Optimal High Dimensional Multiple Testing Under Linear Models

High dimensional multiple testing has many important applications. Motivated by genome-wide association studies (GWAS), we consider the problem of mulitiple testing under high dimensional sparse linear model in order to identify the genetic markers associated with the trait of interest. The model is an extension of the normal mixture model under arbitrary dependence. We propose a multiple testing procedure, which ranks and thresholds the adjusted z-surrogate. It is shown that the procedure can control mfdr level and minimize mfnr level asymptotically among all the methods based on the original data. Numerical results show that our method performs well under linear models with correlated predictors. The procedure is further illustrated through an analysis of a genome-wide association study in hypertension. At mfdr level equal to 0.05, it identifies 11 genetic markers assciated with systolic blood pressure and 11 associated with diastolic blood pressure. Many of the markers are located in the regions associated with human blood pressure based on the Rat Genome Database.