Methods for the Analysis of Contingency Tables with Large and Small Cell Counts

Abstract

Abstract The traditional practice has been to analyze contingency tables using asymptotic χ2 approximations for the tail probability of certain test statistics even when the approximation is known to be poor. For two-dimensional tables Pearson's X 2 statistic is most commonly used, whereas for multidimensional tables the likelihood ratio test is preferred because it makes it easy to test series of nested hypotheses. When asymptotic approximations are not adequate, exact tests are preferred. Recent advances in algorithms have broadened the range of two-dimensional tables for which satisfactory answers can be obtained (Baglivo, Olivier, and Pagano 1985; Mehta and Patel 1983, 1986; Pagano and Taylor-Halvorsen 1981). In this article we propose algorithms that use a mixture of exact and asymptotic techniques to calculate tail probabilities. Our methods can be used with several different test statistics for two-dimensional and multidimensional contingency-table models and hence further broaden the set of tables for which satisfactory answers can be obtained.