CRT*: Conditional Randomization Testing with Heterogeneous External and Unlabeled Data

Abstract

The conditional randomization test (CRT) provides a principled approach to conditional independence (CI) testing, guaranteeing exact type-I error control when the true conditional distribution is known. In practice, however, this distribution must be estimated, and estimation errors can inflate type-I errors, while high dimensionality and limited sample sizes can reduce power. Although external and unlabeled data offer the potential to improve CI testing, naive integration that ignores distributional heterogeneity can compromise type-I error control and fail to enhance power. We propose \textbf{CRT*}, a novel framework that robustly integrates external and unlabeled datasets to enhance CI testing in heterogeneous scenarios. CRT* employs smooth residual-bootstrap (SRB) with transfer learning for conditional distribution estimation, combined with adaptive data fusion via an optimal convex combination of test statistics. We theoretically establish that the SRB-based estimator converges to the true conditional distribution in expected total variation distance. Furthermore, even in high-dimensional regimes, CRT* maintains valid type-I error control and achieves strictly higher power than standard CRT without external data. Simulations and RNA-seq breast cancer data analyses demonstrate that CRT* substantially improves power while maintaining type-I error control in heterogeneous settings.