Statistics for Managers

Using Microsoft® Excel

5th Edition

Chapter 11
Analysis of Variance

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-1
 Learning Objectives
After completing this chapter, you should be able to:
 Recognize situations in which to use analysis of variance
(ANOVA)
 Understand different analysis of variance designs
 Evaluate assumptions of the model
 Perform a single-factor ANOVA and interpret the results
 Conduct and interpret a Tukey-Kramer post-analysis to
determine which means are different
 Analyze two-factor analysis of variance tests
 Conduct and interpret a Tukey-Kramer post-analysis procedure
to determine which factors are different
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-2
 General ANOVA Analysis
 Investigator controls one or more independent variables
 Called factors or treatment variables
 One factor contains three or more levels or groups or
categories/classifications
 Other factors contains two or more levels or groups or
categories/classifications
 Experimental design: the plan used to test the hypothesis

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-3
 One-Factor ANOVA

 Also known as Completely Randomized

Design and One-way ANOVA
 Experimental units (subjects) are assigned
randomly to treatments
 Subjects are assumed homogeneous
 Only one factor or independent variable
 With three or more treatment levels
 Analyzed by one-factor analysis of variance

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-4
 One-Factor Analysis of Variance

Evaluates the difference among the means of three or more

groups
Examples: Accident rates for 1st, 2nd, and 3rd shift
Expected mileage for five brands of tires

Assumptions
 Populations are normally distributed
(test with Box plot or Normal Probability Plot)
 Populations have equal variances
(use Levene’s Test for Homogeneity of Variance)
 Samples are randomly and independently drawn

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-5
 Why Analysis of Variance?

 We could compare the means in pairs using a t test for

difference of means
 Each t test contains Type 1 error
 The total Type 1 error with k pairs of means is 1- (1 - ) k
 If there are 5 means and you use = .05
 Must perform 10 comparisons
 Type I error is 1 – (.95) 10 = .40
 40% of the time you will reject the null hypothesis of equal
means in favor of the alternative even when the null is true!

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-6
 Hypotheses: One-Factor ANOVA

 H0 : μ1  μ2  μ3    μc
 All population means are equal
 i.e., no treatment effect (no variation in means among groups)


H1 : Not all of the population means are the same
 At least one population mean is different
 i.e., there is a treatment (groups) effect
 Does not mean that all population means are different (at
least one of the means is different from the others)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-7
 Hypotheses: One-Factor
ANOVA
H0 : μ1  μ2  μ3    μc
H1 : Not all μ j are the same

All Means are the same:

The Null Hypothesis is True
(No Group Effect)

μ1  μ 2  μ 3
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-8
 Hypotheses: One-Factor
ANOVA
H 0 : μ1  μ 2  μ 3    μ c At least one mean is different:
The Null Hypothesis is NOT true
H1 : Not all μj are the same (Treatment Effect is present)

or

μ1  μ2  μ3 μ1  μ2  μ3
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-9
 One-Factor ANOVA Table
Source of df SS MS P-value F-Ratio
Variation (Variance)
Between c-1 SSA MSA P(X=F) MSA
F
Groups MSW
Within n-c SSW MSW
Groups
Total n-1 SST =
SSA+SSW

c = number of groups
n = sum of the sample sizes from all groups
df = degrees of freedom

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-10
 One-Factor ANOVA
Test Statistic
H0: μ1= μ2 = … = μc
H1: At least two population means are different
 Test statistic
MSA
F
MSW
 MSA is mean squares among variances
 MSW is mean squares within variances
 Degrees of freedom
 df1 = c – 1 (c = number of groups)
 df2 = n – c (n = sum of all sample sizes)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-11
 One-Factor ANOVA
Test Statistic
 The F statistic is the ratio of the among variance to the
within variance
 The ratio must always be positive
 df1 = c -1 will typically be small
 df2 = n - c will typically be large

Decision Rule:
Reject H0 if F > FU,  = .05
otherwise do not reject H0
0 Do not Reject H0
reject H0
FU
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-12
 One-Factor ANOVA
F Test Example
You want to see if three Club 1 Club 2 Club 3
different golf clubs yield 254 234 200
different distances. You 263 218 222
randomly select five 241 235 197
measurements from trials on an 237 227 206
automated driving machine for 251 216 204
each club. At the .05
significance level, is there a
difference in mean driving
distance?

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-13
 One-Way ANOVA
Example
Distance
270
Club 1 Club 2 Club 3 260 •
254 234 200 •• X 1
250
263 218 222
241 235 197 240 •
• ••
237 227 206 230
• X2
251 216 204 220 •• •
X
210
X3
x1  249.2 x 2  226.0 x 3  205.8 ••
200 ••
x  227.0
190 1 2 3
Club
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-14
 ANOVA -- Single Factor:
Excel Output
EXCEL: Tools | Data Analysis | ANOVA: Single Factor
SUMMARY
Groups Count Sum Average Variance
Club 1 5 1246 249.2 108.2
Club 2 5 1130 226 77.5
Club 3 5 1029 205.8 94.2
ANOVA
Source of
SS df MS F P-value F crit
Variation
Between
4716.4 2 2358.2 25.275 4.99E-05 3.89
Groups
Within
1119.6 12 93.3
Groups
Total 5836.0 14        

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-15
 One-Factor ANOVA Example
Solution
H0: μ1 = μ2 = μ3
H1: μi not all equal

 = .05
p-value: 4.99E-05
Decision:
Reject H0 at  = 0.05 There is evidence that
Conclusion: at least one μi differs
from the rest
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-16
 The Tukey-Kramer Procedure
 Tells which population means are significantly different
 e.g.: μ1 = μ2 ≠ μ3
 Done after rejection of equal means in ANOVA
 Allows pair-wise comparisons
 Compare absolute mean differences with critical
range

μ1= μ2 μ3 x

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-17
 Tukey-Kramer Critical Range

MSW  1 1 
Critical Range  QU 
2  n j n j' 

where:
QU = Value from Studentized Range Distribution with c
and n - c degrees of freedom for the desired level
of  (see appendix E.9 table)
MSW = Mean Square Within
nj and nj’ = Sample sizes from groups j and j’

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-18
 The Tukey-Kramer Procedure:
Example
1. PhStat computes the absolute
Club 1 Club 2 Club 3 mean differences:
254 234 200
x1  x 2  249.2  226.0  23.2
263 218 222
241 235 197 x1  x 3  249.2  205.8  43.4
237 227 206
251 216 204 x 2  x 3  226.0  205.8  20.2

2. You find a QU value from the table in appendix E.9 with

c = 3 (across the table) and n – c = 15 – 3 = 12 degrees of freedom (down
the table) for the desired level of  ( = .05 used here):

QU  3.77
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-19
 The Tukey-Kramer Procedure:
Example (continued)
3. PhStat computes the Critical Range:
MSW  1 1  93.3  1 1 
Critical Range  QU   3.77     16.285
 
2  n j n j'  2 5 5

4. Compare:
5. All of the absolute mean differences x1  x 2  23.2
are greater than critical range.
Therefore there is a significant x1  x 3  43.4
difference between each pair of
means at 5% level of significance. x 2  x 3  20.2
PhStat does all the calculations for you
but you must input the Q value
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-20
 Tukey-Kramer in PHStat

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-21
 ANOVA Assumptions
Levene’s Test
 Tests the assumption that the variances of each
group are equal.
 First, define the null and alternative hypotheses:
 H0: σ21 = σ22 = …=σ2c
 H1: Not all σ2j are equal
 Second, compute the absolute value of the difference
between each value and the median of each group.
 Third, perform a one-way ANOVA on these
absolute differences.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-22
 Two-Factor ANOVA

 Examines the effect of

 Two factors of interest on the dependent
variable
 e.g., Percent carbonation and line speed on soft
drink bottling process
 Interaction between the different levels of these
two factors
 e.g., Does the effect of one particular carbonation
level depend on which level the line speed is set?

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-23
 Two-Factor ANOVA

 Assumptions

 Populations are normally distributed

 Populations have equal variances
 Independent random samples are selected

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-24
 Two-Factor ANOVA
Sources of Variation
Two Factors of interest: A and B
r = number of levels of factor A
c = number of levels of factor B
n/ = number of replications for each cell
n = total number of observations in all cells
(n = rcn/)
Xijk = value of the kth observation of level i
of factor A and level j of factor B

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-25
 Two-Factor ANOVA Summary Table
With Replication
Source of Degrees of Sum of Mean F
p-value
Variation Freedom Squares Squares Statistic

Sample
MSA = MSA/
Factor A r–1 SSA f (FA)
SSA/(r – 1) MSE
(Row)
Columns MSB = MSB/
c–1 SSB f (FB)
Factor B SSB/(c – 1) MSE
MSAB/
Interaction MSAB = MSE f (FA&B)
(r – 1)(c – 1) SSAB
(AB) SSAB/ [(r – 1)(c – 1)]

Within MSE =
rc n – 1)’
SSE
Error SSE/[rc n’ – 1)]
Total rc n’ – 1 SST
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-26
 Two-Factor ANOVA
With Replication
As production manager, Box Machine1 Machine2 Machine3
you want to see if 3 filling 1 25.40 23.40 20.00
machines have different 26.40 24.40 21.00
mean filling times when 2 26.31 21.80 22.20
used with 5 types of boxes. 25.90 23.00 22.00
At the .05 level, is there a 3 24.10 23.50 19.75
difference in machines, in 24.40 22.40 19.00
boxes? Is there an
4 23.74 22.75 20.60
interaction?
25.40 23.40 20.00
5 25.10 21.60 20.40
26.20 22.90 21.90
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-28
 Summary Table

Source of Degrees of Sum of Mean F P-Value

Variation Freedom Squares Square
Sample
5-1=4 7.4714 1.8678 3.6868 .0277
(Boxes)
Columns
3-1=2 106.298 53.149 104.908 1.52E-09
(Machines)
Interaction (5-1)(3-1) = 8 9.7032 1.2129 2.3941 .0690

Within 5·3·(2-1)=15 7.5994 .5066

(Error)
Total 3·5·2 -1 = 29 131.0720
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-29
 The Tukey-Kramer Procedure:
With Replication Factor A
MSW
critical range  Qr ,rc ( n ' 1)
cn '
1. No Macro - compute by formula only
2. MSW (Within) from ANOVA Printout
3. Q from Table E.9 in the Book page 860
alpha = .05 or .01
r is the number of levels of factor A (across table)
rc(n’-1) (down the table)
c is the number of levels of factor B
n’ is the number of replications
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-30
 The Tukey-Kramer Procedure:
With Replication Factor B
MSW
critical range  Qc ,rc ( n ' 1)
rn'
1. No Macro compute by formula only
2. MSW (Within) from ANOVA Printout
3. Q from Table E.9 in the Book page 860
alpha = .05 or .01
c is the number of levels of factor B (across table)
rc(n’-1) (down the table)
r is the number of levels of factor A
n’ is the number of replications
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-31
 Chapter Summary
 Described one-factor analysis of variance
 ANOVA assumptions
 ANOVA test for difference in c means
 The Tukey-Kramer procedure for multiple comparisons

 Described two-factor analysis of variance

 Examined effects of multiple factors
 Examined interaction between factors for the model with replicated
observations
 The Tukey-Kramer procedure for multiple comparisons for both
factor A and factor B for the model with replication

Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc. Chap 11-32