File:MARKOV CHAIN MONTE CARLO AND EXACT CONDITIONAL TESTS WITH THREE-WAY CONTINGENCY TABLES (IA markovchainmonte1094559705).pdf

We propose an algorithm modifying a popular exact conditional test involving the goodness-of-fit of contingency tables. This study focuses on improving the efficiency of Markov chain Monte Carlo (MCMC) when sampling three-way contingency tables--defined as log-linear models with three discrete random categorical variables consisting of finite levels--under the no-three-way interaction model. Standard to MCMC, we approximate the null distribution by sampling tables from the conditional distribution. However, our proposal involves expanding the conditional state space to include tables with cell count values of -1. We apply the proposed methodology, described in full detail, to randomly generated sparse and non-sparse data sets. Our results show that traditional asymptotic methods on sparse contingency tables yield inaccurate results. We also prove mathematically that a Markov chain with our proposed method is connected (i.e., ergodic) on the conditional state space for 3x3xK, with K >= 3. The output from applying the proposed methodology provides conclusive evidence that the distribution of the test statistics for sparse data sets does not resemble the asymptotic distribution.

Subjects: Categorical Data Analysis; Markov-chain Monte Carlo; Chi-Square Test for Independence; Kolmogorov-Smirnov Test; three-way contingency tables; no-three-way interaction model; log-linear models; exact conditional test; goodness-of-fit test; sparse data; asymptotic distribution.