Template:Classification of multiple hypothesis tests
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The following table defines the possible outcomes when testing multiple null hypotheses. Suppose we have a number m of null hypotheses, denoted by: H1, H2, ..., Hm. Using a statistical test, we reject the null hypothesis if the test is declared significant. We do not reject the null hypothesis if the test is non-significant. Summing each type of outcome over all Hi yields the following random variables:
| Null hypothesis is true (H0) | Alternative hypothesis is true (HA) | Total | |
|---|---|---|---|
| Test is declared significant | V | S | R |
| Test is declared non-significant | U | T | [math]\displaystyle{ m - R }[/math] |
| Total | [math]\displaystyle{ m_0 }[/math] | [math]\displaystyle{ m - m_0 }[/math] | m |
- m is the total number hypotheses tested
- [math]\displaystyle{ m_0 }[/math] is the number of true null hypotheses, an unknown parameter
- [math]\displaystyle{ m - m_0 }[/math] is the number of true alternative hypotheses
- V is the number of false positives (Type I error) (also called "false discoveries")
- S is the number of true positives (also called "true discoveries")
- T is the number of false negatives (Type II error)
- U is the number of true negatives
- [math]\displaystyle{ R=V+S }[/math] is the number of rejected null hypotheses (also called "discoveries", either true or false)
In m hypothesis tests of which [math]\displaystyle{ m_0 }[/math] are true null hypotheses, R is an observable random variable, and S, T, U, and V are unobservable random variables.
