# Closed testing procedure

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In [statistics](/source/Statistics), the **closed testing procedure**[1] is a general method for performing more than one [hypothesis test](/source/Statistical_hypothesis_testing) simultaneously.

## The closed testing principle

Suppose there are *k* hypotheses *H*1,..., *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; the adjusted p-value is the largest among those hypotheses. It controls the [family-wise error rate](/source/Family-wise_error_rate) for all the *k* hypotheses at level α in the strong sense.

## Example

Suppose there are three hypotheses *H*1,*H*2, and *H*3 to be tested and the overall type I error rate is 0.05. Then *H*1 can be rejected at level α if *H*1 ∩ *H*2 ∩ *H*3, *H*1 ∩ *H*2, *H*1 ∩ *H*3 and *H*1 can all be rejected using valid tests with α = 0.05.

## Special cases

The [Holm–Bonferroni method](/source/Holm%E2%80%93Bonferroni_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](/source/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[2] are a subclass of closed testing procedures.

## See also

- [Multiple comparisons](/source/Multiple_comparisons)

- [Holm–Bonferroni method](/source/Holm%E2%80%93Bonferroni_method)

- [Bonferroni correction](/source/Bonferroni_correction)

## References

1. **[^](#cite_ref-1)** Marcus, R; Peritz, E; [Gabriel, KR](/source/K._Ruben_Gabriel) (1976). "On closed testing procedures with special reference to ordered analysis of variance". *Biometrika*. **63** (3): 655–660. [doi](/source/Doi_(identifier)):[10.1093/biomet/63.3.655](https://doi.org/10.1093%2Fbiomet%2F63.3.655). [JSTOR](/source/JSTOR_(identifier)) [2335748](https://www.jstor.org/stable/2335748).

1. **[^](#cite_ref-2)** Bretz, F; Maurer, W; Brannath, W; Posch, M (2009). "A graphical approach to sequentially rejective multiple test procedures". *[Stat Med](/source/Stat_Med)*. **28** (4): 586–604. [doi](/source/Doi_(identifier)):[10.1002/sim.3495](https://doi.org/10.1002%2Fsim.3495). [S2CID](/source/S2CID_(identifier)) [12068118](https://api.semanticscholar.org/CorpusID:12068118).

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