In statistics, the Bonferroni correction is one of several methods used to counteract the problem of multiple comparisons.
Contents
Background
The Bonferroni correction is named after Italian mathematician Carlo Emilio Bonferroni for its use of Bonferroni inequalities, but modern usage is often credited to Olive Jean Dunn, who described the procedure's application to confidence intervals.
Statistical hypothesis testing is based on rejecting the null hypothesis if the likelihood of the observed data under the null hypotheses is low. If multiple comparisons are done or multiple hypotheses are tested, the chance of a rare event increases, and therefore, the likelihood of incorrectly rejecting a null hypothesis (i.e., making a Type I error) increases.
The Bonferroni correction compensates for that increase by testing each individual hypothesis at a significance level of
Definition
Let
This control does not require any assumptions about dependence among the p-values or about how many of the null hypotheses are true.
Generalization
Rather than testing each hypothesis at the
Confidence intervals
The Bonferroni correction can be used to adjust confidence intervals. If one establishes
Alternatives
There are alternative ways to control the familywise error rate. For example, the Holm–Bonferroni method and the Šidák correction are universally more powerful procedures than the Bonferroni correction, meaning that they are always at least as powerful. Unlike the Bonferroni procedure, these methods do not control the expected number of Type I errors per family (the per-family Type I error rate).
Criticism
With respect to FWER control, the Bonferroni correction can be conservative if there are a large number of tests and/or the test statistics are positively correlated.
The correction comes at the cost of increasing the probability of producing false negatives, i.e., reducing statistical power.
There is not a definitive consensus on how to define a family in all cases, and adjusted test results may vary depending on the number of tests included in the family of hypotheses.
Note that these criticisms apply to FWER control in general, and are not specific to the Bonferroni correction.