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

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

The method is as follows:

• Suppose you have ${\displaystyle m}$ p-values, sorted into order lowest-to-highest ${\displaystyle P_{1},\ldots ,P_{m}}$, and their corresponding hypotheses ${\displaystyle H_{1},\ldots ,H_{m}}$(null hypotheses). You want the FWER to be no higher than a certain pre-specified significance level ${\displaystyle \alpha }$.
• Is ${\displaystyle P_{1}\leq \alpha /m}$? If so, reject ${\displaystyle H_{1}}$ and continue to the next step, otherwise EXIT.
• Is ${\displaystyle P_{2}\leq \alpha /(m-1)}$? If so, reject ${\displaystyle H_{2}}$ also, and continue to the next step, otherwise EXIT.
• And so on: for each P value, test whether ${\displaystyle P_{k}\leq {\frac {\alpha }{m+1-k}}}$. If so, reject ${\displaystyle H_{k}}$ and continue to examine the larger P values, otherwise EXIT.

This method ensures that the FWER is at most ${\displaystyle \alpha }$, in the strong sense.

Rationale

The simple Bonferroni correction rejects only null hypotheses with p-value less than or equal to ${\displaystyle {\frac {\alpha }{m}}}$, 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 ${\displaystyle \alpha }$. 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 ${\displaystyle \alpha }$, 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 ${\displaystyle {\frac {\alpha }{m}}}$ to ${\displaystyle \alpha }$ (respectively), namely the values ${\displaystyle {\frac {\alpha }{m}},{\frac {\alpha }{m-1}},\ldots ,{\frac {\alpha }{2}},{\frac {\alpha }{1}}}$.

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

Proof

Let ${\displaystyle H_{(1)}\ldots H_{(m)}}$ be the family of hypotheses sorted by their p-values ${\displaystyle P_{(1)}\leq P_{(2)}\leq \cdots \leq P_{(m)}}$. Let ${\displaystyle I_{0}}$ be the set of indices corresponding to the (unknown) true null hypotheses, having ${\displaystyle m_{0}}$ members.

Claim: If we wrongly reject some true hypothesis, there is a true hypothesis ${\displaystyle H_{(\ell )}}$ for which ${\displaystyle P_{(\ell )}}$ at most ${\displaystyle {\frac {\alpha }{m_{0}}}}$.

First note that, in this case, there is at least one true hypothesis, so ${\displaystyle m_{0}\geq 1}$. Let ${\displaystyle \ell }$ be such that ${\displaystyle H_{(\ell )}}$ is the first rejected true hypothesis. Then ${\displaystyle H_{(1)},\ldots ,H_{(\ell -1)}}$are all rejected false hypotheses. It follows that ${\displaystyle \ell -1\leq m-m_{0}}$ and, hence, ${\displaystyle {\frac {1}{m-\ell +1}}\leq {\frac {1}{m_{0}}}}$ (1). Since ${\displaystyle H_{(\ell )}}$ is rejected, it must be ${\displaystyle P_{(\ell )}\leq {\frac {\alpha }{m-\ell +1}}}$ by definition of the testing procedure. Using (1), we conclude that ${\displaystyle P_{(\ell )}\leq {\frac {\alpha }{m_{0}}}}$, as desired.

So let us define the random event ${\displaystyle A=\bigcup _{i\in I_{0}}\left\{P_{i}\leq {\frac {\alpha }{m_{0}}}\right\}}$. Note that, for ${\displaystyle i\in I_{o}}$, since ${\displaystyle H_{i}}$ is a true null hypothesis, we have that ${\displaystyle P\left(\left\{P_{i}\leq {\frac {\alpha }{m_{0}}}\right\}\right)={\frac {\alpha }{m_{0}}}}$. Subadditivity of the probability measure implies that ${\displaystyle \Pr(A)\leq \sum _{i\in I_{0}}P\left(\left\{P_{i}\leq {\frac {\alpha }{m_{0}}}\right\}\right)=\sum _{i\in I_{0}}{\frac {\alpha }{m_{0}}}=\alpha }$. Therefore, the probability to reject a true hypothesis is at most ${\displaystyle \alpha }$.

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 ${\displaystyle H_{i}}$ in a family of hypotheses ${\displaystyle H_{1},\ldots ,H_{m}}$ is rejected – while controlling the FWER at level ${\displaystyle \alpha }$ – if and only if all the sub-families of the intersections with ${\displaystyle H_{i}}$ are rejected at level ${\displaystyle \alpha }$.

The Holm–Bonferroni method is a shortcut procedure, since it makes ${\displaystyle m}$ or less comparisons, while the number of all intersections of null hypotheses to be tested is of order ${\displaystyle 2^{m}}$. It controls the FWER in the strong sense.

In the Holm–Bonferroni procedure, we first test ${\displaystyle H_{(1)}}$. If it is not rejected then the intersection of all null hypotheses ${\displaystyle \bigcap \nolimits _{i=1}^{m}H_{i}}$ is not rejected too, such that there exists at least one intersection hypothesis for each of elementary hypotheses ${\displaystyle H_{1},\ldots ,H_{m}}$ that is not rejected, thus we reject none of the elementary hypotheses.

If ${\displaystyle H_{(1)}}$ is rejected at level ${\displaystyle \alpha /m}$ then all the intersection sub-families that contain it are rejected too, thus ${\displaystyle H_{(1)}}$ is rejected. This is because ${\displaystyle P_{(1)}}$ is the smallest in each one of the intersection sub-families and the size of the sub-families is at most ${\displaystyle m}$, such that the Bonferroni threshold larger than ${\displaystyle \alpha /m}$.

The same rationale applies for ${\displaystyle H_{(2)}}$. However, since ${\displaystyle H_{(1)}}$ already rejected, it sufficient to reject all the intersection sub-families of ${\displaystyle H_{(2)}}$ without ${\displaystyle H_{(1)}}$. Once ${\displaystyle P_{(2)}\leq \alpha /(m-1)}$ holds all the intersections that contains ${\displaystyle H_{(2)}}$ are rejected.

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

Example

Consider four null hypotheses ${\displaystyle H_{1},\ldots ,H_{4}}$ with unadjusted p-values ${\displaystyle p_{1}=0.01}$, ${\displaystyle p_{2}=0.04}$, ${\displaystyle p_{3}=0.03}$ and ${\displaystyle p_{4}=0.005}$, to be tested at significance level ${\displaystyle \alpha =0.05}$. Since the procedure is step-down, we first test ${\displaystyle H_{4}=H_{(1)}}$, which has the smallest p-value ${\displaystyle p_{4}=p_{(1)}=0.005}$. The p-value is compared to ${\displaystyle \alpha /4=0.0125}$, the null hypothesis is rejected and we continue to the next one. Since ${\displaystyle p_{1}=p_{(2)}=0.01<0.0167=\alpha /3}$ we reject ${\displaystyle H_{1}=H_{(2)}}$ as well and continue. The next hypothesis ${\displaystyle H_{3}}$ is not rejected since ${\displaystyle p_{3}=p_{(3)}=0.03>0.025=\alpha /2}$. We stop testing and conclude that ${\displaystyle H_{1}}$ and ${\displaystyle H_{4}}$ are rejected and ${\displaystyle H_{2}}$ and ${\displaystyle H_{3}}$ are not rejected while controlling the family-wise error rate at level ${\displaystyle \alpha =0.05}$. Note that even though ${\displaystyle p_{2}=p_{(4)}=0.04<0.05=\alpha }$ applies, ${\displaystyle H_{2}}$ 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 ${\displaystyle {\frac {\alpha }{m}},{\frac {\alpha }{m-1}},\ldots ,{\frac {\alpha }{1}}}$ with:

${\displaystyle 1-(1-\alpha )^{1/m},1-(1-\alpha )^{1/(m-1)},\ldots ,1-(1-\alpha )^{1}}$

resulting in a slightly more powerful test.

Weighted version

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

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

Adjusted p-values

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

${\displaystyle {\widetilde {p}}_{(i)}=\max _{j\leq i}\left\{(m-j+1)p_{(j)}\right\}_{1},{\text{ where }}\{x\}_{1}\equiv \min(x,1).}$

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

Similar adjusted p-values for Holm-Šidák method can be defined recursively as ${\displaystyle {\widetilde {p}}_{(i)}=\max \left\{{\widetilde {p}}_{(i-1)},1-(1-p_{(i)})^{m-i+1}\right\}}$, where ${\displaystyle {\widetilde {p}}_{(1)}=1-(1-p_{(1)})^{m}}$. Due to the inequality ${\displaystyle 1-(1-\alpha )^{1/n}<\alpha /n}$ for ${\displaystyle n\geq 2}$, the Holm-Šidák method will be more powerful than Holm–Bonferroni method.

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

${\displaystyle {\widetilde {p}}_{(i)}=\max _{j\leq i}\left\{{\frac {\sum _{k=j}^{m}{w_{(k)}}}{w_{(j)}}}p_{(j)}\right\}_{1},{\text{ where }}\{x\}_{1}\equiv \min(x,1).}$

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 FWER that are more powerful than Holm–Bonferroni. For instance, in the Hochberg procedure, rejection of ${\displaystyle H_{(1)}\ldots H_{(k)}}$ is made after finding the maximal index ${\displaystyle k}$ such that ${\displaystyle P_{(k)}\leq {\frac {\alpha }{m+1-k}}}$. 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