Marginal regression models for the analysis of positive association of ordinal response variables

Given a set of discrete response variables, some of which are ordinal, and an arbitrary set of discrete explanatory variables, we propose a simple matrix formulation for parameterising the saturated model as in Glonek (1996). This is such that, within a hierarchical structure, marginal logits and log-odds ratios of various possible types, together with the remaining log-linear interactions of high order, may be modelled by equality and inequality constraints. Inequality constraints are particularly relevant for specifying models of positive association. Efficient algorithms are provided for computing maximum likelihood estimates under such constraints. The asymptotic distribution of the likelihood ratio test is derived and an extension of the usual analysis of deviance is outlined which incorporates inequality constraints.