Comparing χ2 Tables for Separability of Distribution and Effect: Meta-Tests for Comparing Homogeneity and Goodness of Fit Contingency Test Outcomes

This paper describes a series of statistical meta-tests for comparing independent contingency tables for different types of significant difference. Recognizing when an experiment obtains a significantly different result and when it does not is frequently overlooked in research publication. Papers are frequently published citing ‘p values’ or test scores suggesting a ‘stronger effect’ substituting for sound statistical reasoning. This paper sets out a series of tests that together illustrate the correct approach to this question. These meta-tests permit us to evaluate whether experiments have failed to replicate on new data; whether a particular data source or subcorpus obtains a significantly different result than another; or whether changing experimental parameters obtains a stronger effect. The meta-tests are derived mathematically from the χ2 test and the Wilson score interval, and consist of pairwise ‘point’ tests, ‘homogeneity’ tests and ‘goodness of fit’ tests. Meta-tests for comparing tests with one degree of freedom (e.g. ‘2 × 1ʹ and ‘2 × 2ʹ tests) are generalized to those of arbitrary size. Finally, we compare our approach with a competing approach offered by Zar, which, while straightforward to calculate, turns out to be both less powerful and less robust. (Note: A spreadsheet including all the tests in this paper is publicly available at www.ucl.ac.uk/english-usage/statspapers/2x2-x2-separability.xls.)