Bayesian methods for contingency tables using Gibbs sampling

Abstract

Cell counts in contingency tables can be smoothed using loglinear models. Recently, sampling-based methods such as Markov chain Monte Carlo (MCMC) have been introduced, making it possible to sample from posterior distributions. The novelty of the approach presented here is that all conditional distributions can be specified directly, so that straight-forward Gibbs sampling is possible. Thus, the model is constructed in a way that makes burn-in and checking convergence a relatively minor issue. The emphasis of this paper is on smoothing cell counts in contingency tables, and not so much on estimation of regression parameters. Therefore, the prior distribution consists of two stages. We rely on a normal nonconjugate prior at the first stage, and a vague prior for hyperparameters at the second stage. The smoothed counts tend to compromise between the observed data and a log-linear model. The methods are demonstrated with a sparse data table taken from a multi-center clinical trial.