Multinomial-Poisson homogeneous models for contingency tables

Abstract

A unified approach to maximum likelihood inference for a broad, new class of contingency table models is presented. The model class comprises multinomial-Poisson homogeneous (MPH) models, which can be characterized by an independent sampling plan and a system of homogeneous constraints, h(m) = 0, where m is the vector of expected table counts. Maximum likelihood (ML) fitting and large-sample inference for MPH models are described. The MPH models are partitioned into well-defined equivalence classes and explicit comparisons of the large-sample behaviors of ML estimators of equivalent models are given. The equivalence theory not only unifies a large collection of previously known results, it also leads to useful generalizations and many new results. The practical, computational implication is that ML fit results for any particular MPH model can be obtained directly from the ML fit results for any conveniently chosen equivalent model. Issues of hypothesis testability and parameter estimability are also addressed. To illustrate, an example based on statistics journal citation patterns is given for which the data can be used to test the hypothesis that a certain model holds, but they cannot be used to estimate any of that model's parameters.