Tests based on divergences for and against ordered alternatives in cubic contingency tables

Abstract

Cubic contingency tables arise frequently in medical sciences when individuals are measured before, during and after the application of some treatment for a given illness, and data are recorded on an ordered categorical scale. By assigning increasing values to the levels of the illness, the efficiency of the medical treatment can be checked by testing for a given ordering of the cell probabilities p(ijk)'s. One possibility is to consider the hypothesis H-1 that p(ijk) less than or equal to p(i'j'f') if and only if (i', j', k') can be obtained from (i, j, k) through successive pairwise interchanges of adjacent components resulting each time in a decreasing order of the two interchanged components. In this paper we introduce two families of divergence statistics to test for and against H-1, and their asymptotic distributions are obtained. It is also shown that likelihood-ratio test statistics of Barmi and Zimmermann [Statist. Prob. Lett. 45 (1999) 1] are included in these families.