Relational models for contingency tables

TL;DR

This paper introduces a broad class of multiplicative models for contingency tables, extending existing models like log-linear, and discusses conditions for maximum likelihood estimates and model properties.

Contribution

It generalizes multiplicative models for contingency tables, providing coordinate-free formulations, existence conditions for MLEs, and insights into model equivalences and properties.

Findings

Conditions for the existence of MLEs are established.

Equivalence between multinomial and Poisson likelihoods depends on the presence of an overall effect.

Examples illustrate the application and properties of the models.

Abstract

The paper considers general multiplicative models for complete and incomplete contingency tables that generalize log-linear and several other models and are entirely coordinate free. Sufficient conditions of the existence of maximum likelihood estimates under these models are given, and it is shown that the usual equivalence between multinomial and Poisson likelihoods holds if and only if an overall effect is present in the model. If such an effect is not assumed, the model becomes a curved exponential family and a related mixed parameterization is given that relies on non-homogeneous odds ratios. Several examples are presented to illustrate the properties and use of such models.