponential probability plot by using a logarithmic y axis.
Use the plotting position estimates for F(t
i
) described above (without the 100
multiplier) to calculate pairs of (x
i
, y
i
) points.
If the data are consistent with an exponential model, the resulting plot will have
points that line up almost as a straight line going through the origin with slope /
ln10.
b) Weibull Model: Rewrite the Weibull CDF as
If we let y = ln [1/{1-F(t)}] and x = t, then log(y) is linear in log(x) with slope .
Thus, we can make a Weibull probability plot using a log-log scale. Use the plotting
position estimates for F(t
i
) (without the 100 multiplier) to calculate pairs of (x
i
,
y
i
) points.
If the data are consistent with a Weibull model, the resulting plot will have points
that line up roughly on a straight line with slope . This line will cross the log x axis
at time t = and the log y axis (i.e., the intercept) at - log .
c) Lognormal Model: Rewrite the lognormal cdf as
with denoting the inverse function for the standard normal distribution (taking
a probability as an argument and returning the corresponding "z" value).
If we let y = t and x = {F(t)}, then log y is linear in x with slope / ln10 and
intercept (when F(t) = 0.5) of log T
50
. We generate a lognormal probability plot
using a logarithmic y axis. Use the plotting position estimates for F(t
i
) (without the
100 multiplier) to calculate pairs of (x
i
, y
i
) points.
If the data are consistent with a lognormal model, the resulting plot will have points
that line up roughly on a straight line with slope / ln10 and intercept T
50
on the log
y axis.
d) Extreme Value Distribution (Type I - for minimum): Rewrite the extreme
value distribution CDF as
If we let y = -ln(1 - F(x)), then ln y is linear in x with slope 1 / and intercept - /
. We plot y versus x where the y axis is base 10 logarithmic. The points should
follow a straight line with a slope of (1 / )ln10 and an intercept of (- / )ln10.
The ln10 factors in the slope and intercept are needed because the plot uses a base
10 logarithmic axis.
Example
A Weibull
example of
probability
plotting
We generated 20 random Weibull failure times with a shape parameter of = 1.5
and
= 500. Assuming a test time of T = 500 hours, only 10 of these failure times
would have been observed. They are, to the nearest hour: 54, 187, 216, 240, 244,
335, 361, 373, 375, and 386. We will compute plotting position CDF estimates
based on these failure times, and then generate a probability plot.
( 1)
Failure
(i)
( 2)
Time of Failure
(x)
(3)
F(t
i
) estimate
(i-0.3)/(20+0.4)
(4)
ln{1/(1-F(t
i
)}
(y)
1 54 .034 .035
2 187 .083 .087
3 216 .132 .142
4 240 .181 .200
5 244 .230 .262
6 335 .279 .328
7 361 .328 .398
8 373 .377 .474
9 375 .426 .556
10 386 .475 .645
We generate a probability plot using column (4) versus column (2) and log-log scale
axes.
Note that the configuration of points appears to have some curvature. This is mostly
due to the very first point on the plot (the earliest time of failure). The first few
points on a probability plot have more variability than points in the central range
and less attention should be paid to them when visually testing for "straightness".
Use of least
squares
(regression)
to fit a line
through the
points on a
Since our data are plotted on a log-log scale, we fit a straight line using log(x) as the
independent variable and log(y) as the dependent variable.
The regression produces a slope estimate of 1.46, which is close to the 1.5 value
used in the simulation. The intercept is -4.114 and setting this equal to - log we
estimate = 657 (the "true" value used in the simulation was 500).
probability
plot
The analyses in this section can can be implemented using both Dataplot code and
R code. Both packages have special functions to automatically generate probability
plots for a wide variety of distributions.
8.2.2.2. Hazard and cumulative hazard plotting

Cumulative
Hazard
Plotting
has the
same
purpose as
probability
plotting
Similar to probability plots, cumulative hazard plots are used for visually
examining distributional model assumptions for reliability data and have
a similar interpretation as probability plots. The cumulative hazard plot
consists of a plot of the cumulative hazard H(t
i
) versus the time t
i
of the
i-th failure. As with probability plots, the plotting positions are
calculated independently of the model and a reasonable straight-line fit
to the points confirms that the chosen model and the data are consistent.
Advantages of Cumulative Hazard Plotting
1. It is much easier to calculate plotting positions for multicensored
data using cumulative hazard plotting techniques.
2. The most common reliability distributions, the exponential and the
Weibull, are easily plotted.
Disadvantages of Cumulative Hazard Plotting
1. It is less intuitively clear just what is being plotted. In a
probability plot, the cumulative percent failed is meaningful and
the resulting straight-line fit can be used to identify times when
desired percentages of the population will have failed. The percent
cumulative hazard can increase beyond 100 % and is harder to
interpret.
2. Normal cumulative hazard plotting techniques require exact times
of failure and running times.
3. Since computers are able to calculate K-M estimates for
probability plotting, the main advantage of cumulative hazard
plotting goes away.
Since probability plots are generally more useful, we will only give a
brief description of hazard plotting.
How to Make Cumulative Hazard Plots
1. Order the failure times and running times for each of the n units on
test in ascending order from 1 to n. The order is called the rank of
the unit. Calculate the reverse rank for each unit (reverse rank = n-
rank +1).
2. Calculate a hazard "value" for every failed unit (do this only for
the failed units). The hazard value for the failed unit with reverse
rank k is just 1/k.
3. Calculate the cumulative hazard values for each failed unit. The
cumulative hazard value corresponding to a particular failed unit is
the sum of all the hazard values for failed units with ranks up to
and including that failed unit.
4. Plot the time of failure versus the cumulative hazard value. Linear
x and y scales are appropriate for an exponential distribution,
while a log-log scale is appropriate for a Weibull distribution.
A life test
cumulative
hazard
plotting
example
Example: Ten units were tested at high stress test for up to 250 hours.
Six failures occurred at 37, 73, 132, 195, 222 and 248 hours. Four units
were taken off test without failing at the following run times: 50, 100,
200 and 250 hours. Cumulative hazard values were computed in the
following table.
(1)
Time of
Event
(2)
1= failure
0=runtime
(3)
Rank
(4)
Reverse
Rank
(5)
Haz Val
(2) x
1/(4)
(6)
Cum Hazard
Value
37 1 1 10 1/10 .10
50 0 2 9
73 1 3 8 1/8 .225
100 0 4 7
132 1 5 6 1/6 .391
195 1 6 5 1/5 .591
200 0 7 4
222 1 8 3 1/3 .924
248 1 9 2 1/2 1.424
250 0 10 1
Next ignore the rows with no cumulative hazard value and plot column
(1) vs column (6).
Plots of
example
data
Exponential and Weibull Cumulative Hazard Plots
The cumulative hazard for the exponential distribution is just H(t) = t,
which is linear in t with an intercept of zero. So a simple linear graph of
y = column (6) versus x = column (1) should line up as approximately a
straight line going through the origin with slope if the exponential
model is appropriate. The cumulative hazard plot for exponential
distribution is shown below.
The cumulative hazard for the Weibull distribution is H(t) = (t / )
, so a
plot of y versus x on a log-log scale should resemble a straight line with
slope if the Weibull model is appropriate. The cumulative hazard plot
for the Weibull distribution is shown below.
A least-squares regression fit of the data (using base 10 logarithms to
transform columns (1) and (6)) indicates that the estimated slope for the
Weibull distribution is 1.27, which is fairly similar to the exponential
model slope of 1. The Weibull fit looks somewhat better than the
exponential fit; however, with a sample of just 10, and only 6 failures, it
is difficult to pick a model from the data alone.
Software The analyses in this section can can be implemented using both Dataplot
code and R code.
8.2.2.3. Trend and growth plotting (Duane plots)

Repair rates
are typically
either nearly
constant
over time or
else
consistently
follow a
good or a
bad trend
Models for repairable systems were described earlier. These models are
for the cumulative number of failuress (or the repair rate) over time. The
two models used with most success throughout industry are the HPP
(constant repair rate or "exponential" system model) and the NHPP
Power Law process (the repair rate is the polynomial m(t) = t
-
).
Before constructing a Duane Plot, there are a few simple trend plots that
often convey strong evidence of the presence or absence of a trend in the
repair rate over time. If there is no trend, an HPP model is reasonable. If
there is an apparent improvement or degradation trend, a Duane Plot will
provide a visual check for whether the NHPP Power law model is
consistent with the data.
A few simple
plots can
help us
decide
whether
trends are
present
These simple visual graphical tests for trends are
1. Plot cumulative failures versus system age (a step function that
goes up every time there is a new failure). If this plot looks linear,
there is no obvious improvement (or degradation) trend. A bending
downward indicates improvement; bending upward indicates
degradation.
2. Plot the inter arrival times between new failures (in other words,
the waiting times between failures, with the time to the first failure
used as the first "inter-arrival" time). If these trend up, there is
improvement; if they trend down, there is degradation.
3. Plot the reciprocals of the inter-arrival times. Each reciprocal is a
new failure rate estimate based only on the waiting time since the
last failure. If these trend down, there is improvement; an upward
trend indicates degradation.
Trend plots
and a Duane
Plot for
actual
Reliability
Improvement
Test data
Case Study 1: Use of Trend Plots and Duane Plots with Reliability
Improvement Test Data
A prototype of a new, complex piece of equipment went through a 1500
operational hours Reliability Improvement Test. During the test there
were 10 failures. As part of the improvement process, a cross functional
Failure Review Board made sure every failure was analyzed down to the
root cause and design and parts selection fixes were implemented on the
prototype. The observed failure times were: 5, 40, 43, 175, 389, 712, 747,
795, 1299 and 1478 hours, with the test ending at 1500 hours. The
reliability engineer on the Failure Review Board first made trend plots as
described above, then made a Duane plot. These plots follow.
Time Cum MTBF
5 5
40 20
43 14.3
175 43.75
389 77.8
712 118.67
747 106.7
795 99.4
1299 144.3
1478 147.8
Comments: The three trend plots all show an improvement trend. The
reason it might be useful to try all three trend plots is that a trend might
show up more clearly on one plot than the others. Formal statistical tests
on the significance of this visual evidence of a trend will be shown in the
section on Trend Tests.
The points on the Duane Plot line up roughly as a straight line, indicating
the NHPP Power Law model is consistent with the data.
Estimates for the reliability growth slope and the MTBF at the end of
this test for this case study will be given in a later section.
8.2.3. How can you test reliability model assumptions?

8.2.3. How can you test reliability model
assumptions?
Models
are
frequently
necessary
- but
should
always be
checked
Since reliability models are often used to project (extrapolate)
failure rates or MTBF's that are well beyond the range of the
reliability data used to fit these models, it is very important to
"test" whether the models chosen are consistent with whatever
data are available. This section describes several ways of
deciding whether a model under examination is acceptable.
These are:
1. Visual Tests
2. Goodness of Fit Tests
3. Likelihood Ratio Tests
4. Trend Tests
8.2.3.1. Visual tests

A visual
test of a
model is a
simple plot
that tells us
at a glance
whether the
model is
consistent
with the
data
We have already seen many examples of visual tests of
models. These were: Probability Plots, Cum hazard Plots,
Duane Plots and Trend Plots. In all but the Trend Plots, the
model was "tested' by how well the data points followed a
straight line. In the case of the Trend Plots, we looked for
curvature away from a straight line (cum repair plots) or
increasing or decreasing size trends (inter arrival times and
reciprocal inter-arrival times).
These simple plots are a powerful diagnostic tool since the
human eye can often detect patterns or anomalies in the data
by studying graphs. That kind of invaluable information
would be lost if the analyst only used quantitative statistical
tests to check model fit. Every analysis should include as
many visual tests as are applicable.
Advantages of Visual Tests
1. Easy to understand and explain.
2. Can occasionally reveal patterns or anomalies in the
data.
3. When a model "passes" a visual test, it is somewhat
unlikely any quantitative statistical test will "reject" it
(the human eye is less forgiving and more likely to
detect spurious trends)
Combine
visual tests
with formal
quantitative
tests for the
"best of
both
worlds"
approach
Disadvantages of Visual Tests
1. Visual tests are subjective.
2. They do not quantify how well or how poorly a model
fits the data.
3. They are of little help in choosing between two or more
competing models that both appear to fit the data.
4. Simulation studies have shown that correct models may
often appear to not fit well by sheer chance - it is hard
to know when visual evidence is strong enough to
reject what was previously believed to be a correct
model.
You can retain the advantages of visual tests and remove
their disadvantages by combining data plots with formal
statistical tests of goodness of fit or trend.
8.2.3.2. Goodness of fit tests

8.2.3.2. Goodness of fit tests
A
Goodness
of Fit test
checks on
whether
your data
are
reasonable
or highly
unlikely,
given an
assumed
distribution
model
General tests for checking the hypothesis that your data are
consistent with a particular model are discussed in Chapter 7.
Details and examples of the Chi-Square Goodness of Fit test
and the Kolmolgorov-Smirnov (K-S) test are given in
Chapter 1. The Chi-Square test can be used with Type I or
Type II censored data and readout data if there are enough
failures and readout times. The K-S test generally requires
complete samples, which limits its usefulness in reliability
analysis.
These tests control the probability of rejecting a valid model
as follows:
the analyst chooses a confidence level designated by
100 (1 - ).
a test statistic is calculated from the data and compared
to likely values for this statistic, assuming the model is
correct.
if the test statistic has a very unlikely value, or less than
or equal to an probability of occurring, where is a
small value like .1 or .05 or even .01, then the model is
rejected.
So the risk of rejecting the right model is kept to or less,
and the choice of usually takes into account the potential
loss or difficulties incurred if the model is rejected.
8.2.3.3. Likelihood ratio tests

Likelihood
Ratio Tests
are a
powerful,
very general
method of
testing
model
assumptions.
However,
they require
special
software,
not always
readily
available.
Likelihood functions for reliability data are described in
Section 4. Two ways we use likelihood functions to choose
models or verify/validate assumptions are:
1. Calculate the maximum likelihood of the sample data
based on an assumed distribution model (the maximum
occurs when unknown parameters are replaced by their
maximum likelihood estimates). Repeat this calculation for
other candidate distribution models that also appear to fit the
data (based on probability plots). If all the models have the
same number of unknown parameters, and there is no
convincing reason to choose one particular model over
another based on the failure mechanism or previous
successful analyses, then pick the model with the largest
likelihood value.
2. Many model assumptions can be viewed as putting
restrictions on the parameters in a likelihood expression that
effectively reduce the total number of unknown parameters.
Some common examples are:
Examples
where
assumptions
can be
tested by the
Likelihood
Ratio Test
i) It is suspected that a type of data, typically
modeled by a Weibull distribution, can be fit
adequately by an exponential model. The
exponential distribution is a special case of the
Weibull, with the shape parameter set to 1. If
we write the Weibull likelihood function for the
data, the exponential model likelihood function
is obtained by setting to 1, and the number of
unknown parameters has been reduced from two
to one.
ii) Assume we have n cells of data from an
acceleration test, with each cell having a
different operating temperature. We assume a
lognormal population model applies in every
cell. Without an acceleration model assumption,
the likelihood of the experimental data would be
the product of the likelihoods from each cell
and there would be 2n unknown parameters (a
different T
50
and for each cell). If we assume
an Arrhenius model applies, the total number of
parameters drops from 2n to just 3, the single
common and the Arrhenius A and H
parameters. This acceleration assumption
"saves" (2n-3) parameters.
iii) We life test samples of product from two
vendors. The product is known to have a failure
mechanism modeled by the Weibull distribution,
and we want to know whether there is a
difference in reliability between the vendors.
The unrestricted likelihood of the data is the
product of the two likelihoods, with 4 unknown
parameters (the shape and characteristic life for
each vendor population). If, however, we
assume no difference between vendors, the
likelihood reduces to having only two unknown
parameters (the common shape and the common
characteristic life). Two parameters are "lost" by
the assumption of "no difference".
Clearly, we could come up with many more examples like
these three, for which an important assumption can be
restated as a reduction or restriction on the number of
parameters used to formulate the likelihood function of the
data. In all these cases, there is a simple and very useful way
to test whether the assumption is consistent with the data.
The Likelihood Ratio Test Procedure
Details of
the
Likelihood
Ratio Test
procedure
In general,
calculations
are difficult
and need to
be built into
the software
you use
Let L
1
be the maximum value of the likelihood of the data
without the additional assumption. In other words, L
1
is the
likelihood of the data with all the parameters unrestricted
and maximum likelihood estimates substituted for these
parameters.
Let L
0
be the maximum value of the likelihood when the
parameters are restricted (and reduced in number) based on
the assumption. Assume k parameters were lost (i.e., L
0
has
k less parameters than L
1
).
Form the ratio = L
0
/L
1
. This ratio is always between 0
and 1 and the less likely the assumption is, the smaller
will be. This can be quantified at a given confidence level as
follows:
1. Calculate = -2 ln . The smaller is, the larger
will be.
2. We can tell when is significantly large by
comparing it to the 100 (1- ) percentile point of a
Chi Square distribution with k degrees of freedom.
has an approximate Chi-Square distribution with k
degrees of freedom and the approximation is usually
good, even for small sample sizes.
3. The likelihood ratio test computes and rejects the
assumption if is larger than a Chi-Square
percentile with k degrees of freedom, where the
percentile corresponds to the confidence level chosen
by the analyst.
Note: While Likelihood Ratio test procedures are very
useful and widely applicable, the computations are difficult
to perform by hand, especially for censored data, and
appropriate software is necessary.
8.2.3.4. Trend tests

Formal
Trend Tests
should
accompany
Trend Plots
and Duane
Plots. Three
are given in
this section
In this section we look at formal statistical tests that can
allow us to quantitatively determine whether or not the
repair times of a system show a significant trend (which
may be an improvement or a degradation trend). The
section on trend and growth plotting contained a discussion
of visual tests for trends - this section complements those
visual tests as several numerical tests are presented.
Three statistical test procedures will be described:
1. The Reverse Arrangement Test (a simple and useful
test that has the advantage of making no assumptions
about a model for the possible trend)
2. The Military Handbook Test (optimal for
distinguishing between "no trend' and a trend
following the NHPP Power Law or Duane model)
3. The Laplace Test (optimal for distinguishing between
"no trend' and a trend following the NHPP
Exponential Law model)
The Reverse
Arrangement
Test (RAT
test) is simple
and makes no
assumptions
about what
model a trend
might follow
The Reverse Arrangement Test
Assume there are r repairs during the observation period
and they occurred at system ages T
1
, T
2
, T
3
, ...T
r
(we set the
start of the observation period to T = 0). Let I
1
= T
1
,
I
2
= T
2
- T
1
, I
3
= T
3
- T
2
, ..., I
r
= T
r
- T
r-1
be the inter-
arrival times for repairs (i.e., the sequence of waiting times
between failures). Assume the observation period ends at
time T
end
>T
r
.
Previously, we plotted this sequence of inter-arrival times
to look for evidence of trends. Now, we calculate how
many instances we have of a later inter-arrival time being
strictly greater than an earlier inter-arrival time. Each time
that happens, we call it a reversal. If there are a lot of
reversals (more than are likely from pure chance with no
trend), we have significant evidence of an improvement
trend. If there are too few reversals we have significant
evidence of degradation.
A formal definition of the reversal count and some
properties of this count are:
count a reversal every time I
j
< I
k
for some j and k
with j < k
this reversal count is the total number of reversals R
for r repair times, the maximum possible number of
reversals is r(r-1)/2
if there are no trends, on the average one would
expect to have r(r-1)/4 reversals.
As a simple example, assume we have 5 repair times at
system ages 22, 58, 71, 156 and 225, and the observation
period ended at system age 300 . First calculate the inter
arrival times and obtain: 22, 36, 13, 85, 69. Next, count
reversals by "putting your finger" on the first inter-arrival
time, 22, and counting how many later inter arrival times
are greater than that. In this case, there are 3. Continue by
"moving your finger" to the second time, 36, and counting
how many later times are greater. There are exactly 2.
Repeating this for the third and fourth inter-arrival times
(with many repairs, your finger gets very tired!) we obtain 2
and 0 reversals, respectively. Adding 3 + 2 + 2 + 0 = 7, we
see that R = 7. The total possible number of reversals is
5x4/2 = 10 and an "average" number is half this, or 5.
In the example, we saw 7 reversals (2 more than average).
Is this strong evidence for an improvement trend? The
following table allows us to answer that at a 90% or 95%
or 99% confidence level - the higher the confidence, the
stronger the evidence of improvement (or the less likely that
pure chance alone produced the result).
A useful table
to check
whether a
reliability
test has
demonstrated
significant
improvement
Value of R Indicating Significant Improvement (One-Sided
Test)
Number
of
Repairs
Minimum R for
90% Evidence
of
Improvement
Minimum R for
95% Evidence
of
Improvement
Minimum R for
99% Evidence
of
Improvement
4 6 6 -
5 9 9 10
6 12 13 14
7 16 17 19
8 20 22 24
9 25 27 30
10 31 33 36
11 37 39 43
12 43 46 50
One-sided test means before looking at the data we
expected improvement trends, or, at worst, a constant repair
rate. This would be the case if we know of actions taken to
improve reliability (such as occur during reliability
improvement tests).
For the r = 5 repair times example above where we had R =
7, the table shows we do not (yet) have enough evidence to
demonstrate a significant improvement trend. That does not
mean that an improvement model is incorrect - it just means
it is not yet "proved" statistically. With small numbers of
repairs, it is not easy to obtain significant results.
For numbers of repairs beyond 12, there is a good
approximation formula that can be used to determine
whether R is large enough to be significant. Calculate
Use this
formula when
there are
more than 12
repairs in the
data set
and if z > 1.282, we have at least 90% significance. If z >
1.645, we have 95% significance, and a z > 2.33 indicates
99% significance since z has an approximate standard
That covers the (one-sided) test for significant
improvement trends. If, on the other hand, we believe there
may be a degradation trend (the system is wearing out or
being over stressed, for example) and we want to know if
the data confirms this, then we expect a low value for R and
we need a table to determine when the value is low enough
to be significant. The table below gives these critical values
for R.
Value of R Indicating Significant Degradation Trend (One-
Sided Test)
Number
of
Repairs
Maximum R
for 90%
Evidence of
Degradation
Maximum R
for 95%
Evidence of
Degradation
Maximum R
for 99%
Evidence of
Degradation
4 0 0 -
5 1 1 0
6 3 2 1
7 5 4 2
8 8 6 4
9 11 9 6
10 14 12 9
11 18 16 12
12 23 20 16
For numbers of repairs r >12, use the approximation
formula above, with R replaced by [r(r-1)/2 - R].
Because of
the success of
the Duane
model with
industrial
improvement
test data, this
Trend Test is
recommended
The Military Handbook Test
This test is better at finding significance when the choice is
between no trend and a NHPP Power Law (Duane) model.
In other words, if the data come from a system following
the Power Law, this test will generally do better than any
other test in terms of finding significance.
As before, we have r times of repair T
1
, T
2
, T
3
, ...T
r
with
the observation period ending at time T
end
>T
r
. Calculate
and compare this to percentiles of the chi-square
distribution with 2r degrees of freedom. For a one-sided
improvement test, reject no trend (or HPP) in favor of an
improvement trend if the chi square value is beyond the 90
(or 95, or 99) percentile. For a one-sided degradation test,
reject no trend if the chi-square value is less than the 10 (or
5, or 1) percentile.
Applying this test to the 5 repair times example, the test
statistic has value 13.28 with 10 degrees of freedom, and
the chi-square percentile is 79%.
The Laplace Test
This test is better at finding significance when the choice is
between no trend and a NHPP Exponential model. In other
words, if the data come from a system following the
Exponential Law, this test will generally do better than any
test in terms of finding significance.
As before, we have r times of repair T
1
, T
2
, T
3
, ...T
r
with
the observation period ending at time T
end
>T
r
. Calculate
and compare this to high (for improvement) or low (for
degradation) percentiles of the standard normal distribution.
Formal tests Case Study 1: Reliability Test Improvement Data
generally
confirm the
subjective
information
conveyed by
trend plots
(Continued from earlier work)
The failure data and Trend plots and Duane plot were
shown earlier. The observed failure times were: 5, 40, 43,
175, 389, 712, 747, 795, 1299 and 1478 hours, with the test
ending at 1500 hours.
Reverse Arrangement Test: The inter-arrival times are: 5,
35, 3, 132, 214, 323, 35, 48, 504 and 179. The number of
reversals is 33, which, according to the table above, is just
significant at the 95% level.
The Military Handbook Test: The Chi-Square test
statistic, using the formula given above, is 37.23 with 20
degrees of freedom and has significance level 98.9%. Since
the Duane Plot looked very reasonable, this test probably
gives the most precise significance assessment of how
unlikely it is that sheer chance produced such an apparent
improvement trend (only about 1.1% probability).
8.2.4. How do you choose an appropriate physical acceleration model?

8.2.4. How do you choose an appropriate
physical acceleration model?
Choosing a
good
acceleration
model is
part science
and part art
- but start
with a good
literature
search
Choosing a physical acceleration model is a lot like choosing
a life distribution model. First identify the failure mode and
what stresses are relevant (i.e., will accelerate the failure
mechanism). Then check to see if the literature contains
examples of successful applications of a particular model for
this mechanism.
If the literature offers little help, try the models described in
earlier sections :
Arrhenius
The (inverse) power rule for voltage
The exponential voltage model
Two temperature/voltage models
The electromigration model
Three stress models (temperature, voltage and
humidity)
Eyring (for more than three stresses or when the above
models are not satisfactory)
The Coffin-Manson mechanical crack growth model
All but the last model (the Coffin-Manson) apply to chemical
or electronic failure mechanisms, and since temperature is
almost always a relevant stress for these mechanisms, the
Arrhenius model is nearly always a part of any more general
model. The Coffin-Manson model works well for many
mechanical fatigue-related mechanisms.
Sometimes models have to be adjusted to include a
threshold level for some stresses. In other words, failure
might never occur due to a particular mechanism unless a
particular stress (temperature, for example) is beyond a
threshold value. A model for a temperature-dependent
mechanism with a threshold at T = T
0
might look like
time to fail = f(T)/(T-T
0
)
for which f(T) could be Arrhenius. As the temperature
decreases towards T
0
, time to fail increases toward infinity
in this (deterministic) acceleration model.
8.2.4. How do you choose an appropriate physical acceleration model?
Models
derived
theoretically
have been
very
successful
and are
convincing
In some cases, a mathematical/physical description of the
failure mechanism can lead to an acceleration model. Some
of the models above were originally derived that way.
Simple
models are
often the
best
In general, use the simplest model (fewest parameters) you
can. When you have chosen a model, use visual tests and
formal statistical fit tests to confirm the model is consistent
with your data. Continue to use the model as long as it gives
results that "work," but be quick to look for a new model
when it is clear the old one is no longer adequate.
There are some good quotes that apply here:
Quotes from
experts on
models
"All models are wrong, but some are useful." - George Box,
and the principle of Occam's Razor (attributed to the 14th
century logician William of Occam who said Entities should
not be multiplied unnecessarily - or something equivalent to
that in Latin).
A modern version of Occam's Razor is: If you have two
theories that both explain the observed facts then you should
use the simplest one until more evidence comes along - also
called the Law of Parsimony.
Finally, for those who feel the above quotes place too much
emphasis on simplicity, there are several appropriate quotes
from Albert Einstein:
"Make your theory as simple as possible, but no
simpler"
"For every complex question there is a simple
and wrong solution."
8.2.5. What models and assumptions are typically made when Bayesian methods are used for reliability evaluation?

8.2.5. What models and assumptions are
typically made when Bayesian methods
are used for reliability evaluation?
The basics of Bayesian methodology were explained earlier,
along with some of the advantages and disadvantages of
using this approach. Here we only consider the models and
assumptions that are commonplace when applying Bayesian
methodology to evaluate system reliability.
Bayesian
assumptions
for the
gamma
exponential
system
model
Assumptions:
1. Failure times for the system under investigation can be
adequately modeled by the exponential distribution. For
repairable systems, this means the HPP model applies and the
system is operating in the flat portion of the bathtub curve.
While Bayesian methodology can also be applied to non-
repairable component populations, we will restrict ourselves
to the system application in this Handbook.
2. The MTBF for the system can be regarded as chosen from
a prior distribution model that is an analytic representation of
our previous information or judgments about the system's
reliability. The form of this prior model is the gamma
distribution (the conjugate prior for the exponential model).
The prior model is actually defined for = 1/MTBF since it
is easier to do the calculations this way.
3. Our prior knowledge is used to choose the gamma
parameters a and b for the prior distribution model for .
There are many possible ways to convert "knowledge" to
gamma parameters, depending on the form of the
"knowledge" - we will describe three approaches.
Several
ways to
choose the
prior
gamma
parameter
values
i) If you have actual data from previous testing done
on the system (or a system believed to have the same
reliability as the one under investigation), this is the
most credible prior knowledge, and the easiest to use.
Simply set the gamma parameter a equal to the total
number of failures from all the previous data, and set
the parameter b equal to the total of all the previous
test hours.
ii) A consensus method for determining a and b that
works well is the following: Assemble a group of
engineers who know the system and its sub-
components well from a reliability viewpoint.

Have the group reach agreement on a reasonable
MTBF they expect the system to have. They
could each pick a number they would be willing
to bet even money that the system would either
meet or miss, and the average or median of these
numbers would be their 50% best guess for the
MTBF. Or they could just discuss even-money
MTBF candidates until a consensus is reached.

Repeat the process again, this time reaching
agreement on a low MTBF they expect the
system to exceed. A "5%" value that they are
"95% confident" the system will exceed (i.e.,
they would give 19 to 1 odds) is a good choice.
Or a "10%" value might be chosen (i.e., they
would give 9 to 1 odds the actual MTBF exceeds
the low MTBF). Use whichever percentile choice
the group prefers.

Call the reasonable MTBF MTBF
50
and the low
MTBF you are 95% confident the system will
exceed MTBF
05
. These two numbers uniquely
determine gamma parameters a and b that have
percentile values at the right locations
We call this method of specifying gamma prior
parameters the 50/95 method (or the 50/90
method if we use MTBF
10
, etc.). A simple way
to calculate a and b for this method is described
below.
iii) A third way of choosing prior parameters starts the
same way as the second method. Consensus is reached
on an reasonable MTBF, MTBF
50
. Next, however, the
group decides they want a somewhatweak prior that
will change rapidly, based on new test information. If
the prior parameter "a" is set to 1, the gamma has a
standard deviation equal to its mean, which makes it
spread out, or "weak". To insure the 50th percentile is
set at
50
= 1/ MTBF
50
, we have to choose b = ln 2
MTBF
50
, which is approximately .6931 MTBF
50
.
Note: As we will see when we plan Bayesian tests, this
weak prior is actually a very friendly prior in terms of
saving test time
Many variations are possible, based on the above three
methods. For example, you might have prior data from
sources that you don't completely trust. Or you might
question whether the data really apply to the system under
investigation. You might decide to "weight" the prior data by
.5, to "weaken" it. This can be implemented by setting a = .5
x the number of fails in the prior data and b = .5 times the
number of test hours. That spreads out the prior distribution
more, and lets it react quicker to new test data.
Consequences
After a new
test is run,
the
posterior
gamma
parameters
are easily
obtained
from the
prior
parameters
by adding
the new
number of
fails to "a"
and the new
test time to
"b"
No matter how you arrive at values for the gamma prior
parameters a and b, the method for incorporating new test
information is the same. The new information is combined
with the prior model to produce an updated or posterior
distribution model for .
Under assumptions 1 and 2, when a new test is run with T
system operating hours and r failures, the posterior
distribution for is still a gamma, with new parameters:
a' = a + r, b' = b + T
In other words, add to a the number of new failures and add
to b the number of new test hours to obtain the new
parameters for the posterior distribution.
Use of the posterior distribution to estimate the system
MTBF (with confidence, or prediction, intervals) is described
in the section on estimating reliability using the Bayesian
gamma model.
Obtaining Gamma Parameters
An example
using the
"50/95"
consensus
method
A group of engineers, discussing the reliability of a new
piece of equipment, decide to use the 50/95 method to
convert their knowledge into a Bayesian gamma prior.
Consensus is reached on a likely MTBF
50
value of 600 hours
and a low MTBF
05
value of 250. RT is 600/250 = 2.4. (Note:
if the group felt that 250 was a MTBF
10
value, instead of a
MTBF
05
value, then the only change needed would be to
replace 0.95 in the B1 equation by 0.90. This would be the
"50/90" method.)
Using software to find the root of a univariate function, the
gamma prior parameters were found to be a = 2.863 and b =
1522.46. The parameters will have (approximately) a
probability of 50% of l being below 1/600 = 0.001667 and a
probability of 95% of being below 1/250 = 0.004. (The
probabilities are based on the 0.001667 and 0.004 quantiles
of a gamma distribution with shape parameter a = 2.863 and
scale parameter b = 1522.46)
The gamma parameter estimates in this example can be
produced using R code.
This example will be continued in Section 3, in which the
Bayesian test time needed to confirm a 500 hour MTBF at
80% confidence will be derived.
8.3. Reliability Data Collection

In order to assess or improve reliability, it is usually necessary
to have failure data. Failure data can be obtained from field
studies of system performance or from planned reliability
tests, sometimes called Life Tests. This section focuses on how
to plan reliability tests. The aim is to answer questions such
as: how long should you test, what sample size do you need
and what test conditions or stresses need to be run?
Detailed
contents of
Section 8.3
The section detailed outline follows.
3. Reliability Data Collection
1. How do you plan a reliability assessment test?
1. Exponential life distribution (or HPP model) tests
2. Lognormal or Weibull tests
3. Reliability growth tests (Duane model)
4. Accelerated life tests
5. Bayesian gamma prior model tests

8.3.1. How do you plan a reliability assessment test?

8.3.1. How do you plan a reliability assessment
test?
The Plan
for a
reliability
test ends
with a
detailed
description
of the
mechanics
of the test
and starts
with stating
your
assumptions
and what
you want to
discover or
prove
Planning a reliability test means:
How long should you test?
How many units have to be put on test?
For repairable systems, this is often limited to 1.
If acceleration modeling is part of the experimental
plan
What combination of stresses and how many
experimental cells?
How many units go in each cell?
The answers to these questions depend on:
What models are you assuming?
What decisions or conclusions do you want to make
after running the test and analyzing the data?
What risks are you willing to take of making wrong
decisions or conclusions?
It is not always possible, or practical, to completely answer
all of these questions for every model we might want to use.
This section looks at answers, or guidelines, for the following
models:
exponential or HPP Model
Weibull or lognormal model
Duane or NHPP Power Law model
acceleration models
Bayesian gamma prior model
8.3.1.1. Exponential life distribution (or HPP model) tests

8.3.1.1. Exponential life distribution (or HPP
model) tests
Using an
exponential
(or HPP)
model to
test whether
a system
meets its
MTBF
requirement
is common
in industry
Exponential tests are common in industry for verifying that
tools, systems or equipment are meeting their reliability
requirements for Mean Time Between Failure (MTBF). The
assumption is that the system has a constant failure (or repair)
rate, which is the reciprocal of the MTBF. The waiting time
between failures follows the exponential distribution model.
A typical test situation might be: a new complex piece of
equipment or tool is installed in a factory and monitored
closely for a period of several weeks to several months. If it
has no more than a pre-specified number of failures during
that period, the equipment "passes" its reliability acceptance
test.
This kind of reliability test is often called a Qualification
Test or a Product Reliability Acceptance Test (PRAT).
Contractual penalties may be invoked if the equipment fails
the test. Everything is pegged to meeting a customer MTBF
requirement at a specified confidence level.
How Long Must You Test A Piece of Equipment or a
System In order to Assure a Specified MTBF at a Given
Confidence?
You start with a given MTBF objective, say M, and a
confidence level, say 100 (1- ). You need one more piece
of information to determine the test length: how many fails
do you want to allow and still "pass" the equipment? The
more fails allowed, the longer the test required. However, a
longer test allowing more failures has the desirable feature of
making it less likely a good piece of equipment will be
rejected because of random "bad luck" during the test period.
The recommended procedure is to iterate on r = the number
of allowable fails until a larger r would require an
unacceptable test length. For any choice of r, the
corresponding test length is quickly calculated by multiplying
M (the objective) by the factor in the table below
corresponding to the r-th row and the desired confidence
level column.
For example, to confirm a 200-hour MTBF objective at 90%
confidence, allowing up to 4 failures on the test, the test
length must be 200 7.99 = 1598 hours. If this is
unacceptably long, try allowing only 3 fails for a test length
of 200 6.68 = 1336 hours. The shortest test would allow no
fails and last 200 2.3 = 460 hours. All these tests guarantee
a 200-hour MTBF at 90% confidence, when the equipment
passes. However, the shorter test are much less "fair" to the
supplier in that they have a large chance of failing a
marginally acceptable piece of equipment.
Use the
Test length
Table to
determine
how long to
test
Test Length Guide Table
NUMBER
OF
FAILURES
ALLOWED
FACTOR FOR GIVEN CONFIDENCE LEVELS
r 50% 60% 75% 80% 90% 95%
0 .693 .916 1.39 1.61 2.30 3.00
1 1.68 2.02 2.69 2.99 3.89 4.74
2 2.67 3.11 3.92 4.28 5.32 6.30
3 3.67 4.18 5.11 5.52 6.68 7.75
4 4.67 5.24 6.27 6.72 7.99 9.15
5 5.67 6.29 7.42 7.90 9.28 10.51
6 6.67 7.35 8.56 9.07 10.53 11.84
7 7.67 8.38 9.68 10.23 11.77 13.15
8 8.67 9.43 10.80 11.38 13.00 14.43
9 9.67 10.48 11.91 12.52 14.21 15.70
10 10.67 11.52 13.02 13.65 15.40 16.96
15 15.67 16.69 18.48 19.23 21.29 23.10
20 20.68 21.84 23.88 24.73 27.05 29.06
The formula to calculate the factors in the table is the
following.
Example: A new factory tool must meet a 400-hour MTBF
requirement at 80% confidence. You have up to two months
of 3-shift operation to decide whether the tool is acceptable.
What is a good test plan?
Two months of around-the-clock operation, with some time
off for maintenance and repairs, amounts to a maximum of
about 1300 hours. The 80% confidence factor for r = 1 is
2.99, so a test of 400 2.99 = about 1200 hours (with up to 1
fail allowed) is the best that can be done.
Shorten
required
test times
by testing
more than
one system
NOTE: Exponential test times can be shortened significantly
if several similar tools or systems can be put on test at the
same time. Test time means the same as "tool hours" and one
tool operating for 1000 hours is equivalent (as far as the
exponential model is concerned) to 2 tools operating for 500
hours each, or 10 tools operating for 100 hours each. Just
count all the fails from all the tools and the sum of the test
hours from all the tools.
8.3.1.2. Lognormal or Weibull tests

Planning
reliability
tests for
distributions
other than
the
exponential
is difficult
and involves
a lot of
guesswork
Planning a reliability test is not simple and straightforward
when the assumed model is lognormal or Weibull. Since
these models have two parameters, no estimates are possible
without at least two test failures, and good estimates require
considerably more than that. Because of censoring, without a
good guess ahead of time at what the unknown parameters
are, any test plan may fail.
However, it is often possible to make a good guess ahead of
time about at least one of the unknown parameters -
typically the "shape" parameter ( for the lognormal or
for the Weibull). With one parameter assumed known, test
plans can be derived that assure the reliability or failure rate
of the product tested will be acceptable.
Lognormal Case (shape parameter known): The
lognormal model is used for many microelectronic wear-out
failure mechanisms, such as electromigration. As a
production monitor, samples of microelectronic chips taken
randomly from production lots might be tested at levels of
voltage and temperature that are high enough to significantly
accelerate the occurrence of electromigration failures.
Acceleration factors are known from previous testing and
range from several hundred to several thousand.
Lognormal
test plans,
assuming
sigma and
the
acceleration
factor are
known
The goal is to construct a test plan (put n units on stress test
for T hours and accept the lot if no more than r failures
occur). The following assumptions are made:
The life distribution model is lognormal
Sigma = is known from past testing and does not
vary appreciably from lot to lot
Lot reliability varies because T
50
's (the lognormal
median or 50th percentile) differ from lot to lot
The acceleration factor from high stress to use stress is
a known quantity "A"
A stress time of T hours is practical as a line monitor
A nominal use T
50
of T
u
(combined with ) produces
an acceptable use CDF (or use reliability function).
This is equivalent to specifying an acceptable use
CDF at, say, 100,000 hours to be a given value p and
0
calculating T
u
via:
where is the inverse of the standard normal
distribution
An unacceptable use CDF of p
1
leads to a "bad" use
T
50
of T
b
, using the same equation as above with p
o
replaced by p
1
The acceleration factor A is used to calculate a "good" or
acceptable proportion of failures p
a
at stress and a "bad" or
unacceptable proportion of fails p
b
:
where is the standard normal CDF. This reduces the
reliability problem to a well-known Lot Acceptance
Sampling Plan (LASP) problem, which was covered in
Chapter 6.
If the sample size required to distinguish between p
a
and p
b
turns out to be too large, it may be necessary to increase T
or test at a higher stress. The important point is that the
above assumptions and equations give a methodology for
planning ongoing reliability tests under a lognormal model
assumption.
Weibull test
plans,
assuming
gamma and
the
acceleration.
factor are
known
Weibull Case (shape parameter known): The assumptions
and calculations are similar to those made for the
lognormal:
The life distribution model is Weibull
Gamma = is known from past testing and does not
vary appreciably from lot to lot
Lot reliability varies because 's (the Weibull
characteristic life or 62.3 percentile) differ from lot to
lot
The acceleration factor from high stress to use stress is
a known quantity "A"
A stress time of T hours is practical as a line monitor
A nominal use of
u
(combined with ) produces
an acceptable use CDF (or use reliability function).
This is equivalent to specifying an acceptable use
CDF at, say, 100,000 hours to be a given value p
0
and
calculating
u
An unacceptable use CDF of p
1
leads to a "bad" use
of , using the same equation as above with p
o
replaced by p
1
The acceleration factor A is used next to calculate a "good"
or acceptable proportion of failures p
a
at stress and a "bad"
or unacceptable proportion of failures p
b
:
This reduces the reliability problem to a Lot Acceptance
Sampling Plan (LASP) problem, which was covered in
Chapter 6.
If the sample size required to distinguish between p
a
and p
b
turns out to be too large, it may be necessary to increase T
or test at a higher stress. The important point is that the
above assumptions and equations give a methodology for
planning ongoing reliability tests under a Weibull model
assumption.
Planning Tests to Estimate Both Weibull or Both
Lognormal Parameters
Rules-of-
thumb for
general
lognormal
or Weibull
life test
planning
All that can be said here are some general rules-of-thumb:
1. If you can observe at least 10 exact times of failure,
estimates are usually reasonable - below 10 failures
the critical shape parameter may be hard to estimate
accurately. Below 5 failures, estimates are often very
inaccurate.
2. With readout data, even with more than 10 total
failures, you need failures in three or more readout
intervals for accurate estimates.
3. When guessing how many units to put on test and for
how long, try various reasonable combinations of
distribution parameters to see if the corresponding
calculated proportion of failures expected during the
test, multiplied by the sample size, gives a reasonable
number of failures.
4. As an alternative to the last rule, simulate test data
from reasonable combinations of distribution
parameters and see if your estimates from the
simulated data are close to the parameters used in the
simulation. If a test plan doesn't work well with
simulated data, it is not likely to work well with real
data.
8.3.1.3. Reliability growth (Duane model)

Guidelines
for
planning
how long to
run a
reliability
growth test
A reliability improvement test usually takes a large resource
commitment, so it is important to have a way of estimating
how long a test will be required. The following procedure
gives a starting point for determining a test time:
1. Guess a starting value for , the growth slope. Some
guidelines were previously discussed. Pick something
close to 0.3 for a conservative estimate (perhaps a new
cross-functional team will be working on the
improvement test or the system to be improved has
many new parts with possibly unknown failure
mechanisms), or close to 0.5 for a more optimistic
estimate.
2. Use current data and engineering estimates to arrive at
a consensus for what the starting MTBF for the system
is. Call this M
1
.
3. Let M
T
be the target MTBF (the customer
requirement). Then the improvement needed on the test
is given by
IM = M
T
/M
1
4. A first pass estimate of the test time needed is
This estimate comes from using the starting MTBF of M
1
as
the MTBF after 1 hour on test and using the fact that the
improvement from 1 hour to T hours is just .
Make sure
test time
makes
engineering
sense
The reason the above is just a first pass estimate is it will give
unrealistic (too short) test times when a high is assumed.
A very short reliability improvement test makes little sense
because a minimal number of failures must be observed
before the improvement team can determine design and parts
changes that will "grow" reliability. And it takes time to
implement these changes and observe an improved repair
rate.
Iterative Simulation methods can also be used to see if a planned test
simulation
is an aid
for test
planning
is likely to generate data that will demonstrate an assumed
growth rate.
8.3.1.4. Accelerated life tests

Accelerated
testing is
needed when
testing even
large sample
sizes at use
stress would
yield few or
no failures
within a
reasonable
time
Accelerated life tests are component life tests with
components operated at high stresses and failure data
observed. While high stress testing can be performed for the
sole purpose of seeing where and how failures occur and
using that information to improve component designs or
make better component selections, we will focus in this
section on accelerated life testing for the following two
purposes:
1. To study how failure is accelerated by stress and fit an
acceleration model to data from multiple stress cells
2. To obtain enough failure data at high stress to
accurately project (extrapolate) what the CDF at use
will be.
If we already know the acceleration model (or the
acceleration factor to typical use conditions from high stress
test conditions), then the methods described two pages ago
can be used. We assume, therefore, that the acceleration
model is not known in advance.
Test
planning
means
picking
stress levels
and sample
sizes and
test times to
produce
enough data
to fit models
and make
projections
Test planning and operation for a (multiple) stress cell life
test experiment consists of the following:
Pick several combinations of the relevant stresses (the
stresses that accelerate the failure mechanism under
investigation). Each combination is a "stress cell".
Note that you are planning for only one mechanism of
failure at a time. Failures on test due to any other
mechanism will be considered censored run times.
Make sure stress levels used are not too high - to the
point where new failure mechanisms that would never
occur at use stress are introduced. Picking a maximum
allowable stress level requires experience and/or good
engineering judgment.
Put random samples of components in each stress cell
and run the components in each cell for fixed (but
possibly different) lengths of time.
Gather the failure data from each cell and use the data
to fit an acceleration model and a life distribution
model and use these models to project reliability at
use stress conditions.
Test planning would be similar to topics already covered in
the chapters that discussed modeling and experimental
design except for one important point. When you test
components in a stress cell for a fixed length test, it is
typical that some (or possibly many) of the components end
the test without failing. This is the censoring problem, and it
greatly complicates experimental design to the point at
which it becomes almost as much of an art (based on
engineering judgment) as a statistical science.
An example will help illustrate the design issues. Assume a
metal migration failure mode is believed to follow the 2-
stress temperature voltage model given by
Normal use conditions are 4 volts and 25 degrees Celsius,
and the high stress levels under consideration are 6, 8,12
volts and 85
o
, 105
o
and 125
o
. It probably would be a waste
of resources to test at (6v, 85
o
), or even possibly (8v, 85
o
)
or (6v,105
o
) since these cells are not likely to have enough
stress acceleration to yield a reasonable number of failures
within typical test times.
If you write all the 9 possible stress cell combinations in a
3x3 matrix with voltage increasing by rows and temperature
increasing by columns, the result would look like the matrix
below:
Matrix Leading to "Backward L Design"
6v, 85
o
6v, 105
o
6v, 125
o
8v, 85
o
8v,105
o
8v,125
o
12v,85
o
12v,105
o
12v,125
o

"Backwards
L" designs
are common
in
accelerated
life testing.
Put more
experimental
units in
lower stress
cells.
The combinations in bold are the most likely design choices
covering the full range of both stresses, but still hopefully
having enough acceleration to produce failures. This is the
so-called "backwards L" design commonly used for
acceleration modeling experiments.
Note: It is good design practice to put more of your test
units in the lower stress cells, to make up for the fact that
these cells will have a smaller proportion of units failing.
Sometimes
simulation is
the best way
to learn
whether a
test plan has
a chance of
working
Design by Simulation:
A lengthy, but better way to choose a test matrix is the
following:
Pick an acceleration model and a life distribution
model (as usual).
Guess at the shape parameter value of the life
distribution model based on literature studies or earlier
experiments. The shape parameter should remain the
same for all stress cells. Choose a scale parameter
value at use so that the use stress CDF exactly meets
requirements (i.e., for the lognormal, pick a use T
50
that gives the desired use reliability - for a Weibull
model choice, do the same for the characteristic life
parameter).
Guess at the acceleration model parameters values (
H and , for the 2-stress model shown above). Again,
use whatever is in the literature for similar failure
mechanisms or data from earlier experiments).
Calculate acceleration factors from any proposed test
cells to use stress and divide the use scale parameter
by these acceleration factors to obtain "trial" cell scale
parameters.
Simulate cell data for each proposed stress cell using
the derived cell scale parameters and the guessed
shape parameter.
Check that every proposed cell has sufficient failures
to give good estimates.
Adjust the choice of stress cells and the sample size
allocations until you are satisfied that, if everything
goes as expected, the experiment will yield enough
data to provide good estimates of the model
parameters.
After you
make
advance
estimates, it
is sometimes
possible to
construct an
optimal
experimental
design - but
software for
this is
scarce
Optimal Designs:
Recent work on designing accelerated life tests has shown it
is possible, for a given choice of models and assumed values
of the unknown parameters, to construct an optimal design
(one which will have the best chance of providing good
sample estimates of the model parameters). These optimal
designs typically select stress levels as far apart as possible
and heavily weight the allocation of sample units to the
lower stress cells. However, unless the experimenter can find
software that incorporates these optimal methods for his or
her particular choice of models, the methods described
above are the most practical way of designing acceleration
experiments.
8.3.1.5. Bayesian gamma prior model

How to
plan a
Bayesian
test to
confirm a
system
meets its
MTBF
objective
Review Bayesian Basics and assumptions, if needed. We start
at the point when gamma prior parameters a and b have
already been determined. Assume we have a given MTBF
objective, M, and a desired confidence level of 100(1- ).
We want to confirm the system will have an MTBF of at least
M at the 100(1- ) confidence level. As in the section on
classical (HPP) test plans, we pick a number of failures, r, that
we can allow on the test. We need a test time T such that we
can observe up to r failures and still "pass" the test. If the test
time is too long (or too short), we can iterate with a different
choice of r.
When the test ends, the posterior gamma distribution will have
(worst case - assuming exactly r failures) new parameters of
a' = a + r, b' = b + T
and passing the test means that the failure rate
1-
, the upper
100(1- ) percentile for the posterior gamma, has to equal
the target failure rate 1/M. But this percentile is, by definition,
G
-1
(1- ; a', b'), with G
-1
denoting the inverse of the gamma
distribution with parameters a', b'. We can find the value of T
that satisfies G
-1
(1- ; a', b') = 1/M by trial and error.
However, based on the properties of the gamma distribution, it
turns out that we can calculate T directly by using
T = M(G
-1
(1- ; a', 1)) - b
Special Case: The Prior Has a = 1 (The "Weak" Prior)
When the
prior is a
weak prior
with a = 1,
the
Bayesian
test is
always
shorter
than the
classical
There is a very simple way to calculate the required Bayesian
test time when the prior is a weak prior with a = 1. Just use
the Test Length Guide Table to calculate the classical test
time. Call this T
c
. The Bayesian test time T is just T
c
minus
the prior parameter b (i.e., T = T
c
- b). If the b parameter was
set equal to (ln 2) MTBF
50
(where MTBF
50
is the consensus
choice for an "even money" MTBF), then
T = T
c
- (ln 2) MTBF
50
This shows that when a weak prior is used, the Bayesian test
test
time is always less than the corresponding classical test time.
That is why this prior is also known as a friendly prior.
Note: In general, Bayesian test times can be shorter, or longer,
than the corresponding classical test times, depending on the
choice of prior parameters. However, the Bayesian time will
always be shorter when the prior parameter a is less than, or
equal to, 1.
Example: Calculating a Bayesian Test Time
Example A new piece of equipment has to meet a MTBF requirement
of 500 hours at 80 % confidence. A group of engineers decide
to use their collective experience to determine a Bayesian
gamma prior using the 50/95 method described in Section 2.
They think 600 hours is a likely MTBF value and they are
very confident that the MTBF will exceed 250. Following the
example in Section 2, they determine that the gamma prior
parameters are a = 2.863 and b = 1522.46.
Now they want to determine an appropriate test time so that
they can confirm a MTBF of 500 with at least 80 %
confidence, provided they have no more than two failures.
We obtain a test time of 1756.117 hours using
500(G
-1
(1-0.2; 2.863+2, 1)) - 1522.46
To compare this result to the classical test time required, use
the Test Length Guide Table. The table factor is 4.28, so the
test time needed is 500 4.28 = 2140 hours for a non-
Bayesian test. The Bayesian test saves about 384 hours, or an
18 % savings. If the test is run for 1756 hours, with no more
than two failures, then an MTBF of at least 500 hours has
been confirmed at 80 % confidence.
If, instead, the engineers had decided to use a weak prior with
an MTBF
50
of 600, the required test time would have been
2140 - 600 ln 2 = 1724 hours
8.4. Reliability Data Analysis

After you have obtained component or system reliability data,
how do you fit life distribution models, reliability growth
models, or acceleration models? How do you estimate failure
rates or MTBF's and project component or system reliability
at use conditions? This section answers these kinds of
questions.
Detailed
outline for
Section 4
The detailed outline for section 4 follows.
4. Reliability Data Analysis
1. How do you estimate life distribution parameters from
censored data?
1. Graphical estimation
2. Maximum Likelihood Estimation (MLE)
3. A Weibull MLE example
2. How do you fit an acceleration model?
1. Graphical estimation
2. Maximum likelihood
3. Fitting models using degradation data instead of
failures
3. How do you project reliability at use conditions?
4. How do you compare reliability between two or more
populations?
5. How do you fit system repair rate models?
1. Constant repair rate (HPP/Exponential) model
2. Power law (Duane) model
3. Exponential law model
6. How do you estimate reliability using the Bayesian
gamma prior model?
8.4.1. How do you estimate life distribution parameters from censored data?

8.4.1. How do you estimate life distribution
parameters from censored data?
Graphical
estimation
methods
(aided by
computer
line fits)
are easy
and quick
Maximum
likelihood
methods
are
usually
more
precise -
but
require
special
software
Two widely used general methods will be described in this
section:
Graphical estimation
Maximum Likelihood Estimation (MLE)
Recommendation On Which Method to Use
Maximum likelihood estimation (except when the failure data
are very sparse - i.e., only a few failures) is a more precise and
flexible method. However, with censored data, the method of
maximum likelihood estimation requires special computer
programs for distributions other than the exponential. This is
no longer an obstacle since, in recent years, many statistical
software packages have added reliability platforms that will
calculate MLE's and most of these packages will estimate
acceleration model parameters and give confidence bounds as
well.
If important business decisions are based on reliability
projections made from life test data and acceleration
modeling, then it pays to obtain state-of-the art MLE
reliability software. Otherwise, for monitoring and tracking
reliability, estimation methods based on computer-augmented
graphical procedures will often suffice.
8.4.1.1. Graphical estimation

The line on a
probability
plot uniquely
identifies
distributional
parameters
Once you have calculated plotting positions from your
failure data, and have generated the probability plot for your
chosen model, parameter estimation follows easily. But
along with the mechanics of graphical estimation, be aware
of both the advantages and the disadvantages of graphical
estimation methods.
Most
probability
plots have
simple
procedures
to calculate
underlying
distribution
parameter
estimates
Graphical Estimation Mechanics:
If you draw a line through points on a probability plot, there
are usually simple rules to find estimates of the slope (or
shape parameter) and the scale parameter. On lognormal
probability plot with time on the x-axis and cumulative
percent on the y-axis, draw horizontal lines from the 34th
and the 50th percentiles across to the fitted line, and drop
vertical lines to the time axis from these intersection points.
The time corresponding to the 50th percentile is the T
50
estimate. Divide T
50
by the time corresponding to the 34th
percentile (this is called T
34
). The natural logarithm of that
ratio is the estimate of sigma, or the slope of the line ( =
ln(T
50
/ T
34
)).
For a Weibull probability plot draw a horizontal line from
the y-axis to the fitted line at the 62.3 percentile point. That
estimation line intersects the line through the points at a
time that is the estimate of the characteristic life parameter
. In order to estimate the slope of the fitted line (or the
shape parameter ), choose any two points on the fitted line
and divide the change in the y variable by the change in x
variable.
Using a
computer
generated
line fitting
routine
removes
subjectivity
and can lead
directly to
computer
To remove the subjectivity of drawing a line through the
points, a least-squares (regression) fit can be performed
using the equations described in the section on probability
plotting. An example of this for the Weibull was also shown
in that section. Another example of a Weibull plot for the
same data appears later in this section.
Finally, if you have exact times and complete samples (no
censoring), many software packages have built-in
Probability Plotting functions. Examples were shown in the
parameter
estimates
based on the
plotting
positions
sections describing various life distribution models.
Do
probability
plots even if
you use some
other method
for the final
estimates
Advantages of Graphical Methods of Estimation:
Graphical methods are quick and easy to use and
make visual sense.
Calculations can be done with little or no special
software needed.
Visual test of model (i.e., how well the points line up)
is an additional benefit.
Disadvantages of Graphical Methods of Estimation
Perhaps the
worst
drawback of
graphical
estimation is
you cannot
get legitimate
confidence
intervals for
the estimates
The statistical properties of graphical estimates (i.e., how
precise are they on average) are not good:
they are biased,
even with large samples, they are not minimum
variance (i.e., most precise) estimates,
graphical methods do not give confidence intervals
for the parameters (intervals generated by a regression
program for this kind of data are incorrect), and
formal statistical tests about model fit or parameter
values cannot be performed with graphical methods.
As we will see in the next section, Maximum Likelihood
Estimates overcome all these disadvantages - at least for
reliability data sets with a reasonably large number of
failures - at a cost of losing all the advantages listed above
for graphical estimation.
8.4.1.2. Maximum likelihood estimation

There is
nothing
visual
about the
maximum
likelihood
method -
but it is a
powerful
method
and, at
least for
large
samples,
very
precise
Maximum likelihood estimation begins with writing a
mathematical expression known as the Likelihood Function
of the sample data. Loosely speaking, the likelihood of a set
of data is the probability of obtaining that particular set of
data, given the chosen probability distribution model. This
expression contains the unknown model parameters. The
values of these parameters that maximize the sample
likelihood are known as the Maximum Likelihood Estimates
or MLE's.
Maximum likelihood estimation is a totally analytic
maximization procedure. It applies to every form of censored
or multicensored data, and it is even possible to use the
technique across several stress cells and estimate acceleration
model parameters at the same time as life distribution
parameters. Moreover, MLE's and Likelihood Functions
generally have very desirable large sample properties:
they become unbiased minimum variance estimators as
the sample size increases
they have approximate normal distributions and
approximate sample variances that can be calculated
and used to generate confidence bounds
likelihood functions can be used to test hypotheses
about models and parameters
With small
samples,
MLE's may
not be very
precise and
may even
generate a
line that
lies above
or below
the data
points
There are only two drawbacks to MLE's, but they are
important ones:
With small numbers of failures (less than 5, and
sometimes less than 10 is small), MLE's can be heavily
biased and the large sample optimality properties do not
apply
Calculating MLE's often requires specialized software
for solving complex non-linear equations. This is less
of a problem as time goes by, as more statistical
packages are upgrading to contain MLE analysis
capability every year.
Additional information about maximum likelihood
estimatation can be found in Chapter 1.
Likelihood
equation
for
censored
data
Likelihood Function Examples for Reliability Data:
Let f(t) be the PDF and F(t) the CDF for the chosen life
distribution model. Note that these are functions of t and the
unknown parameters of the model. The likelihood function for
Type I Censored data is:
with C denoting a constant that plays no role when solving for
the MLE's. Note that with no censoring, the likelihood reduces
to just the product of the densities, each evaluated at a failure
time. For Type II Censored Data, just replace T above by the
random end of test time t
r
.
The likelihood function for readout data is:
with F(T
0
) defined to be 0.
In general, any multicensored data set likelihood will be a
constant times a product of terms, one for each unit in the
sample, that look like either f(t
i
), [F(T
i
)-F(T
i-1
)], or [1-F(t
i
)],
depending on whether the unit was an exact time failure at
time t
i
, failed between two readouts T
i-1
and T
i
, or survived to
time t
i
and was not observed any longer.
The general mathematical technique for solving for MLE's
involves setting partial derivatives of ln L (the derivatives are
taken with respect to the unknown parameters) equal to zero
and solving the resulting (usually non-linear) equations. The
equation for the exponential model can easily be solved,
however.

MLE for
the
exponential
model
parameter
turns
out to be
just (total #
of failures)
divided by
(total unit
test time)
MLE's for the Exponential Model (Type I Censoring):
Note: The MLE of the failure rate (or repair rate) in the
exponential case turns out to be the total number of failures
observed divided by the total unit test time. For the MLE of
the MTBF, take the reciprocal of this or use the total unit test
hours divided by the total observed failures.
There are examples of Weibull and lognormal MLE analysis
later in this section.
8.4.1.3. A Weibull maximum likelihood estimation example

8.4.1.3. A Weibull maximum likelihood estimation
example
Reliability
analysis
using
Weibull
data
We will plot Weibull censored data and estimate parameters using data
from a previous example (8.2.2.1).
The recorded failure times were 54, 187, 216, 240, 244, 335, 361, 373,
375, and 386 hours, and 10 units that did not fail were removed from the
test at 500 hours. The data are summarized in the following table.
Time Censored Frequency
54 0 1
187 0 1
216 0 1
240 0 1
244 0 1
335 0 1
361 0 1
373 0 1
375 0 1
386 0 1
500 1 10
The column labeled "Time" contains failure and censoring times, the
"Censored" column contains a variable to indicate whether the time in
column one is a failure time or a censoring time, and the "Frequency"
column shows how many units failed or were censored at that time.
First, we generate a survival curve using the Kaplan-Meier method and a
Weibull probability plot. Note: Some software packages might use the
name "Product Limit Method" or "Product Limit Survival Estimates"
instead of the equivalent name "Kaplan-Meier".
Next, we perform a regression analysis for a survival model assuming
that failure times have a Weibull distribution. The Weibull characteristic
life parameter ( ) estimate is 606.5280 and the shape parameter ( )
estimate is 1.7208.
The log-likelihood and Akaike's Information Criterion (AIC) from the
model fit are -75.135 and 154.27. For comparison, we computed the AIC
for the lognormal distribution and found that it was only slightly larger
than the Weibull AIC.
Lognormal AIC Weibull AIC
154.39 154.27
When comparing values of AIC, smaller is better. The probability
density of the fitted Weibull distribution is shown below.
Based on the estimates of and , the lifetime expected value and
standard deviation are the following.
The greek letter, , represents the gamma function.
Discussion Maximum likelihood estimation (MLE) is an accurate and easy way to
estimate life distribution parameters, provided that a good software
analysis package is available. The package should also calculate
confidence bounds and log-likelihood values.
The analyses in this section can can be implemented using R code.
8.4.2. How do you fit an acceleration model?

Acceleration
models can
be fit by
either
graphical
procedures
or maximum
likelihood
methods
As with estimating life distribution model parameters, there
are two general approaches for estimating acceleration model
parameters:
Graphical estimation (or computer procedures based
on a graphical approach)
Maximum Likelihood Estimation (an analytic
approach based on writing the likelihood of all the data
across all the cells, incorporating the acceleration
model).
The same comments and recommendations concerning these
methods still apply. Note that it is even harder, however, to
find useful software programs that will do maximum
likelihood estimation across stress cells and fit and test
acceleration models.
Sometimes it
is possible
to fit a
model using
degradation
data
Another promising method of fitting acceleration models is
sometimes possible when studying failure mechanisms
characterized by a stress-induced gradual degradation
process that causes the eventual failure. This approach fits
models based on degradation data and has the advantage of
not actually needing failures. This overcomes censoring
limitations by providing measurement data at consecutive
time intervals for every unit in every stress cell.

This section will discuss the following:
1. How to fit an Arrhenius model with graphical estimation
2. Graphical estimation: an Arrhenius model example
3. Fitting more complicated models
Estimate
acceleration
model
parameters
by
estimating
cell T
50
values (or
values)
and then
using
regression
to fit the
model
across the
cells
How to fit an Arrhenius Model with Graphical Estimation
Graphical methods work best (and are easiest to describe) for a simple
one-stress model like the widely used Arrhenius model
with T denoting temperature measured in degrees Kelvin (273.16 +
degrees Celsius) and k is Boltzmann's constant (8.617 x 10
-5
in eV/K).
When applying an acceleration model to a distribution of failure times,
we interpret the deterministic model equation to apply at any distribution
percentile we want. This is equivalent to setting the life distribution scale
parameter equal to the model equation (T
50
for the lognormal, for the
Weibull and the MTBF or 1/ for the exponential). For the lognormal,
for example, we have
So, if we run several stress cells and compute T
50
values for each cell, a
plot of the natural log of these T
50
values versus the corresponding 1/kT
values should be roughly linear with a slope of H and an intercept of
ln A. In practice, a computer fit of a line through these points is typically
used to obtain the Arrhenius model estimates. Remember that T is in
Kelvin in the above equations. For temperature in Celsius, use the
following for 1/kT:
11605/(t C + 273.16).
An example will illustrate the procedure.
Graphical Estimation: An Arrhenius Model Example:
Arrhenius
model
example
Component life tests were run at three temperatures: 85 C, 105 C and
125 C. The lowest temperature cell was populated with 100
components; the 105 C cell had 50 components and the highest stress
cell had 25 components. All tests were run until either all the units in the
cell had failed or 1000 hours was reached. Acceleration was assumed to
follow an Arrhenius model and the life distribution model for the failure
mode was believed to be lognormal. The normal operating temperature
for the components is 25 C and it is desired to project the use CDF at
100,000 hours.
Test results:
Cell 1 (85 C): 5 failures at 401, 428, 695, 725 and 738 hours. Ninety-
five units were censored at 1000 hours running time.
Cell 2 (105 C): 35 failures at 171, 187, 189, 266, 275, 285, 301, 302,
305, 316, 317, 324, 349, 350, 386, 405, 480, 493, 530, 534, 536, 567,
589, 598, 599, 614, 620, 650, 668, 685, 718, 795, 854, 917, and 926
hours. Fifteen units were censored at 1000 hours running time.
Cell 3 (125 C): 24 failures at 24, 42, 92, 93, 141, 142, 143, 159, 181,
188, 194, 199, 207, 213, 243, 256, 259, 290, 294, 305, 392, 454, 502 and
696. One unit was censored at 1000 hours running time.
Failure analysis confirmed that all failures were due to the same failure
mechanism (if any failures due to another mechanism had occurred, they
would have been considered censored run times in the Arrhenius
analysis).
Steps to Fitting the Distribution Model and the Arrhenius Model:
Do plots for each cell and estimate T
50
and sigma as previously
discussed.
Plot all the cells on the same graph and check whether the lines are
roughly parallel (a necessary consequence of true acceleration).
If probability plots indicate that the lognormal model is appropriate
and that sigma is consistant among cells, plot ln T
50
versus
11605/(t C + 273.16) for each cell, check for linearity and fit a
straight line through the points. Since the points have different
values of precision, due to different numbers of failures in each
cell, it is recommended that the number of failures in each cell be
used as weights in a regression when fitting a line through the
points.
Use the slope of the line as the H estimate and calculate the
Arrhenius A constant from the intercept using A = e
intercept
.
Estimate the common sigma across all the cells by the weighted
average of the individual cell sigma estimates. Use the number of
failures in a cell divided by the total number of failures in all cells
as that cell's weight. This will allow cells with more failures to
play a bigger role in the estimation process.
Solution for
Arrhenius
model
example
Analysis of Multicell Arrhenius Model Data:
The following lognormal probability plot was generated for our data so
that all three stress cells are plotted on the same graph.
Note that the lines are somewhat straight (a check on the lognormal
model) and the slopes are approximately parallel (a check on the
acceleration assumption).
The cell ln T
50
and sigma estimates are obtained from linear regression
fits for each cell using the data from the probability plot. Each fit will
yield a cell A
o
, the ln T
50
estimate, and A
1
, the cell sigma estimate.
These are summarized in the table below.
Summary of Least Squares Estimation of Cell Lognormal
Parameters
Cell Number
ln T
50
Sigma
1 (t C = 85) 8.168 .908
2 (t C = 105) 6.415 .663
3 (t C = 125) 5.319 .805
The three cells have 11605/(t C + 273.16) values of 32.40, 30.69 and
29.15 respectively, in cell number order. The Arrhenius plot is
With only three cells, it is unlikely a straight line through the points will
present obvious visual lack of fit. However, in this case, the points
appear to line up very well.
Finally, the model coefficients are computed from a weighted linear fit
of ln T
50
versus 11605/(t C + 273.16), using weights of 5, 35, and 24
for each cell. This will yield a ln A estimate of -18.312 (A = e
-18.312
=
0.1115x10
-7
) and a H estimate of 0.808. With this value of H, the
acceleration between the lowest stress cell of 85 C and the highest of
125 C is
which is almost 14 acceleration. Acceleration from 125 C to the use
condition of 25 C is 3708. The use T
50
is e
-18.312
x
e
0.808x11605x1/298.16
= e
13.137
= 507383.
A single sigma estimate for all stress conditions can be calculated as a
weighted average of the three sigma estimates obtained from the
experimental cells. The weighted average is (5/64) 0.908 + (35/64)
0.663 + (24/64) 0.805 = 0.74.
The analyses in this section can can be implemented using both Dataplot
code and R code.
Fitting More Complicated models
Models Two stress models, such as the temperature/voltage model given by
involving
several
stresses can
be fit using
multiple
regression
need at least four or five carefully chosen stress cells to estimate all the
parameters. The Backwards L design previously described is an example
of a design for this model. The bottom row of the "backward L" could be
used for a plot testing the Arrhenius temperature dependence, similar to
the above Arrhenius example. The right hand column could be plotted
using y = ln T
50
and x = ln V, to check the voltage term in the model.
The overall model estimates should be obtained from fitting the multiple
regression model
Fitting this model, after setting up the Y, X1 = X
1
, X2 = X
2
data
vectors, provides estimates for b
0
, b
1
and b
2
.
Three stress models, and even Eyring models with interaction terms, can
be fit by a direct extension of these methods. Graphical plots to test the
model, however, are less likely to be meaningful as the model becomes
more complex.
8.4.2.2. Maximum likelihood

The
maximum
likelihood
method can
be used to
estimate
distribution
and
acceleration
model
parameters
at the same
time
The likelihood equation for a multi-cell acceleration model utilizes the likelihood function for
each cell, as described in section 8.4.1.2. Each cell will have unknown life distribution
parameters that, in general, are different. For example, if a lognormal model is used, each cell
might have its own T
50
and sigma.
Under an acceleration assumption, however, all the cells contain samples from populations
that have the same value of sigma (the slope does not change for different stress cells). Also,
the T
50
values are related to one another by the acceleration model; they all can be written
using the acceleration model equation that includes the proper cell stresses.
To form the likelihood equation under the acceleration model assumption, simply rewrite each
cell likelihood by replacing each cell T
50
with its acceleration model equation equivalent and
replacing each cell sigma with the same overall sigma. Then, multiply all these modified cell
likelihoods together to obtain the overall likelihood equation.
Once the overall likelihood equation has been created, the maximum likelihood estimates
(MLE) of sigma and the acceleration model parameters are the values that maximize this
likelihood. In most cases, these values are obtained by setting partial derivatives of the log
likelihood to zero and solving the resulting (non-linear) set of equations.
The method
is
complicated
and
requires
specialized
software
As you can see, the procedure is complicated, computationally intensive, and is only practical
if appropriate software is available. MLE does have many desirable features.
The method can, in theory at least, be used for any distribution model and acceleration
model and type of censored data.
Estimates have "optimal" statistical properties as sample sizes (i.e., numbers of failures)
become large.
Approximate confidence bounds can be calculated.
Statistical tests of key assumptions can be made using the likelihood ratio test. Some
common tests are:
the life distribution model versus another simpler model with fewer parameters
(i.e., a 3-parameter Weibull versus a 2-parameter Weibull, or a 2-parameter
Weibull versus an exponential),
the constant slope from cell to cell requirement of typical acceleration models,
and
the fit of a particular acceleration model.
In general, the recommendations made when comparing methods of estimating life
distribution model parameters also apply here. Software incorporating acceleration model
analysis capability, while rare just a few years ago, is now readily available and many
companies and universities have developed their own proprietary versions.
Steps For Fitting The Arrhenius Model
Use MLE to
fit an
Arrhenius
model to
example
data
Data from the Arrhenius example given in section 8.4.2.1 were analyzed using MLE. The
analyses in this section can can be implemented using R code.
1. We generate survival curves for each cell. All plots and estimates are based on individual
cell data, without the Arrhenius model assumption.
2. The results of lognormal survival regression modeling for the three data cells are shown
below.
Cell 1 - 85 C
Parameter Estimate Stan. Dev z Value
--------- -------- --------- -------
Intercept 8.891 0.890 9.991
ln(scale) 0.192 0.406 0.473

sigma = exp(ln(scale)) = 1.21
ln likelihood = -53.4

Cell 2 - 105 C
--------- -------- --------- -------
Intercept 6.470 0.108 60.14
ln(scale) -0.336 0.129 -2.60


Cell 3 - 125 C
--------- -------- --------- -------
Intercept 5.33 0.163 32.82
ln(scale) -0.21 0.146 -1.44


The cell ln likelihood values are -53.4, -265.2 and -156.5, respectively. Adding them together
yields a total ln likelihood of -475.1 for all the data fit with separate lognormal parameters for
each cell (no Arrhenius model assumption).
3. Fit the Arrhenius model to all data using MLE.
--------- -------- --------- -------
Intercept -19.906 2.3204 -8.58
l/kT 0.863 0.0761 11.34
ln(scale) -0.259 0.0928 -2.79

sigma = exp(ln(scale))Scale = 0.772
4. The likelihood ratio test statistic for the Arrhenius model fit (which also incorporates the
single sigma acceleration assumption) is -2ln , where denotes the ratio of the likelihood
values with (L
0
), and without (L
1
) the Arrhenius model assumption so that
-2ln = -2ln (L
0
/L
1
) = -2(ln L
0
- ln L
1
).
Using the results from steps 2 and 3, we have -2ln = -2(-476.7 - (-475.1)) = 3.2. The
degrees of freedom for the Chi-Square test statistic is 6 - 3 = 3, since six parameters were
reduced to three under the acceleration model assumption. The chance of obtaining a value 3.2
or higher is 36.3% for a Chi-Square distribution with 3 degrees of freedom, which indicates
an acceptable model (no significant lack of fit).
This completes the Arrhenius model analysis of the three cells of data. If different cells of data
have different voltages, then a new variable "ln V" could be added as an effect to fit the
Inverse Power Law voltage model. In fact, several effects can be included at once if more
than one stress varies across cells. Cross product stress terms could also be included by adding
these columns to the spreadsheet and adding them in the model as additional "effects".
Example Comparing Graphical Estimates and MLE
Arrhenius
example
comparing
graphical
and MLE
method
results
The results from the three-stress-cell Arrhenius example using graphical and MLE methods
for estimating parameters are shown in the table below.
Graphical Estimates MLE
ln T
50
Sigma
ln T
50
Sigma
Cell 1 8.17 0.91 8.89 1.21
Cell 2 6.42 0.66 6.47 0.71
Cell 3 5.32 0.81 5.33 0.81
Acceleration Model Overall Estimates
H Sigma ln A
Graphical 0.808 0.74 -18.312
MLE 0.863 0.77 -19.91
Note that when there are a lot of failures and little censoring, the two methods are in fairly
close agreement. Both methods are also in close agreement on the Arrhenius model results.
However, even small differences can be important when projecting reliability numbers at use
conditions. In this example, the CDF at 25 C and 100,000 hours projects to 0.014 using the
graphical estimates and only 0.003 using the MLE.
MLE
method
tests models
and gives
confidence
intervals
The maximum likelihood method allows us to test whether parallel lines (a single sigma) are
reasonable and whether the Arrhenius model is acceptable. The likelihood ratio tests for the
three example data cells indicated that a single sigma and the Arrhenius model are
appropriate. In addition, we can compute confidence intervals for all estimated parameters
based on the MLE results.
8.4.2.3. Fitting models using degradation data instead of failures

8.4.2.3. Fitting models using degradation data instead of
failures
If you can fit
models using
degradation
data, you
don't need
actual test
failures
When failure can be related directly to a change over time in a measurable
product parameter, it opens up the possibility of measuring degradation over
time and using that data to extrapolate when failure will occur. That allows us to
fit acceleration models and life distribution models without actually waiting for
failures to occur.
This overview of degradation modeling assumes you have chosen a life
distribution model and an acceleration model and offers an alternative to the
accelerated testing methodology based on failure data, previously described. The
following topics are covered.
Common assumptions
Advantages
Drawbacks
A simple method
A more accurate approach for a special case
Example
More details can be found in Nelson (1990, pages 521-544) or Tobias and
Trindade (1995, pages 197-203).
Common Assumptions When Modeling Degradation Data
You need a
measurable
parameter
that drifts
(degrades)
linearly to a
critical
failure value
Two common assumptions typically made when degradation data are modeled
are the following:
1. A parameter D, that can be measured over time, drifts monotonically
(upwards, or downwards) towards a specified critical value DF. When it
reaches DF, failure occurs.
2. The drift, measured in terms of D, is linear over time with a slope (or rate
of degradation) R, that depends on the relevant stress the unit is operating
under and also the (random) characteristics of the unit being measured.
Note: It may be necessary to define D as a transformation of some
standard parameter in order to obtain linearity - logarithms or powers are
sometimes needed.
The figure below illustrates these assumptions by showing degradation plots of
five units on test. Degradation readings for each unit are taken at the same four
time points and straight lines fit through these readings on a unit-by-unit basis.
These lines are then extended up to a critical (failure) degradation value. The
projected times of failure for these units are then read off the plot. The are: t , t ,
1 2
...,t
5
.
Plot of
linear
degradation
trends for
five units
read out at
four time
points
In many practical situations, D starts at 0 at time zero, and all the linear
theoretical degradation lines start at the origin. This is the case when D is a "%
change" parameter, or failure is defined as a change of a specified magnitude in
a parameter, regardless of its starting value. Lines all starting at the origin
simplify the analysis since we don't have to characterize the population starting
value for D, and the "distance" any unit "travels" to reach failure is always the
constant DF. For these situations, the degradation lines would look as follows.
Often, the
degradation
lines go
through the
origin - as
when %
change is the
measurable
parameter
increasing to
a failure
level
It is also common to assume the effect of measurement error, when reading
values of D, has relatively little impact on the accuracy of model estimates.
Advantages of Modeling Based on Degradation Data
Modeling
based on
complete
samples of
measurement
data, even
with low
stress cells,
offers many
advantages
1. Every degradation readout for every test unit contributes a data point. This
leads to large amounts of useful data, even if there are very few failures.
2. You don't have to run tests long enough to obtain significant numbers of
failures.
3. You can run low stress cells that are much closer to use conditions and
obtain meaningful degradation data. The same cells would be a waste of
time to run if failures were needed for modeling. Since these cells are
more typical of use conditions, it makes sense to have them influence
model parameters.
4. Simple plots of degradation vs time can be used to visually test the linear
degradation assumption.
Drawbacks to Modeling Based on Degradation Data
Degradation
may not
proceed in a
smooth,
linear
fashion
towards
what the
customer
calls
"failure"
1. For many failure mechanisms, it is difficult or impossible to find a
measurable parameter that degrades to a critical value in such a way that
reaching that critical value is equivalent to what the customer calls a
failure.
2. Degradation trends may vary erratically from unit to unit, with no
apparent way to transform them into linear trends.
3. Sometimes degradation trends are reversible and a few units appear to
"heal themselves" or get better. This kind of behavior does not follow
typical assumptions and is difficult to model.
4. Measurement error may be significant and overwhelm small degradation
trends, especially at low stresses.
5. Even when degradation trends behave according to assumptions and the
chosen models fit well, the final results may not be consistent with an
analysis based on actual failure data. This probably means that the failure
mechanism depends on more than a simple continuous degradation
process.
Because of the last listed drawback, it is a good idea to have at least one high-
stress cell where enough real failures occur to do a standard life distribution
model analysis. The parameter estimates obtained can be compared to the
predictions from the degradation data analysis, as a "reality" check.
A Simple Method For Modeling Degradation Data
A simple
approach is
to extend
each unit's
degradation
line until a
projected
"failure
time" is
obtained
1. As shown in the figures above, fit a line through each unit's degradation
readings. This can be done by hand, but using a least squares regression
program is better.
2. Take the equation of the fitted line, substitute DF for Y and solve for X.
This value of X is the "projected time of fail" for that unit.
3. Repeat for every unit in a stress cell until a complete sample of
(projected) times of failure is obtained for the cell.
4. Use the failure times to compute life distribution parameter estimates for a
cell. Under the fairly typical assumption of a lognormal model, this is very
simple. Take natural logarithms of all failure times and treat the resulting
data as a sample from a normal distribution. Compute the sample mean
and the sample standard deviation. These are estimates of ln T
50
and ,
respectively, for the cell.
5. Assuming there are k cells with varying stress, fit an appropriate
acceleration model using the cell ln T
50
values, as described in the
graphical estimation section. A single sigma estimate is obtained by taking
the square root of the average of the cell
2
estimates (assuming the same
number of units each cell). If the cells have n
j
units on test, where the n
j
values are not all equal, use the pooled sum-of-squares estimate across all
k cells calculated by
A More Accurate Regression Approach For the Case When D = 0 at time 0
and the "Distance To Fail" DF is the Same for All Units
Models can
be fit using
all the
degradation
readings and
linear
regression
Let the degradation measurement for the i-th unit at the j-th readout time in the
k-th stress cell be given by D
ijk
, and let the corresponding readout time be
denoted by t
jk
. That readout gives a degradation rate (or slope) estimate of D
ijk
/
t
jk
. This follows from the linear assumption or:
(Rate of degradation) (Time on test) = (Amount of degradation)
Based on that readout alone, an estimate of the natural logarithm of the time to
fail for that unit is
y
ijk
= ln DF - (ln D
ijk
- ln t
jk
).
This follows from the basic formula connecting linear degradation with failure
time
(rate of degradation) (time of failure) = DF
by solving for (time of failure) and taking natural logarithms.
For an Arrhenius model analysis, with
with the x
k
values equal to 1/KT. Here T is the temperature of the k-th cell,
measured in Kelvin (273.16 + degrees Celsius) and K is Boltzmann's constant
(8.617 10
-5
in eV/ unit Kelvin). Use a linear regression program to estimate a
= ln A and b = H. If we further assume t
f
has a lognormal distribution, the
mean square residual error from the regression fit is an estimate of
2
(with the
lognormal sigma).
One way to think about this model is as follows: each unit has a random rate R
of degradation. Since t
f
= DF/R, it follows from a characterization property of
the normal distribution that if t
f
is lognormal, then R must also have a lognormal
distribution (assuming DF and R are independent). After we take logarithms, ln
R has a normal distribution with a mean determined by the acceleration model
parameters. The randomness in R comes from the variability in physical
characteristics from unit to unit, due to material and processing differences.
Note: The estimate of sigma based on this simple graphical approach might tend
to be too large because it includes an adder due to the measurement error that
occurs when making the degradation readouts. This is generally assumed to have
only a small impact.
Example: Arrhenius Degradation Analysis
An example
using the
regression
approach to
fit an
Arrhenius
model
A component has a critical parameter that studies show degrades linearly over
time at a rate that varies with operating temperature. A component failure based
on this parameter occurs when the parameter value changes by 30% or more.
Fifteen components were tested under 3 different temperature conditions (5 at 65
C, 5 at 85 C and the last 5 at 105 C). Degradation percent values were read
out at 200, 500 and 1000 hours. The readings are given by unit in the following
three temperature cell tables.
65 C
200 hr 500 hr 1000 hr
Unit 1 0.87 1.48 2.81
Unit 2 0.33 0.96 2.13
Unit 3 0.94 2.91 5.67
Unit 4 0.72 1.98 4.28
Unit 5 0.66 0.99 2.14
85 C
200 hr 500 hr 1000 hr
Unit 1 1.41 2.47 5.71
Unit 2 3.61 8.99 17.69
Unit 3 2.13 5.72 11.54
Unit 4 4.36 9.82 19.55
Unit 5 6.91 17.37 34.84
105 C
200 hr 500 hr 1000 hr
Unit 1 24.58 62.02 124.10
Unit 2 9.73 24.07 48.06
Unit 3 4.74 11.53 23.72
Unit 4 23.61 58.21 117.20
Unit 5 10.90 27.85 54.97
Note that one unit failed in the 85 C cell and four units failed in the 105 C
cell. Because there were so few failures, it would be impossible to fit a life
distribution model in any cell but the 105 C cell, and therefore no acceleration
model can be fit using failure data. We will fit an Arrhenius/lognormal model,
using the degradation data.
Solution:
Fit the
model to the
degradation
data
From the above tables, first create a variable (DEG) with 45 degradation values
starting with the first row in the first table and proceeding to the last row in the
last table. Next, create a temperature variable (TEMP) that has 15 repetitions of
65, followed by 15 repetitions of 85 and then 15 repetitions of 105. Finally,
create a time variable (TIME) that corresponds to readout times.
Fit the Arrhenius/lognormal equation, y
ijk
= a + b x
ijk
, where
y
ijk
= ln(30) - (ln(DEG) - ln(TIME))
and
x
ijk
= 100000 / [8.617*(TEMP + 273.16)].
The linear regression results are the following.
Parameter Estimate Stan. Dev t Value
--------- -------- --------- -------
a -18.94337 1.83343 -10.33
b 0.81877 0.05641 14.52

Residual degrees of freedom = 45
The Arrhenius model parameter estimates are: ln A = -18.94; H = 0.82. An
estimate of the lognormal sigma is = 0.56.
The analyses in this section can can be implemented using both Dataplot code
and R code.
8.4.3. How do you project reliability at use conditions?

8.4.3. How do you project reliability at use
conditions?
When
projecting
from high
stress to use
conditions,
having a
correct
acceleration
model and
life
distribution
model is
critical
General Considerations
Reliability projections based on failure data from high stress
tests are based on assuming we know the correct acceleration
model for the failure mechanism under investigation and we
are also using the correct life distribution model. This is
because we are extrapolating "backwards" - trying to
describe failure behavior in the early tail of the life
distribution, where we have little or no actual data.
For example, with an acceleration factor of 5000 (and some
are much larger than this), the first 100,000 hours of use life
is "over" by 20 hours into the test. Most, or all, of the test
failures typically come later in time and are used to fit a life
distribution model with only the first 20 hours or less being
of practical use. Many distributions may be flexible enough
to adequately fit the data at the percentiles where the points
are, and yet differ from the data by orders of magnitude in
the very early percentiles (sometimes referred to as the early
"tail" of the distribution).
However, it is frequently necessary to test at high stress (to
obtain any failures at all!) and project backwards to use.
When doing this bear in mind two important points:
Project for
each failure
mechanism
separately
Distribution models, and especially acceleration
models, should be applied only to a single failure
mechanism at a time. Separate out failure mechanisms
when doing the data analysis and use the competing
risk model to build up to a total component failure rate
Try to find theoretical justification for the chosen
models, or at least a successful history of their use for
the same or very similar mechanisms. (Choosing
models solely based on empirical fit is like
extrapolating from quicksand to a mirage.)
How to Project from High Stress to Use Stress
Two types of use-condition reliability projections are
common:
1. Projection to use conditions after completing a multiple
stress cell experiment and successfully fitting both a
life distribution model and an acceleration model
2. Projection to use conditions after a single cell at high
stress is run as a line reliability monitor.
Arrhenius
model
projection
example
The Arrhenius example from the graphical estimation and the
MLE estimation sections ended by comparing use projections
of the CDF at 100,000 hours. This is a projection of the first
type. We know from the Arrhenius model assumption that the
T
50
at 25 C is just
Using the graphical model estimates for ln A and we have
T
50
at use = e
-18.312
e
0.808 11605/298.16

= e
13.137
= 507383
and combining this T
50
with the estimate of the common
sigma of 0.74 allows us to easily estimate the CDF or failure
rate after any number of hours of operation at use conditions.
In particular, the CDF value of a lognormal at T/T
50
(where
time T = 100,000, T
50
= 507383, and sigma = 0.74) is 0.014,
which matches the answer given in the MLE estimation
section as the graphical projection of the CDF at 100,000
hours at a use temperature of 25 C.
If the life distribution model had been Weibull, the same type
of analysis would be performed by letting the characteristic
life parameter vary with stress according to the acceleration
model, while the shape parameter is constant for all stress
conditions.
The second type of use projection was used in the section on
lognormal and Weibull tests, in which we judged new lots of
product by looking at the proportion of failures in a sample
tested at high stress. The assumptions we made were:
we knew the acceleration factor between use and high
stress
the shape parameter (sigma for the lognormal, gamma
for the Weibull) is also known and does not change
significantly from lot to lot.
With these assumptions, we can take any proportion of
failures we see from a high stress test and project a use CDF
or failure rate. For a T-hour high stress test and an
acceleration factor of A from high stress to use stress, an
observed proportion p is converted to a use CDF at 100,000
hours for a lognormal model using:
T
50Stress
= TG
-1
(p, 0, )
CDF = G((100000/(AT
50Stress
)), 0, ).
where G(q, , ) is the lognormal distribution function with
mean and standard deviation .
If the model is Weibull, we can find the use CDF or failure
rate with:
A
Stress
= TW
-1
(p, , 1)
CDF = W((100000/(AA
Stress
)), , 1).
where W(q, , ) is the Weibull distribution function with
shape parameter and scale parameter .
The analyses in this section can can be implemented using
8.4.4. How do you compare reliability between two or more populations?

8.4.4. How do you compare reliability between
two or more populations?
Several
methods
for
comparing
reliability
between
populations
are
described
Comparing reliability among populations based on samples of
failure data usually means asking whether the samples came
from populations with the same reliability function (or CDF).
Three techniques already described can be used to answer
this question for censored reliability data. These are:
Comparing sample proportion failures
Likelihood ratio test comparisons
Lifetime regression comparisons
Comparing Sample Proportion Failures
Assume each sample is a random sample from possibly a
different lot, vendor or production plant. All the samples are
tested under the same conditions. Each has an observed
proportion of failures on test. Call these sample proportions of
failures p
1
, p
2
, p
3
, ...p
n
. Could these all have come from
equivalent populations?
This is a question covered in Chapter 7 for two populations,
and for more than two populations, and the techniques
described there apply equally well here.
Likelihood Ratio Test Comparisons
The Likelihood Ratio test was described earlier. In this
application, the Likelihood ratio has as a denominator the
product of all the Likelihoods of all the samples assuming
each population has its own unique set of parameters. The
numerator is the product of the Likelihoods assuming the
parameters are exactly the same for each population. The test
looks at whether -2ln is unusually large, in which case it is
unlikely the populations have the same parameters (or
reliability functions).
This procedure is very effective if, and only if, it is built into
the analysis software package being used and this software
covers the models and situations of interest to the analyst.
Lifetime Regression Comparisons
Lifetime regression is similar to maximum likelihood and
8.4.4. How do you compare reliability between two or more populations?
likelihood ratio test methods. Each sample is assumed to have
come from a population with the same shape parameter and a
wide range of questions about the scale parameter (which is
often assumed to be a "measure" of lot-to-lot or vendor-to-
vendor quality) can be formulated and tested for significance.
For a complicated, but realistic example, assume a company
manufactures memory chips and can use chips with some
known defects ("partial goods") in many applications.
However, there is a question of whether the reliability of
"partial good" chips is equivalent to "all good" chips. There
exists lots of customer reliability data to answer this question.
However the data are difficult to analyze because they contain
several different vintages with known reliability differences
as well as chips manufactured at many different locations.
How can the partial good vs all good question be resolved?
A lifetime regression model can be constructed with variables
included that change the scale parameter based on vintage,
location, partial versus all good, and any other relevant
variables. Then, a good lifetime regression program will sort
out which, if any, of these factors are significant and, in
particular, whether there is a significant difference between
"partial good" and "all good".
8.4.5. How do you fit system repair rate models?

8.4.5. How do you fit system repair rate
models?
Fitting
models
discussed
earlier
This subsection describes how to fit system repair rate models
when you have actual failure data. The data could come from
from observing a system in normal operation or from running
tests such as Reliability Improvement tests.
The three models covered are the constant repair rate
(HPP/exponential) model, the power law (Duane) model and
the exponential law model.
8.4.5.1. Constant repair rate (HPP/exponential) model

This
section
covers
estimating
MTBF's
and
calculating
upper and
lower
confidence
bounds
The HPP or exponential model is widely used for two reasons:
Most systems spend most of their useful lifetimes operating in the flat
constant repair rate portion of the bathtub curve
It is easy to plan tests, estimate the MTBF and calculate confidence
intervals when assuming the exponential model.
This section covers the following:
1. Estimating the MTBF (or repair rate/failure rate)
2. How to use the MTBF confidence interval factors
3. Tables of MTBF confidence interval factors
4. Confidence interval equation and "zero fails" case
5. Calculation of confidence intervals
6. Example
Estimating the MTBF (or repair rate/failure rate)
For the HPP system model, as well as for the non repairable exponential
population model, there is only one unknown parameter (or equivalently,
the MTBF = 1/ ). The method used for estimation is the same for the HPP
model and for the exponential population model.
The best
estimate of
the MTBF
is just
"Total
Time"
divided by
"Total
Failures"
The estimate of the MTBF is
This estimate is the maximum likelihood estimate whether the data are
censored or complete, or from a repairable system or a non-repairable
population.
Confidence
Interval
Factors
multiply
the
estimated
How To Use the MTBF Confidence Interval Factors
1. Estimate the MTBF by the standard estimate (total unit test hours
divided by total failures)
2. Pick a confidence level (i.e., pick 100x(1- )). For 95%, = .05; for
90%, = .1; for 80%, = .2 and for 60%, = .4
MTBF to
obtain
lower and
upper
bounds on
the true
MTBF
3. Read off a lower and an upper factor from the confidence interval
tables for the given confidence level and number of failures r
4. Multiply the MTBF estimate by the lower and upper factors to obtain
MTBF
lower
and MTBF
upper
5. When r (the number of failures) = 0, multiply the total unit test hours
by the "0 row" lower factor to obtain a 100 (1- /2)% one-sided
lower bound for the MTBF. There is no upper bound when r = 0.
6. Use (MTBF
lower
, MTBF
upper
) as a 100(1- )% confidence interval
for the MTBF (r > 0)
7. Use MTBF
lower
as a (one-sided) lower 100(1- /2)% limit for the
MTBF
8. Use MTBF
upper
as a (one-sided) upper 100(1- /2)% limit for the
MTBF
9. Use (1/MTBF
upper
, 1/MTBF
lower
) as a 100(1- )% confidence
interval for
10. Use 1/MTBF
upper
as a (one-sided) lower 100(1- /2)% limit for
11. Use 1/MTBF
lower
as a (one-sided) upper 100(1- /2)% limit for
Tables of MTBF Confidence Interval Factors
Confidence
bound
factor
tables for
60, 80, 90
and 95%
confidence
Confidence Interval Factors to Multiply MTBF Estimate
60% 80%
Num
Fails r
Lower for
MTBF
Upper for
MTBF
Lower for
MTBF
Upper for
MTBF
0 0.6213 - 0.4343 -
1 0.3340 4.4814 0.2571 9.4912
2 0.4674 2.4260 0.3758 3.7607
3 0.5440 1.9543 0.4490 2.7222
4 0.5952 1.7416 0.5004 2.2926
5 0.6324 1.6184 0.5391 2.0554
6 0.6611 1.5370 0.5697 1.9036
7 0.6841 1.4788 0.5947 1.7974
8 0.7030 1.4347 0.6156 1.7182
9 0.7189 1.4000 0.6335 1.6567
10 0.7326 1.3719 0.6491 1.6074
11 0.7444 1.3485 0.6627 1.5668
12 0.7548 1.3288 0.6749 1.5327
13 0.7641 1.3118 0.6857 1.5036
14 0.7724 1.2970 0.6955 1.4784
15 0.7799 1.2840 0.7045 1.4564
20 0.8088 1.2367 0.7395 1.3769
25 0.8288 1.2063 0.7643 1.3267
30 0.8436 1.1848 0.7830 1.2915
35 0.8552 1.1687 0.7978 1.2652
40 0.8645 1.1560 0.8099 1.2446
45 0.8722 1.1456 0.8200 1.2280
50 0.8788 1.1371 0.8286 1.2142
75 0.9012 1.1090 0.8585 1.1694
100 0.9145 1.0929 0.8766 1.1439
500 0.9614 1.0401 0.9436 1.0603
Confidence Interval Factors to Multiply MTBF Estimate
90% 95%
Num
Fails
Lower for
MTBF
Upper for
MTBF
Lower for
MTBF
Upper for
MTBF
0 0.3338 - 0.2711 -
1 0.2108 19.4958 0.1795 39.4978
2 0.3177 5.6281 0.2768 8.2573
3 0.3869 3.6689 0.3422 4.8491
4 0.4370 2.9276 0.3906 3.6702
5 0.4756 2.5379 0.4285 3.0798
6 0.5067 2.2962 0.4594 2.7249
7 0.5324 2.1307 0.4853 2.4872
8 0.5542 2.0096 0.5075 2.3163
9 0.5731 1.9168 0.5268 2.1869
10 0.5895 1.8432 0.5438 2.0853
11 0.6041 1.7831 0.5589 2.0032
12 0.6172 1.7330 0.5725 1.9353
13 0.6290 1.6906 0.5848 1.8781
14 0.6397 1.6541 0.5960 1.8291
15 0.6494 1.6223 0.6063 1.7867
20 0.6882 1.5089 0.6475 1.6371
25 0.7160 1.4383 0.6774 1.5452
30 0.7373 1.3893 0.7005 1.4822
35 0.7542 1.3529 0.7190 1.4357
40 0.7682 1.3247 0.7344 1.3997
45 0.7800 1.3020 0.7473 1.3710
50 0.7901 1.2832 0.7585 1.3473
75 0.8252 1.2226 0.7978 1.2714
100 0.8469 1.1885 0.8222 1.2290
500 0.9287 1.0781 0.9161 1.0938
Confidence Interval Equation and "Zero Fails" Case
Formulas
for
confidence
bound
factors -
Confidence bounds for the typical Type I censoring situation are obtained
from chi-square distribution tables or programs. The formula for calculating
confidence intervals is:
even for
"zero fails"
case
In this formula,
2
/2,2r
is a value that the chi-square statistic with 2r
degrees of freedom is less than with probability /2. In other words, the
left-hand tail of the distribution has probability /2. An even simpler
version of this formula can be written using T = the total unit test time:
These bounds are exact for the case of one or more repairable systems on
test for a fixed time. They are also exact when non repairable units are on
test for a fixed time and failures are replaced with new units during the
course of the test. For other situations, they are approximate.
When there are zero failures during the test or operation time, only a (one-
sided) MTBF lower bound exists, and this is given by
MTBF
lower
= T/(-ln )
The interpretation of this bound is the following: if the true MTBF were
any lower than MTBF
lower
, we would have seen at least one failure during
T hours of test with probability at least 1-. Therefore, we are 100(1-) %
confident that the true MTBF is not lower than MTBF
lower
.
Calculation
of
confidence
limits
A one-sided, lower 100(1-/2) % confidence bound for the MTBF is given
by
LOWER = 2T/G
-1
(1-/2, [2(r+1)])
where T is the total unit or system test time, r is the total number of
failures, and G(q,) is the
2
distribution function with shape parameter .
A one-sided, upper 100(1-/2) % confidence bound for the MTBF is given
by
UPPER = 2T/G
-1
(/2, [2r])
The two intervals together, (LOWER, UPPER), are a 100(1-) % two-sided
confidence interval for the true MTBF.
Please use caution when using CDF and inverse CDF functions in
commercial software because some functions require left-tail probabilities
and others require right-tail probabilities. In the left-tail case, /2 is used
for the upper bound because 2T is being divided by the smaller percentile,
and 1-/2 is used for the lower bound because 2T is divided by the larger
percentile. For the right-tail case, 1-/2 is used to compute the upper bound
and /2 is used to compute the lower bound. Our formulas for G
-1
(q,)
assume the inverse CDF function requires left-tail probabilities.
Example
Example
showing
how to
calculate
confidence
limits
A system was observed for two calendar months of operation, during which
time it was in operation for 800 hours and had 2 failures.
The MTBF estimate is 800/2 = 400 hours. A 90 %, two-sided confidence
interval is given by (4000.3177, 4005.6281) = (127, 2251). The same
interval could have been obtained using
LOWER = 1600/G
-1
(0.95,6)
UPPER = 1600/G
-1
(0.05,4)
Note that 127 is a 95 % lower limit for the true MTBF. The customer is
usually only concerned with the lower limit and one-sided lower limits are
often used for statements of contractual requirements.
Zero fails
confidence
limit
calculation
What could we have said if the system had no failures? For a 95 % lower
confidence limit on the true MTBF, we either use the 0 failures factor from
the 90 % confidence interval table and calculate 800 0.3338 = 267, or we
use T/(ln ) = 800/(ln 0.05) = 267.
The analyses in this section can can be implemented using both Dataplot
code and R code.
8.4.5.2. Power law (Duane) model

The Power
Law
(Duane)
model has
been very
successful in
modeling
industrial
reliability
improvement
data
Brief Review of Power Law Model and Duane Plots
Recall that the Power Law is a NHPP with the expected number of
fails, M(t), and the repair rate, M'(t) = m(t), given by:
The parameter = 1-b is called the Reliability Growth Slope and
typical industry values for growth slopes during reliability
improvement tests are in the .3 to .6 range.
If a system is observed for a fixed time of T hours and failures
occur at times t
1
, t
2
, t
3
, ..., t
r
(with the start of the test or
observation period being time 0), a Duane plot is a plot of (t
i
/ i)
versus t
i
on log-log graph paper. If the data are consistent with a
Power Law model, the points in a Duane Plot will roughly follow a
straight line with slope and intercept (where t = 1 on the log-log
paper) of -log
10
a.
MLE's for
the Power
Law model
are given
Estimates for the Power Law Model
Computer aided graphical estimates can easily be obtained by
doing a regression fit of Y = ln (t
i
/ i) vs X = ln t
i
. The slope is
the estimate and e
-intercept
is the a estimate. The estimate of b is
1- .
However, better estimates can easily be calculated. These are
modified maximum likelihood estimates (corrected to eliminate
bias). The formulas are given below for a fixed time of T hours,
and r failures occurring at times t
1
, t
2
, t
3
, ..., t
r
.
The estimated MTBF at the end of the test (or observation) period
is
Approximate
confidence
bounds for
the MTBF at
end of test
are given
Approximate Confidence Bounds for the MTBF at End of Test
We give an approximate 100(1-) % confidence interval (M
L
,
M
U
) for the MTBF at the end of the test. Note that M
L
is a
100(1-/2) % one-sided lower confidence bound and M
U
is a
100(1-/2) % one-sided upper confidence bound. The formulas
are:
where z
1-/2
is the 100(1-/2) percentile point of the standard
Case Study 1: Reliability Improvement Test Data Continued
Fitting the
power law
model to
case study 1
failure data
This case study was introduced in section 2, where we did various
plots of the data, including a Duane Plot. The case study was
continued when we discussed trend tests and verified that
significant improvement had taken place. Now we will complete
the case study data analysis.
The observed failure times were: 5, 40, 43, 175, 389, 712, 747, 795,
1299 and 1478 hours, with the test ending at 1500 hours. We
estimate , a, and the MTBF at the end of test, along with a
100(1-) % confidence interval for the true MTBF at the end of
test (assuming, of course, that the Power Law model holds). The
parameters and confidence intervals for the power law model were
estimated to be the following.
Estimate of = 0.5165
Estimate of a = 0.2913
Estimate of MTBF at the end of the test = 310.234
80 % two-sided confidence interval:
(157.7139 , 548.5565)
90 % one-sided lower confidence limit = 157.7139
8.4.5.3. Exponential law model

8.4.5.3. Exponential law model
Estimates
of the
parameters
of the
Exponential
Law model
can be
obtained
from either
a graphical
procedure
or
maximum
likelihood
estimation
Recall from section 1 that the Exponential Law refers to a
NHPP process with repair rate M'(t) = m(t) = . This
model has not been used nearly as much in industrial
applications as the Power Law model, and it is more difficult
to analyze. Only a brief description will be given here.
Since the expected number of failures is given by
M(t) = and ln M(t) = , a plot of the
cum fails versus time of failure on a log-linear scale should
roughly follow a straight line with slope . Doing a
regression fit of y = ln cum fails versus x = time of failure
will provide estimates of the slope and the intercept - ln
.
Alternatively, maximum likelihood estimates can be obtained
from the following pair of equations:
The first equation is non-linear and must be solved iteratively
to obtain the maximum likelihood estimate for . Then, this
estimate is substituted into the second equation to solve for
the maximum likelihood estimate for .
8.4.6. How do you estimate reliability using the Bayesian gamma prior model?

8.4.6. How do you estimate reliability using the Bayesian gamma
prior model?
The Bayesian paradigm was introduced in Section 1 and Section 2 described the assumptions
underlying the gamma/exponential system model (including several methods to transform
prior data and engineering judgment into gamma prior parameters "a" and "b"). Finally, we
saw in Section 3 how to use this Bayesian system model to calculate the required test time
needed to confirm a system MTBF at a given confidence level.
Review of
Bayesian
procedure
for the
gamma
exponential
system
model
The goal of Bayesian reliability procedures is to obtain as accurate a posterior distribution as
possible, and then use this distribution to calculate failure rate (or MTBF) estimates with
confidence intervals (called credibility intervals by Bayesians). The figure below
summarizes the steps in this process.
How to
estimate
the MTBF
with
bounds,
based on
the
Once the test has been run, and r failures observed, the posterior gamma parameters are:
a' = a + r, b' = b + T
and a (median) estimate for the MTBF is calculated by
1 / G
-1
(0.5, a', (1/b'))
posterior
distribution
where G(q, , ) represents the gamma distribution with shape parameter , and scale
parameter . Some people prefer to use the reciprocal of the mean of the posterior distribution
as their estimate for the MTBF. The mean is the minimum mean square error (MSE)
estimator of , but using the reciprocal of the mean to estimate the MTBF is always more
conservative than the "even money" 50% estimator.
A lower 80% bound for the MTBF is obtained from
1 / G
-1
(0.8, a', (1/b'))
and, in general, a lower 100(1-) % lower bound is given by
1 / G
-1
((1-), a', (1/b')).
A two-sided 100(1-) % credibility interval for the MTBF is
[1 / G
-1
((1-/2), a', (1/b')),
1 / G
-1
((/2), a', (1/b'))].
Finally, the G((1/M), a', (1/b')) calculates the probability that MTBF is greater than M.
Example
A Bayesian
example to
estimate
the MTBF
and
calculate
upper and
lower
bounds
A system has completed a reliability test aimed at confirming a 600 hour MTBF at an 80%
confidence level. Before the test, a gamma prior with a = 2, b = 1400 was agreed upon, based
on testing at the vendor's location. Bayesian test planning calculations, allowing up to 2 new
failures, called for a test of 1909 hours. When that test was run, there actually were exactly
two failures. What can be said about the system?
The posterior gamma CDF has parameters a' = 4 and b' = 3309. The plot below shows CDF
values on the y-axis, plotted against 1/ = MTBF, on the x-axis. By going from probability, on
the y-axis, across to the curve and down to the MTBF, we can estimate any MTBF percentile
point.
The MTBF values are shown below.
1 / G
-1
(0.9, 4, (1/3309))
= 495 hours
1 / G
-1
(0.8, 4, (1/3309))
= 600 hours (as expected)
1 / G
-1
(0.5, 4, (1/3309))
= 901 hours
1 / G
-1
(0.1, 4, (1/3309))
= 1897 hours
The test has confirmed a 600 hour MTBF at 80 % confidence, a 495 hour MTBF at 90 %
confidence and (495, 1897) is a 90 % credibility interval for the MTBF. A single number
(point) estimate for the system MTBF would be 901 hours. Alternatively, you might want to
use the reciprocal of the mean of the posterior distribution (b'/a') = 3309/4 = 827 hours as a
single estimate. The reciprocal mean is more conservative, in this case it is a 57 % lower
bound (G((4/3309), 4, (1/3309))).
8.4.7. References For Chapter 8: Assessing Product Reliability

8.4.7. References For Chapter 8: Assessing
Product Reliability
Aitchison, J. and Brown, J. A. C.,(1957), The Log-normal distribution,
Cambridge University Press, New York and London.
Ascher, H. (1981), "Weibull Distribution vs Weibull Process,"
Proceedings Annual Reliability and Maintainability Symposium, pp. 426-
431.
Ascher, H. and Feingold, H. (1984), Repairable Systems Reliability,
Marcel Dekker, Inc., New York.
Bain, L.J. and Englehardt, M. (1991), Statistical Analysis of Reliability
and Life-Testing Models: Theory and Methods, 2nd ed., Marcel Dekker,
New York.
Barlow, R. E. and Proschan, F. (1975), Statistical Theory of Reliability
and Life Testing, Holt, Rinehart and Winston, New York.
Birnbaum, Z.W., and Saunders, S.C. (1968), "A Probabilistic
Interpretation of Miner's Rule," SIAM Journal of Applied Mathematics,
Vol. 16, pp. 637-652.
Birnbaum, Z.W., and Saunders, S.C. (1969), "A New Family of Life
Distributions," Journal of Applied Probability, Vol. 6, pp. 319-327.
Cox, D.R. and Lewis, P.A.W. (1966), The Statistical Analysis of Series
of Events, John Wiley & Sons, Inc., New York.
Cox, D.R. (1972), "Regression Models and Life Tables," Journal of the
Royal Statistical Society, B 34, pp. 187-220.
Cox, D. R., and Oakes, D. (1984), Analysis of Survival Data, Chapman
and Hall, London, New York.
Crow, L.H. (1974), "Reliability Analysis for Complex Repairable
Systems", Reliability and Biometry, F. Proschan and R.J. Serfling, eds.,
SIAM, Philadelphia, pp 379- 410.
Crow, L.H. (1975), "On Tracking Reliability Growth," Proceedings
Annual Reliability and Maintainability Symposium, pp. 438-443.
Crow, L.H. (1982), "Confidence Interval Procedures for the Weibull
Process With Applications to Reliability Growth," Technometrics,
24(1):67-72.
Crow, L.H. (1990), "Evaluating the Reliability of Repairable Systems,"
Proceedings Annual Reliability and Maintainability Symposium, pp. 275-
279.
Crow, L.H. (1993), "Confidence Intervals on the Reliability of
Repairable Systems," Proceedings Annual Reliability and Maintainability
Symposium, pp. 126-134
Duane, J.T. (1964), "Learning Curve Approach to Reliability
Monitoring," IEEE Transactions On Aerospace, 2, pp. 563-566.
Gumbel, E. J. (1954), Statistical Theory of Extreme Values and Some
Practical Applications, National Bureau of Standards Applied
Mathematics Series 33, U.S. Government Printing Office, Washington,
D.C.
Hahn, G.J., and Shapiro, S.S. (1967), Statistical Models in Engineering,
John Wiley & Sons, Inc., New York.
Hoyland, A., and Rausand, M. (1994), System Reliability Theory, John
Wiley & Sons, Inc., New York.
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994), Continuous
Univariate Distributions Volume 1, 2nd edition, John Wiley & Sons, Inc.,
New York.
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995), Continuous
Univariate Distributions Volume 2, 2nd edition, John Wiley & Sons, Inc.,
New York.
Kaplan, E.L., and Meier, P. (1958), "Nonparametric Estimation From
Incomplete Observations," Journal of the American Statistical
Association, 53: 457-481.
Kalbfleisch, J.D., and Prentice, R.L. (1980), The Statistical Analysis of
Failure Data, John Wiley & Sons, Inc., New York.
Kielpinski, T.J., and Nelson, W.(1975), "Optimum Accelerated Life-Tests
for the Normal and Lognormal Life Distributins," IEEE Transactions on
Reliability, Vol. R-24, 5, pp. 310-320.
Klinger, D.J., Nakada, Y., and Menendez, M.A. (1990), AT&T Reliability
Manual, Van Nostrand Reinhold, Inc, New York.
Kolmogorov, A.N. (1941), "On A Logarithmic Normal Distribution Law
Of The Dimensions Of Particles Under Pulverization," Dokl. Akad Nauk,
USSR 31, 2, pp. 99-101.
Kovalenko, I.N., Kuznetsov, N.Y., and Pegg, P.A. (1997), Mathematical
Theory of Reliability of Time Dependent Systems with Practical
Applications, John Wiley & Sons, Inc., New York.
Landzberg, A.H., and Norris, K.C. (1969), "Reliability of Controlled
Collapse Interconnections." IBM Journal Of Research and Development,
Vol. 13, 3.
Lawless, J.F. (1982), Statistical Models and Methods For Lifetime Data,
John Wiley & Sons, Inc., New York.
Leon, R. (1997-1999), JMP Statistical Tutorials on the Web at
http://www.nist.gov/cgi-bin/exit_nist.cgi?
url=http://web.utk.edu/~leon/jmp/.
Mann, N.R., Schafer, R.E. and Singpurwalla, N.D. (1974), Methods For
Statistical Analysis Of Reliability & Life Data, John Wiley & Sons, Inc.,
New York.
Martz, H.F., and Waller, R.A. (1982), Bayesian Reliability Analysis,
Krieger Publishing Company, Malabar, Florida.
Meeker, W.Q., and Escobar, L.A. (1998), Statistical Methods for
Reliability Data, John Wiley & Sons, Inc., New York.
Meeker, W.Q., and Hahn, G.J. (1985), "How to Plan an Accelerated Life
Test - Some Practical Guidelines," ASC Basic References In Quality
Control: Statistical Techniques - Vol. 10, ASQC , Milwaukee,
Wisconsin.
Meeker, W.Q., and Nelson, W. (1975), "Optimum Accelerated Life-Tests
for the Weibull and Extreme Value Distributions," IEEE Transactions on
Reliability, Vol. R-24, 5, pp. 321-322.
Michael, J.R., and Schucany, W.R. (1986), "Analysis of Data From
Censored Samples," Goodness of Fit Techniques, ed. by D'Agostino,
R.B., and Stephens, M.A., Marcel Dekker, New York.
MIL-HDBK-189 (1981), Reliability Growth Management, U.S.
Government Printing Office.
MIL-HDBK-217F ((1986), Reliability Prediction of Electronic
Equipment, U.S. Government Printing Office.
MIL-STD-1635 (EC) (1978), Reliability Growth Testing, U.S.
Government Printing Office.
Nelson, W. (1990), Accelerated Testing, John Wiley & Sons, Inc., New
York.
Nelson, W. (1982), Applied Life Data Analysis, John Wiley & Sons, Inc.,
New York.
O'Connor, P.D.T. (1991), Practical Reliability Engineering (Third
Edition), John Wiley & Sons, Inc., New York.
Peck, D., and Trapp, O.D. (1980), Accelerated Testing Handbook,
Technology Associates and Bell Telephone Laboratories, Portola, Calif.
Pore, M., and Tobias, P. (1998), "How Exact are 'Exact' Exponential
System MTBF Confidence Bounds?", 1998 Proceedings of the Section on
Physical and Engineering Sciences of the American Statistical
Association.
SEMI E10-0701, (2001), Standard For Definition and Measurement of
Equipment Reliability, Availability and Maintainability (RAM),
Semiconductor Equipment and Materials International, Mountainview,
CA.
Tobias, P. A., and Trindade, D. C. (1995), Applied Reliability, 2nd
edition, Chapman and Hall, London, New York.

Engineering Statistics Handbook

Uploaded by

Copyright:

Available Formats

Engineering Statistics Handbook

Uploaded by

Document Information

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Engineering Statistics Handbook

Uploaded by

Copyright:

Available Formats

1.

Exploratory Data Analysis

= -2). Using the formula above we

= -2). Our initial guess regarding

For factor variables coded with + and - settings, the

and the corresponding value of C

for each value of degrees

is closest to is the correct

for a given value of . The procedure is:

as the (median) scale

You might also like