The conditional permutation test for independence while controlling for confounders
Thomas B. Berrett, Yi Wang, Rina Foygel Barber, Richard J. Samworth
Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2020 · Volume 82 · Issue 1
Abstract
We propose a general new method, theconditional permutation test, for testing the conditional independence of variablesXandYgiven a potentially high dimensional random vectorZthat may contain confounding factors. The test permutes entries ofXnon-uniformly, to respect the existing dependence betweenXandZand thus to account for the presence of these confounders. Like the conditional randomization test of Candès and co-workers in 2018, our test relies on the availability of an approximation to the distribution ofX|Z—whereas their test uses this estimate to draw newX-values, for our test we use this approximation to design an appropriate non-uniform distribution on permutations of theX-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 ofX|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.
Keywords
Conditional independence testing, Exchangeability, Model misspecification, Model-X knockoffs, Permutation tests, Semisupervised learning
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