Analytic P-value calculation for the higher criticism test in finite d problems

Abstract

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.

Keywords: Empirical process; Genome-wide association studies; Higher criticism; Multiple hypothesis testing; Signal detection.

Fig. 1

An example c(t | h) = h[2d Φ̄ (t){1 − 2 Φ̄ (t)}]^1/2 + 2dΦ̄ (t) is plotted with d = 6 and h = 2.4. The partition given by Lemma 1 is labeled on the t-axis.

Fig. 2

The power for various methods are compared. The higher criticism applied to the decorrelated test statistics is represented by the solid line. The likelihood ratio test is represented by the dotted line. The sequence kernel association test is represented by the dashed line. Power is simulated for each ρ that is a nonnegative multiple of 0·05 and the smoothed results are displayed. Signal sparsity is 5% with β = 0.11 in the left plot with signal sparsity 10% with β = 0.08 in the right plot.