Improving Tests for Discrete Small Sample Data

This book provides a guide for improving test statistics that are based on phi-divergence for discrete models, which include various kinds of independence models of contingency tables as well as generalized linear models of binary data. The improvements are based on the theory of asymptotic expansion and lead to correct conclusions of a test even when sample sizes are not large. Without such an improvement, there is a risk that the results of a test will lead to the opposite conclusion, as a limiting distribution is used for an approximated distribution of test statistics. Mainly, for the phi-divergence family of statistics that include Pearson’s chi-square statistic, the log-likelihood ratio statistic, and the power divergence family of statistics as a special case, the book derives the improvement of statistics as transformed statistics. This accomplishment is achieved by using the expression of approximation of the distribution of original phi-divergence statistics based on Edgeworth expansion. For an independence model of a contingency table, a complete independence model, an independence model among a group of factors, and a conditional independence model are considered. The test statistics of a contingency table for a log-linear model are also presented for consideration. Additionally, the selection of statistics for which the distribution is close to the limiting distribution is discussed using the evaluation of second-order correction of moments.