USP: an independence test that improves on Pearson's chi-squared and the $G$-test

TL;DR

The paper introduces the USP independence test, which outperforms traditional chi-squared and G-tests by controlling test size, handling small counts, and detecting subtle dependencies, with proven statistical properties and practical implementation.

Contribution

The USP test is a novel independence test based on a U-statistic that guarantees size control and improved power over existing tests, with a unique unbiased estimator and R implementation.

Findings

USP test controls size at all sample sizes.

USP has higher power than chi-squared and G-tests.

USP performs well on both simulated and real data.

Abstract

We present the $U$-Statistic Permutation (USP) test of independence in the context of discrete data displayed in a contingency table. Either Pearson's chi-squared test of independence, or the $G$-test, are typically used for this task, but we argue that these tests have serious deficiencies, both in terms of their inability to control the size of the test, and their power properties. By contrast, the USP test is guaranteed to control the size of the test at the nominal level for all sample sizes, has no issues with small (or zero) cell counts, and is able to detect distributions that violate independence in only a minimal way. The test statistic is derived from a $U$-statistic estimator of a natural population measure of dependence, and we prove that this is the unique minimum variance unbiased estimator of this population quantity. The practical utility of the USP test is demonstrated on both simulated data, where its power can be dramatically greater than those of Pearson's test and the $G$-test, and on real data. The USP test is implemented in the R package USP.