Template:Classification of multiple hypothesis tests
Revision as of 01:36, 19 November 2016 by 23.242.207.48 (talk) (not all of the outcomes described are "errors" (specifically, correct rejections and correct retentions of null hypothesis are not errors))
The following table defines the possible outcomes committed when testing multiple null hypotheses. Suppose we have a number m of multiple 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 the test results over Hi will give us the following table and related random variables:
| Null hypothesis is true (H0) | Alternative hypothesis is true (HA) | Total | |
|---|---|---|---|
| Test is declared significant | <math>V</math> | <math>S</math> | <math>R</math> |
| Test is declared non-significant | <math>U</math> | <math>T</math> | <math>m - R</math> |
| Total | <math>m_0</math> | <math>m - m_0</math> | <math>m</math> |
- <math>m</math> is the total number hypotheses tested
- <math>m_0</math> is the number of true null hypotheses, an unknown parameter
- <math>m - m_0</math> is the number of true alternative hypotheses
- <math>V</math> is the number of false positives (Type I error) (also called "false discoveries")
- <math>S</math> is the number of true positives (also called "true discoveries")
- <math>T</math> is the number of false negatives (Type II error)
- <math>U</math> is the number of true negatives
- <math>R=V+S</math> is the number of rejected null hypotheses (also called "discoveries", either true or false)
In <math>m</math> hypothesis tests of which <math>m_0</math> are true null hypotheses, <math>R</math> is an observable random variable, and <math>S</math>, <math>T</math>, <math>U</math>, and <math>V</math> are unobservable random variables.
