Analysis of Variance ANOVA for Linear Models

In Section 6.4, it was shown for replicate experiments at one factor level that the sum of squares of residuals, SS can be partitioned into a sum of squares due to purely experimental uncertainty, SS, and a sum of squares due to lack of fit, SSi f. Each sum of squares divided by its associated degrees of freedom gives an estimated variance. Two of these variances, and were used to calculate a Fisher F-ratio from which the significance of the lack of fit could be estimated. 

In this chapter we examine these and other sums of squares and resulting variances in greater detail. This general area of investigation is called the analysis of variance (ANOVA) applied to linear models [Scheff6 (1953), Dunn and Clark (1987), Alius, Brereton, and Nickless (1989), and Neter, Wasserman, and Kutner (1990)]. 

There is an old paradox that suggests that it is not possible to walk from one side of the room to the other because you must first walk halfway across the room, then halfway across the remaining distance, then halfway across what still remains, and so on because it takes an infinite number of these steps, it is supposedly not possible to reach the other side. This seeming paradox is, of course, false, but the idea of breaking up a continuous journey into a (finite) number of discrete steps is useful for understanding the analysis of variance applied to linear models. 

In this section we will develop matrix representations of these distances, show simple matrix calculations for associated sums of squares, and demonstrate that certain of these sums of squares are additive. 

The individual responses in a data set are conveniently collected in a matrix of measured responses, Y 

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Response Surfaces. 3. Basic Statistics. 4. One Experiment. 5. Two Experiments. 6. Hypothesis Testing. 7. The Variance-Covariance Matrix. 8. Three Experiments. 9. Analysis of Variance (ANOVA) for Linear Models. 10. A Ten-Experiment Example. 11. Approximating a Region of a Multifactor Response Surface. 12. Additional Multifactor Concepts and Experimental Designs. Append- ices Matrix Algebra. Critical Values of t. Critical Values of F, a = 0.05. Index. 

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