Bonferroni Procedure

The Bonferroni procedure is a statistical adjustment to the significance level of hypothesis tests when multiple tests are being performed. The purpose of an adjustment such as the Bonferroni procedure is to reduce the probability of identifying significant results that do not exist, that is, to guard against making Type I errors (rejecting null hypotheses when they are true) in the testing process. This potential for error increases with an increase in the number of tests being performed in a given study and is due to the multiplication of probabilities across the multiple tests. The Bonferroni procedure is often used as an adjustment in multiple comparisons after a significant finding in an analysis of variance (ANOVA) or when constructing simultaneous confidence intervals for several population parameters, but more broadly, it can be used in any situation that involves multiple tests. The Bonferroni procedure is one of the more commonly used procedures in multiple testing situations, primarily because it is an easy adjustment to make. A strength of the Bonferroni procedure is its ability to maintain Type I error rates at or below a nominal value. A weakness of the Bonferroni procedure is that it often overcorrects, making testing results too conservative because of a decrease in statistical power.

A variety of other procedures have been developed to control the overall Type I error level when multiple tests are performed. Some of these other multiple comparison and multiple testing procedures, including the Student-Newman-Keuls procedure, are derivatives of the Bonferroni procedure, modified to make the procedure less conservative without sacrificing Type I error control. Other multiple comparison and multiple testing procedures are simulation based and are not directly related to the Bonferroni procedure.

This entry describes the procedure's background, explains the procedure, and provides an example. This entry also presents applications for the procedure and examines recent research.

Explanation

The Bonferroni procedure is an application of the Bonferroni inequality to the probabilities associated with multiple testing. It prescribes using an adjustment to the significance level for individual tests when simultaneous statistical inference for several tests is being performed. The adjustment can be used for bounding simultaneous confidence intervals, as well as for simultaneous testing of hypotheses.

The Bonferroni inequality states the following:

1. Let Ai, i = 1 to k, represent k events. Then,

where Āi is the complement of the event Ai.

2. Consider the mechanics of the Bonferroni inequality,

3. and rewrite the inequality as follows:

Now, consider Ā as a Type I error in the zth test in a collection of k hypothesis tests. Then

represents the probability that no Type I errors occur in the k hypothesis tests, and

represents the probability that at least one Type I error occurs in the k hypothesis tests. P(Āi) represents the probability of a Type I error in the zth test, and we can label this probability as αi = P (Āi). So Bonferroni's inequality implies that the probability of at least one Type I error occurring in k hypothesis tests is

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