Holm–Bonferroni method


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 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.

Motivation

When considering several hypotheses, the problem of multiplicity arises: the more hypotheses are checked, the higher the probability of obtaining Type I errors. The Holm–Bonferroni method is one of many approaches for controlling the family-wise error rate by adjusting the rejection criteria for each of the individual hypotheses.

Formulation

The method is as follows:
This method ensures that, where FWER is the family-wise error rate.

Rationale

The simple Bonferroni correction rejects only null hypotheses with p-value less than, in order to ensure that the risk of rejecting one or more true null hypotheses 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.
The Holm–Bonferroni method also controls the maximum family-wise error rate 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 , namely the values.
Holm–Bonferroni controls the FWER as follows. Let be a family of hypotheses, and be the sorted p-values. Let be the set of indices corresponding to the true null hypotheses, having members.
Let us assume that we wrongly reject a true hypothesis. We have to prove that the probability of this event is at most. Let be the first rejected true hypothesis. Then are all rejected false hypotheses and. From there, we get . Since is rejected we have by definition of the test. Using, the right hand side is at most. Thus, if we wrongly reject a true hypothesis, there has to be a true hypothesis with P-value at most.
So let us define the random variable. Whatever the set of true hypotheses is, we have . Therefore, the probability to reject a true hypothesis is at most.

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 family-wise 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 or less, while the number of all intersections of null hypotheses to be tested is of order.
The closure principle states that a hypothesis in a family of hypotheses is rejected – while controlling the family-wise error rate of – if and only if all the sub-families of the intersections with are controlled at level of family-wise error rate of.
In Holm–Bonferroni procedure, we first test. If it is not rejected then the intersection of all null hypotheses is not rejected too, such that there exist at least one intersection hypothesis for each of elementary hypotheses that is not rejected, thus we reject none of the elementary hypotheses.
If is rejected at level then all the intersection sub-families that contain it are rejected too, thus is rejected.
This is because is the smallest in each one of the intersection sub-families and the size of the sub-families is the most, such that the Bonferroni threshold larger than.
The same rationale applies for. However, since already rejected, it sufficient to reject all the intersection sub-families of without. Once holds all the intersections that contains are rejected.
The same applies for each.

Example

Consider four null hypotheses with unadjusted p-values,, and, to be tested at significance level. Since the procedure is step-down, we first test, which has the smallest p-value. The p-value is compared to, the null hypothesis is rejected and we continue to the next one. Since we reject as well and continue. The next hypothesis is not rejected since. We stop testing and conclude that and are rejected and and are not rejected while controlling the family-wise error rate at level. Note that even though applies, is not rejected. This is because the testing procedure stops once a failure to reject occurs.

Extensions

Holm–Šidák method

When the hypothesis tests are not negatively dependent, it is possible to replace with:
resulting in a slightly more powerful test.

Weighted version

Let be the ordered unadjusted p-values. Let, correspond to. Reject as long as

Adjusted ''p''-values

The adjusted p-values for Holm–Bonferroni method are:
In the earlier example, the adjusted p-values are,, and. Only hypotheses and are rejected at level.
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, meaning that it is always at least as powerful.
There are other methods for controlling the family-wise error rate that are more powerful than Holm–Bonferroni. For instance, in the Hochberg procedure, rejection of is made after finding the maximal index such that. Thus, The Hochberg procedure is uniformly more powerful than the Holm procedure. However, the Hochberg procedure requires the hypotheses to be independent or under certain forms of positive dependence, whereas Holm–Bonferroni can be applied without such assumptions. A similar step-up procedure is the Hommel procedure, which is uniformly more powerful than the Hochberg 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."