ANOVA tables

The problem now is to decide if the model as a whole is significant, and if so, which coefficients are significant. This is most easily done in an analysis of variance table (ANOVA). 

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. 

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). 

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)...

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...

Subroutine ANOVA from program MULTI is given in Table 5.18. 

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

One can also state that the log double-centered biplot shows interactions between the rows and columns of the table. In the context of analysis of variance (ANOVA), interaction is the variance that remains in the data after removal of the main effects produced by the rows and columns of the table [12], This is precisely the effect of double-centering (eq. (31.49)). 

Given an estimate of the time to completion and precision of the analysis, one can temporarily eliminate time as a variable and construct an analysis of variance (ANOVA) to examine the effects of pH and temperature. A simple ANOVA would consist of four groups, with several replicates in each group, as shown in Table 4. 

The OUTSTAT= output data set pvalue contains the p-value in the PROB variable. If you have multiple predictor variables, you need to use the PROC GLM ODS data set OverallANOVA to get the overall model p-value from the ProbF variable. These output data sets contain other variables, such as the degrees of freedom, sum of squares, mean square, and F statistic, if you need them for an ANOVA table presentation. 

Table 5.3. Evaluation scheme of two-way ANOVA (ab model single measurements in each point the index point marks that levels over which is actually added up or averaged, respectively)...

If possible, two-way ANOVA should be applied doing repetitions at each level. In case of double measurements the 2ab model represented in Tables 5.5 and 5.6 is taken as the basis of evaluation and variance decomposition. 

Table 5.7. Variance components in three-way ANOVA (abc model with repetitions)... 

Table 10-5 Part A - ANOVA for the errorless data from Table 10-4...

Table 10-5 Part B - RESIDUALS for ANOVA from Table 10-4 after correcting for row and column means...

When we subject this data to the same ANOVA calculations as the errorless data, we arrive at the following results (Table 10-8) ... 

Table 10-8 Part A - DATA ANOVA for the hypothetical data containing error with mean equal 0 and standard deviation (S) equal to unity...

Table 35-1 illustrates the ANOVA results for each individual sample in our hypothetical study. This test indicates whether any of the reported results from the analytical methods or locations is significantly different from the others. From the table it can be observed that statistically significant variation in the reported analytical results is to be expected based on these data. However, there is no apparent pattern in the method or location most often varying from the others. Thus, this statistical test is inconclusive and further investigation is warranted. 

Table 35-3 illustrates the ANOVA results comparing laboratories (i.e., different locations) performing the same METHOD A for analysis. This statistical test indicates that for the mid-level concentration spiked samples (i.e. 4 and 4 at 3.40 and 3.61% levels, respectively) difference in reported average values occurred. However, this trend did not continue for the highest concentration sample (i.e., Sample No. 6) with a concentration of 3.80%. The Lab 1 was slightly lower in reported value for Samples 4 and 5. There is no significant systematic error observed between laboratories using the METHOD A. 

Table 35-4 reports ANOVA comparing the METHOD B procedure to the METHOD A procedure for combined laboratories. Thus the combined METHOD B analyses for each sample were compared to the combined METHOD A analyses for the same sample. This statistical test indicates whether there is a significant bias in the reported results for each method, irrespective of operator or location. An apparent trend is indicated using this statistical analysis, that trend being a positive bias for METHOD B as compared to... 

Table 35-3 vs. Lab 2) ANOVA comparing laboratories for METHOD A spectrophotometry (Lab 1 ... 

Table 35-4 ANOVA comparing methods for combined laboratories and operators, all Method B vs. all Method A...

---