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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; the adjusted p-value is the largest among those hypotheses. It controls the family-wise error rate for all the k hypotheses at level α in the strong sense.

Example

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 α.

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 family-wise error rate for all the k hypotheses at level α in the strong sense.

Multiple test procedures developed using the graphical approach for constructing and illustrating multiple test procedures² are a subclass of closed testing procedures.

See also

• Multiple comparisons
• Holm–Bonferroni method
• Bonferroni correction

References

1. Marcus, R; Peritz, E; Gabriel, KR (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. JSTOR 2335748. /wiki/K._Ruben_Gabriel ↩

2. 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. S2CID 12068118. /wiki/Stat_Med ↩