Comparing measures of association in 2x2 probability tables

TL;DR

This paper compares various measures of association in 2x2 probability tables, introducing a new measure that balances the influence of small entries and skewed margins using a geometric approach.

Contribution

It develops a geometric framework to visualize and compare association measures, and proposes a novel measure based on relative entropy for better weighting of tables.

Findings

The coordinates reveal how measures depend on margins and odds-ratio.

The new measure up-weights tables with small entries and down-weights skewed margins.

Comparison shows differences in limit behavior and weighting properties of measures.

Abstract

Measures of association play a role in selecting 2x2 tables exhibiting strong dependence in high-dimensional binary data. Several measures are in use differing on specific tables and in their dependence on the margins. We study a 2-dimensional group of margin transformations on the 3-dimensional manifold T of all 2x2 probability tables. The margin transformations allow introducing natural coordinates that identify T with the real 3-space such that the x-axis corresponds to log(sqrt(odds-ratio)) and margins vary on planes x=const. We use these coordinates to visualise and compare measures of association with respect to their dependence on the margins given the odds-ratio, their limit behaviour when cells approach zero and their weighting properties. We propose a novel measure of association in which tables with single small entries are up-weighted but those with skewed margins are down-weighted according to the relative entropy among the tables of the same odds-ratio.