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.

Motivation

When considering several hypotheses, the problem of multiplicity arises: the more hypotheses are tested, 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 criterion for each of the individual hypotheses.

Formulation

The method is as follows: This method ensures that the FWER is at most \alpha, in the strong sense.

Rationale

The simple Bonferroni correction rejects only null hypotheses with p-value less than or equal to, 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 \alpha. 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 \alpha, 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 \alpha (respectively), namely the values.

Proof

Let be the family of hypotheses sorted by their p-values. Let I_0 be the set of indices corresponding to the (unknown) true null hypotheses, having m_0 members. Claim: If we wrongly reject some true hypothesis, there is a true hypothesis H_{(\ell)} for which P_{(\ell)} at most. First note that, in this case, there is at least one true hypothesis, so m_0 \geq 1. Let \ell be such that H_{(\ell)} is the first rejected true hypothesis. Then are all rejected false hypotheses. It follows that and, hence, (1). Since H_{(\ell)} is rejected, it must be by definition of the testing procedure. Using (1), we conclude that, as desired. So let us define the random event. Note that, for i \in I_o, since H_i is a true null hypothesis, we have that. Subadditivity of the probability measure implies that. Therefore, the probability to reject a true hypothesis is at most \alpha.

Alternative proof

The Holm–Bonferroni method can be viewed as a closed testing procedure, with the Bonferroni correction applied locally on each of the intersections of null hypotheses. The closure principle states that a hypothesis H_i in a family of hypotheses is rejected – while controlling the FWER at level \alpha – if and only if all the sub-families of the intersections with H_i are rejected at level \alpha. The Holm–Bonferroni method is a shortcut procedure, since it makes m or less comparisons, while the number of all intersections of null hypotheses to be tested is of order 2^m. It controls the FWER in the strong sense. In the Holm–Bonferroni procedure, we first test H_{(1)}. If it is not rejected then the intersection of all null hypotheses is not rejected too, such that there exists at least one intersection hypothesis for each of elementary hypotheses that is not rejected, thus we reject none of the elementary hypotheses. If H_{(1)} is rejected at level \alpha/m then all the intersection sub-families that contain it are rejected too, thus H_{(1)} is rejected. This is because P_{(1)} is the smallest in each one of the intersection sub-families and the size of the sub-families is at most m, such that the Bonferroni threshold larger than \alpha/m. The same rationale applies for H_{(2)}. However, since H_{(1)} already rejected, it sufficient to reject all the intersection sub-families of H_{(2)} without H_{(1)}. Once holds all the intersections that contains H_{(2)} are rejected. The same applies for each.

Example

Consider four null hypotheses with unadjusted p-values p_1=0.01, p_2=0.04, p_3=0.03 and p_4=0.005, to be tested at significance level \alpha=0.05. Since the procedure is step-down, we first test H_4=H_{(1)}, 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 H_1=H_{(2)} as well and continue. The next hypothesis H_3 is not rejected since. We stop testing and conclude that H_1 and H_4 are rejected and H_2 and H_3 are not rejected while controlling the family-wise error rate at level \alpha=0.05. Note that even though applies, H_2 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 H_{(i)}, correspond to P_{(i)}. Reject H_{(i)} 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 H_1 and H_4 are rejected at level \alpha=0.05. Similar adjusted p-values for Holm-Šidák method can be defined recursively as, where. Due to the inequality for n \geq 2, the Holm-Šidák method will be more powerful than Holm–Bonferroni method. 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 FWER that are more powerful than Holm–Bonferroni. For instance, in the Hochberg procedure, rejection of is made after finding the maximal index k 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."