In statistics, the closed testing procedure is a general method for performing more than one hypothesis test simultaneously.
Suppose there are k hypotheses H<sub>1</sub>,..., H<sub>k</sub> to be tested and the overall type I error rate is ñ. 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 ñ 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 ñ in the strong sense.
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 ñ if H<sub>1</sub> â© H<sub>2</sub> â© H<sub>3</sub>, H<sub>1</sub> â© H<sub>2</sub>, H<sub>1</sub> â© H<sub>3</sub> and H<sub>1</sub> can all be rejected using valid tests with ñ = 0.05.
The HolmâÂÂBonferroni 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 ñ 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.