Simultaneously Modeling Joint and Marginal Distributions of Multivariate Categorical Responses

Abstract We discuss model-fitting methods for analyzing simultaneously the joint and marginal distributions of multivariate categorical responses. The models are members of a broad class of generalized logit and loglinear models. We fit them by improving a maximum likelihood algorithm that uses Lagrange's method of undetermined multipliers and a Newton-Raphson iterative scheme. We also discuss goodness-of-fit tests and adjusted residuals, and give asymptotic distributions of model parameter estimators. For this class of models, inferences are equivalent for Poisson and multinomial sampling assumptions. Simultaneous models for joint and marginal distributions may be useful in a variety of applications, including studies dealing with longitudinal data, multiple indicators in opinion research, cross-over designs, social mobility, and inter-rater agreement. The models are illustrated for one such application, using data from a recent General Social Survey regarding opinions about various types of government spending.