Gart's test of interaction in a 2 × 2 × 2 contingency table for small samples

Abstract

SUMMARY For the one-response, two-factor design with categorical data, Gart (1972) proposed a statistic to test the hypothesis of zero second-order interaction. It can be shown that unconditionally the statistic approaches infinity with probability one when the interaction effect is zero and the row and column effects are not zero. Thus the null hypothesis is rejected with probability one and the statistic is not valid for the hypothesis, unless certain other special conditions are true. Because the sample size must be extremely large for the above conclusion to hold, merit for Gart's test might still be found when the samples are small; the statistic was developed by approximating the exact probability. Such a conjecture is examined by a simulation in which Gart's statistic is compared to Wald's statistic in the 2 × 2 × 2 case, both as to respective ability to approximate the exact test and as to their power. Gart's statistic is the better in approximating the inference of the exact test; this is because of the conservative nature of both the exact and Gart's tests. But this makes Gart's test less powerful than Wald's.