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 chance 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 the number of true null hypotheses. The familywise 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 determined 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 (not to be confused with the Dunn procedure for rank-based analysis of variance) can be used to adjust confidence intervals. If one establishes confidence intervals, and wishes to have an overall confidence level of , each individual confidence interval can be adjusted to the level of .
When searching for a signal in a continuous parameter space there can also be a problem of multiple comparisons, or look-elsewhere effect. For example, a physicist might be looking to discover a particle of unknown mass by considering a large range of masses; this was the case during the Nobel Prize winning detection of the Higgs boson. In such cases, one can apply a continuous generalization of the Bonferroni correction by employing Bayesian logic to relate the effective number of trials, , to the prior-to-posterior volume ratio.
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).
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. Such criticisms apply to FWER control in general, and are not specific to the Bonferroni correction.
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