Regression Graph Models for Categorical Data

Table of Contents

1. Frontmatter

2. Chapter 1. Regression Graph Models

   Monia Lupparelli, Giovanni Maria Marchetti, Claudia Tarantola

   Abstract

   This chapter introduces a class of graphical Markov models broadly called regression graph models. In general all graphical models are defined by specific conditional independence constraints in a system of random variables, constraints that have a precise graph representation. These models are useful to specify stepwise data generating processes and research hypotheses in cohort studies. We give some definitions and properties common to all graphical models, and then we discuss some examples. We will explain in some detail the case of a system of Gaussian variables.

3. Chapter 2. Multivariate Logistic Regression Models

   Monia Lupparelli, Giovanni Maria Marchetti, Claudia Tarantola

   Abstract

   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.

4. Chapter 3. Maximum Likelihood Inference

   Monia Lupparelli, Giovanni Maria Marchetti, Claudia Tarantola

   Abstract

   This chapter is concerned with maximum likelihood inference of discrete regression graph models. An optimization procedure is discussed based on a gradient-ascent algorithm to maximize the log-likelihood function where independence constraints are specified through Lagrange multipliers. The algorithm can be recursively applied to the sequence of multivariate regression models induced by the basic factorization of a regression graph. Model selection based on a structural learning of the DAG of the chain components is also discussed with reference to some illustrative applications.

5. Chapter 4. Bayesian Inference

   Monia Lupparelli, Giovanni Maria Marchetti, Claudia Tarantola

   Abstract

   Bayesian inference for discrete regression chain graph models is discussed, specifically for graphs that are Markov equivalent to bi-directed graphs, that is they define the same set of independencies for the joint probability distribution as illustrated in Chap. 2. Conjugate analysis is applicable only to specific graphical configurations, requiring Markov Chain Monte Carlo techniques for broader applications. The discussion will encompass model specification and estimation, focusing on probability and marginal log-linear parameterizations of the model. The previous issues will be illustrated through real data applications. We additionally discuss the complexities involved in prior specification that is an essential aspect of the Bayesian framework.

6. Backmatter