Closed testing procedure
The closed testing principle
Suppose there are k hypotheses H1,..., Hk 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 Hi, if all possible intersection hypotheses involving Hi 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.
Suppose there are three hypotheses H1,H2, and H3 to be tested and the overall type I error rate is 0.05. Then H1 can be rejected at level α if H1 ∩ H2 ∩ H3, H1 ∩ H2, H1 ∩ H3 and H1 can all be rejected using valid tests with level 0.05.
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.
- Marcus, R; Peritz, E; Gabriel, KR (1976). "On closed testing procedures with special reference to ordered analysis of variance". Biometrika. 63: 655–660. doi:10.1093/biomet/63.3.655. JSTOR 2335748.
- Bretz, F; Maurer, W; Brannath, W; Posch, M (2009). "A graphical approach to sequentially rejective multiple test procedures". Stat Med. 28 (4): 586–604. doi:10.1002/sim.3495.