Associated Hypotheses in Linear Models for Unbalanced Data

Abstract

When looking at factorial experiments there are several natural hypotheses that can be tested. In a two-factor or a by b design, the three null hypotheses of greatest interest are the absence of each main effect and the absence of interaction. There are two ways to construct the numerator sum of squares for testing these, namely either adjusted or sequential sums of squares (also known as type I and type III in SAS). Searle has pointed out that, for unbalanced data, a sequential sum of squares for one of these hypotheses is equal (with probability 1) to an adjusted sum of squares for a non-standard associated hypothesis. In his view, then, sequential sums of squares may test the wrong hypotheses. We give an exposition of this topic to show how to derive the hypothesis associated to a given sequential sum of squares.