Anatomy of pearson’s chi-square statistic in three-way contingency tables

Abstract

We consider orthogonal decompositions of Pearson’s chi-square statistic in three-way contingency tables. We derive algebraic formulae for the decompositions, conditionally on given marginal frequencies. Results indicate that the order in which various effects are taken into account plays a crucial role. This is analogous to multiple regression analysis with correlated predictor variables. Because of their orthogonality, terms in the decompositions follow independent asymptotic chi-square distributions under suitable null hypotheses. We also compare our results with partitions of the log likelihood ratio (LR) chi-square associated with log linear models for contingency tables.