Control of the mean number of false discoveries, Bonferroni and stability of multiple testing

TL;DR

This paper challenges the belief that Bonferroni is overly conservative, demonstrating its stability and effectiveness in controlling false discoveries and maintaining stability in large-scale testing, especially with correlated data.

Contribution

It shows that Bonferroni controls the per family error rate effectively and exhibits superior stability compared to other procedures, particularly in correlated data scenarios.

Findings

Bonferroni controls the PFER effectively.

Bonferroni exhibits superior stability in discoveries.

Stability is maintained even with correlated p-values.

Abstract

The Bonferroni multiple testing procedure is commonly perceived as being overly conservative in large-scale simultaneous testing situations such as those that arise in microarray data analysis. The objective of the present study is to show that this popular belief is due to overly stringent requirements that are typically imposed on the procedure rather than to its conservative nature. To get over its notorious conservatism, we advocate using the Bonferroni selection rule as a procedure that controls the per family error rate (PFER). The present paper reports the first study of stability properties of the Bonferroni and Benjamini--Hochberg procedures. The Bonferroni procedure shows a superior stability in terms of the variance of both the number of true discoveries and the total number of discoveries, a property that is especially important in the presence of correlations between individual $p$-values. Its stability and the ability to provide strong control of the PFER make the Bonferroni procedure an attractive choice in microarray studies.