Anatomy of Pearson’s Chi-square Statistic in Three-way Contingency Tables Yoshio Takane and Lixing Zhou Abstract We consider orthogonal decompositions of Pearson’s chi-square statistic in three-way contingency tables. We derive algebraic formulae for the decomposi- tions, conditionally on given marginal frequencies. Results indicate that the order in which various effects are taken into account play a crucial role. This is analogous to multiple regression analysis with correlated predictor variables. Because of their or- thogonality, terms in the decompositions follow independent asymptotic chi-square distributions under suitable null hypotheses. We also compare our results with par- titions of the log likelihood ratio (LR) chi-square associated with log linear models for contingency tables. 1 Introduction Research in psychology and other social sciences often involves discrete multivari- ate data. Such data are conveniently summarized in the form of contingency tables. There have been two widely used classes of techniques for analysis of such tables. One is log linear models (e.g.,Andersen, 1980; Bishop et al., 1975), and the other is correspondence analysis (CA; e.g., Greenacre, 1984; Nishisato, 1980). The for- mer allow ANOVA-like decompositions of the log likelihood ratio (LR) statistic (also known as the deviance statistic or the Kullback-Leibler (1951) divergence). This statistic measures the difference in log likelihood between the saturated and independence models. When the latter model is correct, it follows the asymptotic Yoshio Takane Department of Psychology, University of Victoria, P. O. Box 3050 Victoria, BC V8W 3P5 Canada, e-mail: takane@uvic.ca Lixing Zhou Department of Psychology, McGill University, 1205 Dr. Penfield Ave. Montreal QC H3A 1B1 Canada, e-mail: lixing.zhou@mail.mcgill.ca 1