The Asymptotic Proportion of Subdivisions of a 2×2 Table that Result in Simpson's Paradox

The asymptotic proportion of the subdivisions of the original 2×2 table such that SP occurs is calculated and it is shown that this asymPTotic proportion is bounded above by 1/12.

Abstract

A 2×2 table of nonnegative integers is subdivided additively into n 2×2 subtables of nonnegative integers in an arbitrary way. The first case of Simpson's Paradox (SP) occurs when the determinant of the original table is less than zero, but the determinant of each of the n 2×2 subtables is greater than or equal to zero. The second case of SP occurs when the previous inequalities hold with ‘less than’ and ‘greater than or equal’ replaced by ‘greater than’ and ‘less than or equal’, respectively. For the case n =2, this paper calculates the asymptotic proportion of the subdivisions of the original 2×2 table such that SP occurs. It is shown that this asymptotic proportion is bounded above by 1/12.