Closed testing procedure

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

The closed testing principle[edit]

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.

Example[edit]

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 H1H2H3, H1H2, H1H3 and H1 can all be rejected using valid tests with level 0.05.

Special cases[edit]

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.

Multiple test procedures developed using the graphical approach for constructing and illustrating multiple test procedures[2] are a subclass of closed testing procedures.

See also[edit]

References[edit]

  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. 
  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.