Preface to First Edition - lib

88 ANALYSIS OF VARIANCEA and B there is a proportion of the variance of the response variable thatcan be attributed to either A or B. The consequence is that A and B togetherexplain less of the variation of the dependent variable than the sum of whicheach explains alone. The result is that the sum of squares corresponding toa factor depends on which other terms are currently in the model for theobservations, so the sums of squares depend on the order in which the factorsare considered and represent a comparison of models. For example, for theorder a,b,a × b, the sums of squares are such that• SSa: compares the model containing only the a main effect with one containingonly the overall mean.• SSb|a: compares the model including both main effects, but no interaction,with one including only the main effect of a.• SSab|a,b: compares the model including an interaction and main effectswith one including only main effects.The use of these sums of squares (sometimes known as Type I sums ofsquares) in a series of tables in which the effects are considered in differentorders provides the most appropriate approach to the analysis of unbalanceddesigns.We can derive the two analyses of variance tables for the foster feedingexample by applying the R codeR> summary(aov(weight ~ litgen * motgen, data = foster))to giveDf Sum Sq Mean Sq F value Pr(>F)litgen 3 60.16 20.05 0.3697 0.775221motgen 3 775.08 258.36 4.7632 0.005736litgen:motgen 9 824.07 91.56 1.6881 0.120053Residuals 45 2440.82 54.24and then the codeR> summary(aov(weight ~ motgen * litgen, data = foster))to giveDf Sum Sq Mean Sq F value Pr(>F)motgen 3 771.61 257.20 4.7419 0.005869litgen 3 63.63 21.21 0.3911 0.760004motgen:litgen 9 824.07 91.56 1.6881 0.120053Residuals 45 2440.82 54.24There are (small) differences in the sum of squares for the two main effectsand, consequently, in the associated F-tests and p-values. This would not betrue if in the previous example in Subsection 5.3.1 we had used the codeR> summary(aov(weightgain ~ type * source, data = weightgain))instead of the code which produced Figure 5.2 (readers should confirm thatthis is the case).Although for the foster feeding data the differences in the two analyses ofvariance with different orders of main effects are very small, this may not© 2010 by Taylor and Francis Group, LLC