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

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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.

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Takane, Y., & Zhou, L. (2013). Anatomy of pearson’s chi-square statistic in three-way contingency tables. In Springer Proceedings in Mathematics and Statistics (Vol. 66, pp. 41–57). Springer New York LLC. https://doi.org/10.1007/978-1-4614-9348-8_4

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