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

Generalized log-linear models can be used to describe the association struc ture and/or the marginal distributions of multivariate categorical response s. We simultaneously model the association structure and marginal distribut ions 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 the responses; and a margi nal (M) model, which describes the marginal distributions of the responses. Because the model's composite link function is not required to be invertib le, a large class of models can be entertained and model specification is t ypically straightforward. We propose a "mixed freedom/constraint" parameter ization that exploits the special structure of an AM model. Using this para meterization, maximum likelihood 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 o f the problems associated with sampling 0's are avoided. We compare the asy mptotic behavior of AM model parameter estimators assuming product-multinom ial and Poisson sampling. For computational convenience, the product-multin omial variances are obtained by adjusting the Poisson variances. We propose a conditional score statistic for AM model assessment. The proposed maximu m likelihood methods are illustrated through an analysis of marijuana use d ata from five waves of the National Youth Survey.