The Analysis of Ratios of Odds Ratios in Stratified Contingency Tables

Abstract

The ratio of odds ratios in a 2 x 2 x 2 contingency table is applicable to both cohort and case-control studies as a measure of the interaction effect (on the logit scale) of two dichotomous exposure factors on a dichotomous response. Analysis of this parameter is here extended by the introduction of a fourth factor for stratification of the table, in order to control for confounding or to study homogeneity of the effects of interest. The exact conditional distribution of the sufficient statistic is outlined, and this leads to exact methods for testing homogeneity of ratios of odds ratios over strata, and for inference concerning a common parameter value. Large-sample methods, based on the asymptotic distribution and on Mantel-Haenszel principles, are developed and compared with log-linear model maximum likelihood and arcsin transformation methods. A case-control study of cervical dysplasia in relation to two common sexually-transmitted pathogens serves as an example. The study of main effects of each exposure in such a setting is also discussed. Consideration of a ratio of odds ratios arises naturally in studying relationships between three dichotomous factors. Bartlett's (1935) condition for no three-factor interaction in a 2 x 2 x 2 contingency table is equivalent to the assertion that the true ratio of odds ratios equals one (provided that its value exists). The parameter has a variety of potential applications. In one usage (a), given two treatments (Factor 2) and a dichotomous outcome or response (Factor 1), it is of interest to assess homogeneity of the relative efficacy of the two treatments under two different environmental conditions (Factor 3). This situation was discussed by Zacks and Solomon (1976), who reduced the problem to inference concerning a ratio of odds ratios-specifically, the ratio of outcome-to-Factor 2 odds ratios, with the component odds ratios conditioned on different levels of Factor 3. (Note that the value of the resultant ratio does not depend on the order of conditioning by the two factors.) This application is a special case of a more general usage (b), in which there are four cohorts of observational units defined by observed or experimentallyapplied levels of two dichotomous factors (Factors 2 and 3), a dichotomous outcome is observed, and it is of interest to assess the interaction effect of the two factors on outcome. A third potential usage (c) is in case-control studies involving two exposure factors. In such studies, it is possible to estimate Factor 2-to-outcome odds ratios conditional on each level of the second exposure factor, Factor 3. A well-known fact, central to the interpretation of results from case-control studies, is that odds ratios are invariant under interchange of roles of the two dichotomies on which they are defined. The ratio of these two