Phase transition and regularized bootstrap in large-scale $t$-tests with false discovery rate control

Abstract

Applying the Benjamini and Hochberg (B–H) method to multiple Student’s $t$ tests is a popular technique for gene selection in microarray data analysis. Given the nonnormality of the population, the true $p$-values of the hypothesis tests are typically unknown. Hence it is common to use the standard normal distribution $N(0,1)$, Student’s $t$ distribution $t_{n-1}$ or the bootstrap method to estimate the $p$-values. In this paper, we prove that when the population has the finite 4th moment and the dimension $m$ and the sample size $n$ satisfy $\log m=o(n^{1/3})$, the B–H method controls the false discovery rate (FDR) and the false discovery proportion (FDP) at a given level $\alpha$ asymptotically with $p$-values estimated from $N(0,1)$ or $t_{n-1}$ distribution. However, a phase transition phenomenon occurs when $\log m\geq c_{0}n^{1/3}$. In this case, the FDR and the FDP of the B–H method may be larger than $\alpha$ or even converge to one. In contrast, the bootstrap calibration is accurate for $\log m=o(n^{1/2})$ as long as the underlying distribution has the sub-Gaussian tails. However, such a light-tailed condition cannot generally be weakened. The simulation study shows that the bootstrap calibration is very conservative for the heavy tailed distributions. To solve this problem, a regularized bootstrap correction is proposed and is shown to be robust to the tails of the distributions. The simulation study shows that the regularized bootstrap method performs better than its usual counterpart.