6Analysis of Variance: Estimation of Variance Components (Model II of the Analysis of Variance)
6.1 Introduction: Linear Models with Random Effects
In this chapter models of the analysis of variance (ANOVA) where all factors are random are considered; we call the model in this case model II. Our aim in such models is not only as in Chapter 5 the testing of particular hypotheses but also the methods of estimating the components of variance. For the latter we first of all consider the best elaborated case of the one‐way analysis of variance. We again use the notation of Section 5.1 and consider formally the same models as in Chapter 5. The difference between Chapters 5 and 6 is that the effects of model II are random. We assume that, for instance, for a factor A, say, exactly a levels are randomly selected from a universe PA of (infinite) levels of the factor A so that α1, … , αa; the effects of these levels are random variables.
The terms main effect and interaction effect are defined analogously as in Chapter 5, but these effects are now random variables and not parameters that could be estimated.
Models, in which some effects are fixed and other are random, are discussed in Chapter 7. In Chapter 6 some terms defined in Chapter 5 are used, without defining them once more.