Deploying the Conditional Randomization Test in High Multiplicity Problems

This paper introduces the sequential CRT, which is a variable selection procedure that combines the conditional randomization test (CRT) and Selective SeqStep+. Valid p-values are constructed via the flexible CRT, which are then ordered and passed through the selective SeqStep+ filter to produce a list of discoveries. We develop theory guaranteeing control on the false discovery rate (FDR) even though the p-values are not independent. We show in simulations that our novel procedure indeed controls the FDR and are competitive with -- and sometimes outperform -- state-of-the-art alternatives in terms of power. Finally, we apply our methodology to a breast cancer dataset with the goal of identifying biomarkers associated with cancer stage.