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 familywise error rate and offers a simple test uniformly more powerful than the Bonferroni correction. It is one of the earliest usages of stepwise algorithms in simultaneous inference. It is named after Sture Holm, who codified the method, and Carlo Emilio Bonferroni.
Contents
Motivation
When considering several hypotheses, the problem of multiplicity arises: the more hypotheses we check, the higher the probability of a Type I error (false positive). The Holm–Bonferroni method is one of many approaches that control the family-wise error rate (the probability that one or more Type I errors will occur) by adjusting the rejection criteria of each of the individual hypotheses or comparisons.
Formulation
The method is as follows:
The Holm–Bonferroni method ensures that this method will control the
Proof
Holm-Bonferroni controls the FWER as follows. Let
Let us assume that we wrongly reject a true hypothesis. We have to prove that the probability of this event is at most
So let us define
Alternative proof
The Holm–Bonferroni method can be viewed as closed testing procedure, with Bonferroni method applied locally on each of the intersections of null hypotheses. As such, it controls the familywise error rate for all the k hypotheses at level α in the strong sense. Each intersection is tested using the simple Bonferroni test.
It is a shortcut procedure since practically the number of comparisons to be made equal to
The closure principle states that a hypothesis
In Holm-Bonferroni procedure, we first test
If
The same rationale applies for
The same applies for each
Example
Consider four null hypotheses
Holm–Šidák method
When the hypothesis tests are not negatively dependent, it is possible to replace
resulting in a slightly more powerful test.
Weighted version
Let
Adjusted p-values
The adjusted p-values for Holm–Bonferroni method are:
In the earlier example, the adjusted p-values are
The weighted adjusted p-values are:
A hypothesis is rejected at level α if and only if its adjusted p-value is less than α. In the earlier example using equal weights, the adjusted p-values are 0.03, 0.06, 0.06, and 0.02. This is another way to see that using α = 0.05, only hypotheses one and four are rejected by this procedure.
Alternatives and usage
The Holm–Bonferroni method is uniformly more powerful than the classic Bonferroni correction. There are other methods for controlling the family-wise error rate that are more powerful than Holm-Bonferroni.
In the Hochberg procedure, rejection of
A similar step-up procedure is the Hommel procedure.
Naming
Carlo Emilio Bonferroni did not take part in inventing the method described here. Holm originally called the method the "sequentially rejective Bonferroni test", and it became known as Holm-Bonferroni only after some time. Holm's motives for naming his method after Bonferroni are explained in the original paper: "The use of the Boole inequality within multiple inference theory is usually called the Bonferroni technique, and for this reason we will call our test the sequentially rejective Bonferroni test."