Maxway CRT: Improving the Robustness of Model-X Inference
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The model-X conditional randomization test (CRT) proposed by Candès et al. (2018) is known as a flexible and powerful testing procedure for the conditional independence hypothesis: X is independent of Y conditional on Z. Though having many attractive properties, the model-X CRT relies on the model-X assumption that we have access to perfect knowledge of the distribution of X conditional on Z. If there is a specification error in modeling the distribution of X conditional on Z, this approach may lose its validity. This problem is even more severe when the adjustment covariates Z are of high dimensionality, in which situation precise modeling of X against Z can be hard. In response to this, we propose the Maxway (Model and Adjust X With the Assistance of Y) CRT, a more robust inference approach for conditional independence when the conditional distribution of X is unknown and needs to be estimated from the data. Besides the distribution of X | Z, the Maxway CRT also learns the distribution of Y | Z, using it to calibrate the resampling distribution of X to gain robustness to the error in modeling X. We show that the type-I error inflation of the Maxway CRT can be controlled by the learning error for the low-dimensional adjusting model plus the product of learning errors for the distribution of X | Z and the distribution of Y | Z. This result can be interpreted as an "almost doubly robust" property of the Maxway CRT. Through extensive simulation studies, we demonstrate that the Maxway CRT achieves significantly better type-I error control than existing model-X inference approaches while having similar power. Finally, we apply our methodology to the UK biobank dataset with the goal of studying the relationship between the functional SNP of statins and the risk for type II diabetes mellitus.
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