Analysis of Variance and Covariance

Every model in Chapters 2 and 3 has one or more equivalents without full replication. For model 2.1 it is 1.1, for 2.2 it is 2.1, for 3.1 it is 4.1 or 6.1, for 3.2 it is 4.2 or 6.2, for 3.3 it is 5.6 or 6.3, and for 3.4 it is 3.1. Here we give two further versions of factorial models 3.1 and 3.2 without full replication. The lack of replicated sampling units means that at least one of the factors must be random, as demonstrated by model 7.1(i) below in comparison to (ii) and (iii). Factorial designs that lack full replication must further assume that there are no significant higher-order interactions between factors, which cannot be tested by the model since there is no measure of the residual error among replicate observations (subjects). This is problematic because lower-order effects can only be interpreted fully with respect to their higher-order interactions (chapter 3). Falsely assuming an absence of higher-order interactions will cause tests of lower-order effects to overestimate the Type I error (rejection of a true null hypothesis) and to underestimate the Type II error (acceptance of a false null hypothesis). Without testing for interactions, causality cannot be attributed to significant main effects, and no conclusion can be drawn about non-significant main effects. For some analyses, the existence of a significant main effect when levels of an orthogonal random block are pooled together may hold interest regardless of whether or not the effect also varies with block; the main effect indicates an overall trend averaged across levels of the random factor.