Bonferroni correction

The method is named for its use of the Bonferroni inequalities.[1] An extension of the method to confidence intervals was proposed by Olive Jean Dunn.[2]

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.[3]

The Bonferroni correction compensates for that increase by testing each individual hypothesis at a significance level of ${\displaystyle \alpha /m}$, where ${\displaystyle \alpha }$ is the desired overall alpha level and ${\displaystyle m}$ is the number of hypotheses.[4] For example, if a trial is testing ${\displaystyle m=20}$ hypotheses with a desired ${\displaystyle \alpha =0.05}$, then the Bonferroni correction would test each individual hypothesis at ${\displaystyle \alpha =0.05/20=0.0025}$. Likewise, when constructing multiple confidence intervals the same phenomenon appears.

Let ${\displaystyle H_{1},\ldots ,H_{m}}$ be a family of hypotheses and ${\displaystyle p_{1},\ldots ,p_{m}}$ their corresponding p-values. Let ${\displaystyle m}$ be the total number of null hypotheses, and let ${\displaystyle m_{0}}$ 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 ${\displaystyle H_{i}}$, that is, of making at least one type I error. The Bonferroni correction rejects the null hypothesis for each ${\displaystyle p_{i}\leq {\frac {\alpha }{m}}}$, thereby controlling the FWER at ${\displaystyle \leq \alpha }$. Proof of this control follows from Boole's inequality, as follows:

${\displaystyle {\text{FWER}}=P\left\{\bigcup _{i=1}^{m_{0}}\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq \sum _{i=1}^{m_{0}}\left\{P\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}=m_{0}{\frac {\alpha }{m}}\leq \alpha .}$

This control does not require any assumptions about dependence among the p-values or about how many of the null hypotheses are true.[5]

There are alternative ways to control the family-wise 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).[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]

The correction comes at the cost of increasing the probability of producing false negatives, i.e., reducing statistical power.[10][9] 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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