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 familywise 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 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.

Formulation

The method is as follows:

Let $H_{{1}},...,H_{{m}}$ be a family of $m$ null hypotheses and $P_{{1}},...,P_{{m}}$ the corresponding p-values.
Start by ordering the p-values (from lowest to highest) $P_{(1)} \ldots P_{(m)}$ and let the associated hypotheses be $H_{{(1)}}\ldots H_{{(m)}}$
For a given significance level $\alpha$ , let $k$ be the minimal index such that $P_{{(k)}}>{\frac {\alpha }{m+1-k}}$
Reject the null hypotheses $H_{{(1)}}\ldots H_{{(k-1)}}$ and do not reject $H_{{(k)}}\ldots H_{{(m)}}$
If $k=1$ then do not reject any of the null hypotheses and if no such $k$ exist then reject all of the null hypotheses.

The Holm–Bonferroni method ensures that this method will control the $FWER\leq \alpha$ , where $FWER$ is the familywise error rate

Rationale

The simple Bonferroni correction rejects only null hypotheses with p-value less than ${\frac {\alpha }{m}}$ , in order to ensure that the risk of rejecting one or more false 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 accepting one or more false null hypotheses (i.e., of committing one or more type II errors).

The Holm-Bonferroni method also controls the maximum familywise error rate 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 ${\frac {\alpha }{m}}$ to $\alpha$ (respectively), namely the values ${\frac {\alpha }{m}},{\frac {\alpha }{m-1}},...,{\frac {\alpha }{2}},{\frac {\alpha }{1}}$ .

The index $k$ identifies the first p-value that is not low enough to validate rejection. Therefore, the null hypotheses $H_{(1)},...,H_{(k-1)}$ are rejected, while the null hypotheses $H_{(k)},...,H_{(m)}$ are accepted (not rejected).
If $k=1$ then no p-values were low enough for rejection, therefore no null hypotheses are rejected (i.e., all null hypotheses are accepted).
If no such index $k$ could be found then all p-values were low enough for rejection, therefore all null hypotheses are rejected (none are accepted).

Proof

Holm-Bonferroni controls the FWER as follows. Let $H_{{(1)}}\ldots H_{{(m)}}$ be a family of hypotheses, and $P_{{(1)}}\leq P_{{(2)}}\leq \ldots \leq P_{{(m)}}$ be the sorted p-values. Let $I_{0}$ be the set of indices corresponding to the (unknown) true null hypotheses, having $m_{{0}}$ members.

Let us assume that we wrongly reject a true hypothesis. We have to prove that the probability of this event is at most $\alpha$ . Let $h$ be the first rejected true hypothesis (first in the ordering given by the Bonferroni–Holm test). Then $H_{(1)}\ldots H_{(h-1)}$ are all rejected false hypotheses and $h-1\leq m-m_{0}$ . From there, we get ${\frac {1}{m-h+1}}\leq {\frac {1}{m_{0}}}$ (1). Since $h$ is rejected we have $P_{(h)}\leq {\frac {\alpha }{m-h+1}}$ by definition of the test. Using (1), the right hand side is at most ${\frac {\alpha }{m_{0}}}$ . Thus, if we wrongly reject a true hypothesis, there has to be a true hypothesis with P-value at most ${\frac {\alpha }{m_{0}}}$ .

So let us define $A=\left\{P_{i}\leq {\frac {\alpha }{m_{{0}}}}{\text{ for some }}i\in I_{{0}}\right\}$ . Whatever the (unknown) set of true hypotheses $I_{0}$ is, we have $\Pr(A)\leq \alpha$ (by the Bonferroni inequalities). Therefore, the probability to reject a true hypothesis is at most $\alpha$ .

Alternative proof

The Holm–Bonferroni method can be viewed as closed testing procedure,^[2] 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 $m$ or less, while the number of all intersections of null hypotheses to be tested is of order $2^{m}$ .

The closure principle states that a hypothesis $H_{i}$ in a family of hypotheses $H_{1},...,H_{m}$ is rejected - while controlling the family-wise error rate of $\alpha$ - if and only if all the sub-families of the intersections with $H_{i}$ are controlled at level of family-wise error rate of $\alpha$ .

In Holm-Bonferroni procedure, we first test $H_{{(1)}}$ . If it is not rejected then the intersection of all null hypotheses $\bigcap \nolimits _{{i=1}}^{m}{{H_{i}}}$ is not rejected too, such that there exist at least one intersection hypothesis for each of elementary hypotheses $H_{1},...,H_{m}$ 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 the 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 $P_{{(2)}}\leq \alpha /(m-1)$ holds all the intersections that contains $H_{{(2)}}$ are rejected.

The same applies for each $1\leq i\leq m$ .

Example

Consider four null hypotheses $H_{1},...,H_{4}$ 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 $p_{4}=p_{{(1)}}=0.005$ . The p-value is compared to $\alpha /4=0.0125$ , the null hypothesis is rejected and we continue to the next one. Since $p_{1}=p_{{(2)}}=0.01<0.0167=\alpha /3$ we reject $H_{1}=H_{{(2)}}$ as well and continue. The next hypothesis $H_{3}$ is not rejected since $p_{3}=p_{{(3)}}=0.03>0.025=\alpha /2$ . 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 familywise error rate at level $\alpha =0.05$ . Note that even though $p_{2}=p_{{(4)}}=0.04<0.05=\alpha$ 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 ${\frac {\alpha }{m}},{\frac {\alpha }{m-1}},...,{\frac {\alpha }{1}}$ with:

1-(1-\alpha )^{{1/m}},1-(1-\alpha )^{{1/(m-1)}},...,1-(1-\alpha )^{{1}}

resulting in a slightly more powerful test.

Weighted version

Let $P_{{(1)}},...,P_{{(m)}}$ be the ordered unadjusted p-values. Let $H_{(i)}$ , $0\leq w_{{(i)}}$ correspond to $P_{{(i)}}$ . Reject $H_{(i)}$ as long as

P_{{(j)}}\leq {\frac {w_{{(j)}}}{\sum _{{k=j}}^{m}{w_{{(k)}}}}}\alpha ,\quad j=1,...,i

Adjusted p-values

The adjusted p-values for Holm–Bonferroni method are:

\widetilde {p}_{{(i)}}=\max _{{j\leq i}}\left\{(m-j+1)p_{{(j)}}\right\}_{{1}}

, where

\{x\}_{{1}}\equiv \min(x,1)

.

In the earlier example, the adjusted p-values are $\widetilde {p}_{1}=0.03$ , $\widetilde {p}_{2}=0.06$ , $\widetilde {p}_{3}=0.06$ and $\widetilde {p}_{4}=0.02$ . Only hypotheses $H_{1}$ and $H_{4}$ are rejected at level $\alpha =0.05$ .

The weighted adjusted p-values are:

\widetilde {p}_{{(i)}}=\max _{{j\leq i}}\left\{{\frac {\sum _{{k=j}}^{m}{w_{{(k)}}}}{w_{{(j)}}}}p_{{(j)}}\right\}_{{1}}

, where

\{x\}_{{1}}\equiv \min(x,1)

.

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 $H_{{(1)}}\ldots H_{{(k)}}$ is made after finding the maximal index $k$ such that $P_{{(k)}}\leq {\frac {\alpha }{m+1-k}}$ . 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

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

↑ Holm, S. (1979). "A simple sequentially rejective multiple test procedure". Scandinavian Journal of Statistics. 6 (2): 65&ndash, 70. JSTOR 4615733. MR 0538597.
↑ 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.
↑ 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. ISSN 0006-3444.

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[1] Holm, S. (1979). "A simple sequentially rejective multiple test procedure". Scandinavian Journal of Statistics. 6 (2): 65&ndash, 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.

[Hommel1988-3] 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. ISSN 0006-3444.