General Random Models
This chapter considers more complicated models which have random effects. Two-factor random models can be viewed as beginning with a two-factor fixed effects model with all levels of the two factor populations. In the random model, the chapter draws random samples of levels from each factor. When the design is unbalanced, the sum of the squares for main effects and for interaction is no longer independent. The chapter examines the general random model, in which all effects are random but the design may be rather unbalanced. The mean response can be partitioned into four parts, the grand mean plus two terms involving each main effect plus an interaction. Inference about random effects can be made in terms of quadratic forms under suitable conditions. An understanding of general quadratic forms for the partitioned random model developed above can offer insight into how to address inferential questions about variance components.