# Phase Transition and Regularized Bootstrap in Large Scale t-tests with False Discovery Rate Control

arXiv: https://arxiv.org/abs/1310.4371 · HTML: https://mlsanity.com/paper/1310.4371/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2013.

## Built on by (most cited first, 3 of 3)

- [Directional FDR Control for Sub-Gaussian Sparse GLMs](https://mlsanity.com/paper/2105.00393/) (2021)
- [Error-based Knockoffs Inference for Controlled Feature Selection](https://mlsanity.com/paper/2203.04483/) (2022)
- [Robust High-dimensional Tuning Free Multiple Testing](https://mlsanity.com/paper/2211.11959/) (2022)

## Abstract (arXiv)

Applying Benjamini and Hochberg (B-H) method to multiple Student's t tests is a popular technique in gene selection in microarray data analysis. Because of the non-normality 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 tn-1 or the bootstrap method to estimate the p-values. In this paper, we first study N(0,1) and tn-1 calibrations. We prove that, when the population has the finite 4-th moment and the dimension m and the sample size n satisfy log m=o(n1/3), B-H method controls the false discovery rate (FDR) at a given level α asymptotically with p-values estimated from N(0,1) or tn-1 distribution. However, a phase transition phenomenon occurs when log m≥ c0n1/3. In this case, the FDR of B-H method may be larger than α or even tends to one. In contrast, the bootstrap calibration is accurate for log m=o(n1/2) as long as the underlying distribution has the sub-Gaussian tails. However, such light tailed condition can not be weakened in general. The simulation study shows that for the heavy tailed distributions, the bootstrap calibration is very conservative. In order 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 the usual bootstrap method.