Asymptotics of Bonferroni for dependent normal test statistics

The Bonferroni adjustment is sometimes used to control the familywise error rate (FWE) when the number of comparisons is huge. In genome wide association studies, researchers compare cases to controls with respect to thousands of single nucleotide polymorphisms. It has been claimed that the Bonferroni adjustment is only slightly conservative if the comparisons are nearly independent. We show that the veracity of this claim depends on how one defines “nearly”. Specifically, if the test statistics’ pairwise correlations converge to 0 as the number of tests tend to ∞, the conservatism of the Bonferroni procedure depends on their rate of convergence. The type I error rate of Bonferroni can tend to 0 or 1−exp(−α)≈α, depending on that rate. We show using elementary probability theory what happens to the distribution of the number of errors when using Bonferroni, as the number of dependent normal test statistics gets large. We also use the limiting behavior of Bonferroni to shed light on properties of other commonly used test statistics.