A TEST OF HOMOGENEITY OF ODDS RATIOS AGAINST ORDER RESTRICTIONS

This article deals with testing the homogeneity of the odds ratios psi-1, ..., psi-k, taken relative to the first column of a given 2 x (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 N21, ..., N2k, 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 N2j'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 chi-BAR2 distribution. The test is applied to a numerical example and its power is compared with Mantel's test and the ordinary chi-2 test. In practice, many odds ratios exhibit a trend. For example, there is usually a simple order on the odds ratios: 1 less-than-or-equal-to psi-1 less-than-or-equal-to ... less-than-or-equal-to psi-k. In the study of dose-response relationships, a unimodal trend may be considered, that is, there is a positive integer p such that 1 less-than-or-equal-to psi-1 less-than-or-equal-to ... less-than-or-equal-to psi-p greater-than-or-equal-to ... greater-than-or-equal-to psi-k, which is said to be the umbrella order by Mack and Wolfe (1981) and includes the simple order with p = k. A simple tree order may be denoted by 1 less-than-or-equal-to psi-1 less-than-or-equal-to psi-j for j = 2, ..., k. The test proposed in this article can be applied to test homogeneity of the odds ratios: psi-1 = ... = psi-k = 1 against the simple or any other partial order restricted alternatives.