Bonferroni Correction

Abstract

In statistics, the Bonferroni correction (perhaps more accurately described as the Dunn-Bonferroni correction) is a method used to address the problem of multiple comparisons. It was developed and introduced by Olive Jean Dunn. The correction is based on the idea that if an experimenter is testing n dependent or independent hypotheses on a set of data, then one way of maintaining the familywise error rate is to test each individual hypothesis at a statistical significance level of 1/n times what it would be if only one hypothesis were tested. So if one wants the significance level for the whole family of tests to be (at most) α, then the Bonferroni correction would be to test each of the individual tests at a significance level of α/n. Statistically significant simply means that a given result is unlikely to have occurred by chance assuming the null hypothesis is actually correct (i.e., no difference among groups, no effect of treatment, no relation among variables). The Dunn-Bonferroni correction is derived by observing Boole's inequality. If you perform n tests, each of them significant with probability β, (where β is unknown) then the probability that at least one of them comes out significant is (by Boole's inequality) ≤ n⋅β. Now we want this probability to equal α, the significance level for the entire series of tests. By solving for β, we get β = α/n. This result does not require that the tests be independent.