The quantification of Simpsons paradox and other contributions to contingency table theory

TL;DR

This paper advances contingency table analysis by quantifying Simpson's paradox, introducing new tests for partial interactions, clarifying the relation between interaction measures, and improving methods for fixing cell counts and simulating tables.

Contribution

It provides novel formulas and tests for analyzing three-way interactions and partial interactions, filling gaps in contingency table theory.

Findings

Quantifies Simpson's paradox with a simple formula.

Introduces a test to determine if partial interactions of a variable agree.

Addresses fixed cell counts and limitations in simulating contingency tables.

Abstract

The analysis of contingency tables is a powerful statistical tool used in experiments with categorical variables. This study improves parts of the theory underlying the use of contingency tables. Specifically, the linkage disequilibrium parameter as a measure of two-way interactions applied to three-way tables makes it possible to quantify Simpsons paradox by a simple formula. With tests on three-way interactions, there is only one that determines whether the partial interactions of all variables agree or whether there is at least one variable whose partial interactions disagree. To date, there has been no test available that determines whether the partial interactions of a certain variable agree or disagree, and the presented work closes this gap. This work reveals the relation of the multiplicative and the additive measure of a three-way interaction. Another contribution addresses the question of which cells in a contingency table are fixed when the first- and second-order marginal totals are given. The proposed procedure not only detects fixed zero counts but also fixed positive counts. This impacts the determination of the degrees of freedom. Furthermore, limitations of methods that simulate contingency tables with given pairwise associations are addressed.