Bonferroni correction

The Bonferroni correction is a simple method for counteracting the multiple comparisons problem in statistics, named after the mathematician Carlo Emilio Bonferroni. When many null hypotheses are tested at once, the chance of wrongly rejecting at least one true hypothesis grows, so the correction tests each individual hypothesis at a significance level of alpha divided by m, where alpha is the desired overall level and m is the number of hypotheses. With twenty hypotheses and an overall level of 0.05, each test is judged at 0.0025. The same decisions result if each p-value is instead multiplied by the number of tests, capped at 1, and compared with the unchanged alpha. The procedure rests on Boole's inequality and controls the family-wise error rate at or below alpha without any assumption about dependence among the p-values or about how many null hypotheses are true. It extends to unequal splits of alpha fixed before looking at the data and to simultaneous confidence intervals. It is conservative when there are many tests or the statistics are positively correlated, and like other corrections it lowers statistical power. The Holm-Bonferroni method and the Sidak correction are uniformly more powerful alternatives.