Difference between revisions of "Template:Classification of multiple hypothesis tests"
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(not all of the outcomes described are "errors" (specifically, correct rejections and correct retentions of null hypothesis are not errors)) |
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| − | The following table defines the possible outcomes | + | The following table defines the possible outcomes when testing multiple null hypotheses. |
Suppose we have a number ''m'' of multiple null hypotheses, denoted by: ''H''<sub>1</sub>, ''H''<sub>2</sub>, ..., ''H''<sub>''m''</sub>. | Suppose we have a number ''m'' of multiple null hypotheses, denoted by: ''H''<sub>1</sub>, ''H''<sub>2</sub>, ..., ''H''<sub>''m''</sub>. | ||
Using a [[Statistical hypothesis testing|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. | Using a [[Statistical hypothesis testing|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. | ||
Revision as of 01:36, 19 November 2016
The following table defines the possible outcomes 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.
