Association-marginal modeling of multivariate categorical responses: A maximum likelihood approach

Generalized log-linear models can be used to describe the association structure and marginal distribution using association-marginal (AM) models, which are specially formulated generalized log-linear models that combine two models: an association (A) model, which describes the association among all responses; and a marginal (M) model, which describes the marginal distributions of the responses. Because the model's composite link function is nor required to be invertible, a large class of models can be entertained and model specification is typically straightforward. We proposed a "mixed freedom/constraint" parameterization that exploits the special structure of an AM model. Using this parameterization, maximum likehood fitting is straightforward and typically feasible for large , sparse tables. When a parsimonious association model is used , the size of the fitting problem is substantially reduced, and some of the problems associated with sampling 0's are avoided. We compare the asymptotic behavior of AM model parameter estimators assuming product-multinomial and Poisson variances. We propose a conditional score atatistic for AM model assessment. The proposed maximum likelihhood methods are illustrated through an analysis of marijuana use data from five waves of the National Youth Service.