A program for the weighted linear least-squares regression of

A program for the weighted linear least-squares regression of unbalanced response arrays Elio Desimoni* DIFCA, Università degli Studi di Milano, Via Celoria 2, 20133 Milan, Italy. E-mail: [email protected] Received 22nd March 1999, Accepted 28th June 1999

A program is described to establish calibration diagrams by weighted, linear, least-squares regression of unbalanced response arrays. Whatever the confidence level, the number of analysed standard solutions and of their available replicates, the program allows (i) testing for scedasticity, linearity, outliers and normality, (ii) evaluation of slope and intercept of the calibration function and their confidence interval and (iii) evaluation of an unknown concentration and its confidence interval by interpolation/extrapolation. For negative results of the linearity test, the program structure allows a rapid evaluation of data to be discarded to attempt entering the linear range. The program was validated by analysing response arrays obtained by adding a Gaussian noise to known response/concentration functional relationships. The results of validation tests led to the implementation of an empirical but efficient way to correct regression results when certain experimental situations lead to unjustified over-weighting of some responses. The analysis of some real calibration data sets allows the versatility of the software to be evaluated.

Introduction Calibration graphs in instrumental analysis can be obtained by ordinary, linear, least-squares regression (LLSR) if (i) errors occur only in the y-direction, (ii) errors are normally distributed, (iii) variance does not change with concentration and, of course, (iv) responses are linearly related to concentration.1–5 The first assumption is usually valid, unless the insufficient quality of standard solutions used to establish the calibration graph suggests the application of more suitable regression techniques.1,5,6 The second assumption is also usually valid, unless when working, for example, close to physical limits, as with measurements at the lowest detectable concentrations. The third assumption has always to be tested because, when data are heteroscedastic, weighted LLSR (wLLSR) is mandatory. The last assumption has always to be tested also, because deviations from linearity are also frequent. Of course, exponentials, logarithms, parabolas, etc., are perfectly valid (and often theoretically appropriate) calibration relationships but, when choosing linear least-squares regression, testing for linearity is mandatory. Recommended tests of normality, scedasticity and linearity require repeated observations of the standard solutions used in establishing the calibration graph. Depending on the verdict of these tests, the analysis of regression is a creative and interactive process, which needs to be quality-controlled.7 Surprisingly, as underlined in a recent IUPAC Technical Report,8 the analytical literature rarely describes important statistical details related to the calibration design and to the processing of data in the regression analysis. Also, many popular statistical packages do not always warn the user to perform scedasticity and linearity tests before LLSR and/or rarely offer wLLSR. Moreover, they do not easily allow processing of unbalanced data arrays, e.g. those containing a different number of observations of some tested solutions. Similar arrays of responses can result, for example, after elimination of outliers from balanced data arrays acquired by modern processor-driven instruments. In these cases, adequate weighting is necessary.8 Because of this situation, some workers developed in-house programs. An example is the program CALWER.9 Developed in Excel from Microsoft® and proposed to perform weighted

regression, CALWER allows the use of several types of calibration functions and variance models and, moreover, it permits a traceable link between raw data and reported values. More recently,10 a program developed in Mathcad 7 Professional from Mathsoft®, LLSRR, was specifically aimed to test automatically scedasticity and linearity of unbalanced response arrays by specific F-tests, but when performing ordinary LLSR only. This paper describes a new version of LLSRR, namely uwLLSR, suitable for performing interactively the weighted LLSR of unbalanced, heteroscedastic data in the light of internal routines for testing linearity, scedasticity, outliers and normality. The approach is as pragmatic as possible, avoiding the use of sophisticated or robust techniques, to allow an easy understanding by analysts interested in improving the quality of their data processing without a full immersion in high level statistics. As was previously done,10 the proposed worksheet is validated by analysing noised data whose true calibration function and signal-to-noise ratio are a priori known.

Experimental uwLLSR was prepared as a template in Mathcad 7.02a Professional (MathSoft Inc.®, Cambridge, MA, USA). Since the software always maintains 15 digits of precision internally, even if displayed data can be rounded as convenient or logical, people processing the same response arrays by using a different precision might obtain results slightly different from those reported in this paper. Unbalanced arrays of noised responses used in validation tests were obtained by: (i) choosing a response/concentration functional relationship S(C) = b.C + a (1) and a standard deviation/concentration functional relationship s(C) = p.C + q s(C) =

Ap2·C2

s(C) =

p.Ck

+

(2) q2

+q

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(3) (4) 1191

[even though eqn. (3) is the most correct,11,12 in this work all relationships were used in turn to add a Gaussian noise to responses obtained from eqn. (1)]; (ii) generating a balanced array of noised responses, Si,j, by the Mathcad function Si,j = rnorm(jj,S(Ci), s(Ci))

(5)

which, for 1@i@ii, returns a vector of jj random responses from normal distributions with means S(Ci) and standard deviations s(Ci), respectively (ii and jj are the maximum values of row and column range variables, i and j, respectively); (iii) cutting-off the 15-digit noised responses obtained from eqn. (5) at the second decimal place by the Mathcad function

oor(102 ◊ Si, j )

(6)

10 2

(iv) and, finally, arbitrarily dropping some of the noised responses to obtain an unbalanced, noised data set: for example, after discarding S2,3, S2,4 and S4,4 the original balanced 4 3 4 array

ÈS1,1 ÍS 2,1 S := Í ÍS3,1 Í ÍÎS4,1

S1,2 S2,2 S3,2

S1,3 S2,3 S3,3

S4,2

S4,3

S1,4 ˘ ÈS1,1 ÍS S2,4 ˙˙ 2,1 Æ S := Í ÍS3,1 S3,4 ˙ Í ˙ S4,4 ˙˚ ÍÎS4,1

S1,2 S2,2 S3,,2 S4,2

S1,3 m S3,3 S4,3

S1,4 ˘ m ˙˙ S3,4 ˙ ˙ m ˙˚

turns into an unbalanced array characterised by the 4,2,4,3 repetition design. Once an experimental situation has been defined by choosing proper values of a, b, p, q, k [if necessary, as in eqn. (4)] and a repetition design, one can generate and analyse as many data sets as necessary by simply clicking eqn. (5) in the Mathcad template.

Results and discussion Description of the worksheet The program runs nine subsequent sections. Section 1: data input. Here one enters the selected level of significance, a, (from which any critical value necessary to tand F-tests is obtained through the Mathcad software probability density functions), the number of analysed solutions, ii, and the maximum available number of replicates, jj. Next, one enters concentration and response data in the default 9 3 1 and, respectively, 9 3 8 arrays. This is because uwLLSR is developed to analyse repeated observations of at least some of the standard solutions used to evaluate the calibration graph. The above-mentioned dimensions of the default arrays were chosen since, according to literature suggestions,13,14, the calibration is often unstable when the number of calibration points is less than six, while additional points beyond 20 do not give much further help. Moreover, the minimum number of observations necessary to obtain a useful estimate of standard deviation is about six.15 On considering that in most routine operations these numbers can barely be considered realistic, default arrays should suffice in almost all cases (if necessary, any other array dimension can be set). By using repeated observations, uwLLSR differs from CALWER.9 Eventual missing data in the arrays are entered as ‘m’, so that the program can ignore them in the following calculations and in the final plots. Finally, one has also to set the values of Force and K parameters. The use of Force is explained in the section dealing 1192

Analyst, 1999, 124, 1191–1196

with validation tests (see below). K is used to select the calibration graph or the standard additions mode (K = 1 or 0, respectively). When choosing the calibration graph mode, one has also to enter the number of observations of the unknown, M, and, in a specific array, the unknown response(s). Section 2: Bartlett test for homogeneity of variances. Among those available,1,3,16 the Bartlett test is the only test suitable for comparing more than two variances whose degrees of freedom (dof) are not equal,1 i.e. for managing unbalanced data arrays. The program estimates the standard deviations of all sets of repeated observations, and uses them to calculate and compare the experimental F value with the critical (1-tail) value relevant to the selected confidence level and to proper dof. If the experimental value of F is larger than the critical value, in other words if the VAR parameter displayed by the worksheet is higher than unity (VAR = test value/critical value), variances are not homogeneous. Even though not essential, since the worksheet always performs a weighted regression, the Bartlett test helps in understanding the actual degree of heteroscedasticity. Section 3: linear regression of standard deviation. The functional relationship between standard deviation and concentration is necessary to calculate weights, and consequently to evaluate confidence intervals (CIs), over the explored concentration range. Since usually only a few repeated measurements are available, standard deviation estimates are likely to be very poor. As suggested by the Analytical Methods Committee of the Royal Society of Chemistry,13 under the assumption that standard deviation can be approximated to a linear function of concentration, uwLLSR fits raw, experimental standard deviations by ordinary LLSR and plots the standard deviation functional relationship dv(C) = p·C + q

(7)

On knowing p and q, the program can calculate weights, wi, at any concentration by the equation4

wi =

ii ◊ dv(Ci )-2

Â

dv(Ci )-2

(8)

i

Per cent. weights are also calculated and displayed. By using experimental standard deviation values, uwLLSR differs from CALWER,9, which applies all the available variance models to single response arrays and selects the best one by comparing method performances via the logarithmic likelihood. Weights obtained by eqn. (8) cannot be used after data transformation, such as when attempting to obtain linear data from non-linear data. In these cases transformation-dependent weights should replace those defined above.4,17,18 Section 4: the line of regression of y on x. Since the lack-offit test13,19 needs the functional relationship between response and concentration, the weighted regression is performed by a priori hypothesising a linear relationship. Weighted correlation (r) and determination (r2) coefficients are evaluated by a properly modified equation, suitable for working with unbalanced data arrays. Standard equations of slope (b), intercept (a) and of their standard deviations (sb and sa, respectively) were also modified to manage unbalanced data arrays. Finally, the program returns the CI of slope and intercept, t.sb and t.sa, respectively (t is the two-tail, critical value at the desired confidence level and N 2 2 dof; N is the number of responses used in establishing the diagram).

Section 5: lack-of-fit test. The program performs the lackof-fit test by analysing the residual variance.13,19. Also, in this case the usual equations are conveniently modified to manage unbalanced arrays. If the experimental value of F is lower than the critical, one-tail value, that is, if the LIN parameter (LIN = test value/critical value) is lower than 1.0, the hypothesis of linearity is not disproved: linearity can never be proved.19,20 Should linearity be disproved, transformations or curvilinear, spline functions or other non-linear regression methods would be mandatory. A simpler approach,2 involving the LLSR of data arrays in which the highest, next-highest, etc. point is omitted, can be easily attempted by simply changing ii to ii 2 1, ii 2 2, etc. As soon as this is done, uwLLSR returns updated results (see under Validation of the program). Section 6: evaluation of unknown concentrations. If linearity is not disproved, the regression line can be used to interpolate (calibration graph) or extrapolate (standard additions method) the unknown concentration, Cuk. uwLLSR estimates the standard deviation, s(Cuk), by using an equation which holds even when the slope is not highly significant,21 and the relevant CI, t.s(Cuk). When using the standard additions method, whatever the value of Suk eventually entered by the operator, Cuk is set to b/a and the equation of s(Cuk) is automatically corrected to take into account that no experimental response is used.2,4 When using the calibration graph method and M is greater than 1, the average unknown signal is used to interpolate the unknown concentration. Section 7: plotting regression line and confidence intervals of interpolated unknown. In this section, uwLLSR plots the calibration graph, with original data points, weighted regression line and CIs of interpolated unknowns. Underneath, it shows the plot of weighted residuals. Horizontal lines in Fig. 1 are drawn at ±sy/x and ±z(1-a/2).sy/x (sy/x is the residual standard deviation). Since weighted residuals can be considered a random sample from a normal distribution with mean zero and standard deviation sy/x, data outside the ±z(1-a/2).sy/x band in the

plot of weighted residuals are suspect outliers, having only an a probability to pertain to the distribution of weighted residuals. Section 8: outliers test. In this section, eventual outliers can be checked by an F-test.3 To perform the test, one has to enter the actual values of the dof and of sy/x relevant to the response/ concentration functional relationship, and to go back to Section 1, where he/she can drop the suspect outlier from the response matrix by substituting it with m (false missing value). For the program to be running in the right way, the remaining data must occupy the leftmost places of the row. As soon as he/she returns to Section 8 after dropping the suspect response, the program recalculates all data and performs the F-test by using old and new sy/x and dof values:3 if the OUT parameter (OUT = test value/critical 1-tail value) is larger than 1, the suspect response can be permanently discarded. It should be noted that discarding outliers is controversial, because of the a risk of stating that the point is an outlier when, in fact, it is not. However, one should also consider that the lower is the selected a value, the lower the risk. Section 9: Shapiro–Wilk test. The Shapiro–Wilk test implemented in uwLLSR allows for testing normality of not less than three replicates.22 After choosing the specific significance level (0.5, 0.1, 0.05 or 0.01) and the row index, i, of the concentration whose data have to be tested, the worksheet calculates the NOR parameter (NOR = critical value/test value): if NOR is larger than unity, normality is not verified. However, because of the limited number of replicates usually available, testing normality for calibration graphs obtained by instrumental analyses is a challenging task: if normality is to be tested, a conveniently large number of replicates is necessary. If normality is not verified under real analytical conditions (e.g. when processing data sets containing few responses) one can also choose to use Gaussian statistics as a fairly good approximation, because alternatives usually require much effort. Below Section 9, the program summarises some regression results (LIN, VAR, correlation and determination coefficients, sy/x, b, a, Cuk and relevant CIs, etc.). As in CALWER,9 under this section the operator can save, as a text region, all information necessary to ensure a full traceability of the calibration procedure. Validation of the program

Fig. 1 Example of calibration diagram obtained during run B; the upper plot displays original responses, weighted regression line and confidence band of interpolated values. The lower plot displays weighted residuals and ±sy/x and ±z(1-a/2).sy/x bands.

Four experimental situations (A?D) were tested. Each was characterised by different functional relationship, RSD%, slopes, intercepts, calibration mode, dof, repetition designs, etc. These parameters, together with the theoretical unknown concentration value, Cuk, relevant to an arbitrarily selected response, Suk, simulating the results of the analysis of an unknown sample, are reported in Table 1. By choosing K equal to 0 or 1 (see Section 1), the standard additions or, respectively, the calibration graph mode was selected. Twenty-five data sets were repeatedly generated and analysed in each of the four situations. The results were averaged over the 25 runs. Average coefficient of determination, rm2, slope, bm, and intercept, am, of the response functional relationship [eqn. (1)] are summarised in Table 2. Cukm in Table 2 is the average value of the interpolated, or extrapolated, unknown concentration. Table 2 also shows the average CIs of bm, am and Cukm (in all cases, the confidence level was chosen equal to 0.95). According to suggestions,23225 data are rounded to the second uncertain figure of their uncertainty (that is of their CIs). One can compare these results with the true data of Table 1. Outb, Outa and Outc in Table 2 indicate the number of times the true value of a parameter, (slope, intercept and Cuk values of Table 1) was found outside the relevant CI. During run D, Cuk Analyst, 1999, 124, 1191–1196

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Table 1

Parameters selected for the validation tests (all parameters are in arbitrary units)

S(C) = b.C + a C values Repetition design (ri) N s = f(C) s range RSD% range Suk/Cuk Experimental method

Run A

Run B

Run C

Run D

Run E

15.0.C + 20.0 0.0–10.0–20.0– 30.0–40.0–50.0 4.2.1.1.3.4

-2.5.C-1.0 0.0–50–100–150

4.0.C + 16 0.0–4.0–8.0–12

3.3.1.3

5.0.C + 10.0 0.0–2.0–6.0– 8.0–10.0–12.0 4.3.1.4.3.4

5.3.1.5

5.0.C+3 10,20,30,40,50, 60,70,80 3.3.3.3.3.3.3.3

15 0.8.C + 0.5 0.50?40.50 2.50?5.26 0.0/1.33 Standard additions

10 0.05.C1.1 + 0.03 0.03?12.41 –2.96?–3.30 –200/79.6 Calibration graph

19 0.5.C + 0.3 0.30?6.30 3.00?9.00 45.0/7.0 Calibration graph

14 A0.07·C2 + 0.8 0.89?3.30 4.33?5.15 0.0/4.0 Standard additions

24 A0.04·C2 + 0.05 0.22?12.60 3.96?7.35 52.3/9.86 Calibration graph

was found outside its experimental CI seven times. The average distance of these seven Cuk values from the closest limit of their experimental CI was 0.12 arbitrary units (a.u.). Nevertheless, the average CI over the 25 runs, 4.02 ± 0.22 a.u., embraced the true CI (4.0 a.u.). Sometimes the addition of a Gaussian noise to responses obtained from theoretically linear relationships led to non-linear data sets (as revealed by the lack-of-fit test). During validation runs A–D, no attempt was made to enter the linear range by data reduction or other methods, so that average rm2, bm, am and Cukm values are those obtained whatever the response of the lack-of-fit test. The efficiency of the lack-of-fit test was checked by run E (see Table 1), simulating the establishment of a precision calibration graph by analysing in triplicate eight standard solutions: in this case the true response of the solutions having the two highest concentrations (70 and 80 a.u. in Table 1) was decreased by 4.25 and 8.68%, respectively, before adding the Gaussian noise. This allowed the simulation of a net deviation from linearity at the upper end of the explored concentration range. After the addition of Gaussian noise according to the selected standard deviation functional relationship (see Table 1), 25 data sets were generated and analysed. In nine cases the response/concentration functional relationship was found to be linear, in 13 cases non-linearity was avoided by dropping data relevant to C = 80 a.u., and in three cases non-linearity was avoided by dropping data relevant to C = 80 a.u. and C = 70 a.u. The smoothing of raw standard deviations by ordinary LLSR13 is not problem-free. Because of the unfavourable experimental situation C, the intercept of the standard deviation regression function [q in eqn. (7)] was frequently found to be much too low. The closer to zero it is, the higher the weight assigned to responses of the relevant concentration. If weights at the lowest concentration are overestimated, uwLLSR returns an abnormally wide and unjustifiable CI of interpolated values (see Fig. 2). Should this happen in real cases, additional measurements should be attempted to by-pass the problem. Alternatively, uwLLSR allows the standard deviation regression line to be forced through the first point, s1/C1, by simply changing the Force parameter in Section 1 from 0 (default) to 1: this

forcing (empirical, but standard deviation cannot be zero) usually suffices to increase slightly the standard deviation intercept and to obtain reasonably narrow CIs (see Fig. 3). In the course of all the 100 simulations in runs A–D, the above situation was met only six times, always when analysing data sets of situation C. In these cases, the results of forced standard deviation regression were retained for calculating average values displayed in Table 2. As evident in Fig. 2 and 3, two points lie outside the ±2sy/x band. After their elimination, in agreement with the results of the outlier test, the results displayed in Fig. 4 were obtained. Table 3 shows weights, determination coefficients and standard deviations of interpolated concentration obtained in Fig. 2–4: the best sy/x and sCuk values were obtained after forcing and eliminating the two outliers. Fig. 2–4 represent a good example of creative and interactive analysis.7 Examples of real calibration data sets The program is being routinely used in the author’s laboratory. Table 4 presents a real calibration data set, relevant to the quantification of sulfides by flow injection analysis with amperometric detection at a palladium–vitreous carbon chemically modified electrode.26 The program outputs when choos-

Table 2 Average results of validation tests (all parameters are in arbitrary units) Run A

Run B

Run C

Run D

rm2 bm am Cukm Outb Outa OutC

0.9952 14.98 ± 0.55 20.1 ± 2.4 1.34 ± 0.18 5 0 1

0.9991 –2.500 ± 0.056 –1.1 ± 1.4 79.6 ± 4.7 1 0 1

0.9798 4.98 ± 0.32 10.08 ± 0.41 7.0 ± 1.8 3 3 –

0.9921 3.99 ± 0.21 16.02 ± 0.77 4.02 ± 0.22 3 4 7

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Fig. 2

Example of over-weighting in analysing situation C.

ing different a risks are summarised in Table 5. It can be observed that the raw data set results are non-linear, even when choosing a very low a risk (0.005). The curvature of the calibration plot can be eliminated by discarding data relevant to the two most concentrated standard solutions. After this, a suspect outlier can be identified when choosing a = 0.05. However, no outlier is evident when choosing a = 0.01 or lower. The use of a = 0.01 and of responses of only the first six standard solutions allows the coefficient of determination to be

Table 3 Per cent. weights, coefficients of determination and sCuk values relevant to Fig. 2 (before forcing), 3 (after forcing) and 4 (after forcing and elimination of two outliers) Fig. 2 w1 w2 w3 w4 w5 w6 r2 sCuk

Fig. 3

99.88% 0.095% 0.011% 0.0062% 0.0040% 0.0028% 0.8928 0.1774

Fig. 4

96.61% 2.58% 0.36% 0.21% 0.14% 0.096% 0.9836 0.0654

98.94% 0.83% 0.10% 0.060% 0.039% 0.027% 0.9860 0.0635

Table 4 Calibration data set of sulfides in flow injection analysis at a palladium–vitreous carbon chemically modified electrode Concentration/mM

Response/mA

0.88 1.77 3.55 7.10 14.2 28.4 56.8 81.2

0.170 0.341 0.663 1.37 2.60 5.29 10.83 15.85

0.211 0.349 0.684 1.34 2.58 5.24 11.10 15.75

0.179 0.353 0.656 1.39 2.64 5.26 11.05 15.94

Table 5 Effect of the selected a level on Bartlett, lack–of–fit and outlier tests No. of standard solutions a = 0.05 n = 8 n = 7 n = 6 Fig. 3 Same example as in Fig. 1 but after forcing the standard deviation/ concentration functional relationship through the first data point.

a = 0.01

a = 0.005

Heteroscedastic, Heteroscedastic, Heteroscedastic, non–linear non–linear non–linear Heteroscedastic, Heteroscedastic, Heteroscedastic, non–linear non-linear non-linear Homoscedastic, Homoscedastic, Homoscedastic, non–linear linear linear 1 suspect outlier

increased from 0.9990 to 0.9997 and sy/x to be lowered from 0.0496 to 0.0267. The final result is displayed in Fig. 5. Two additional examples of real calibration data sets are supplied as Electronic Supplementary Information.†

Conclusions The results of validation tests confirmed that uwLLSR allows one to evaluate slope and intercept and their CIs, to interpolate/ extrapolate the unknown concentration and to evaluate its CI whatever the actual experimental conditions (scedasticity, number of standards, number of replicate observations of each standard and of the unknown). The available (facultative) outliers and normality tests, the possibility to reduce the explored concentration range easily (by changing ii to ii 2 1, ii 2 2, and so on) and to adjust weights by forcing the standard deviation regression line, permit fully interactive operations. The program can be used to process homoscedastic data arrays also, since weighted regression always produces superior results than ordinary regression.2,4,9 Of course, uwLLSR can be used to process balanced data arrays also, since they are particular cases of unbalanced data arrays. When using a Pentium II IBM-compatible PC, results are returned about 10–20 s after the necessary inputs. Fig. 4 Same example as in Fig. 3 but after elimination of the lowest and highest responses of the lowest concentration.

† Available as Electronic Supplementary Material; see http://www.rsc.org/ suppdata/an/1999/1191.

Analyst, 1999, 124, 1191–1196

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Ministero dell’Università e della Ricerca Scientifica (MURST, Rome).

References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Fig. 5 Calibration graph relevant to the quantification of sulfides by flow injection analysis with amperometric detection at a palladium–vitreous carbon chemically modified electrode. Calibration function: S(C) = (0.1841 ± 0.0025).C+(0.024 ± 0.024); a = 0.01; n = 16.

15 16 17 18 19

The described worksheet can be obtained free from the author by E-mail.

20 21 22

Acknowledgements This work was carried out with the financial support of the Italian National Research Council (C.N.R., Rome) and of

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