FWER Definition
Suppose we have m null hypotheses, denoted by: H1, H2, ..., Hm.
Using a statistical test, each hypothesis is declared significant/non-significant.
Summing the test results over Hi will give us the following table and related random variables:
| Null hypothesis is True | Alternative hypothesis is True | Total | |
|---|---|---|---|
| Declared significant | |||
| Declared non-significant | |||
| Total |
- is the number of true null hypotheses, an unknown parameter
- is the number of true alternative hypotheses
- is the number of false positives (Type I error)
- is the number of true positives
- is the number of false negatives (Type II error)
- is the number of true negatives
- is the number of rejected null hypotheses
- is an observable random variable, while, and are unobservable random variables.
The FWER is the probability of making even one type I error In the family,
or equivalently,
Thus, by assuring, the probability of making even one type I error in the family is controlled at level .
A procedure controls the FWER in the weak sense if the FWER control at level is guaranteed only when all null hypotheses are true (i.e. when = so the global null hypothesis is true)
A procedure controls the FWER in the strong sense if the FWER control at level is guaranteed for any configuration of true and non-true null hypotheses (including the global null hypothesis)
Read more about this topic: Familywise Error Rate
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