- Holm-Bonferroni method
In
statistics , the Holm-Bonferroni method [Holm, S (1979): "A simple sequentially rejective multiple test procedure", "Scandinavian Journal of Statistics", 6:65-70] performs more than one hypothesis test simultaneously. It is named after Sture Holm andCarlo Emilio Bonferroni .Suppose there are "k" hypotheses to be tested and the overall type 1 error rate is α. Start by ordering the
p-value s and comparing the smallest p-value to α/"k". If that p-value is less than α/"k", then reject that hypothesis and start all over with the same α and test the remaining "k" - 1 hypothesis, i.e. order the "k" - 1 remaining p-values and compare the smallest one to α/("k" - 1). Continue doing this until the hypothesis with the smallest p-value cannot be rejected. At that point, stop and accept all hypotheses that have not been rejected at previous steps.Here is an example. Four hypotheses are tested with α = 0.05. The four unadjusted p-values are 0.01, 0.03, 0.04, and 0.005. The smallest of these is 0.005. Since this is less than 0.05/4, hypothesis four is rejected. The next smallest p-value is 0.01, which is smaller than 0.05/3. So, hypothesis one is also rejected. The next smallest p-value is 0.03. This is not smaller than 0.05/2. Therefore, hypotheses one and four are rejected while hypotheses two and three are not rejected.
The Holm-Bonferroni method is an example of a closed test procedure [Marcus R, Peritz E, Gabriel KR (1976): "On closed testing procedures with special reference to ordered analysis of variance", "Biometrika" 63: 655-660] . 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 also possible to define a weighted version. Let "p"1,..., "p""k" be the unadjusted p-values and let "w"1,..., "w""k"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 "p"1/"w"1 ≤ "p"2/"w"2 ≤ ... ≤ "p""k"/"w""k". The adjusted p-value for the first hypothesis is "q"1 = min{1,"p"1/"w"1}. Inductively, define the adjusted p-value for hypothesis "i" by "q""i"=min{1,max{"q""i"-1,("w""i" + ... + "w""k")×"p""i"/"w""i". 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.
References
ee also
*
Multiple comparisons
Wikimedia Foundation. 2010.
