Reconciling model-X and doubly robust approaches to conditional independence testing

TL;DR

This paper explores the robustness of model-X and doubly robust methods for conditional independence testing, demonstrating that the distilled conditional randomization test (dCRT) remains valid under certain estimation conditions and comparing it to the GCM test.

Contribution

It shows that the dCRT is doubly robust, proves asymptotic equivalence with the GCM test, and evaluates their performance through extensive simulations.

Findings

dCRT maintains Type-I error control if the outcome mean is well estimated

dCRT and GCM have similar power and error rates, with dCRT sometimes better in error control

Post-lasso statistics improve Type-I error control for both tests

Abstract

Model-X approaches to testing conditional independence between a predictor and an outcome variable given a vector of covariates usually assume exact knowledge of the conditional distribution of the predictor given the covariates. Nevertheless, model-X methodologies are often deployed with this conditional distribution learned in sample. We investigate the consequences of this choice through the lens of the distilled conditional randomization test (dCRT). We find that Type-I error control is still possible, but only if the mean of the outcome variable given the covariates is estimated well enough. This demonstrates that the dCRT is doubly robust, and motivates a comparison to the generalized covariance measure (GCM) test, another doubly robust conditional independence test. We prove that these two tests are asymptotically equivalent, and show that the GCM test is optimal against (generalized) partially linear alternatives by leveraging semiparametric efficiency theory. In an extensive simulation study, we compare the dCRT to the GCM test. These two tests have broadly similar Type-I error and power, though dCRT can have somewhat better Type-I error control but somewhat worse power in small samples or when the response is discrete. We also find that post-lasso based test statistics (as compared to lasso based statistics) can dramatically improve Type-I error control for both methods.