Bonferroni correction

In statistics, the Bonferroni correction is a method to counteract the problem of multiple comparisons. Bonferroni correction is the simplest method for counteracting the multiple comparisons problem; however, it is a conservative method that gives greater chance of failure to reject a false null hypothesis than other methods, as it ignores potentially valuable information, such as the distribution of p-values across all comparisons (which, if the null hypothesis is correct for all comparisons, is expected to take uniform distribution).

Of note, the Bonferroni correction controls for family-wise error rate (i.e. relating to the null hypothesis being true for all comparisons simultaneously; alternatively that the null hypothesis is false for at least one test). Alternative approaches, such as the Benjamini–Hochberg procedure, are explicitly designed to control error rate when detecting individual "discoveries", and are thus more appropriate in many scenarios, such as the detection of differentially expressed genes in bioinformatics.

Background

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 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 $\alpha /m$ , where $\alpha$ is the desired overall alpha level and $m$ is the number of hypotheses. For example, if a trial is testing $m=20$ hypotheses with a desired $\alpha =0.05$ , then the Bonferroni correction would test each individual hypothesis at $\alpha =0.05/20=0.0025$ . Likewise, when constructing multiple confidence intervals the same phenomenon appears.

Definition

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

${\text{FWER}}=P\left\{\bigcup _{i=1}^{m_{0}}\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}\leq \sum _{i=1}^{m_{0}}\left\{P\left(p_{i}\leq {\frac {\alpha }{m}}\right)\right\}=m_{0}{\frac {\alpha }{m}}\leq \alpha .$ This control does not require any assumptions about dependence among the p-values or about how many of the null hypotheses are true.

Extensions

Generalization

Rather than testing each hypothesis at the $\alpha /m$ level, the hypotheses may be tested at any other combination of levels that add up to $\alpha$ , provided that the level of each test is decided before looking at the data. For example, for two hypothesis tests, an overall $\alpha$ of 0.05 could be maintained by conducting one test at 0.04 and the other at 0.01.

Confidence intervals

The procedure proposed by Dunn can be used to adjust confidence intervals. If one establishes $m$ confidence intervals, and wishes to have an overall confidence level of $1-\alpha$ , each individual confidence interval can be adjusted to the level of $1-{\frac {\alpha }{m}}$ .

Continuous problems

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, $m$ , to the prior-to-posterior volume ratio.

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.[citation needed] Such criticisms apply to FWER control in general, and are not specific to the Bonferroni correction.