The Analysis of Two-Factor Interactions in Fixed Effects Linear Models

Abstract

This article considers two related issues concerning the analysis of interactions in complex linear models. The first issue concerns the omnibus test for interaction. Apparently, it is not well known that the usual F test for interaction can be replaced, in many applications, by a test that is more powerful against a certain class of alternatives. The competing test is based on the maximal product interaction contrast F statistic and achieves its power advantage by focusing solely on product contrasts. The maximal product interaction F test is reviewed and three new results are reported: (a) An extended table of exact critical values is computed, (b) a table of moment functions useful for approximating the p-value corresponding to an observed maximal F statistic is computed, and (c) a simulation study concerning the null distribution of the maximal F statistic when data are unbalanced or covariates are present is reported. It is conjectured that lack of balance or presence of covariates has no effect on the null distribution. The simulation results support the conjecture. The second issue concerns follow-up tests when the omnibus test is significant. It appears that researchers, in general, do not perform coherent follow-up tests on interactions. To make it easier for researchers to do so, an exposition on the use of product interaction contrasts and partial interactions in complex fixed-effects models is provided. The recommended omnibus and follow-up tests are illustrated on an educational data set analyzed using SAS (SAS Institute, 1988) andSPSS (1990).