On the power of conditional independence testing under model-X

For testing conditional independence (CI) of a response Y and a predictor X given covariates Z, the model-X (MX) framework has been the subject of active methodological research, especially in the context of MX knockoffs and their application to genome-wide association studies. In this paper, we study the power of MX CI tests, yielding quantitative insights into the role of machine learning and providing evidence in favor of using likelihood-based statistics in practice. Focusing on the conditional randomization test (CRT), we find that its conditional mode of inference allows us to reformulate it as testing a point null hypothesis involving the conditional distribution of X. The Neyman-Pearson lemma implies that a likelihood-based statistic yields the most powerful CRT against a point alternative. We obtain a related optimality result for MX knockoffs. Switching to an asymptotic framework with arbitrarily growing covariate dimension, we derive an expression for the power of the CRT against local semiparametric alternatives in terms of the prediction error of the machine learning algorithm on which its test statistic is based. Finally, we exhibit a resampling-free test with uniform asymptotic Type-I error control under the assumption that only the first two moments of X given Z are known.