Residual Pattern Based Test for Interactions in Two‐Way ANOVA

Abstract

This article proposes a new test to detect interactions in replicated two-way ANOVA models, more powerful than the classical F -test and more general than the test of Terbeck and Davies (1998, Annals of Statistics 26, 1279-1305) developed for the case with unconditionally identifiable interaction pattern. We use the parameterization without the conventional restrictions on the interaction terms and base our test on the maximum of the standardized disturbance estimates. We show that our test is unbiased and consistent, and discuss how to estimate the p -value of the test. In a 3 x 3 case, which is our main focus in this article, the exact p -value can be computed by using four-dimensional integrations. For a general I x J case which requires an (I - 1) x (J - 1) dimensional integration for a numerical evaluation of the exact p -value, we propose to use an improved Bonferroni inequality to estimate an upperbound of the p -value and simulations indicate a reasonable accuracy of the upperbound. Via simulations, we show that our test is more powerful than the classical F -test and also that it can deal with both situations: unconditionally identifiable and non-unconditionally identifiable cases. An application to genetic data is presented in which the new test is significant, while the classical F -test failed to detect interactions.