Analysis of variance tests

Part 2 Analysis of Variance testing for both locations and analytical methods to determine if an overall bias exists for location or analytical method ... 

Shapiro SS, Wilk MB (1965) An analysis of variance test for normality (Complete samples). Biometrika 52 591-611 Sidak Z (1967) Rectangular confidence regions for the means of multivariate normal distributions. J Am Statist Assoc 62 626-631... 

This test wilt have robustness properties similar to analysis of variance tests. 

For the six types of combinations, a t-test and a correlation analysis were conducted. Using LDA as a classifier, the third and fourth combinations showed a relatively low p-value (0.09) with a correlation value of 0.77. According to an analysis of variance test with the six types of combinations and 10 subjects, the accuracy trends of the subjects for the six types of combinations showed low statistical significance. However, the distributions of the six combinations for each subject showed very high statistical significance (P = 4.2E-09), indicating that the characteristics of each subject are feasible for application to biometric-method-based EEG [5]. 

Once a significant difference has been demonstrated by an analysis of variance, a modified version of the f-test, known as Fisher s least significant difference, can be used to determine which analyst or analysts are responsible for the difference. The test statistic for comparing the mean values Xj and X2 is the f-test described in Chapter 4, except that Spool is replaced by the square root of the within-sample variance obtained from an analysis of variance. 

Collaborative testing provides a means for estimating the variability (or reproducibility) among analysts in different labs. If the variability is significant, we can determine that portion due to random errors traceable to the method (Orand) and that due to systematic differences between the analysts (Osys). In the previous two sections we saw how a two-sample collaborative test, or an analysis of variance can be used to estimate Grand and Osys (or oJand and Osys). We have not considered, however, what is a reasonable value for a method s reproducibility. 

Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section Two-Population Test of Hypothesis for Means can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged as shown in the table several measurements for each treatment. The goal is to see if the treatments differ significantly from each other that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. 

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). 

FIGURE 11.4 Two-way analysis of variance. Arrangement of data in rows and columns such that each row of the cell culture plate (shown at the top of the figure) defines a single dose-response curve to the agonist. Also, data is arranged by plate in that each plate defines eight dose-response curves and the total data set is comprised of 32 dose-response curves. The possible effect of location with respect to row on the plate and/or which plate (order of plate analysis) can be tested with the two-way analysis of variance. 

In analysis of variance, the variance due to each source of variation is systematically isolated. A test of significance, the E-test, is then applied to establish roughly how seriously one must regard each source of variation. The interested reader is urged to consult books on statistics14 for discussions of this valuable statistical method. 

For work of the highest precision, it is highly advisable to carry through an analysis of variance together with suitable tests of significance, not only to establish what the precision is, but also to uncover individual sources of error so that they can be made less serious. How this is done for instrumental and manipulative errors has been demonstrated in this chapter. 

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. 

A 2 X 2 contingency table was used to evaluate the frequency of anomalies and resorptions within the fetal population and between litters. Body weight and body measurements were statistically analyzed by an Analysis of Variance and Tukey s test (13). In all cases, the level of significance was P < 0.05. 

Significantly different from control by an Analysis of Variance and Tukey s test (measurements) or the 2X2 contingency table (resorptions), P <0.05. 

The results of such multiple paired comparison tests are usually analyzed with Friedman s rank sum test [4] or with more sophisticated methods, e.g. the one using the Bradley-Terry model [5]. A good introduction to the theory and applications of paired comparison tests is David [6]. Since Friedman s rank sum test is based on less restrictive, ordering assumptions it is a robust alternative to two-way analysis of variance which rests upon the normality assumption. For each panellist (and presentation) the three products are scored, i.e. a product gets a score 1,2 or 3, when it is preferred twice, once or not at all, respectively. The rank scores are summed for each product i. One then tests the hypothesis that this result could be obtained under the null hypothesis that there is no difference between the three products and that the ranks were assigned randomly. Friedman s test statistic for this reads... 

When there are many samples and many attributes the comparison of profiles becomes cumbersome, whether graphically or by means of analysis of variance on all the attributes. In that case, PCA in combination with a biplot (see Sections 17.4 and 31.2) can be a most effective tool for the exploration of the data. However, it does not allow for hypothesis testing. Figure 38.8 shows a biplot of the panel-average QDA results of 16 olive oils and 7 appearance attributes. The biplot of the... 

A central concept of statistical analysis is variance,105 which is simply the average squared difference of deviations from the mean, or the square of the standard deviation. Since the analyst can only take a limited number n of samples, the variance is estimated as the squared difference of deviations from the mean, divided by n - 1. Analysis of variance asks the question whether groups of samples are drawn from the same overall population or from different populations.105 The simplest example of analysis of variance is the F-test (and the closely related t-test) in which one takes the ratio of two variances and compares the result with tabular values to decide whether it is probable that the two samples came from the same population. Linear regression is also a form of analysis of variance, since one is asking the question whether the variance around the mean is equivalent to the variance around the least squares fit. 

The dry weights (104 C, 48 hr) of ten plants from each treatment group were taken at the termination of each experiment in order to compare growth effects with plant water status. Dry weight data were analyzed using analysis of variance (ANOVA) and Duncan s multiple-range test. Diffusive resistance and water potential were evaluated using the t-test. Each of these and subsequent experiments was replicated. 

The general principles of testing chemical homogeneity of solids are given e.g. by Malissa [1973], Cochran [1977], and Danzer et al. [1979]. The terms of variation o20tal and o2nal can be separated by analysis of variance (Sect. 5.1.1). According to Danzer and Kuchler [1977] there exists an exponential dependence between the total variance and the reciprocal sample mass... 

On this basis, an analysis of variance (ANOVA) can be carried out to test the significance of the variations e, = SAiXj, ej = IA)Xj, or more in detail, eB = SabXB, ec = SACxc etc. 

The total error has to be calculated here in different way compared with robustness. Whereas in (i) of-A is the variance within a laboratory, o kA is the variance between laboratories plus that within the laboratories, °ijkA = °k + °ijA- The interpretation is analogous to (i), if F < F( a>Vl>V2, then the null hypothesis cannot be rejected and the procedure can be considered as to be rugged. Advantageously, the test can be carried out by schemes of ANOVA (analysis of variance) as given in Sect. 5.1.1. 

For the basic evaluation of a linear calibration line, several parameters can be used, such as the relative process standard deviation value (Vxc), the Mandel-test, the Xp value [28], the plot of response factor against concentration, the residual plot, or the analysis of variance (ANOVA). The lowest concentration that has been used for the calibration curve should not be less than the value of Xp (see Fig. 4). Vxo (in units of %) and Xp values of the linear regression line Y = a + bX can be calculated using the following equations [28] ... 

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