Abstract: Motivated by applications in phenome-wide association studies (Phe-WAS), we consider in this paper simultaneous testing of columns of high-dimensional cross-covariance matrices and develop a multiple testing procedure with theoretical guarantees. It is shown that the proposed testing procedure maintains a desired false discovery rate (FDR) and false discovery proportion (FDP) under mild regularity conditions. We also provide results on the magnitudes of the signals that can be detected with high power. Simulation studies demonstrate that the proposed procedure can be substantially more powerful than existing FDR controlling procedures in the presence of correlation of unknown structure. The proposed multiple testing procedure is applied to a PheWAS of two auto-immune genetic markers using a rheumatoid arthritis patient cohort constructed from the electronic medical records of Partners Healthcare System.

Key words and phrases: Covariance, false discovery rate, multiple responses, multiple testing, PheWAS.