Bonferroni correction

In statistics, the Bonferroni correction is a method used to counteract the problem of multiple comparisons. It is considered the simplest and most conservative method to control the familywise error rate.

It is named after Italian mathematician Carlo Emilio Bonferroni[1] for the use of Bonferroni inequalities, but modern usage is credited to Olive Jean Dunn, who first used it in a pair of articles written in 1959[2] and 1961.[3]

Informal introduction

Statistical inference logic is based on rejecting the null hypotheses if the likelihood of the observed data under the null hypotheses is low. The problem of multiplicity arises from the fact that as we increase the number of hypotheses in a test, we also increase the likelihood of witnessing a rare event, and therefore, the chance to reject the null hypotheses when it's true (type I error).

Bonferroni correction is the most naive way to address this issue. The correction is based on the idea that if an experimenter is testing n dependent or independent hypotheses on a set of data, then one way of maintaining the familywise error rate (FWER) is to test each individual hypothesis at a statistical significance level of 1/n times what it would be if only one hypothesis were tested.

So, if the desired significance level for the whole family of tests should be (at most) α, then the Bonferroni correction would test each individual hypothesis at a significance level of α/n. For example, if a trial is testing eight hypotheses with a desired α = 0.05, then the Bonferroni correction would test each individual hypothesis at α = 0.05/8 = 0.00625.

Statistically significant simply means that a given result is unlikely to have occurred by chance assuming the null hypothesis is actually correct (i.e., no difference among groups, no effect of treatment, no relation among variables).

Definition

[dubious ]

Let $H_{1},...,H_{m}$ be a family of hypotheses and $p_{1},...,p_{m}$ the corresponding p-values. Let $I_{0}$ be the subset of the (unknown) true null hypotheses, having $m_{0}$ members.

The familywise error rate is the probability of rejecting at least one of the members in $I_{0}$; that is, to make one or more type I error. The Bonferroni Correction states that choosing all $p_{i}\leq\frac{\alpha}{m}$ will control the $\mathit{FWER}\leq\alpha$. The proof follows from Boole's inequality: $\mathit{FWER}=\mathit{Pr}\left\{ \bigcup_{I_{o}}\left(p_{i}\leq\frac{\alpha}{m}\right)\right\} \leq\sum_{I_{o}}\left\{\mathit{Pr}\left(p_{i}\leq\frac{\alpha}{m}\right)\right\}\leq m_{0}\frac{\alpha}{m}\leq m\frac{\alpha}{m}=\alpha$

This result does not require that the tests be independent.

Modifications

Generalization

We have used the fact that $\sum_{i=1}^{n}\frac{\alpha}{n}=\alpha$, but the correction can be generalized and applied to any $\sum_{i=1}^{n}a_{i}=\alpha$, as long as the weights are defined prior to the test.

Confidence intervals

Bonferroni correction can be used to adjust confidence intervals. If we are forming $m$ confidence intervals, and wish to have overall confidence level of $1-\alpha$, then adjusting each individual confidence interval to the level of $1-\frac{\alpha}{m}$ will be the analog confidence interval correction.

Simultaneous inference and selective inference

Bonferroni correction is the basic type of simultaneous inference, that aims to control the familywise error rate. Significant statistical research was done in the field from early 1960s until late '90s, and many improvements were offered. Most notably are the Holm–Bonferroni method, which offers a uniformly more powerful test procedure (i.e., more powerful regardless of the values of the unobservable parameters), and the Hochberg (1988) method, guaranteed to be no less powerful and is in many cases more powerful when the tests are independent (and also under some forms of positive dependence).

In 1995 Benjamini and Hochberg suggested to control the false discovery rate instead of the familywise error rate and do selective inference corrections. This approach addresses the technological improvements that occurred at the end of the century, and provides the researcher with better tools to do large-scale inferences, which was considered one of the weak points of simultaneous inferences methodologies.

Alternatives

This list only include some of the alternatives that control the familywise error rate.

Holm–Bonferroni method

A uniformly more powerful test procedure (i.e., more powerful regardless of the values of the unobservable parameters) is the Holm–Bonferroni method.

Šidák correction

Main article: Šidák correction

A more powerful test procedure than the Bonferroni correction, which is suited when the individual tests are independent.

False discovery rate

Main article: False discovery rate

A less restrictive criterion that does not control the familywise error rate is the approximate false discovery rate that does require an ordering the p-values, then using different criteria for each test.

Criticisms

The Bonferroni correction can be somewhat conservative if there are a large number of tests and/or the test statistics are positively correlated. Bonferroni correction controls the probability of false positives only. The correction ordinarily comes at the cost of increasing the probability of producing false negatives, and consequently reducing statistical power. When testing a large number of hypotheses, this can result in large critical values.

Another criticism concerns the concept of a family of hypotheses. The statistical community has not reached a consensus on how to define such a family. Currently it is defined subjectively per test. As there is no standard definition, test results may change dramatically, only by modifying the way we consider the hypotheses families.

In addition, in certain situations where one wants to retain, not reject, the null hypothesis, then Bonferroni correction is non-conservative.