The Decomposability and Monotonicity of Pearson's Chi-Square for Collapsed Contingency Tables with Applications

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Abstract
The Pearson chi-square statistic for a contingency table is decomposed into a sum of components. Each component involves the Pearson chi-square statistic for tables that are formed by collapsing (i.e., combining) rows and/or columns of the original table. Since the terms in the decomposition are necessarily nonnegative, this leads to a monotonicity in the statistic; the Pearson chi-square value for any table N is never smaller than the corresponding values for tables formed by collapsing N. The decomposition is shown to be applicable to tests for quasi-independence or quasi-homogeneity, and the monotonicity provides a coherent simultaneous test procedure. The setting for the proofs provides a probabilistic interpretation for the Pearson chi-square statistic and suggests future areas of research.

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