The Multinomial-Poisson Transformation

Abstract

SUMMARY The multinomial-Poisson (MP) transformation simplifies maximum likelihood estimation in a wide variety of models for multinomial data. On the basis of specialized derivations, investigators have applied the MP transformation to various models. Here we present a general derivation, which is simpler than the specialized derivations and allows investigators to use the MP transformation readily in new models. We also show how the MP transformation can accommodate incomplete multinomial data and how it can assist in finding closed form maximum likelihood estimates and variances. Previous applications include log-linear models, capture-recapture models, proportional hazards models with categorical covariates and generalizations of the Rasch model. New applications include computing the variance of the logarithm of the odds ratio, a model for voter plurality, conditional logistic regression for matched sets and two-stage case-control studies.