Miller 1991
Miller 1991
Miller 1991
Summary of Contents
Linear calibration
Co rreIat io n coefficient
’Least squares’ line
Errors and confidence limits
Method of standard additions
Limit of detection and sensitivity
Intersection o f t w o straight lines
Residuals in regression analysis
Regression techniques in the comparison o f analytical methods
Robust and non-parametric regression methods
Analysis of variance i n linear regression
Weighted linear regression methods
Partly straight, partly curved calibration plots
Treatment of non-linear data by transformations
Curvilinear regression
Spline functions and other robust non-linear regression methods
Keywords: Analytical calibration method; statistics and rectilinear graph; curve fitting method; robust and
non -parametric method; review
ment of new methods, and in any other case where there is the of many instrumental methods; flow injection analysis, for
least uncertainty, the assumption of linearity must be carefully example, shows many examples of RSDs of 0.5% or less.4 In
investigated. It is always valuable to inspect the calibration such cases, it may be necessary either to abandon assumption
graph visually on graph paper o r on a computer monitor, as (i) (again, suitable statistical methods are available-see
gentle curvature that might otherwise go unnoticed is often below), o r to maintain the validity of the assumption by
detected in this way (see below). Here, and in many other making up the standards gravimetrically rather than volu-
aspects of calibration statistics, the low-cost computer pro- metrically, i.e., with an even greater accuracy than usual. If
grams available for most personal computers are very valu- the assumption is valid, the line calculated as shown below, is
able. As will be seen, it is important to plot the graph with the called the line of regression of y on x, and has the general
instrument response on the y-axis and the concentrations of formula y = bx + a , where b and a are, respectively, its slope
the standards on the x-axis. One of the calibration points and intercept. This line is calculated by minimizing the sums of
should normally be a ‘blank’, i.e., a sample containing all the the squares of the distances between the standard points and
reagents, solvents, etc., present in the other standards, but no the line in the y-direction. (Hence the term ‘least squares’ for
analyte. It is poor practice to subtract the blank signal from this method.) It is important to note that the line of regression
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those of the other standards before plotting the graph. The of x on y would seek to minimize the squares of x-direction
blank point is subject to errors as are all the other points and errors, and therefore would be entirely inappropriate when
should be treated in the same way. As shown in Part 1 of this the signal is plotted on the y-axis. (The two lines are not the
review,’ if two results, x1and x2, have random errors el and e2, same except in the hypothetical situation when all the points
then the random error in x1 - x2 is not el - e2. Thus, lie exactly on a straight line.) The y-direction distances
subtraction of the blank seriously complicates the proper between each calibration point and the point on the calculated
estimation of the random errors of the calibration graph. line at the same value of x are known as the y-residuals and are
Moreover, even if the blank signal is subtracted from the other of great importance in several calculations, as will be shown
measurements, the resulting graph may not pass exactly later in this paper.
through the origin. Assumption (iii), that the y-direction errors are equal, is
Linearity is often tested using the correlation coefficient, r . also open to comment. In statistical terms it means that all the
This quantity, whose full title is the ‘product-moment correla- points on the graph are of equal weight, i. e., equal importance
tion coefficient’, is given by in the calculation of the best line-hence the term ‘un-
weighted’ least squares. In recent years this assumption has
been tested for several different types of instrumental
analysis, and in many cases it is found that the y-direction
errors tend to increase as x increases, though not necessarily in
where the points on the graph are (xl,yl), ( x 2 ,y 2 ) , . . (xi,yi), linear proportion. Such findings should encourage the use of
. . . (xn,yn), and X and J are, as usual, the mean values of xi and weighted least squares methods, in which greater weight is
yi respectively. It may be shown that -1 d r d + l . In the given to those points with the smallest experimental errors.
hypothetical situation when r = - 1, all the points on the These points are discussed further in a later section.
graph would lie on a perfect straight line of negative slope; if r If assumptions (i)-(iii) are accepted then the slope, b , and
= +1, all the points would lie exactly on a line of positive
intercept, a , of the unweighted least squares line are found
slope; and r = 0 indicates no linear correlation between x and from
y . Even rather ‘poor’ calibration graphs, i.e., with significant
y-direction errors, will have r values close to 1 (or - l), values
of Irl< about 0.98 being unusual. Worse, points that clearly lie
on a gentle curve can easily give high values of I T -(. So the
magnitude of r , considered alone, is a poor guide to linearity.
A study of the ‘y-residuals’ (see below) is a simple and a=J-bx (3)
instructive test of whether a linear plot is appropriate. A The equations show that, when b has been determined, a can
recent report of the Analytical Methods Committee3 provides be calculated by using the fact that the fitted line passes
a useful critique of the uses of r , and suggests an alternative through the centroid, (X, J ) . These results are proved in
method of testing linearity, based on the weighted least reference 5 , a classic text on the mathematics of regression
squares method (see below). methods. The values of a and 6 can be simply applied to the
determination of the concentration of a test sample from the
corresponding instrument output.
‘Least Squares’ Line
If a linear plot is valid, the analyst must plot the ‘best’ straight Errors and Confidence Limits
line through the points generated by the standard solutions.
The common approach t o this problem (not necessarily the The concentration value for a test sample calculated by
best!) is to use the unweighted linear least squares method, interpolation from the least squares line is of little value unless
which utilizes three assumptions. These are (i) that all the it is accompanied by an estimate of its random variation. T o
errors occur in the y-direction, i.e., that errors in making up understand how such error estimates are made, it is first
the standards are negligible compared with the errors in important to appreciate that analytical scientists use the line of
measuring instrument signals, (ii) that the y-direction errors regression of y on x in an unusual and complex way. This is
are normally distributed, and (iii) that the variation in the best appreciated by considering a conventional application of
y-direction errors is the same at all values of x. Assumption (ii) the line in a non-chemical field. Suppose that the weights of a
is probably justified in most experiments (although robust and series of infants are plotted against their ages. In this case the
non-parametric calibration methods which minimize its sig- weights would be subject to measurement errors and to
nificance are available, see below), but the other two inter-individual variations (e.g., all 3 month old infants would
assumptions merit closer examination. not weigh the same), so would be correctly plotted on the
The assumption that errors only occur in the y-direction is y-axis: the infants’ ages, which would presumably be known
effectively valid in many experiments; errors in instrument exactly, would be plotted on the x-axis. The resulting plot
signals are often at least 2-3% [relative standard deviation would be used to predict the average weight (y) of a child of
(RSD)], whereas the errors in making up the standards should given age (x). That is, the graph would be used to estimate a
be not more than one-tenth of this. However, modern y-value from an input x-value. The y-value obtained would of
automatic techniques are dramatically improving the precision course be subject to error, because the least squares line itself
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is subject to uncertainty. The graph would not normally be Fig. 1, the confidence interval for this xo value results from the
used to estimate the age of a child from its weight! uncertainty in the measurement of yo, combined with the
In analytical work, however, the calibration graph is used in confidence interval for the regression line at that yo value. The
the inverse way-an experimental value of y ('yo,the instru- standard deviation sxo is given by
ment signal for a test sample) is input, and the corresponding
value of x (xo, the concentration of the test sample) is
determined by interpolation. The important difference is that (7)
xo is subject to error for two reasons, (1) the errors in the
calibration line, as in the weight versus age example, and (2) It can be shown5 that this equation is an approximation that is
the random error in the input yo value. Error calculations only valid when the function
involving this 'inverse regression' methods are thus far from t2
simple and indeed involve approximations (see below).
First, we must estimate the random errors of the slope and
intercept of the regression line itself. These involve the
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performed, the sum of the first two components of the y-axis intercept and the slope of the calibration line, calculated
bracketed term in (7) and (9) is minimized by setting rn = n. using equations (2) and ( 3 ) , i.e. ,
However, small values of n are to be avoided for a separate
reason, viz., that the use of n - 2 degrees of freedom then x, = alb (10)
leads to very large values of t and correspondingly wide The standard deviation of x,, sxe, is given by a modified form
confidence intervals. Calculation shows that, in the simple of equation (7):
case where yo = y , then for any given values of sylx and 6 , the
priority (at the 95% confidence level) is to avoid values of n <
5 because of the high values o f t associated with <3 degrees of
freedom. When n 3 5 , maximum precision from a fixed
number of measurements is obtained when rn = n. This standard deviation can as always be converted into a
The last bracketed term in equations (7) and (9) shows that confidence interval using the appropriate t value. It might be
precision (for fixed rn and n) is maximized when yo is as close expected that such confidence intervals would be wider for this
as possible to 7 (this is expected in view of the confidence extrapolation method than for a conventional interpolation
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interval variation shown in Fig. l), and when 7 (xi - X ) 2 is as method. In reality, however, this is not so, as the uncertainty
in the value of x, derives only from the random errors of the
large as possible. The latter finding suggests ;hat calibration regression line itself, the corresponding value of y being fixed
graphs might best be plotted with a cluster of points near the at zero in this case. The real disadvantages of the method of
origin, and another cluster at the upper limit of the linear standard additions are that each calibration line is valid for
range of interest [Fig. 2(a)]. If n calibration points are only a single test sample, larger amounts of the test sample
determined in two clusters of n/2 points at the extremes of a may be needed and automation is difficult.
straight line, the value of the term 7 (xi - X)’ is increased by a The slope of a standard additions plot is normally different
factor [3(n - l)/(n + l)]compared Lith the case in which then from that of the conventional calibration plot for the same
points are equally spaced along the same line [Fig. 2(b)]. In sample. The slope ratio is a measure of the proportional
practice it is usual to use a calibration graph with points systematic error produced by the matrix effect, a principle
roughly equally distributed over the concentration range of used in many ‘recovery’ experiments.’” The use of the
interest. The use of two clusters of points gives no assurance of conventional standard additions method has been discussed at
the linearity of the plot between the two extreme x values; length by Cardone. 11,12 The generalized standard additions
moreover, the term [ ( y o - 7)2/b2 ?(xi - i ) 2 ] is often the method (GSAM)13 is applicable to multicomponent analysis
I problems, but belongs to the realm of chemometrics.*4
smallest of the three bracketed terms in equations (7) and (9),
so reducing its value further may have only a marginal over-all
effect on the precision of no. Limit of Detection and Sensitivity
The ability to detect minute amounts of analyte is a feature of
Method of Standard Additions many instrumental techniques and is often the major reason
for their use. Moreover, the concept of a limit of detection
In several analytical methods (e.g., potentiometry, atomic (LOD) seems obvious: it is the least amount of material
and molecular spectroscopy) matrix effects on the measured the analyst can detect because it yields an instrument response
signal demand the use of the method of standard additions. significantly greater than a blank. Nonetheless, the definition
Known amounts of analyte are added (with allowance for any and measurement of LODs has caused great controversy in
dilution effects) to aliquots of the test sample itself, and the recent years, with additional and considerable confusion over
calibration graph (Fig. 3) shows the variation of the measured nomenclature, and there have been many publications by
signal with the amount of analyte added. In this way some statutory bodies and official committees in efforts to clarify the
matrix effects are equalized between the sample and the situation. Ironically, the significance of LODs, at least in the
standards. The concentration of the test sample, x,, is given by strict quantitative sense, is probably overestimated. There is
the intercept on the x-axis, which is clearly the ratio of the clearly a need for a means of expressing that (for example)
spectrofluorimetry at its best is capable of determining lower
amounts of analytes than absorptiometry, and the principal
use of LODs in the literature appears to be to show that a
newly discovered method is indeed ‘better’ than its predeces-
sors. But there are many reasons why the LOD of a particular
method will be different in different laboratories, when
Fig. 2 Calibration graphs with: ( a ) clusters of standards at high and Fig. 3 Calibration graph for the method of standard additions. The
low concentrations; and ( b ) equally spaced standards point 0 is due to the original sample; for details see text
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This equation yields the confidence limits for the true mean critical examination and that it is not a straightforward matter
value of y at any given value of x . The t value is taken at the to decide whether a straight line or a curve should be drawn
desired confidence level (usually 95%) and nl - 2 degrees of through a set of calibration points. Important additional
freedom. A similar equation applies to the line y = b2x + a2. information on both these topics can be obtained from the
One reasonable definition for the lower confidence limit for y-residuals, the 0,- 9 ) values which represent the differences
xx, ( x L ) ,is the abscissa value of the point of intersection of the between the experimental y-values and the fitted y-values.
upper confidence limit of line 1and the lower confidence limit The residuals thus represent the random experimental errors
of line 2 (Fig. 5 ) . At this point in the measurements of y, if the statistical model used (the
unweighted regression line of y on x ) is correct. Many
statistical tests can be applied to these residuals (a comprehen-
sive survey is given in reference 5) but for routine work it is
often sufficient to plot the individual residuals against 9 or
against x . Many regression programs for personal computers
offer this facility and some provide additional refinements,
e.g., the inclusion of lines showing the standard deviations of
which can be solved for xL. An analogous equation can be the residuals.
written for xu, which is similarly defined by the intersection of It can be shown that, if the calibration line is calculated from
the lower confidence limit for line 1and the upper confidence the equation y = bx + a (but not if it is forced through the
limit for line 2 origin by using the form y = bx), the residuals always total
zero, allowing for rounding errors. As already noted, the
residuals are assumed to be normally distributed. Fig. 6(a)
shows the form that the residuals should thus take if the
unweighted regression line is a good model for the experimen-
tal data. Fig. 6(b) and ( c ) indicates possible results if the
unweighted regression line is inappropriate. If the residuals
As the confidence intervals for lines 1 and 2 may be of tend to become larger as y (or x ) increases, the use of a
different width, and as the two lines may interesect at any weighted regression line (see below) is indicated, and if the
angle, the confidence limits for x, may not be symmetrical residuals tend to fall on a curve, the use of a curved calibration
about x, itself. It should also be noted that the confidence graph rather than a linear one is desirable. In the latter case
limits for x, derived from (for example) the 95% confidence the signs (+ or -) of the residuals, which should be in random
limits for the two separate lines are not necessarily the 95% order if an appropriate statistical model has been used, will
confidence limits for x,. As the estimation method used above tend to occur in sequence (‘runs’); in the example given, there
assumes the worst case in combining the random errors of the is clearly a sequence of positive residuals, followed by a
two lines, the derived confidence limits are on the pessimistic sequence of negative ones followed by a second positive
(ie., realistic!) side. Finally it is important to note that the sequence. The number of ‘runs’ (three in the example given) is
practical applications of this method utilize extrapolations of thus significantly less than if the signs of the residuals had been
the two straight lines to the intersection point. These
extrapolations are generally short, and care is usually taken to
+ and - in random order. The Wald-Wolfowitz method tests
for the significance of the number of runs in a set of data5.22 by
perform the experiments in conditions where the extrapola- comparing the observed number of runs with tabulated data,23
tions are believed to be valid. However, if this belief is but it cannot be used if there are fewer than nine points in the
erroneous (e.g., in studies of drug-protein binding where calibration graph. Like the other residual diagnostic methods
there is more than one class of binding site instead of the single described here, the test is thus of restricted value in
class often assumed), even the best statistical methods cannot instrumental analysis, where the number of calibration points
produce chemically valid results. is frequently less than this. Tests on residuals are not,
however, limited to linear regression plots: they can also be
Residuals in Regression Statistics applied to non-linear plots, and indeed to any situation in
which experimental data are fitted to a statistical model and
Previous sections of this review have shown that the un- some unexplained variations occur.
weighted regression methods in common use in analytical Examination of the residuals may shed light on a further
chemistry are based on several assumptions which merit problem, that of outliers among the data. The first part of this
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as just one function of the yi values, i.e., the slope, 6 , will other variations at much higher concentrations. In some cases
calculate C(j+ - j ) 2 from b2 Z(xi - X ) 2 . The second source of the standard deviation is expected to rise in proportion to the
variation is the SS about regression, i. e., the variation due to concentration, i.e., the RSD is approximately constant,39
deviations from the calibration line, each term in the SS being while in other cases the standard deviation rises, though less
of the form (yi - j i ) 2 . This SS has ( n - 2) DF, reflecting the rapidly than the concentration.40 Many attempts have been
fact that the residuals come from a model requiring the made to formulate rules and equations for this concentration
estimation of two parameters, a and b. In accordance with the related behaviour of the standard deviation for different
additivity principle described above, it is possible to show5 methods.41.42 In practice, however, it will frequently be better
that to rely on the analyst’s experience of a particular method,
instrument, etc. in this respect.
Total SS about J = SS due to regression +
SS about If experience suggests that the standard deviation of
regression (20) replicate measurements does indeed vary significantly with x
Moreover, the number of DF for the total SS is ( n - 2) + 1 = (heteroscedastic data) , a weighted regression line should be
( n - 1).This result is expected, as only ( n - 1) yi values are plotted. The equations for this line differ from equations (2)-
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needed to determine the total SS about y , as Z(yi - 7 ) = 0 by (7) because a weighting factor, wi, must be associated with
definition. A typical ANOVA table for a linear regression plot each calibration point xi, yi. This factor is inversely propor-
is shown in Table 1. This is a one-way ANOVA calculation, tional to the variance of y i , si2, and must either be estimated
there being only one source of variation in addition to the from a suitable model (see above), or determined directly
inevitable experimental error. The significance of the correla- from replicate measurements of yi:
tion can be tested by using the F-test, i.e., by calculating
wi = si-2/(Csi-2/n)
I (23)
F1, ( n - 2) = MSreg/MSres (21) Equation (23) conveniently scales the weighting factors so that
In practice this is rarely necessary (though readily available their sum is equal to n , the number of xi values. The slope and
in software packages), as the F-values are generally vastly intercept of the weighted regression line are then given,
greater than the critical values. A more common estimate of respectively, by
the goodness of fit is given by the statistic R2, sometimes C wix,yi- njiwJw
known as the (multiple) coefficient of determination or the
(multiple) correlation coefficient. The prefix ‘multiple’ occurs b = Zwixi2 - n(xw)2 (24)
1
because R2 can also be used in curvilinear regression (see
below). If the regression line (straight or curved) is to be a a = jjw- bXW (25)
good fit to the experimental points, the SS due to regression
Both these equations use the coordinates of the weighted
should be a high proportion of the total SS about J . This is
centroid, (Xw, Jw), given by X, = zw,xi/n and J w = Cwiyi/n,
expressed quantitatively using the equation 1 I
respectively; the weighted regression line must pass through
R2 = SS due to regressiordtotal SS about J (22) this point. The standard deviation, sxow, and hence a con-
R2 clearly lies between 0 and 1(although it can never reach 1 if fidence interval of a concentration estimated from a weighted
there are multiple determinations of yi at given xi valuess), regression line is given by
however, it is often alternatively expressed as a percentage-
the percentage of goodness of fit provided by a regression
equation. It can be shown5 that, for a straight line plot, R2 =
r2, the square of the product-moment correlation coefficient.
The application of R2 to non-linear regression methods is where wo is an interpolated weight appropriate to the
considered further below. experimental yo value, and s(ylx)wis given by
Degrees
of
Source of variation freedom Sum of squares Mean square (MS)
n
Regression 1 c (jji
i = 1
- y)2
PI bi - pi)*
About regression ( i . e . , n-2 ? (ri - pip
1 = 1 $Ix = -2
residual)
n
Total n-1 ,bi - J)2
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the observed data. This is most commonly attempted by using regarded as an optimization problem and can be tackled by
a polynomial equation of the form methods such as simplex optimization.51 This approach offers
no particular benefit in cases where exact solutions are also
y = a + bx + cx2 + dx3 + . . . (29) available by matrix manipulation, such as the polynomial
The advantage of this type of equation is that, after the equation (29), but may be very advantageous in other cases
number of terms has been established, matrix manipulation where exact solutions are not accessible. Again commercially
allows an exact solution for a, b, c, etc. if the least-squares available software is plentiful. A recent paper52 describes the
fitting criterion (see above) is used. Most computer packages calculation of confidence limits for calibration lines calculated
offer such polynomial curve-fitting programs, so, in practice using the simplex method.
the major problem for the experimental scientist is to decide ?Vhichevermodel and calculation method is chosen to plot a
on the appropriate number of terms to be used in equation calibration graph, it is desirable to examine the residuals
(29). The number of terms must clearly be <n for the equation generated and use them to study the validity of the chosen
to have any physical meaning and common sense suggests that model. If the latter is suitable the residuals should show no
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the least number of terms providing a satisfactory fit to the marked trend in sign, spread or value when plotted against
data should always be used (quadratic or cubic fits are corresponding x or y values. As for linear regression, outlying
frequently excellent). points can also be studied.
Several approaches to this problem are available, the
simplest (though probably not the best) being the use of the Spline Functions and Other Robust Non-linear
coefficient of determination, R2. As described above, this Regression Methods
coefficient expresses the extent to which the total SS about jj
can be explained by the regression equation under scrutiny. As non-linear calibration plots often arise from a combination
Values of R2 close to 1 (or 100%+omputer packages often of physico-chemical effects (see above) and failure of mathem-
present the result in this form) are thus apparently indicative atical approximations etc., it is perhaps unrealistic to expect
of a good fit between the chosen equation and the experimen- that any single curve will adequktely describe such data. It
tal data. In practice we would thus examine in turn the values may thus be better to plot a curve which is a continuous series
of R2 obtained for the quadratic, cubic, quartic, etc. fits, and of shorter curved portions. The most popular approach of this
then make a judgement on the most appropriate polynomial. type is the cubic spline method, which seeks to fit the
This method is open to two objections. The first is that, like experimental data with a series of curved portions each of
the related correlation coefficient, r (see above), R2 can take cubic form, y = a + bx + cx2 + dx3. These portions are
very high values (>> 0.9) even when visual inspectinn shows connected at points called ‘knots’, at each of which the two
that the fit is obviously indifferent. More seriously still, it may linked cubic functions and their first two derivatives must be
be shown that R2 always increases as successive terms are continuous. In practice, the knots may coincide with
added to the polynomial equation, even if the latter are of no experimental calibration points, but this is not essential and a
real value. (Draper and Smiths point out that this presents variety of approaches to the selection of the number and
particular dangers if the data are grouped, i.e., if we have positions of the knots is available. Spline function calculations
several y-values at each of only a few x-values. The number of are provided by several software packages, and their applica-
terms in the polynomial must then be less than the number of tion to analytical problems has been reviewed by Wold53 and
x-values.) Thus, if this method is to be used, it is essential to by Weg~cheider.5~
attach little importance to absolute R2 values and to continue A group of additional regression methods now attracting
adding terms to the polynomial only if this leads to substantial considerable attention relies on the use of fitting criteria other
increases in R2. than or in addition to the least squares approach. In particular
An alternative and probably more satisfactory curve-fitting the ‘reweighted least squares’ method described in detail by
criterion is the use of the ‘adjusted R2’ statistic, given by50 Rousseeuw and Leroy55 utilizes the least median of squares
criterion (i.e., minimization of the median of the squared
R2 (adjusted) = 1 - (residual MS/total MS) (30) residuals) to identify large residuals. The least squares curve is
The use of the mean square (MS) instead of SS terms, allows then fitted to the remaining points. These modern robust
the number of degrees of freedom ( n - p ) , and hence the methods can of course be applied to straight line graphs as well
number of fitted parameters (p),to be taken into account. For as curves, and despite their requirement for more advanced
any given data set and polynomial function, adjusted R2 computer programs they are already attracting the attention of
always has a lower value than R2 itself; many computer analytical scientists (e.g., reference 56). Such developments
packages provide both calculations. provide timely reminders that the apparently simple task of
Among several other methods for establishing the best fitting a straight line or a curve to a set of analytical data is still
polynomial fit537 the simple use of the F-test1 has much to provoking much original research.
commend it. In this application (often referred to as a partial
F-test), Fis used to test the null hypothesis that the addition of I thank Dr. Jane C. Miller for many invaluable discussions on
an extra polynomial term to equation (29) does not signifi- the material of this review.
cantly improve the goodness of fit of the curve, when
compared with the curve obtained without the extra term.
Thus References
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F= (31) 2 Martens, H., and Naes, T., Multivariate Calibration, Wiley,
residual MS for n-order model
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Fl,n - p at the desired probability level; p is again the number of 4 RGiiEka, J., and Hansen, E. H., Flow Injection Analysis, Wiley,
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known, the calculation of the parameter values can be PWS-Kent, Boston, 2nd edn., 1988.
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