Holm–Bonferroni method explained

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

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:

m

p-values, sorted into order lowest-to-highest

P1,\ldots,Pm

, and their corresponding hypotheses

H1,\ldots,Hm

(null hypotheses). You want the FWER to be no higher than a certain pre-specified significance level

\alpha

.

P1\leq\alpha/m

? If so, reject

H1

and continue to the next step, otherwise EXIT.

P2\leq\alpha/(m-1)

? If so, reject

H2

also, and continue to the next step, otherwise EXIT.

Pk\leq

\alpha
m+1-k
. If so, reject

Hk

and continue to examine the larger P values, otherwise EXIT.

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

\alpha
m
, 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
\alpha
m
to

\alpha

(respectively), namely the values
\alpha
m

,

\alpha
m-1

,\ldots,

\alpha
2

,

\alpha
1
.

k

identifies the first p-value that is not low enough to validate rejection. Therefore, the null hypotheses

H(1),\ldots,H(k-1)

are rejected, while the null hypotheses

H(k),...,H(m)

are not rejected.

k=1

then no p-values were low enough for rejection, therefore no null hypotheses are rejected.

k

could be found then all p-values were low enough for rejection, therefore all null hypotheses are rejected (none are accepted).

Proof

Let

H(1)\ldotsH(m)

be the family of hypotheses sorted by their p-values

P(1)\leqP(2)\leq\leqP(m)

. Let

I0

be the set of indices corresponding to the (unknown) true null hypotheses, having

m0

members.

Claim: If we wrongly reject some true hypothesis, there is a true hypothesis

H(\ell)

for which

P(\ell)

at most
\alpha
m0
.

First note that, in this case, there is at least one true hypothesis, so

m0\geq1

. Let

\ell

be such that

H(\ell)

is the first rejected true hypothesis. Then

H(1),\ldots,H(\ell-1)

are all rejected false hypotheses. It follows that

\ell-1\leqm-m0

and, hence,
1
m-\ell+1

\leq

1
m0
(1). Since

H(\ell)

is rejected, it must be

P(\ell)\leq

\alpha
m-\ell+1
by definition of the testing procedure. Using (1), we conclude that

P(\ell)\leq

\alpha
m0
, as desired.

So let us define the random event

A=

cup
i\inI0

\left\{Pi\leq

\alpha
m0

\right\}

. Note that, for

i\inIo

, since

Hi

is a true null hypothesis, we have that

P\left(\left\{Pi\leq

\alpha
m0

\right\}\right)=

\alpha
m0
. Subadditivity of the probability measure implies that
\Pr(A)\leq\sum
i\inI0

P\left(\left\{Pi\leq

\alpha
m0

\right\}\right)=

\sum
i\inI0
\alpha
m0

=\alpha

. 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,[2] with the Bonferroni correction applied locally on each of the intersections of null hypotheses.

The closure principle states that a hypothesis

Hi

in a family of hypotheses

H1,\ldots,Hm

is rejected – while controlling the FWER at level

\alpha

– if and only if all the sub-families of the intersections with

Hi

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

2m

.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
m
cap\nolimits
i=1

Hi

is not rejected too, such that there exists at least one intersection hypothesis for each of elementary hypotheses

H1,\ldots,Hm

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

P(2)\leq\alpha/(m-1)

holds all the intersections that contains

H(2)

are rejected.

The same applies for each

1\leqi\leqm

.

Example

Consider four null hypotheses

H1,\ldots,H4

with unadjusted p-values

p1=0.01

,

p2=0.04

,

p3=0.03

and

p4=0.005

, to be tested at significance level

\alpha=0.05

. Since the procedure is step-down, we first test

H4=H(1)

, which has the smallest p-value

p4=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

p1=p(2)=0.01<0.0167=\alpha/3

we reject

H1=H(2)

as well and continue. The next hypothesis

H3

is not rejected since

p3=p(3)=0.03>0.025=\alpha/2

. We stop testing and conclude that

H1

and

H4

are rejected and

H2

and

H3

are not rejected while controlling the family-wise error rate at level

\alpha=0.05

. Note that even though

p2=p(4)=0.04<0.05=\alpha

applies,

H2

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

\alpha,
m
\alpha
m-1

,\ldots,

\alpha
1
with:

1-(1-\alpha)1/m,1-(1-\alpha)1/(m-1),\ldots,1-(1-\alpha)1

resulting in a slightly more powerful test.

Weighted version

Let

P(1),\ldots,P(m)

be the ordered unadjusted p-values. Let

H(i)

,

0\leqw(i)

correspond to

P(i)

. Reject

H(i)

as long as

P(j)\leq

w(j)
m
\sumw(k)
k=j

\alpha,j=1,\ldots,i

Adjusted p-values

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

\widetilde{p}(i)=maxj\leq\left\{(m-j+1)p(j)\right\}1,where\{x\}1\equivmin(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

H1

and

H4

are rejected at level

\alpha=0.05

.

Similar adjusted p-values for Holm-Šidák method can be defined recursively as

\widetilde{p}(i)=max\left\{\widetilde{p}(i-1),1-(1-p(i))m-i+1\right\}

, where

\widetilde{p}(1)=1-(1-p(1))m

. Due to the inequality

1-(1-\alpha)1/n<\alpha/n

for

n\geq2

, the Holm-Šidák method will be more powerful than Holm–Bonferroni method.

The weighted adjusted p-values are:

\widetilde{p}(i)=maxj\leq\left\{

m
\sum{w(k)
k=j
} p_\right\}_1, \text \_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 FWER that are more powerful than Holm–Bonferroni. For instance, in the Hochberg procedure, rejection of

H(1)\ldotsH(k)

is made after finding the maximal index

k

such that

P(k)\leq

\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

  1. Holm . S.. 1979. A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics. 6 . 2 . 65 - 70. 538597 . 4615733.
  2. Marcus . R. . Peritz . E. . Gabriel . K. R. . 1976 . On closed testing procedures with special reference to ordered analysis of variance . . 63 . 3 . 655–660 . 10.1093/biomet/63.3.655 .
  3. Hommel. G.. A stagewise rejective multiple test procedure based on a modified Bonferroni test. Biometrika. 75. 2. 1988. 383–386. 0006-3444. 10.1093/biomet/75.2.383. 2027.42/149272. free.