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Concept
Bonferroni correction
Summary
In statistics, the Bonferroni correction is a method to counteract the multiple comparisons problem.
The method is named for its use of the Bonferroni inequalities.
An extension of the method to confidence intervals was proposed by Olive Jean Dunn.
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 hypotheses are tested, the probability of observing 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 , where is the desired overall alpha level and is the number of hypotheses. For example, if a trial is testing hypotheses with a desired , then the Bonferroni correction would test each individual hypothesis at . Likewise, when constructing multiple confidence intervals the same phenomenon appears.
Let be a family of hypotheses and their corresponding p-values. Let be the total number of null hypotheses, and let be the number of true null hypotheses (which is presumably unknown to the researcher). The family-wise error rate (FWER) is the probability of rejecting at least one true , that is, of making at least one type I error. The Bonferroni correction rejects the null hypothesis for each , thereby controlling the FWER at . Proof of this control follows from Boole's inequality, as follows:
This control does not require any assumptions about dependence among the p-values or about how many of the null hypotheses are true.
Rather than testing each hypothesis at the level, the hypotheses may be tested at any other combination of levels that add up to , provided that the level of each test is decided before looking at the data. For example, for two hypothesis tests, an overall of 0.05 could be maintained by conducting one test at 0.04 and the other at 0.01.
The procedure proposed by Dunn can be used to adjust confidence intervals.
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