The moments of the Pearson chi-squared statistic and the minimum expected value in two-way tables

Abstract

SUMMARY Explicit expressions for the first three moments of Pearson's X 2 statistic for a two-way contingency table are used to investigate the accuracy of the χ 2 approximation to its distribution. The usual ’minimum expected value' criterion is shown to give a global bound on the accuracy of the moments, but to be too conservative. This is illustrated by examples. It is shown that the adequacy of the approximation depends on the significance level that is to be used as well as the accuracy of the moments. When the χ 2 2 approximation is inadequate, corrections based on the moments can be used. Numerical studies of the usefulness of these corrections are presented.