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Concept
Holm–Bonferroni method
Summary
In statistics, the Holm–Bonferroni method, also called the Holm method or Bonferroni–Holm method, is used to counteract the problem of multiple comparisons. It is intended to control the family-wise error rate (FWER) and offers a simple test uniformly more powerful than the Bonferroni correction. It is named after Sture Holm, who codified the method, and Carlo Emilio Bonferroni.
When considering several hypotheses, the problem of multiplicity arises: the more hypotheses are checked, the higher the probability of obtaining Type I errors (false positives). The Holm–Bonferroni method is one of many approaches for controlling the FWER, i.e., the probability that one or more Type I errors will occur, by adjusting the rejection criteria for each of the individual hypotheses.
The method is as follows:
Suppose you have p-values, sorted into order lowest-to-highest , and their corresponding hypotheses (null hypotheses). You want the FWER to be no higher than a certain pre-specified significance level .
Is ? If so, reject and continue to the next step, otherwise EXIT.
Is ? If so, reject also, and continue to the next step, otherwise EXIT.
And so on: for each P value, test whether . If so, reject and continue to examine the larger P values, otherwise EXIT.
This method ensures that the FWER is at most , in the strong sense.
The simple Bonferroni correction rejects only null hypotheses with p-value less than , in order to ensure that the FWER, i.e., the risk of rejecting one or more true null hypotheses (i.e., of committing one or more type I errors) is at most . The cost of this protection against type I errors is an increased risk of failing to reject one or more false null hypotheses (i.e., of committing one or more type II errors).
The Holm–Bonferroni method also controls the FWER at , but with a lower increase of type II error risk than the classical Bonferroni method. The Holm–Bonferroni method sorts the p-values from lowest to highest and compares them to nominal alpha levels of to (respectively), namely the values .
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