In statistics, the Bonferroni correction is a method to counteract the multiple comparisons problem.
[1] Application of the method to confidence intervals was described by Olive Jean Dunn.
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.
[3] The Bonferroni correction compensates for that increase by testing each individual hypothesis at a significance level of
, then the Bonferroni correction would test each individual hypothesis at
The Bonferroni correction can also be applied as a p-value adjustment: Using that approach, instead of adjusting the alpha level, each p-value is multiplied by the number of tests (with adjusted p-values that exceed 1 then being reduced to 1), and the alpha level is left unchanged.
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
The Bonferroni correction rejects the null hypothesis for each
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.
, provided that the level of each test is decided before looking at the data.
The procedure proposed by Dunn[2] can be used to adjust confidence intervals.
, each individual confidence interval can be adjusted to the level of
[2] 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,
[7] There are alternative ways to control the family-wise error rate.
But 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).
[8] 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.
[9] Multiple-testing corrections, including the Bonferroni procedure, increase the probability of Type II errors when null hypotheses are false, i.e., they reduce statistical power.