Using power-divergence statistics to test for homogeneity in product-multinomial distributions
Testing for homogeneity in the product-multinomial distribution, where the hypotheses are hierarchical, uses maximum likelihood estimation and the loglikelihood ratio statistic G 2. We extend these ideas to the power-divergence family of test statistics, which is a one-parameter family of goodness-of-fit statistics that includes the loglikelihood ratio statistic G 2, Pearson's X 2, the Freeman-Tukey statistic, the modified loglikelihood ratio statistic, and the Neyman-modified chi-squared statistic. Explicit minimum-divergence estimators can be obtained for all members of the one-parameter family, which allows a straightforward analysis of divergence. An analysis of fourteen retrospective studies on the association between smoking and lung cancer demonstrates the ease of interpretation of the resulting analysis of divergence.