Maximal Simpson-disaggregations of 2×2 tables

Abstract

A 2 × 2 table of frequencies can be expressed as the sum of two other tables corresponding to the levels of a real or potential ‘masked’ factor in which the direction of the apparent association between the marginal factors is opposite that of the original. This paper analyses the counterpart of this disaggregation obtained by replacing frequencies by probabilities. It is shown that such a disaggregation cannot occur if the original experiment is sufficiently well-balanced with respect to this masked factor. When the class of such disaggregations is nonempty that of its elements which maximizes the average absolute differences of certain marginal probabilities in the resulting tables is found.