ANOVA table construction

As has been already pointed out, of course, the constructed data would only be appropriate for the model that reflected the way in which the experiment was carried out. However, to illustrate the analysis the same data from Table 2.1 will be used. The ANOVA table for the tablet formulation data, assuming that the experiment was run according to arrangement (II), is shown in Table 2.19. 

From these simple calculations the ANOVA table can be constructed together with the HORRAT ratio. An important difference from the conventional ANOVA table is that it allows for the sample variation to be accounted for. The general ANOVA table and calculated values for S samples are shown in Tables 28 and 29 but the example uses only two. However, if more than two samples are desired/available the procedure can be extended. 

Determine the sum of square replicate error, and so, from question 4, die sum of square lack-of-fit error. Divide die sum of square residual, lack-of-fit and replicate errors by dieir appropriate degrees of freedom and so construct a simple ANOVA table widi diese three errors, and compute die F-ratio. 

Table 9 shows the construction of the ANOVA table. If the variance estimate of a class variable MS variabie deviates significantly from that obtained by that for random error MSettot, then the null hypothesis that the means at the different levels for that variable are equal is rejected. In other words, the classification of data by that variable is explanatory of the variation observed in the data. We conduct the test by using the variance ratio test F = MSvariabie/AfSError, with... 

These equations are illustrated graphically in Figure 5. As before, an ANOVA table can be constructed for each model and the significance of each term estimated by sums of squares decomposition and comparison of standard regression coefficients. 

Construct the ANOVA table Having calculated the total sums of squares from all sources of variation, along with their degrees of freedom, we can now start to construct the ANOVA table. The only other calculations required are the mean squares for among-samples and within-samples (divide each sums of squares by its associated df) and the test statistic, F (divide among-samples mean square by within-samples mean square). All of this information is shown in the partial ANOVA table presented as Table 11.3. Determine if the test statistic is in the rejection region As always, we need to determine if the test statistic F falls in the rejection region. So far, we have not determined the... 

The sums of squares are an important component of the ANOVA table that we shall construct. Another important component is known as the degrees of freedom (DF). The total degrees of freedom (DFT) is given by n - 1, because if we know y and n - 1 other of the y values, we can calculate the remaining y value. Likewise, there are m - 1 degrees of freedom for the model (DFM) because y is determined from m parameters (i.e., the model coefficients) and we must subtract 1 DF for y. Finally, there are n - m degrees of freedom in the residual (DFR) because the last term comprise n values of y and m values to determine y. We are now ready to build our ANOVA (Table 3.3). 

The six-step procedure can be easily applied to the regression ANOVA for determining if /3i = 0. Let us now use the data in Example 2.1 to construct an ANOVA table. 

After the model has been constructed and the parameters estimated, it is crucial to evaluate it. In this book, we describe two methods for evaluating models residual plots and lack of fit. Residual plots are discussed in Section 7.8.1 and lack of fit is described in the context of the analysis of variance (ANOVA) table. The error is defined as... 

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. 

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