Association-Marginal Modeling of Multivariate Categorical Responses: A Maximum Likelihood Approach

Abstract

Generalized log-linear models can be used to describe the association structure and/or the marginal distributions of multivariate categorical responses. We simultaneously model the association structure and marginal distributions 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 marginal (M) model, which describes the marginal distributions of the responses. Because the model's composite link function is not required to be invertible, a large class of models can be entertained and model specification is typically straightforward. We propose a “mixed freedom/constraint” parameterization that exploits the special structure of an AM model. Using this parameterization, 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 of the problems associated with sampling O's are avoided. We compare the asymptotic behavior of AM model parameter estimators assuming product-multinomial and Poisson sampling. For computational convenience, the product-multinomial variances are obtained by adjusting the Poisson variances. We propose a conditional score statistic for AM model assessment. The proposed maximum likelihood methods are illustrated through an analysis of marijuana use data from five waves of the National Youth Survey.

Key Words:

• Association model
• Composite link function
• Conditional score statistic
• Correlated responses
• Generalized log-linear model
• Marginal model
• Partitioning of chi-square.