A note on the rank transform for interactions

SUMMARY The asymptotic properties of the rank transform statistic for testing for interaction in a balanced two-way classification are studied. Necessary and sufficient conditions are obtained for the asymptotic distribution of this rank transform statistic to be chi-squared under the null hypothesis of no interaction. It is shown that the rank transform test statistic for interaction is asymptotically chi-squared, divided by its degrees of freedom, when there are exactly two levels of both main effects or when there is only one main effect. In the latter case, the test detects a nested effect instead of an interaction. In all other two-way layouts, there exist values for the main effects such that, under the null hypothesis of no interaction, the expected value of the rank transform test statistic goes to infinity as the sample size increases.