a type i error

8/3/2019 A type I error

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The desired (i.e., non-erroneous) outcomes of the test are called true positive meaning "rejectingnull hypothesis, when it is false" and true negative meaning "not rejecting null hypothesis, whenit is true". A statistical test can either reject (prove false) or fail to reject (fail to prove false) anull hypothesis, but never prove it true (i.e., failing to reject a null hypothesis does not prove ittrue).

In colloquial usage type I error might be called "failing to believe the truth" and type II error"believing the falsehood". In light of hypothesis testing, however, type II error means more of"staying with falsehood for lack of better alternative", than active belief in it. Cautiousness isadvised, when applying statistical tests to philosophical and ill-defined problems, in which thenotion oftruthor "real state of things" is relative, as this might lead to confusion (see:type IIIerror).

Contents

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1 Consequences of type I and type II errors 2 Statistical error

o 2.1 Type I erroro 2.2 Type II erroro 2.3 Understanding Type I and Type II errors

3 Etymology 4 Related terms

o 4.1 False positive rateo 4.2 False negative rateo 4.3 The null hypothesiso 4.4 Bayes' theorem

5 Various proposals for further extensiono 5.1 Systems Theoryo 5.2 Davido 5.3 Mostellero 5.4 Kaisero 5.5 Kimballo 5.6 Mitroff and Featheringhamo 5.7 Raiffao 5.8 Marascuilo and Levin

6 Usage exampleso 6.1 Inventory Controlo 6.2 Computers

6.2.1 Computer security 6.2.2 Spam filtering 6.2.3 Malware 6.2.4 Optical character recognition (OCR) 6.2.5 Security screening 6.2.6 Biometrics
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o 6.3 Medical screeningo 6.4 Medical testingo 6.5 Paranormal investigation

7 See also 8 Notes 9 References 10 External links

[edit] Consequences of type I and type II errors

Both types of errors are problems for individuals, corporations, anddata analysis. A falsepositive (with null hypothesis of health) in medicine causes unnecessary worry or treatment,while a false negative gives the patient the dangerous illusion of good health and the patientmight not get an available treatment. A false positive in manufacturingquality control(with anull hypothesis of a product being well-made), discards a product, which is actually well-made,while a false negative stamps a broken product as operational. A false positive (with nullhypothesis of no effect) in scientific research suggest an effect, which is not actually there, whilea false negative fails to detect an effect that is there.

Based on the real-life consequences of an error, one type may be more serious than the other. Forexample,NASAengineers would prefer to throw out an electronic circuit that is really fine (nullhypothesis: not broken; reality: not broken; action: thrown out; error: type I, false positive) thanto use one on a spaceship that is actually broken (null hypothesis: not broken; reality: broken;action: use it; error: type II, false negative). In that situation a type I error raises the budget, but atype II error would risk the entire mission.

Alternately, criminal courts set highbar for proof and procedureand sometimes release someonewho is guilty (null hypothesis: innocent; reality: guilty; test find: not guilty; action: release; error:type I, false positive) rather than convict someone who is innocent (null hypothesis: innocent;reality: not guilty; test find: guilty; action: convict; error: type II, false negative). Each systemmakes its own choice regarding where to draw the line.

Minimizing errors of decision is not a simple issue; for any givensample sizethe effort to reduceone type of error generally results in increasing the other type of error. The only way to minimizeboth types of error, without just improving the test, is to increase the sample size, and this may ormay not be feasible.

[edit] Statistical error

The notion of statistical error is integral part ofhypothesis testing. The test requires anunambiguous statement of a null hypothesis, which usually corresponds to a default "state ofnature", for example "this person is healthy", "this accused is not guilty" or "this product is notbroken". An alternative hypothesis is the negation of null hypothesis, for example, "this person isnot healthy", "this accused is guilty" or "this product is broken". What we actually call type I or
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type II error depends directly on null hypothesis. Negation of null hypothesis causes type I andtype II errors to switch places. The goal of the test is to determine, if the null hypothesis can berejected.

The result of the test may be negative, relative to null hypothesis (not healthy, guilty, broken) or

positive (healthy, not guilty, not broken). If the result of the test corresponds with reality, then acorrect decision has been made. However, if the result of the test does not correspond withreality, then an error has occurred.

[edit] Type I error

Type I error, also known as an error of the first kind, an error or a false positive is the error of

rejecting a truenull hypothesis(H0). An example of this would be if a test shows that a woman ispregnant (H0: she is not) when in reality she is not, or telling a patient he is sick (H0: he is not),when in fact he is not . Type I error can be viewed as the error of excessive credulity[1]. In termsof folk tales, an investigator may be "crying wolf" (setting a false alarm) without a wolf in sight

(H0: no wolf).

[edit] Type II error

Type II error, also known as an error of the second kind, a error or a false negative is the errorof failing to reject a false null hypothesis. An example of this would be if a test shows that awoman is not pregnant (H0: she is not), when in reality, she is. Type II error can be viewed as theerror of excessiveskepticism[2]. In terms of folk tales, an investigator may fail to see the wolf(failing to set an alarm, seeAesop's story ofThe Boy Who Cried Wolf).

Tabelarized relations between truth/falseness of the null hypothesis and outcomes of the test:

Null hypothesis (H0) is true Null hypothesis (H0) is false

Reject null hypothesisType I error

False positiveCorrect outcome

True Positive

Fail to reject null hypothesisCorrect outcomeTrue Negative

Type II errorFalse negative

[edit] Understanding Type I and Type II errors

From the Bayesian point of view, a type I error is one that looks at information that should not

substantially change one's prior estimate of probability, but does. A type II error is that one looksat information which should change one's estimate, but does not. (Though the null hypothesis isnot quite the same thing as one's prior estimate, it is, rather, one's pro forma prior estimate.)

Hypothesis testing is the art of testing whether a variation between two sample distributions canbe explained by chance or not. In many practical applications type I errors are more delicate thantype II errors. In these cases, care is usually focused on minimizing the occurrence of thisstatistical error. Suppose, the probability for a type I error is 1% , then there is a 1% chance that
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the observed variation is not true. This is called the level of significance, denoted with the Greekletter alpha (). While 1% might be an acceptable level of significance for one application, a

different application can require a very different level. For example, the standard goal ofsixsigmais to achieve precision to 4.5 standard deviations above or below the mean. This meansthat only 3.4 parts per million are allowed to be deficient in a normally distributed process.

[edit] Etymology

In 1928,Jerzy Neyman(18941981) andEgon Pearson(18951980), both eminentstatisticians,discussed the problems associated with "deciding whether or not a particular sample may bejudged as likely to have been randomly drawn from a certain population"[3]p. 1: and, asFlorenceNightingale Davidremarked, "itis necessary to remember the adjective random [in the termrandom sample] should apply to the method of drawing the sample and not to the sample

itself".[4]

They identified "two sources of error", namely:

(a) the error of rejecting a hypothesis that should have been accepted, and(b) the error of accepting a hypothesis that should have been rejected.[3]p.31

In 1930, they elaborated on these two sources of error, remarking that:

...in testing hypotheses two considerations must be kept in view, (1) we must be able to

reduce the chance of rejecting a true hypothesis to as low a value as desired; (2) the test

must be so devised that it will reject the hypothesis tested when it is likely to be false.[5]

In 1933, they observed that these "problems are rarely presented in such a form that we can

discriminate with certainty between the true and false hypothesis" (p.187). They also noted that,in deciding whether to accept or reject a particular hypothesis amongst a "set of alternativehypotheses" (p.201), it was easy to make an error:

...[and] these errors will be of two kinds:(I) we rejectH0[i.e., the hypothesis to be tested] when it is true,(II) we acceptH0when some alternativehypothesisHiis true.

[6]p.187

In all of the papers co-written by Neyman and Pearson the expression H0always signifies "thehypothesis to be tested" (see, for example,[6]p. 186).

In the same paper[6]p. 190 they call these two sources of error, errors of type I and errors of typeII respectively

False positive rate

The false positive rate is the proportion of absent events that yield positive test outcomes, i.e., theconditional probability of a positive test result given an absent event.
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The false positive rate is equal to the significance level. The specificity of the test is equal to 1minus the false positive rate.

Instatistical hypothesis testing, this fraction is given the Greek letter, and 1 is defined asthespecificityof the test. Increasing thespecificityof the test lowers the probability oftype I

errors, but raises the probability oftype II errors (false negatives that reject the alternativehypothesis when it is true).[Note 1]

[edit] False negative rate

The false negative rate is the proportion of events that are being tested for which yield negativetest outcomes with the test, i.e., the conditional probability of a negative test result given that theevent being looked for has taken place.

Instatistical hypothesis testing, this fraction is given the letter. The "power" (or the"sensitivity") of the test is equal to 1minus .

[edit] The null hypothesis

Main article:Null hypothesis

It is standard practice for statisticians to conducttestsin order to determine whether or not a"speculativehypothesis" concerning the observed phenomena of the world (or its inhabitants)can be supported. The results of such testing determine whether a particular set of results agreesreasonably (or does not agree) with the speculated hypothesis.

On the basis that it is always assumed, by statistical convention, that the speculated hypothesis is

wrong, and the so-called "null hypothesis" that the observed phenomena simply occur by chance(and that, as a consequence, the speculated agent has no effect)the test will determinewhether this hypothesis is right or wrong. This is why the hypothesis under test is often calledthe null hypothesis (most likely, coined by Fisher (1935, p. 19)), because it is this hypothesis thatis to be either nullified or not nullified by the test. When the null hypothesis is nullified, it ispossible to conclude that data support the "alternative hypothesis" (which is the originalspeculated one).

The consistent application by statisticians of Neyman and Pearson's convention of representing"the hypothesis to be tested" (or "the hypothesis to be nullified") with the expression H0 has ledto circumstances where many understand the term "the null hypothesis" as meaning "thenil

hypothesis"a statement that the results in question have arisen through chance. This is notnecessarily the casethe key restriction, as per Fisher (1966), is that "the null hypothesis mustbe exact, that is free from vagueness and ambiguity, because it must supply the basis of the

'problem of distribution,' of which the test of significance is the solution."[7]As a consequence ofthis, in experimental science the null hypothesis is generally a statement that a particulartreatment has no effect; in observational science, it is that there is no difference between thevalue of a particular measured variable, and that of an experimental prediction.
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The extent to which the test in question shows that the "speculated hypothesis" has (or has not)been nullified is called itssignificance level; and the higher the significance level, the less likelyit is that the phenomena in question could have been produced by chance alone. BritishstatisticianSir Ronald Aylmer Fisher(18901962) stressed that the "null hypothesis":

...is never proved or established, but is possibly disproved, in the course of experimentation.Every experiment may be said to exist only in order to give the facts a chance of disproving the

null hypothesis. (1935, p.19)

[edit] Bayes' theorem

The probability that an observed positive result is a falsepositive (as contrasted with an observedpositive result being a truepositive) may be calculated usingBayes' theorem.

The key concept of Bayes' theorem is that the true rates offalse positives and false negatives arenot a function of theaccuracyof the test alone, but also the actual rate orfrequencyof

occurrence within the test population; and, often, the more powerful issue is the actual rates ofthe condition within the sample being tested.

[edit] Various proposals for further extension

Since the paired notions ofType I errors (or "false positives") and Type II errors (or "falsenegatives") that were introduced by Neyman and Pearson are now widely used, their choice ofterminology ("errors of the first kind" and "errors of the second kind"), has led others tosuppose that certain sorts of mistake that they have identified might be an "error of the thirdkind", "fourth kind", etc.[Note 2]

None of these proposed categories have met with any sort of wide acceptance. The following is abrief account of some of these proposals.

[edit] Systems Theory

Insystems theoryan additional type III error is often defined[9]: Type III (): asking the wrongquestion and using the wrongnull hypothesis

[edit] David

Florence Nightingale David (19091993)[4]a sometime colleague of both Neyman and Pearsonat theUniversity College London, making a humorous aside at the end of her 1947 paper,suggested that, in the case of her own research, perhaps Neyman and Pearson's "two sources oferror" could be extended to a third:

I have been concerned here with trying to explain what I believe to be the basic ideas [of my

"theory of the conditional power functions"], and to forestall possible criticism that I am falling
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into error (of the third kind) and am choosing the test falsely to suit the significance of the

sample. (1947, p.339)

[edit] Mosteller

In 1948,Frederick Mosteller(19162006)[Note 3]argued that a "third kind of error" was requiredto describe circumstances he had observed, namely:

Type I error: "rejecting the null hypothesis when it is true". Type II error: "accepting the null hypothesis when it is false". Type III error: "correctly rejecting the null hypothesis for the wrong reason". (1948, p. 61)[Note 4]

[edit] Kaiser

According to Henry F. Kaiser (19271992), in his 1966 paper extended Mosteller's classificationsuch that an error of the third kindentailed an incorrect decision of direction following a rejected

two-tailed test of hypothesis. In his discussion (1966, pp. 162163), Kaiser also speaks oferrors, errors, and errors for type I, type II and type III errors respectively (C.O.Dellomos).

[edit] Kimball

In 1957, Allyn W. Kimball, a statistician with theOak Ridge National Laboratory, proposed adifferent kind of error to stand beside "the first and second types of error in the theory of testinghypotheses". Kimball defined this new "error of the third kind" as being "the error committed bygiving the right answer to the wrong problem" (1957, p. 134).

MathematicianRichard Hamming(19151998) expressed his view that "It is better to solve theright problem the wrong way than to solve the wrong problem the right way".

Harvard economistHoward Raiffadescribes an occasion when he, too, "fell into the trap ofworking on the wrong problem" (1968, pp. 264265).[Note 5]

[edit] Mitroff and Featheringham

In 1974, Ian Mitroff and Tom Featheringham extended Kimball's category, arguing that "one ofthe most important determinants of a problem's solution is how that problem has been

represented or formulated in the first place".

They defined type III errors as either "the error... of having solved the wrong problem... whenone should have solved the right problem" or "the error... [of] choosing the wrong problemrepresentation... when one should have... chosen the right problem representation" (1974),p. 383.

In 2009,dirty rotten strategiesbyIan I. MitroffandAbraham Silverswas published regardingtype III and type IV errors providing many examples of both developing good answers to the
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wrong questions (III) and deliberately selecting the wrong questions for intensive and skilledinvestigation (IV). Most of the examples have nothing to do with statistics, many being problemsof public policy or business decisions.[10]

[edit] Raiffa

In 1969, the Harvard economist Howard Raiffa jokingly suggested "a candidate for the error ofthe fourth kind: solving the right problem too late" (1968, p. 264).

[edit] Marascuilo and Levin

In 1970, L. A. Marascuilo and J. R. Levin proposed a "fourth kind of error"a "Type IVerror"which they defined in a Mosteller-like manner as being the mistake of "the incorrectinterpretation of a correctly rejected hypothesis"; which, they suggested, was the equivalent of"a physician's correct diagnosis of an ailment followed by the prescription of a wrong medicine"(1970, p. 398).

[edit] Usage examples

Statistical tests always involve a trade-off between:

(a) the acceptable level offalse positives (in which a non-match is declared to be a match) and

(b) the acceptable level offalse negatives (in which an actual match is not detected).

A threshold value can be varied to make the test more restrictive or more sensitive; with themore restrictive tests increasing the risk of rejecting true positives, and the moresensitive testsincreasing the risk of accepting false positives.

[edit] Inventory Control

An automated inventory control system that rejects high-quality goods of a consignmentcommits a Type I Errorwhile a system that accepts low-quality goods commits a Type II Error.

[edit] Computers

The notions of "false positives" and "false negatives" have a wide currency in the realm of

computers and computer applications.

[edit]Computer security

Security vulnerabilities are an important consideration in the task of keeping all computer datasafe, while maintaining access to that data for appropriate users (seecomputer security,computerinsecurity). Moulton (1983), stresses the importance of:
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avoiding the type I errors (or false positive) that classify authorized users as imposters. avoiding the type II errors (or false negatives) that classify imposters as authorized users (1983,

p. 125).

[edit]Spam filtering

A false positive occurs when "spam filtering" or "spam blocking" techniques wrongly classify alegitimate email message as spam and, as a result, interferes with its delivery. While most anti-spam tactics can block or filter a high percentage of unwanted emails, doing so without creatingsignificant false-positive results is a much more demanding task.

A false negative occurs when aspamemail is not detected as spam, but is classified as "non-spam". A low number of false negatives is an indicator of the efficiency of"spam filtering"methods.

[edit]Malware

The term false positive is also used whenantivirussoftware wrongly classifies an innocuous fileas avirus. The incorrect detection may be due toheuristicsor to an incorrectvirus signaturein adatabase. Similar problems can occur withantitrojanorantispywaresoftware.

[edit]Optical character recognition (OCR)

Detectionalgorithmsof all kinds often create false positives.Optical character recognition(OCR) software may detect an "a" where there are only some dots that appearto be an "a" to thealgorithm being used.

[edit]Security screening

False positives are routinely found every day inairport security screening, which are ultimatelyvisual inspectionsystems. The installed security alarms are intended to prevent weapons beingbrought onto aircraft; yet they are often set to such highsensitivitythat they alarm many times aday for minor items, such as keys, belt buckles, loose change, mobile phones, and tacks in shoes(seeexplosive detection,metal detector.)

The ratio offalse positives (identifying an innocent traveller as a terrorist) to true positives(detecting a would-be terrorist) is, therefore, very high; and because almost every alarm is a falsepositive, thepositive predictive valueof these screening tests is very low.

The relative cost of false results determines the likelihood that test creators allow these events tooccur. As the cost of a false negative in this scenario is extremely high (not detecting a bombbeing brought onto a plane could result in hundreds of deaths) whilst the cost of a false positiveis relatively low (a reasonably simple further inspection) the most appropriate test is one with ahigh statisticalsensitivitybut low statisticalspecificity(one that allows minimal false negativesin return for a high rate of false positives).
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[edit]Biometrics

Biometricmatching, such as forfingerprint,facial recognitionoririsrecognition, is susceptibleto type I and type II errors. The null hypothesis is that the input does identify someone in thesearched list of people, so:

the probability of type I errors is called the "False Reject Rate" (FRR) or False Non-match Rate(FNMR),

while the probability of type II errors is called the "False Accept Rate" (FAR) or False Match Rate(FMR).

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If the system is designed torarely match suspectsthen the probability of type II errors can becalled the "False AlarmRate". On the other hand, if the system is used for validation (andacceptance is the norm) then the FAR is a measure of system security, while the FRR measuresuser inconvenience level.

[edit] Medical screening

In the practice of medicine, there is a significant difference between the applications ofscreeningandtesting:

Screening involves relatively cheap tests that are given to large populations, none of whommanifest any clinical indication of disease (e.g.,Pap smears).

Testing involves far more expensive, often invasive, procedures that are given only to those whomanifest some clinical indication of disease, and are most often applied to confirm a suspected

diagnosis.

For example, most States in the USA require newborns to be screened forphenylketonuriaandhypothyroidism, among othercongenital disorders. Although they display a high rate offalsepositives, the screening tests are considered valuable because they greatly increase the likelihoodof detecting these disorders at a far earlier stage.[Note 6]

The simple blood tests used to screen possibleblood donorsforHIVandhepatitishave asignificant rate offalse positives; however, physicians use much more expensive and far moreprecise tests to determine whether a person is actually infected with either of these viruses.

Perhaps the most widely discussed false positives in medical screening come from the breastcancer screening proceduremammography. The US rate offalse positive mammograms is up to15%, the highest in world. One consequence of the high false positive rate in the US is that, inany 10 year period, half of the American women screened receive a false positive mammogram.False positive mammograms are costly, with over $100 million spent annually in the US onfollow-up testing and treatment. They also cause women unneeded anxiety. As a result of thehigh false positive rate in the US, as many as 90-95% of women who get a positive mammogramdo nothave the condition. The lowest rate in the world is in theNetherlands, 1%. Thelowest rates are generally in Northern Europe where mammography films are read twice and ahigh threshold for additional testing is set (the high threshold decreases thepowerof the test).
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The ideal population screening test would be cheap, easy to administer, and producezero false-negatives, if possible. Such tests usually produce more false-positives, which can subsequentlybe sorted out by more sophisticated (and expensive) testing.

[edit] Medical testing

False negatives and False positives are significant issues inmedical testing.

False negatives may provide a falsely reassuring message to patients and physicians that diseaseis absent, when it is actually present. This sometimes leads to inappropriate or inadequatetreatment of both the patient and their disease. A common example is relying oncardiac stressteststo detect coronaryatherosclerosis, even thoughcardiac stress testsare known to only detectlimitations ofcoronary arteryblood flow due to advancedstenosis.

False negatives produce serious and counter-intuitive problems, especially when the conditionbeing searched for is common. If a test with a falsenegative rate of only 10%, is used to test a

population with a trueoccurrence rate of70%, many of the "negatives" detected by the test willbe false. (SeeBayes' theorem)

False positives can also produce serious and counter-intuitive problems when the conditionbeing searched for is rare, as inscreening. If a test has a falsepositive rate of one in tenthousand, but only one in a million samples (or people) is a truepositive, most of the "positives"detected by that test will be false. The probability that an observed positive result is a falsepositive may be calculated usingBayes' theorem.

[edit] Paranormal investigation

The notion of a false positive is common in cases ofparanormalorghostphenomena seen inimages and such, when there is another plausible explanation. When observing a photograph,recording, or some other evidence that appears to have a paranormal originin this usage, afalse positive is a disproven piece of media "evidence" (image, movie, audio recording, etc.) thatactually has a normal explanation.[Note 7]

Sensitivity and specificity are statistical measures of the performance of abinary classificationtest, also known in statistics asclassification function.Sensitivity (also called recall rate insome fields) measures the proportion of actual positives which are correctly identified as such(e.g. the percentage of sick people who are correctly identified as having the condition).Specificity measures the proportion of negatives which are correctly identified (e.g. the

percentage of healthy people who are correctly identified as not having the condition). These twomeasures are closely related to the concepts oftype I and type II errors. A theoretical, optimalprediction aims to achieve 100% sensitivity (i.e. predict all people from the sick group as sick)and 100% specificity (i.e. not predict anyone from the healthy group as sick), howevertheoretically any predictor will possess a minimumerror boundknown as theBayes error rate.

For any test, there is usually a trade-off between the measures. For example: in anairportsecuritysetting in which one is testing for potential threats to safety, scanners may be set to
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trigger on low-risk items like belt buckles and keys (low specificity), in order to reduce the riskof missing objects that do pose a threat to the aircraft and those aboard (high sensitivity). Thistrade-off can be represented graphically as anROC curve.

Contents

[hide]

1 Definitionso 1.1 Sensitivityo 1.2 Specificityo 1.3 Graphical illustration

2 Medical example 3 Worked example 4 Terminology in information retrieval 5 See also 6 Further reading 7 External links 8 References

[edit] Definitions

Imagine a study evaluating a new test that screens people for a disease. Each person taking thetest either has or does not have the disease. The test outcome can be positive (predicting that theperson has the disease) or negative (predicting that the person does not have the disease). The

test results for each subject may or may not match the subject's actual status. In that setting:

True positive: Sick people correctly diagnosed as sick False positive: Healthy people incorrectly identified as sick True negative: Healthy people correctly identified as healthy False negative: Sick people incorrectly identified as healthy.

[edit] Sensitivity

Sensitivity relates to the test's ability to identify positive results.

Again, consider the example of the medical test used to identify a disease. The sensitivity of atest is the proportion of people who have the disease who test positive for it. This can also bewritten as:
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If a test has high sensitivity then a negative result would suggest the absence of disease.[1]Forexample, a sensitivity of 100% means that the test recognizes all actual positivesi.e. all sickpeople are recognized as being ill. Thus, in contrast to a high specificity test, negative results in ahigh sensitivity testare used to rule outthe disease.[1]

From a theoretical point of view, a 'bogus' test kit which always indicates positive, regardless ofthe disease status of the patient, will achieve 100% sensitivity. Therefore the sensitivity alonecannot be used to determine whether a test is useful in practice.

Sensitivity is not the same as theprecisionorpositive predictive value(ratio of true positives tocombined true and false positives), which is as much a statement about the proportion of actualpositives in the population being tested as it is about the test.

The calculation of sensitivity does not take into account indeterminate test results. If a test cannotbe repeated, the options are to exclude indeterminate samples from analysis (but the number ofexclusions should be stated when quoting sensitivity), or, alternatively, indeterminate samples

can be treated as false negatives (which gives the worst-case value for sensitivity and maytherefore underestimate it).

A test with a high sensitivity has a lowtype II errorrate.

[edit] Specificity

Specificity relates to the ability of the test to identify negative results.

Consider the example of the medical test used to identify a disease. The specificity of a test isdefined as the proportion of patients who do not have the disease who will test negative for it.

This can also be written as:

If a test has high specificity, a positive result from the test means a high probability of thepresence of disease.[1]

From a theoretical point of view, a 'bogus' test kit which always indicates negative, regardless ofthe disease status of the patient, will achieve 100% specificity. Therefore the specificity alonecannot be used to determine whether a test is useful in practice.

A test with a high specificity has a lowtype I errorrate.

[edit] Graphical illustration
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High sensitivity and low specificity

Low sensitivity and high specificity

[edit] Medical example

In medical diagnostics, test sensitivity is the ability of a test to correctly identify those with thedisease (true +ve rate), whereas test specificity is the ability of the test to correctly identify thosewithout the disease (true -ve rate).[2]

If 100 patients known to have a disease were tested, and 43 test positive, then the test has 43%sensitivity. If 100 with no disease are tested and 96 return a negative result, then the test has 96%specificity.

A highly specific test is unlikely to give a false positive result: a positive result should thus beregarded as a true positive. A sensitive test rarely misses a condition, so a negative result shouldbe reassuring (the disease tested for is absent). A

SPIN and SNOUT are commonly used mnemonics which says: A highly SPecific test, whenPositive, rules IN disease (SP-P-IN), and a highly 'SeNsitive' test, when Negative rules OUTdisease (SN-N-OUT).

[edit] Worked example

Relationships among termsview talk edit

Condition

(as determined by "Gold standard")
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Positive Negative

Test

outcome

Positive True PositiveFalse Positive(Type I error)

Positive predictive value

=

Negative

False Negative

(Type II error) True Negative

Negative predictive value

=

Sensitivity

=

Specificity

=

A worked exampleThefecal occult blood(FOB) screen test was used in 2030 people to look for bowelcancer:

Patients withbowel cancer

(as confirmed onendoscopy)

Positive Negative

Fecal

occult

blood

screen

test

outcome

PositiveTrue Positive

(TP) = 20False Positive

(FP) = 180

Positive predictive value= TP / (TP + FP)= 20 / (20 + 180)

= 20 / 200= 10%

NegativeFalse Negative

(FN) = 10True Negative

(TN) = 1820

Negative predictive value= TN / (FN + TN)

= 1820 / (10 + 1820)= 1820 / 1830

99.5%

Sensitivity

= TP / (TP + FN)= 20 / (20 + 10)

= 20 / 30 66.67%

Specificity

= TN / (FP + TN)= 1820 / (180 + 1820)

= 1820 / 2000= 91%

Related calculations

False positive rate () = FP / (FP + TN) = 180 / (180 + 1820) = 9% = 1 specificity

False negative rate () = FN / (TP + FN) = 10 / (20 + 10) = 33% = 1 sensitivity Power= sensitivity = 1 Likelihood ratiopositive = sensitivity / (1 specificity) = 66.67% / (1 91%) = 7.4 Likelihood ratio negative = (1 sensitivity) / specificity = (1 66.67%) / 91% = 0.37

Hence with large numbers of false positives and few false negatives, a positive FOB screen testis in itself poor at confirming cancer (PPV = 10%) and further investigations must be undertaken,it did, however, correctly identify 66.7% of all cancers (the sensitivity). However as a screening
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a type i error

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