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Closed Testing Procedure





THE CLOSED TESTING PRINCIPLE

Suppose there are ''k'' hypotheses ''H''1,..., ''H''''k'' to be tested and the overall type I error rate is α. The closed testing principle allows the rejection of any one of these elementary hypotheses, say ''H''''i'', if all possible intersection hypotheses involving ''H''''i'' can be rejected by using valid local level α tests. It controls the Familywise Error Rate for all the ''k'' hypotheses at level α in the strong sense.


EXAMPLE

Suppose there are three hypotheses ''H''1,..., ''H''3 to be tested and the overall type I error rate is 0.05. Then ''H''1 can be rejected at level α if ''H''1 ∩ ''H''2 ∩ ''H''3, ''H''1 ∩ ''H''2, ''H''1 ∩ ''H''3 and ''H''1 can all be rejected using valid tests with level 0.05.


SPECIAL CASES

The Holm-Bonferroni Method is a special case of a closed test procedure for which each intersection null hypothesis is tested using the simple Bonferroni test. As such, it controls the Familywise Error Rate for all the ''k'' hypotheses at level α in the strong sense.


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