ANOVA

A variety of statistical methods may be used to compare three or more sets of data. The most commonly used method is an analysis of variance (ANOVA). In its simplest form, a one-way ANOVA allows the importance of a single variable, such as the identity of the analyst, to be determined. The importance of this variable is evaluated by comparing its variance with the variance explained by indeterminate sources of error inherent to the analytical method. 

Let s use a simple example to develop the rationale behind a one-way ANOVA calculation. The data in Table 14.7 show the results obtained by several analysts in determining the purity of a single pharmaceutical preparation of sulfanilamide. Each column in this table lists the results obtained by an individual analyst. For convenience, entries in the table are represented by the symbol where i identifies the analyst and j indicates the replicate number thus 3 5 is the fifth replicate for the third analyst (and is equal to 94.24%). The variability in the results shown in Table 14.7 arises from two sources indeterminate errors associated with the analytical procedure that are experienced equally by all analysts, and systematic or determinate errors introduced by the analysts. 

A number of statistics and spreadsheet software packages are available to perform ANOVA calculations. 

This value of fexp is compared with the critical value for f(a, v), where the significance level is the same as that used in the ANOVA calculation, and the degrees of freedom is the same as that for the within-sample variance. Because we are interested in whether the larger of the two means is significantly greater than the other mean, the value of f(a, v) is that for a one-tail significance test. 

An analysis of variance can be extended to systems involving more than a single variable. For example, a two-way ANOVA can be used in a collaborative study to determine the importance to an analytical method of both the analyst and the instrumentation used. The treatment of multivariable ANOVA is beyond the scope of this text, but is covered in several of the texts listed as suggested readings at the end of the chapter. 

The following texts provide additional information about ANOVA calculations, including discussions of two-way analysis of variance. Graham, R. C. Data Analysis for the Chemical Sciences. VCH Publishers New York, 1993. 

FIGURE 11.3 One-way ANOVA (analysis of variance). One-way analysis of variance of basal rates of metabolism in melanophores (as measured by spontaneous dispersion of pigment due to G,.-protein activation) for four experiments. Cells were transiently transfected with cDNA for human calcitonin receptor (8 j-ig/ml) on four separate occasions to induce constitutive receptor activity. The means of the four basal readings for the cells for each experiment (see Table 11.4) are shown in the histogram (with standard errors). The one-way analysis of variance is used to determine whether there is a significant effect of test occasion (any one of the four experiments is different with respect to level of constitutive activity). 

Analysis of variance (ANOVA), a statistical procedure that quantifies differences between means of samples and the extent of variances within and between those means to determine the probability of there being a difference in the samples. 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

It should be noted that in this example the performance of only one variable, the three analysts, is investigated and thus this technique is called a one-way ANOVA. If two variables, e.g. the three analysts with four different titration methods, were to be studied, this would require the use of a two-way ANOVA. Details of suitable texts that provide a solution for this type of problem and methods for multivariate analysis are to be found in the Bibliography, page 156. 

Comparisons between data sets (r test, multiple range test, F test, simple ANOVA),... 

Do the simple ANOVA test (Section 1.5.6) to detect variability between the group means in excess of what is expected due to chance alone. 

Analysis of variance (ANOVA) tests whether one group of subjects (e.g., batch, method, laboratory, etc.) differs from the population of subjects investigated (several batches of one product different methods for the same parameter several laboratories participating in a round-robin test to validate a method, for examples see Refs. 5, 9, 21, 30. Multiple measurements are necessary to establish a benchmark variability ( within-group ) typical for the type of subject. Whenever a difference significantly exceeds this benchmark, at least two populations of subjects are involved. A graphical analogue is the Youden plot (see Fig. 2.1). An additive model is assumed for ANOVA. 

For the data given in Table 1.15, an ANOVA calculation yields the results shown in Table 1.16. 

Table 1.15. Raw Data and Intermediate Results of an ANOVA Test for Simulated Data. (Eq. 1.30)...

Other forms of ANOVA The simple ANOVA set out above tests for the presence of one unknown factor that produces (additive) differences between outwardly similar groups of measurements. Extensions of the concept allow one to simultaneously test for several factors (two-way ANOVA, etc.). The... 

A one-way (simple) ANOVA with six replicates can be conducted by either regarding each titration technique or each batch as a group, and looking for differences between groups. 

A two-way ANOVA (not discussed here) would combine the two approaches under 2. 

Because of the observed homoscedacity, a simple ANOVA-test (see Table 4.8) can be applied to determine whether the means all belong to the same population. If there was any indication of differences among the means, this would mean that the conditioner worked in a position-sensitive mode and would have to be mechanically modified. 

Table 4.38. Effect of Raw Data Rounding on Bartlett and ANOVA Tests...

The standard deviations are not distinguishable (Bartlett test). Conclusions are valid for all three data sets. All means belong to the same population (ANOVA test). Overall result 97.5 3.2 (compound assay). 

The six data sets do not differ in variance (Bartlett test) or in means (ANOVA), so there is no way to group them using the multiple range test. This being so, the data were pooled for compounds A and B, yielding the two columns at right (data in CU-Assay2.dat). 

Subroutine ANOVA from program MULTI is given in Table 5.18. 

Table 5.18. BASIC Code for the Core of Subroutine ANOVA...

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