en>Mrkonium7: /* See also */

2015-01-04T15:48:33Z

en>Fraggle81: Reverted 1 good faith edit by 120.29.78.226 using STiki

2014-03-04T18:23:58Z

Reverted 1 good faith edit by 120.29.78.226 using STiki

139.184.30.134: /* The Marginal Revolution and Marxism */

2014-01-24T11:17:43Z

The Marginal Revolution and Marxism

← Older revision		Revision as of 13:17, 24 January 2014
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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]].

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

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			Suppose we have ''m'' null hypotheses, denoted by: ''H''<sub>1</sub>, ''H''<sub>2</sub>, ..., ''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]]

112.120.54.131: /* Market price and diminishing marginal utility */

2012-08-21T07:59:18Z

Market price and diminishing marginal utility

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Marginal utility - Revision history

en>Mrkonium7: /* See also */

en>Fraggle81: Reverted 1 good faith edit by 120.29.78.226 using STiki

139.184.30.134: /* The Marginal Revolution and Marxism */

112.120.54.131: /* Market price and diminishing marginal utility */