Difference between revisions of "Template:Classification of multiple hypothesis tests"

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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>,&nbsp;''H''<sub>2</sub>,&nbsp;...,&nbsp;''H''<sub>''m''</sub>.
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Suppose we have a number ''m'' of null hypotheses, denoted by: {{math|''H''<sub>1</sub>,&nbsp;''H''<sub>2</sub>,&nbsp;...,&nbsp;''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>''&nbsp; yields the following random variables:
 
Summing each type of outcome over all ''H<sub>i</sub>''&nbsp; yields the following random variables:
  
 
{|class="wikitable"
 
{|class="wikitable"
! |
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| |
 
! 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
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! {{rh}} | Test is declared significant
| <math>V</math>
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| {{mvar|V}}
| <math>S</math>
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| {{mvar|S}}
| <math>R</math>
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| {{mvar|R}}
 
|- align="center"
 
|- align="center"
! | Test is declared non-significant
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! {{rh}} | Test is declared non-significant
| <math>U</math>
+
| {{mvar|U}}
| <math>T</math>
+
| {{mvar|T}}
 
| <math>m - R</math>
 
| <math>m - R</math>
 
|- align="center"
 
|- align="center"
! Total
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! {{rh}} | Total
 
| <math>m_0</math>
 
| <math>m_0</math>
 
| <math>m - m_0</math>
 
| <math>m - m_0</math>
| <math>m</math>
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| {{mvar|m}}
 
|}
 
|}
  
* <math>m</math> is the total number hypotheses tested
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* {{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]]
* <math>V</math> is the number of [[Type I and type II errors|false positives (Type I error)]] (also called "false discoveries")
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* {{mvar|V}} is the number of [[Type I and type II errors|false positives (Type I error)]] (also called "false discoveries")
* <math>S</math> is the number of [[Type I and type II errors|true positives]] (also called "true discoveries")
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* {{mvar|S}} is the number of [[Type I and type II errors|true positives]] (also called "true discoveries")
* <math>T</math> is the number of [[Type I and type II errors|false negatives (Type II error)]]
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* {{mvar|T}} is the number of [[Type I and type II errors|false negatives (Type II error)]]
* <math>U</math> is the number of [[Type I and type II errors|true negatives]]
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* {{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 <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 variable]]s.
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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: H1H2, ..., 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

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.