## The decomposability and monotonicity of Pearson’s chi-square for collapsed contingency tables with applications.(English)Zbl 0546.62030

Summary: 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.

### MSC:

 62H17 Contingency tables 62H10 Multivariate distribution of statistics 62H15 Hypothesis testing in multivariate analysis
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