Beyond Simpson's paradox: A descriptive approach

Abstract

In considering the strength of association of particular variables, we cannot ignore the effects of confounding factors that cause Simpson's paradox. Many methods for adjusting these effects have been proposed, and a great deal of effort has been devoted to statistical tests. Apart from the statistical tests, the aim of the present study is to examine the strength of association of two categorical variables without reference to any explicit confounding factors. In other words, our aim is to specify the conditions under which Simpson's paradox does not occur, where the idea of classifying the original universe into groups is adopted. Let us begin by focusing our attention on a 2 × 2 contingency table (cross-classification table) and considering the association of X with Y, where X and Y denote dichotomous variables with classes A and B for X and classes + and − for Y. To examine the strength of association between these variables, the index k = q/p is used, where p denotes the proportion of A + in A and q denotes that of B + in B. Using the maximum and minimum values of the index k obtained by numerical calculation, the strength of association is examined. The results are discussed and examples given. © 1996 John Wiley & Sons, Ltd.