Testing ANOVA effects: A resolution for unbalanced models

ANOVA effects can be included in multiple linear regression models in multiple ways, most simply with dummy variables, but then the model has containment issues. Contrast coding is also widely used, mainly to provide a full-column-rank copy of the same model. Then a natural sum of squares to test an effect is the extra SSE due to omitting its terms from the model. In general, in unbalanced models, it is cumbersome to establish exactly what such extra SSEs test – that is, what their non-centrality parameters are as functions of the cell means. Determining and testing only estimable parts may require extra analysis and computing. And, after all that, there is the question of whether another numerator sum of squares might be better. It is established here that, with contrast coding, the model formed by deleting an ANOVA effect’s terms is the correct restricted model for that effect; that the non-centrality parameter of the resulting extra SSE is zero if and only if all estimable linear functions of the effect are zero; and that no other numerator sum of squares that tests the effect has a greater non-centrality parameter or lesser numerator degrees of freedom.