Statistica Sinica 13(2003), 189-206

BAYESIAN COMPUTATION FOR CONTINGENCY TABLES

WITH INCOMPLETE CELL-COUNTS

Guo-Liang Tian$^{*,\dag}$, Kai Wang Ng$^\ddag$ and Zhi Geng$^*$

$^*$Peking University, $^\ddag$The University of Hong Kong

and $^\dag$St.$\!$ Jude Children's Research Hospital

Abstract: This article studies Bayesian analysis of contingency tables (or multinomial data) where the cell counts are not fully observed due to reasons such as nonresponse and misclassification, and derives the posterior distributions of the unknown cell probabilities in terms of various types of generalized Dirichlet distributions. For some special situations such as grouped and nested Dirichlet distributions, the posterior means of the unknown cell probabilities can be obtained in closed form by using inverse Bayes formulae and/or stochastic representation. When closed-form expressions do not exist, we suggest using importance sampling with a feasible proposal density to approximately compute the posterior quantities, and propose a procedure for choosing an effective proposal density. Applications are illustrated by sample surveys with nonresponse, crime survey data, death penalty attitude data, and misclassified multinomial data.

Key words and phrases: Bayesian inference, grouped and nested Dirichlet distributions, incomplete data, inverse Bayes formulae, stochastic representation.