Association and Simpson conversion in $2 \times 2 \times 2$ contingency tables

TL;DR

This paper explores the generalization of Simpson's paradox to three-dimensional contingency tables, characterizing when such reversals can occur and proposing a conjecture based on computational experiments.

Contribution

It provides a geometric characterization of Simpson reversal in $2 imes 2 imes 2$ tables and introduces a conjecture on the likelihood of these events.

Findings

Characterization of Simpson reversal cases

Two combinatorial-geometric lemmas

A conjecture on the likelihood of Simpson reversals

Abstract

We study a generalisation of Simpson reversal (also known as Simpson's paradox or the Yule-Simpson effect) to $2 \times 2 \times 2$ contingency tables and characterise the cases for which it can and cannot occur with two combinatorial-geometric lemmas. We also present a conjecture based on some computational experiments on the expected likelihood of such events.