The conditional permutation test for independence while controlling for confounders

TL;DR

This paper introduces the conditional permutation test, a new method for assessing the independence of variables while controlling for confounders, especially in high-dimensional settings, using non-uniform permutations based on conditional distributions.

Contribution

The paper presents a novel conditional permutation test that accounts for confounders by permuting data non-uniformly, along with an efficient MCMC implementation and theoretical bounds on Type I error inflation.

Findings

The test effectively controls Type I error under approximation errors.

Experimental validation shows accurate independence testing on simulated and real data.

The method outperforms traditional tests in high-dimensional confounding scenarios.

Abstract

We propose a general new method, the conditional permutation test, for testing the conditional independence of variables $X$ and $Y$ given a potentially high-dimensional random vector $Z$ that may contain confounding factors. The proposed test permutes entries of $X$ non-uniformly, so as to respect the existing dependence between $X$ and $Z$ and thus account for the presence of these confounders. Like the conditional randomization test of Cand\`es et al. (2018), our test relies on the availability of an approximation to the distribution of $X \mid Z$. While Cand\`es et al. (2018)'s test uses this estimate to draw new $X$ values, for our test we use this approximation to design an appropriate non-uniform distribution on permutations of the $X$ values already seen in the true data. We provide an efficient Markov Chain Monte Carlo sampler for the implementation of our method, and establish bounds on the Type I error in terms of the error in the approximation of the conditional distribution of $X\mid Z$, finding that, for the worst case test statistic, the inflation in Type I error of the conditional permutation test is no larger than that of the conditional randomization test. We validate these theoretical results with experiments on simulated data and on the Capital Bikeshare data set.