Estimating cell probabilities in contingency tables with constraints on marginals/conditionals by geometric programming with applications

Abstract

Contingency tables are often used to display the multivariate frequency distribution of variables of interest. Under the common multinomial assumption, the first step of contingency table analysis is to estimate cell probabilities. It is well known that the unconstrained maximum likelihood estimator (MLE) is given by cell counts divided by the total number of observations. However, in the presence of (complex) constraints on the unknown cell probabilities or their functions, the MLE or other types of estimators may often have no closed form and have to be obtained numerically. In this paper, we focus on finding the MLE of cell probabilities in contingency tables under two common types of constraints: known marginals and ordered marginals/conditionals, and propose a novel approach based on geometric programming. We present two important applications that illustrate the usefulness of our approach via comparison with existing methods. Further, we show that our GP-based approach is flexible, readily implementable, effort-saving and can provide a unified framework for various types of constrained estimation of cell probabilities in contingency tables.