Difference between revisions of "Template:Classification of multiple hypothesis tests"
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imported>Cedar101 m ({{rh}}, {{mvar}}) |
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<!-- see https://en.wikipedia.org/wiki/Special:WhatLinksHere/Template:Classification_of_multiple_hypothesis_tests --> | <!-- see https://en.wikipedia.org/wiki/Special:WhatLinksHere/Template:Classification_of_multiple_hypothesis_tests --> | ||
The following table defines the possible outcomes when testing multiple null hypotheses. | The following table defines the possible outcomes when testing multiple null hypotheses. | ||
| − | Suppose we have a number ''m'' of null hypotheses, denoted by: ''H''<sub>1</sub>, ''H''<sub>2</sub>, ..., ''H''<sub>''m''</sub>. | + | Suppose we have a number ''m'' of null hypotheses, denoted by: {{math|''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. | ||
Summing each type of outcome over all ''H<sub>i</sub>'' yields the following random variables: | Summing each type of outcome over all ''H<sub>i</sub>'' yields the following random variables: | ||
{|class="wikitable" | {|class="wikitable" | ||
| − | + | | | | |
! Null hypothesis is true (H<sub>0</sub>) | ! Null hypothesis is true (H<sub>0</sub>) | ||
! Alternative hypothesis is true (H<sub>A</sub>) | ! Alternative hypothesis is true (H<sub>A</sub>) | ||
! | Total | ! | Total | ||
|- align="center" | |- align="center" | ||
| − | ! | Test is declared significant | + | ! {{rh}} | Test is declared significant |
| − | | | + | | {{mvar|V}} |
| − | | | + | | {{mvar|S}} |
| − | | | + | | {{mvar|R}} |
|- align="center" | |- align="center" | ||
| − | ! | Test is declared non-significant | + | ! {{rh}} | Test is declared non-significant |
| − | | | + | | {{mvar|U}} |
| − | | | + | | {{mvar|T}} |
| <math>m - R</math> | | <math>m - R</math> | ||
|- align="center" | |- align="center" | ||
| − | ! Total | + | ! {{rh}} | Total |
| <math>m_0</math> | | <math>m_0</math> | ||
| <math>m - m_0</math> | | <math>m - m_0</math> | ||
| − | | | + | | {{mvar|m}} |
|} | |} | ||
| − | * | + | * {{mvar|m}} is the total number hypotheses tested |
* <math>m_0</math> is the number of true [[null hypothesis|null hypotheses]], an unknown parameter | * <math>m_0</math> is the number of true [[null hypothesis|null hypotheses]], an unknown parameter | ||
* <math>m - m_0</math> is the number of true [[alternative hypothesis|alternative hypotheses]] | * <math>m - m_0</math> is the number of true [[alternative hypothesis|alternative hypotheses]] | ||
| − | * | + | * {{mvar|V}} is the number of [[Type I and type II errors|false positives (Type I error)]] (also called "false discoveries") |
| − | * | + | * {{mvar|S}} is the number of [[Type I and type II errors|true positives]] (also called "true discoveries") |
| − | * | + | * {{mvar|T}} is the number of [[Type I and type II errors|false negatives (Type II error)]] |
| − | * | + | * {{mvar|U}} is the number of [[Type I and type II errors|true negatives]] |
* <math>R=V+S</math> is the number of rejected null hypotheses (also called "discoveries", either true or false) | * <math>R=V+S</math> is the number of rejected null hypotheses (also called "discoveries", either true or false) | ||
| − | In | + | In {{mvar|m}} hypothesis tests of which <math>m_0</math> are true null hypotheses, {{mvar|R}} is an observable random variable, and {{mvar|S}}, {{mvar|T}}, {{mvar|U}}, and {{mvar|V}} are unobservable [[random variable]]s. |
Latest revision as of 08:10, 17 December 2018
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 | |
|---|---|---|---|
| Template:Rh | Test is declared significant | Template:Mvar | Template:Mvar | Template:Mvar |
| Template:Rh | Test is declared non-significant | Template:Mvar | Template:Mvar | <math>m - R</math> |
| Template:Rh | Total | <math>m_0</math> | <math>m - m_0</math> | Template:Mvar |
- Template:Mvar 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
- Template:Mvar is the number of false positives (Type I error) (also called "false discoveries")
- Template:Mvar is the number of true positives (also called "true discoveries")
- Template:Mvar is the number of false negatives (Type II error)
- Template:Mvar 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 Template:Mvar hypothesis tests of which <math>m_0</math> are true null hypotheses, Template:Mvar is an observable random variable, and Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar are unobservable random variables.