Analytical p-value calculation for the higher criticism test in finite-d problems

The higher criticism is effective for testing a joint null hypothesis against a sparse alternative, e.g., for testing the effect of a gene or a genetic pathway that consists of d genetic markers. Accurate p-value calculations for the higher criticism based on the asymptotic distribution require a very large d , which is not the case for the number of genetic variants in a gene or a pathway. In this paper we propose an analytic method that accurately computes the p-value of the higher criticism test for finite d problems. Unlike previous treatments, this method does not rely on asymptotics in d or simulation, and is exact for arbitrary d when the test statistics are normally distributed. The method is particularly computationally advantageous when d is not large. We illustrate the proposed method with a case-control genome-wide association study of lung cancer and compare its power to competing methods through simulations.