View source for False positive rate

{{Short description|Chance of wrongly rejecting the null hypothesis}}
{{one source|date=July 2016}}
In [[statistics]], when performing [[multiple comparisons]], a '''false positive ratio''' (also known as '''fall-out''' or '''false alarm rate'''<ref name="cawcr">{{cite web |title=Forecast Verification methods Across Time and Space Scales |url=https://www.cawcr.gov.au/projects/verification/#Contingency_table |website=WWRP/WGNE Joint Working Group on Forecast Verification Research |access-date=8 January 2025 |url-status=live |archive-url=https://web.archive.org/web/20241227151212/https://www.cawcr.gov.au/projects/verification/#Contingency_table | archive-date=27 December 2024|language=en}}</ref> ) is the [[probability]] of falsely rejecting the [[null hypothesis]] for a particular [[hypothesis test|test]]. The false positive rate is calculated as the ratio between the number of negative events wrongly categorized as positive ([[false positive]]s) and the total number of actual negative events (regardless of classification).

The false positive '''rate''' (or "false alarm rate") usually refers to the '''expectancy''' of the false positive '''ratio'''.

== Definition ==
The false positive rate (false alarm rate) is <math>\boldsymbol{\mathrm{FPR}} =\frac{\mathrm{FP}}{\mathrm{FP} + \mathrm{TN}}</math> <ref name="cawcr"></ref>

where <math> \mathrm{FP} </math> is the number of false positives, <math> \mathrm{TN} </math> is the number of true negatives and <math> N=\mathrm{FP}+\mathrm{TN} </math> is the total number of ground truth negatives.

The [[significance level]] used to test each hypothesis is set based on the form of inference ([[family-wise error rate#Alternative approaches|simultaneous inference vs. selective inference]]) and its supporting criteria (for example [[family-wise 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 rate''', as opposed false alarm '''ratio - FAR''' ) 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 ''<math>m_0</math>'' 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 | issue= 15 | pages = 961–964 | doi=10.1056/NEJM198810133191501 | pmid=3419477}}</ref> <math>V/m_0</math> serves as the false positive "rate" rather than as its "ratio".

=== Classification of multiple hypothesis tests ===
{{Main|Classification of multiple hypothesis tests}}
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{{Classification of multiple hypothesis tests}}

== Comparison with other error rates ==
{{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-priori''' 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 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 [[family-wise error rate]], which is defined as <math> \boldsymbol{\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>.

== See also ==
* [[False positives and false negatives]]
* [[False coverage rate]]
* [[False discovery rate]]
* [[Sensitivity and specificity]]
* [[Confusion matrix]]
* [[Contingency table]]

== References ==
<references />

[[Category:Multiple comparisons]]
[[Category:Statistical hypothesis testing]]
[[Category:Analysis of variance]]
[[Category:Statistical ratios]]