Proportional subclass numbers in two-factor ANOVA

The between-within split of total sum of squares in one-way analysis of variance (ANOVA) is intuitively appealing and computationally simple, whether balanced or not. In the balanced two-factor setting, the same heuristic and computations apply to analyse treatment sum of squares into main effects and interaction effects sums of squares. Accomplishing the same in unbalanced settings is more difficult, requiring development of tests of general linear hypotheses. However, textbooks treat unbalanced settings with proportional subclasss numbers (psn) as essentially equivalent to balanced settings. It is shown here that, while psn permit an ANOVA-like partition of sums of squares, test statistics for main effects of the two factors generally test the wrong hypotheses when the model includes interaction effects.