View source for Closed testing procedure

{{primary source|date=June 2012}}
In [[statistics]], the '''closed testing procedure'''<ref>{{cite journal | doi = 10.1093/biomet/63.3.655 | last1 = Marcus | first1 = R | last2 = Peritz | first2 = E | last3 = Gabriel | first3 = KR | author3-link=K. Ruben Gabriel | year = 1976 | title = On closed testing procedures with special reference to ordered analysis of variance | journal = Biometrika | volume = 63 | issue = 3 | pages = 655–660 |jstor=2335748}}</ref> is a general method for performing more than one [[Statistical hypothesis testing|hypothesis test]] simultaneously.

==The closed testing principle==
Suppose there are ''k'' hypotheses ''H''<sub>1</sub>,..., ''H''<sub>''k''</sub> 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''<sub>''i''</sub>, if all possible intersection hypotheses involving ''H''<sub>''i''</sub> 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 ''H''<sub>1</sub>,''H''<sub>2</sub>, and ''H''<sub>3</sub> to be tested and the overall type I error rate is 0.05. Then ''H''<sub>1</sub> can be rejected at level α if ''H''<sub>1</sub> ∩ ''H''<sub>2</sub> ∩ ''H''<sub>3</sub>,  ''H''<sub>1</sub> ∩ ''H''<sub>2</sub>,  ''H''<sub>1</sub> ∩ ''H''<sub>3</sub> and  ''H''<sub>1</sub> 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 procedures<ref>{{cite journal | doi = 10.1002/sim.3495 | last1 = Bretz | first1 = F | last2 = Maurer | first2 = W | last3 = Brannath | first3 = W | last4 = Posch | first4 = M | year = 2009 | title = A graphical approach to sequentially rejective multiple test procedures | journal = [[Stat Med]] | volume = 28 | issue = 4 | pages = 586–604 | s2cid = 12068118 }}</ref> are a subclass of closed testing procedures.

==See also==
* [[Multiple comparisons]]
* [[Holm–Bonferroni method]]
* [[Bonferroni correction]]

==References==
{{Reflist}}

[[Category:Multiple comparisons]]