2. Multivariate Logistic Regression Models

In the this chapter we introduce graphical models for categorical data based on the multinomial distributions. Gaussian-based models have some similarities with multinomial models (both families are closed under the operations of marginalization and conditioning), but have also many important differences. The multinomial does not depend only on a mean vector and a covariance matrix like the Gaussian, but contains also higher order parameters called interactions. Moreover, the types of parameterization (i.e., a 1-1 transformation of the original parameters ) are crucial in order to get interesting properties. One example is the logistic transformation used in Chap. 11 that will be generalized in this chapter. We introduce the multivariate logistic transformation and other generalizations called marginal parameterizations that are particularly beneficial in the context of regression graph models for categorical data. These parameterizations, based on transformations of the joint probabilities related of a contingency table, provide directly log-linear measures of association and independence constraints on the parameter space. The results concerning marginal models are illustrated through examples and case studies related to the interpretation of the regression chain graph models.