Holm–Bonferroni method

In statistics, the Holm–Bonferroni method,[1] 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 edit

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.[citation needed]

Formulation edit

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.

Rationale edit

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  .

  • The index   identifies the first p-value that is not low enough to validate rejection. Therefore, the null hypotheses   are rejected, while the null hypotheses   are not rejected.
  • If   then no p-values were low enough for rejection, therefore no null hypotheses are rejected.
  • If no such index   could be found then all p-values were low enough for rejection, therefore all null hypotheses are rejected (none are accepted).

Proof edit

Let   be the family of hypotheses sorted by their p-values  . Let   be the set of indices corresponding to the (unknown) true null hypotheses, having   members.

Claim: If we wrongly reject some true hypothesis, there is a true hypothesis   for which   at most  .

First note that, in this case, there is at least one true hypothesis, so  . Let   be such that   is the first rejected true hypothesis. Then  are all rejected false hypotheses. It follows that   and, hence,   (1). Since   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  , since   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  .

Alternative proof edit

The Holm–Bonferroni method can be viewed as a closed testing procedure,[2] with the Bonferroni correction applied locally on each of the intersections of null hypotheses.

The closure principle states that a hypothesis   in a family of hypotheses   is rejected – while controlling the FWER at level   – if and only if all the sub-families of the intersections with   are rejected at level  .

The Holm-Bonferroni method is a shortcut procedure, since it makes   or less comparisons, while the number of all intersections of null hypotheses to be tested is of order  . It controls the FWER in the strong sense.

In the 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 exists 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 at 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 edit

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 edit

Holm–Šidák method edit

When the hypothesis tests are not negatively dependent, it is possible to replace   with:


resulting in a slightly more powerful test.

Weighted version edit

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


Adjusted p-values edit

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  .

Similar adjusted p-values for Holm-Šidák method can be defined recursively as  , where  . Due to the inequality   for  , the Holm-Šidák method will be more powerful than Holm-Bonferroni method.

The weighted adjusted p-values are:[citation needed]


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 edit

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   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.[3]

Naming edit

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."

References edit

  1. ^ Holm, S. (1979). "A simple sequentially rejective multiple test procedure". Scandinavian Journal of Statistics. 6 (2): 65–70. JSTOR 4615733. MR 0538597.
  2. ^ Marcus, R.; Peritz, E.; Gabriel, K. R. (1976). "On closed testing procedures with special reference to ordered analysis of variance". Biometrika. 63 (3): 655–660. doi:10.1093/biomet/63.3.655.
  3. ^ Hommel, G. (1988). "A stagewise rejective multiple test procedure based on a modified Bonferroni test". Biometrika. 75 (2): 383–386. doi:10.1093/biomet/75.2.383. hdl:2027.42/149272. ISSN 0006-3444.