Distribution-free test procedures in two-way layouts; a concept of rank-interaction

Summary  In this paper the concept of 'rank-interaction' is introduced and a distribution-free method for testing against the presence of 'rank-interaction' is suggested in the case of a two-way layout (classification) with m (> 1) observations per cell. Roughly speaking rank-interaction can be understood as the phenomenon at which the ranks of the levels of some relevant variable are different for different classes of the other factor. The exact null distribution of the test statistic has been computed in some cases. The asymptotic distribution under the null hypothesis has been derived. A test suggested by J.V. B radley in his book 'Distribution-free Statistical Tests' 2] is discussed. In the opinion of the authors it is doubtful whether the asymptotic distribution of the test statistic under the null hypothesis, as given by B radley , is correct. The test of B radley was intended to be sensitive to the presence of interactions defined in the usual way and hence not only to 'rank-interaction'. The same applies to methods proposed by some other authors. We claim that situations exist where one should test against rank-interaction and not against the usual more general alternative.