Analyzying Bivariate Ordinal Polytomous Data: A Marginal Multinomial Logistic Approach

Abstract

The multinomial cell counts based likelihood and the generalized estimating equations (GEE) approaches are widely used for analysis of bivariate ordinal categorical responses. In both of these approaches, the joint cell probabilities are usually modeled in terms of a global odds ratio as a measure of association and the marginal probabilities for each of the two ordered response variables. These methods utilize the stochastic ordering of the responses by modelling the cumulative margins with certain suitable link functions so that the link function of a cumulative margin is linear in covariates and an intercept representing the ordinal category. This type of modelling, therefore, requires suitable order restricted inference for the cutpoints (intercepts) separating the ordinal categories. These cutpoints are, however, frequently estimated in traditional ways without challenging their order restrictions. In this paper, we distinguish the ordinal categories in a general way so that the covariate effects are generally different under different ordinal categories. This allows one to model the cumulative margins through certain non-linear regression functions which does not require any introduction of the cutpoints.