A Test of Homogeneity of Odds Ratios against Order Restrictions

Abstract

Abstract This article deals with testing the homogeneity of the odds ratios Ψ 1, …, Ψ k, taken relative to the first column of a given 2 × (k + 1) cross-classification table of ordinal variables, against a partial order restriction. The inference of these odds ratios is considered on an extended hypergeometric distribution, a conditional distribution of cell frequencies N 2l , …, N 2k , say, given both marginal totals. Take a transformation such that the order restriction on the odds ratios tends to be in some linear inequalities restriction on means of the N 2j 's based on the conditional distribution. A test is proposed from the transformation as a one-sided likelihood ratio test in the normal case and its asymptotic null distribution is the χ 2 distribution. The test is applied to a numerical example and its power is compared with Mantel's test and the ordinary χ 2 test. In practice, many odds ratios exhibit a trend. For example, there is usually a simple order on the odds ratios: 1 ⩽ Ψ 1 ⩽ Ψk . In the...