Graphical and Recursive Models for Contingency Tables

TLDR

The paper introduces two classes of contingency‑table models—graphical (undirected) and recursive (directed)—defined by conditional independence restrictions. The authors derive decomposable/multiplicative models as the intersection of these two classes. These decomposable models enable exploratory analysis and interpretation through graph representations of the data.

Abstract

SUMMARY We discuss two classes of models for contingency tables, graphical and recursive models, both of which arise from restrictions that are expressible as conditional independencies of variable pairs. The first of these is a subclass of hierarchical log linear models. Each of its models can be represented by an undirected graph. In the second class each model corresponds to a particular kind of a directed graph instead and can be characterized by a nontrivial factorization of the joint distribution in terms of response variables. We derive decomposable or multiplicative models as the intersecting class. This result has useful consequences for exploratory types of analysis as well as for the model interpretation: we can give an aid for detecting well-fitting decomposable models in a transformation of the observed contingency table and each decomposable model may be interpreted with the help of an undirected or directed graph.