Holm–Bonferroni method

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In statistics, the Holm–Bonferroni method [1] is a method 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 one of the earliest usage of stepwise algorithms in simultaneous inference.

It is named after Sture Holm who invented the method in 1978 and Carlo Emilio Bonferroni.

Introduction[edit]

When considering several hypotheses in the same test the problem of multiplicity arises. Intuitively, the more hypotheses we check, the higher the probability to witness a rare result. With 10 different hypotheses and significance level of 0.05, there is more than 40% chance of having one or more type I errors. Holm–Bonferroni method is one of the many ways to address this issue. It modifies the rejection criteria in order to enable us to control the overall probability of witnessing one or more type I errors (also called the familywise error rate) at a predetermined level.

Formulation[edit]

The method algorithm is as follows:

  • Let H_{1},...,H_{m} be a family of 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 don't reject any of the hypotheses and if no such k exist then reject all hypotheses.

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

Proof that Bonferroni-Holm controls the FWER[edit]

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 k be the first rejected true hypothesis (first in the ordering given by the Bonferroni–Holm test). Necessarily, k \leq m-m_0+1 (because, since k is the first, there are m_0-1 true hypotheses coming between k and m). Since k is rejected we have P_{(k)} \leq \frac{\alpha}{m+1-k} by definition of the test. Using k \leq m-m_0+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.

Example[edit]

Four null hypotheses are tested with α = 0.05. The four unadjusted p-values are 0.01, 0.04, 0.03 and 0.005. The smallest of these is 0.005. Since this is less than 0.05/4, null hypothesis four is rejected (meaning some alternative hypothesis likely explains the data). The next smallest p-value is 0.01, which is smaller than 0.05/3. So the null hypothesis is also rejected. The next smallest p-value is 0.03. This is not smaller than 0.05/2, so you fail to reject this hypothesis (meaning you have not seen evidence to conclude an alternative hypothesis is preferable to the level of α = 0.05). As soon as that happens, you stop, and therefore, also fail to reject the remaining hypothesis that has a p-value of 0.04. Therefore, hypotheses one and four are rejected while hypotheses two and three are not rejected. Applying the approximate false discovery rate produces the same result without requiring ordering the p-values, then using different criteria for each test.

Extensions[edit]

The Holm–Bonferroni method is an example of a closed test procedure.[2] 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.

Adjusted P-value[edit]

The adjusted P-values for Holm–Bonferroni method are: \widetilde{p}_{(i)}=\max_{j\leq i}\left\{ (N-j+1)p_{(j)}\right\} _{1}, where \{x\}_{1}\equiv \min(x,1).

Šidák version[edit]

Main article: Šidák correction

When hypotheses are independent, 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[edit]

Let p1,..., pk be the unadjusted p-values and let w1,..., wk be a set of corresponding positive weights that add to 1. Without loss of generality, assume the p-values and the weights are all ordered such that p1/w1p2/w2 ≤ ... ≤ pk/wk.[citation needed] The adjusted p-value for the first hypothesis is q1 = min{1,p1/w1}. Inductively, define the adjusted p-value for hypothesis i by qi = min{1,max{qi−1,(wi + ... + wkpi/wi. 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]

Holm-Bonferroni is more powerful than the regular Bonferroni, and can always be used as a substitute. With that said, it is not the best simultaneous inference test available. There are many other tests that intend to control the familywise error rate, many of them are more powerful than Holm-Bonferroni. Among those tests there is the Hochberg procedure (1988). If we replace the rejection criteria and look for the maximal index k such that P_{(k)} \leq \frac{\alpha}{m+1-k}, then reject H_{(1)} \ldots H_{(k)}, we will get the Hochberg method, which is guaranteed to be no less powerful and is in many cases more powerful then Holm-Bonferroni. With that said, Hochberg method require the hypotheses to be independent (and also under some forms of positive dependence), while Holm-Bonferroni can be applied with no conditions.

Bonferroni contribution[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."

See also[edit]

References[edit]

  1. ^ Holm, S. (1979). "A simple sequentially rejective multiple test procedure". Scandinavian Journal of Statistics 6 (2): 65–70. JSTOR 4615733. MR 538597. 
  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.