Asymptotics of Bonferroni for Dependent Normal Test Statistics

Area of Science:

• Statistical genetics
• Genomics
• Biostatistics

Background:

• The Bonferroni adjustment is frequently used to control the familywise error rate (FWE) in large-scale studies.
• Genome-wide association studies (GWAS) involve numerous statistical comparisons, making Bonferroni adjustment a common, yet debated, method.

Purpose of the Study:

• To investigate the conservatism of the Bonferroni adjustment under varying degrees of dependence between test statistics.
• To clarify the conditions under which the Bonferroni adjustment maintains its error rate control in large-scale hypothesis testing.

Main Methods:

• Analysis of the type I error rate distribution of the Bonferroni procedure for large numbers of dependent normal test statistics.
• Application of elementary probability theory to model the behavior of errors under Bonferroni correction.
• Examination of the rate of convergence of pairwise correlations of test statistics to zero.

Main Results:

• The conservatism of the Bonferroni adjustment is critically dependent on the rate at which test statistic correlations converge to zero.
• The type I error rate can approach zero or the target alpha level (1 - exp(-α)) based on this convergence rate.
• The study provides a theoretical framework for understanding Bonferroni's performance with dependent data.

Conclusions:

• The claim that Bonferroni is only slightly conservative for nearly independent tests requires precise definition of "nearly independent."
• The rate of correlation convergence significantly impacts the actual familywise error rate achieved by Bonferroni.
• Findings offer insights into the behavior of other statistical tests in the context of dependent data.