Closed testing procedure
# Closed testing procedure

In statistics, the closed testing procedure is a general method for performing more than one hypothesis test 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 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 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 are a subclass of closed testing procedures.

