Step-Wise Goodness-of-Fit Testing for High-Dimensional Cross-Classified Tables

The Pearson and likelihood ratio statistics are commonly used to test goodness of fit for models applied to data from a multinomial distribution. The goodness-of-fit test based on Pearson's chi-squared statistic is sometimes considered to be a global test that gives little guidance to the source of poor fit when the null hypothesis is rejected, and it has also been recognized that the global test can often be outperformed in terms of power by focused or directional tests. In this paper, a sequence of test statistics is obtained by decomposing the Pearson statistic from the full table into orthogonal components defined on marginal distributions. These statistics are asymptotic independent and can used in a step-wise procedure, starting with higher-order marginals and proceeding to lower-order marginals. For a five-way table, there would be a test statistic for the 5th-order marginal, then fourth-order, then third-order and finally-second order. Although this step-wise procedure involves multiple testing, it can be shown to have higher power than the ordinary Pearson goodness-of-fit statistic, especially Higher-order marginals might be combined. An asymptotic distribution can be invoked for the statistics on lower-order marginals, but higher-order tables are sparse and a bootstrap method might be needed for the test on higher-order marginals. Since the test statistics are orthogonal components of the Pearson statistic, a test statistic for higher-order marginals can be obtained by subtracting the sum of the lower-order test statistics from the Pearson statistic. An application from epidemiology and psychometrics is presented.