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In [[statistics]], when performing [[multiple comparisons]], the term '''false positive ratio''', also known as the '''false alarm ratio''', usually refers to the probability of falsely rejecting the [[null hypothesis]] for a particular [[hypothesis test|test]].
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The false positive '''rate''' (or "false alarm rate") usually refers to the '''expectancy''' of the false positive '''ratio'''.
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== Definition ==
 
Suppose we have ''m'' null hypotheses, denoted by: ''H''<sub>1</sub>,&nbsp;''H''<sub>2</sub>,&nbsp;...,&nbsp;''H''<sub>''m''</sub>.<br />
Using a [[Statistical hypothesis testing|statistical test]], each hypothesis is declared significant/non-significant.<br />
Summing the test results over <math>H_i</math> will give us the following table and related random variables:
 
{|class="wikitable"
! |
! Null hypothesis is True
! Alternative hypothesis is True
! | Total
|- align="center"
! | Declared significant
| <math>V</math>
| <math>S</math>
| <math>R</math>
|- align="center"
! | Declared non-significant
| <math>U</math>
| <math>T</math>
| <math>m - R</math>
|- align="center"
! Total
| <math>m_0</math>
| <math>m - m_0</math>
| <math>m</math>
|}
 
* <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>V</math> is the number of [[Type I and type II errors|false positives (Type I error)]]
* <math>S</math> is the number of [[Type I and type II errors|true positives]]
* <math>T</math> 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]]
* <math>R</math> is the number of rejected null hypotheses. It is an observable [[random variable]], while <math>S</math>, <math>T</math>, <math>U</math>, and <math>V</math> are unobservable [[random variable]]s.
 
The level of significance that is used to test each hypothesys is set based on the form of inference ([[Familywise error rate#Simultaneous inference vs. selective inference|simultaneous inference vs. selective inference]]) and its supporting criteria (for example [[Familywise error rate|FWER]] or [[False discovery rate|FDR]]), that were pre-determined by the researcher.
 
When performing [[multiple comparisons]] in a [[statistics|statistical]] framework such as above, the '''false positive ratio''' (also known as the '''false alarm ratio''', as opposed to false positive '''rate''' / false alarm '''rate''' ) usually refers to the probability of falsely rejecting the [[null hypothesis]] for a particular [[hypothesis test|test]]. Using the terminology suggested here, it is simply <math>V/m_0</math>.
 
Since ''V'' is a random variable and ''m_0'' is a constant (<math> V \leq m_0 </math>), the false positive '''ratio''' is also a random variable, ranging between 0-1.
<br />
The '''false positive rate''' (or "false alarm rate") usually refers to the '''expectancy of the false positive ratio''', expressed by <math>E(V/m_0)</math>.
 
It is worth noticing that the two definitions ("false positive ratio" / "false positive rate") are somewhat interchangeable. For example, in the referenced article<ref name=Burke.at.all1988>{{cite journal | last1 = Burke | first1= Donald | first2= John | last2= Brundage | last3= Redfield | first3= Robert | year = 1988 | title = Measurement of the False Positive Rate in a Screening Program for Human Immunodeficiency Virus Infections | journal = [[The New England Journal of Medicine]] | volume = 319 | pages = 961–964 | url = http://www.nejm.org/doi/pdf/10.1056/NEJM198810133191501 | doi=10.1056/NEJM198810133191501}}</ref> <math>V/m_0</math> serves as the false positive "rate" rather than as its "ratio".
 
== The difference between "false positive rate", "type I error rate" and other close terms ==
{{original research section|date=February 2013}}
While the false positive rate is mathematically equal to the [[type I error]] rate, it is viewed as a separate term for the following reasons:{{citation needed|date=February 2013}}
 
* The [[type I error]] rate is often associated with the '''a-prior''' setting of the [[significance level]] by the researcher: the significance level represents an acceptable error rate '''considering that all null hypotheses are true''' (the "global null" hypothesis). the choice of a significance level may thus be somewhat arbitrary (i.e. setting 10% (0.1), 5% (0.05), 1% (0.01) etc.)
 
: As opposed to that, the false positive rate is associated with a '''post-prior''' result, which is the expected number of false positives divided by the total number of hypotheses under the '''real''' combination of true and non-true null hypotheses (disregarding the "global null" hypothesis). Since the false positive rate is a parameter that is not controlled by the researcher, it cannot be identified with the significance level.
 
* Moreover, false positive rate is a term usually used regarding a medical test or diagnostic device (i.e. "the false positive rate of a certain diagnostic device is 1%"), while type I error is a term associated with statistical tests, where the meaning of the word "positive" is not as clear (i.e. "the type I error of a test is 1%").
 
The false positive rate should also not be confused with the [[familywise error rate]], which is defined as <math> \mathrm{FWER} = \Pr(V \ge 1)\,</math>. As the number of tests grows, the familywise error rate usually converges to 1 while the false positive rate remains fixed.
 
Lastly, it is important to note the profound difference between the false positive rate and the [[false discovery rate]]: while the first is defined as <math>E(V/m_0)</math>, the second is defined as <math>E(V/R)</math>.
 
== References ==
 
<references />
 
[[Category:Multiple comparisons]]
[[Category:Statistical tests]]
[[Category:Analysis of variance]]

Latest revision as of 16:48, 4 January 2015

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