Closed testing procedure

In statistics, the closed testing procedure^[1] is a general method for performing more than one hypothesis test simultaneously.

The closed testing principle

Suppose there are k hypotheses H₁,..., 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 H₁,H₂, and H₃ are 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.

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.

References

1. ^ 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. 

See also

• Multiple comparisons
• Holm–Bonferroni method
• Bonferroni correction