Sensitivity and specificity

Sensitivity and specificity are statistical measures of the performance of a binary classification test, also known in statistics as classification function. Specificity (also called the true positive rate, or the recall rate in some 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), and is complementary to the false negative rate. Sensitivity (sometimes called the true negative rate) measures the proportion of negatives which are correctly identified as such (e.g., the percentage of healthy people who are correctly identified as not having the condition), and is complementary to the false positive rate.

A perfect predictor would be described as 100% sensitive (e.g., all sick are identified as sick) and 100% specific (e.g., all healthy are identified as healthy); however, theoretically any predictor will possess a minimum error bound known as the Bayes error rate.

For any test, there is usually a trade-off between the measures. For instance, in an airport security setting in which one is testing for potential threats to safety, scanners may be set to trigger on low-risk items like belt buckles and keys (low specificity), in order to reduce the risk of missing objects that do pose a threat to the aircraft and those aboard (high sensitivity). This trade-off can be represented graphically as a receiver operating characteristic curve.

Definitions

Terminology and derivations
from a confusion matrix
true positive (TP) eqv. with hit true negative (TN) eqv. with correct rejection false positive (FP) eqv. with false alarm, Type I error false negative (FN) eqv. with miss, Type II error sensitivity or true positive rate (TPR) eqv. with hit rate, recall ${\mathit {TPR}}={\mathit {TP}}/P={\mathit {TP}}/({\mathit {TP}}+{\mathit {FN}})$ specificity (SPC) or true negative rate ${\mathit {SPC}}={\mathit {TN}}/N={\mathit {TN}}/({\mathit {FP}}+{\mathit {TN}})$ precision or positive predictive value (PPV) ${\mathit {PPV}}={\mathit {TP}}/({\mathit {TP}}+{\mathit {FP}})$ negative predictive value (NPV) ${\mathit {NPV}}={\mathit {TN}}/({\mathit {TN}}+{\mathit {FN}})$ fall-out or false positive rate (FPR) ${\mathit {FPR}}={\mathit {FP}}/N={\mathit {FP}}/({\mathit {FP}}+{\mathit {TN}})=1-{\mathit {SPC}}$ false discovery rate (FDR) ${\mathit {FDR}}={\mathit {FP}}/({\mathit {TP}}+{\mathit {FP}})=1-{\mathit {PPV}}$ accuracy (ACC) ${\mathit {ACC}}=({\mathit {TP}}+{\mathit {TN}})/(P+N)$ F1 score is the harmonic mean of precision and sensitivity ${\mathit {F1}}=2{\mathit {TP}}/(2{\mathit {TP}}+{\mathit {FP}}+{\mathit {FN}})$ Matthews correlation coefficient (MCC) ${\frac {{\mathit {TP}}\times {\mathit {TN}}-{\mathit {FP}}\times {\mathit {FN}}}{\sqrt {({\mathit {TP}}+{\mathit {FP}})({\mathit {TP}}+{\mathit {FN}})({\mathit {TN}}+{\mathit {FP}})({\mathit {TN}}+{\mathit {FN}})}}}$ Informedness ${\mathit {TPR}}+{\mathit {SPC}}-1$ Markedness ${\mathit {PPV}}+{\mathit {NPV}}-1$ Source: Fawcett (2006).^[1]

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

In general, Positive = identified and negative = rejected. Therefore:

True positive = correctly identified
False positive = incorrectly identified
True negative = correctly rejected
False negative = incorrectly rejected

Let us define an experiment from P positive instances and N negative instances for some condition. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix, as follows:

This Wikipedia page has been superseded by template:diagnostic_testing_diagram and is retained primarily for historical reference.

		True condition
	Total population	Condition positive	Condition negative	Prevalence = ⁠Σ Condition positive/Σ Total population⁠		Accuracy (ACC) = ⁠Σ True positive + Σ True negative/Σ Total population⁠
Predicted condition	Predicted condition positive	True positive	False positive, Type I error	Positive predictive value (PPV), Precision = ⁠Σ True positive/Σ Predicted condition positive⁠		False discovery rate (FDR) = ⁠Σ False positive/Σ Predicted condition positive⁠
	Predicted condition negative	False negative, Type II error	True negative	False omission rate (FOR) = ⁠Σ False negative/Σ Predicted condition negative⁠		Negative predictive value (NPV) = ⁠Σ True negative/Σ Predicted condition negative⁠
view talk edit		True positive rate (TPR), Recall, Sensitivity (SEN), probability of detection, Power = ⁠Σ True positive/Σ Condition positive⁠	False positive rate (FPR), Fall-out, probability of false alarm = ⁠Σ False positive/Σ Condition negative⁠	Positive likelihood ratio (LR+) = ⁠TPR/FPR⁠	Diagnostic odds ratio (DOR) = ⁠LR+/LR−⁠	Matthews correlation coefficient (MCC) = √TPR·TNR·PPV·NPV − √FNR·FPR·FOR·FDR	F₁ score = ⁠2 · PPV · TPR/PPV + TPR⁠ = ⁠2 · Precision · Recall/Precision + Recall⁠
		False negative rate (FNR), Miss rate = ⁠Σ False negative/Σ Condition positive⁠	Specificity (SPC), Selectivity, True negative rate (TNR) = ⁠Σ True negative/Σ Condition negative⁠	Negative likelihood ratio (LR−) = ⁠FNR/TNR⁠

Sensitivity

Sensitivity relates to the test's ability to identify a condition correctly. Consider the example of a medical test used to identify a disease. Sensitivity of the test is the proportion of people known to have the disease, who test positive for it. Mathematically, this can be expressed as:

{\begin{aligned}{\text{sensitivity}}&={\frac {\text{number of true positives}}{{\text{number of true positives}}+{\text{number of false negatives}}}}\\\\&={\frac {\text{number of true positives}}{\text{total number of sick individuals in population}}}\\\\&={\text{probability of a positive test, given that the patient is ill}}\end{aligned}}

A negative result in a test with high sensitivity is useful for ruling out disease. A high sensitivity test is reliable when its result is negative, since it rarely misdiagnoses those who have the disease. A test with 100% sensitivity will recognize all patients with the disease by testing positive. A negative test result would definitively rule out presence of the disease in a patient.

A positive result in a test with high sensitivity is not useful for ruling in disease. Suppose a 'bogus' test kit is designed to show only one reading, positive. When used on diseased patients, all patients test positive, giving the test 100% sensitivity. However, sensitivity by definition does not take into account false positives. The bogus test also returns positive on all healthy patients, giving it a false positive rate of 100%, rendering it useless for diagnosing or "ruling in" the disease.

Sensitivity is not the same as the precision or positive predictive value (ratio of true positives to combined true and false positives), which is as much a statement about the proportion of actual positives 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 cannot be repeated, indeterminate samples either should be excluded from the analysis (the number of exclusions should be stated when quoting sensitivity) or can be treated as false negatives (which gives the worst-case value for sensitivity and may therefore underestimate it).

A test with high sensitivity has a low type II error rate. In non-medical contexts, sensitivity is sometimes called recall.

Specificity

Specificity relates to the test's ability to exclude a condition correctly. Consider the example of a medical test for diagnosing a disease. Specificity of a test is the proportion of healthy patients known not to have the disease, who will test negative for it. Mathematically, this can also be written as:

{\begin{aligned}{\text{specificity}}&={\frac {\text{number of true negatives}}{{\text{number of true negatives}}+{\text{number of false positives}}}}\\\\&={\frac {\text{number of true negatives}}{\text{total number of well individuals in population}}}\\\\&={\text{probability of a negative test given that the patient is well}}\end{aligned}}

Positive result in a test with high specificity is useful for ruling in disease. The test rarely gives positive results in healthy patients. A test with 100% specificity will read negative, and accurately exclude disease from all healthy patients. A positive result will highlight a high probability of the presence of disease.^[2]

Negative result in a test with high specificity is not useful for ruling out disease. Assume a 'bogus' test is designed to read only negative. This is administered to healthy patients, and reads negative on all of them. This will give the test a specificity of 100%. Specificity by definition does not take into account false negatives. The same test will also read negative on diseased patients, therefore it has a false negative rate of 100%, and will be useless for ruling out disease.

A test with a high specificity has a low type I error rate.

Graphical illustration

High sensitivity and low specificity
Low sensitivity and high specificity

Medical examples

In medical diagnostics, test sensitivity is the ability of a test to correctly identify those with the disease (true positive rate), whereas test specificity is the ability of the test to correctly identify those without the disease (true negative rate). 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. Sensitivity and specificity are prevalence-independent test characteristics, as their values are intrinsic to the test and do not depend on the disease prevalence in the population of interest.^[3] Positive and negative predictive values, but not sensitivity or specificity, are values influenced by the prevalence of disease in the population that is being tested.

Misconceptions

It is often claimed that a highly specific test is effective at ruling in a disease when positive, while a highly sensitive test is deemed effective at ruling out a disease when negative.^[4]^[5] This has led to the widely used mnemonics SPIN and SNOUT, according to which a highly SPecific test, when Positive, rules IN disease (SP-P-IN), and a highly 'SeNsitive' test, when Negative rules OUT disease (SN-N-OUT). Both rules of thumb are, however, inferentially misleading, as the diagnostic power of any test is determined by both its sensitivity and its specificity.^[6]^[7]^[8]

Sensitivity index

The sensitivity index or d' (pronounced 'dee-prime') is a statistic used in signal detection theory. It provides the separation between the means of the signal and the noise distributions, compared against the standard deviation of the noise distribution. For normally distributed signal and noise with mean and standard deviations $\mu _{S}$ and $\sigma _{S}$ , and $\mu _{N}$ and $\sigma _{N}$ , respectively, d' is defined as:

d'={\frac {\mu _{S}-\mu _{N}}{\sqrt {{\frac {1}{2}}(\sigma _{S}^{2}+\sigma _{N}^{2})}}}

^[9]

An estimate of d' can be also found from measurements of the hit rate and false-alarm rate. It is calculated as:

d' = Z(hit rate) - Z(false alarm rate),^[10]

where function Z(p), p ∈ [0,1], is the inverse of the cumulative Gaussian distribution.

d' is a dimensionless statistic. A higher d' indicates that the signal can be more readily detected.

Worked example

A worked example: A diagnostic test with sensitivity 67% and specificity 91% is applied to 2030 people to look for a disorder with a population prevalence of 1.48%

		Fecal occult blood screen test outcome		^{view talk edit}
	Total population (pop.) = 2030	Test outcome positive	Test outcome negative	Accuracy (ACC) = (TP + TN) / pop. = (20 + 1820) / 2030 ≈ 90.64%	F₁ score = 2 × ⁠precision × recall/precision + recall⁠ ≈ 0.174
Patients with bowel cancer (as confirmed on endoscopy)	Actual condition positive (AP) = 30 (2030 × 1.48%)	True positive (TP) = 20 (2030 × 1.48% × 67%)	False negative (FN) = 10 (2030 × 1.48% × (100% − 67%))	True positive rate (TPR), recall, sensitivity = TP / AP = 20 / 30 ≈ 66.7%	False negative rate (FNR), miss rate = FN / AP = 10 / 30 ≈ 33.3%
Patients with bowel cancer (as confirmed on endoscopy)	Actual condition negative (AN) = 2000 (2030 × (100% − 1.48%))	False positive (FP) = 180 (2030 × (100% − 1.48%) × (100% − 91%))	True negative (TN) = 1820 (2030 × (100% − 1.48%) × 91%)	False positive rate (FPR), fall-out, probability of false alarm = FP / AN = 180 / 2000 = 9.0%	Specificity, selectivity, true negative rate (TNR) = TN / AN = 1820 / 2000 = 91%
	Prevalence = AP / pop. = 30 / 2030 ≈ 1.48%	Positive predictive value (PPV), precision = TP / (TP + FP) = 20 / (20 + 180) = 10%	False omission rate (FOR) = FN / (FN + TN) = 10 / (10 + 1820) ≈ 0.55%	Positive likelihood ratio (LR+) = ⁠TPR/FPR⁠ = (20 / 30) / (180 / 2000) ≈ 7.41	Negative likelihood ratio (LR−) = ⁠FNR/TNR⁠ = (10 / 30) / (1820 / 2000) ≈ 0.366
		False discovery rate (FDR) = FP / (TP + FP) = 180 / (20 + 180) = 90.0%	Negative predictive value (NPV) = TN / (FN + TN) = 1820 / (10 + 1820) ≈ 99.45%	Diagnostic odds ratio (DOR) = ⁠LR+/LR−⁠ ≈ 20.2

Related calculations

False positive rate (α) = type I error = 1 − specificity = FP / (FP + TN) = 180 / (180 + 1820) = 9%
False negative rate (β) = type II error = 1 − sensitivity = FN / (TP + FN) = 10 / (20 + 10) ≈ 33%
Power = sensitivity = 1 − β
Positive likelihood ratio = sensitivity / (1 − specificity) ≈ 0.67 / (1 − 0.91) ≈ 7.4
Negative likelihood ratio = (1 − sensitivity) / specificity ≈ (1 − 0.67) / 0.91 ≈ 0.37
Prevalence threshold = $PT={\frac {{\sqrt {TPR(-TNR+1)}}+TNR-1}{(TPR+TNR-1)}}$ ≈ 0.2686 ≈ 26.9%

This hypothetical screening test (fecal occult blood test) correctly identified two-thirds (66.7%) of patients with colorectal cancer.^[a] Unfortunately, factoring in prevalence rates reveals that this hypothetical test has a high false positive rate, and it does not reliably identify colorectal cancer in the overall population of asymptomatic people (PPV = 10%).

On the other hand, this hypothetical test demonstrates very accurate detection of cancer-free individuals (NPV ≈ 99.5%). Therefore, when used for routine colorectal cancer screening with asymptomatic adults, a negative result supplies important data for the patient and doctor, such as reassuring patients worried about developing colorectal cancer.

Estimation of errors in quoted sensitivity or specificity

Sensitivity and specificity values alone may be highly misleading. The 'worst-case' sensitivity or specificity must be calculated in order to avoid reliance on experiments with few results. For example, a particular test may easily show 100% sensitivity if tested against the gold standard four times, but a single additional test against the gold standard that gave a poor result would imply a sensitivity of only 80%. A common way to do this is to state the binomial proportion confidence interval, often calculated using a Wilson score interval.

Confidence intervals for sensitivity and specificity can be calculated, giving the range of values within which the correct value lies at a given confidence level (e.g., 95%).^[13]

Terminology in information retrieval

In information retrieval, the positive predictive value is called precision, and sensitivity is called recall.

The F-score can be used as a single measure of performance of the test. The F-score is the harmonic mean of precision and recall:

F=2\times {\frac {{\text{precision}}\times {\text{recall}}}{{\text{precision}}+{\text{recall}}}}

In the traditional language of statistical hypothesis testing, the sensitivity of a test is called the statistical power of the test, although the word power in that context has a more general usage that is not applicable in the present context. A sensitive test will have fewer Type II errors.

References

^ Fawcelt, Tom (2006). "An Introduction to ROC Analysis". Pattern Recognition Letters. 27 (8): 861–874. doi:10.1016/j.patrec.2005.10.010.
^ "SpPins and SnNouts". Centre for Evidence Based Medicine (CEBM). Retrieved 26 December 2013.
^ Mangrulkar, Rajesh. "Diagnostic Reasoning I and II". Retrieved 24 January 2012.
^ Michigan State University Evidence Based Medicine resource
^ Emory University Medical School Evidence Based Medicine course
^ Baron, JA (Apr–Jun 1994). "Too bad it isn't true...". Medical decision making : an international journal of the Society for Medical Decision Making. 14 (2): 107. doi:10.1177/0272989X9401400202. PMID 8028462.
^ Boyko, EJ (Apr–Jun 1994). "Ruling out or ruling in disease with the most sensitive or specific diagnostic test: short cut or wrong turn?". Medical decision making : an international journal of the Society for Medical Decision Making. 14 (2): 175–179. doi:10.1177/0272989X9401400210. PMID 8028470.
^ Pewsner, D; Battaglia, M; Minder, C; Marx, A; Bucher, HC; Egger, M (Jul 24, 2004). "Ruling a diagnosis in or out with "SpPIn" and "SnNOut": a note of caution". BMJ (Clinical research ed.). 329 (7459): 209–13. doi:10.1136/bmj.329.7459.209. PMC 487735. PMID 15271832.
^ Gale, SD; Perkel, DJ (Jan 20, 2010). "A basal ganglia pathway drives selective auditory responses in songbird dopaminergic neurons via disinhibition". The Journal of neuroscience : the official journal of the Society for Neuroscience. 30 (3): 1027–1037. doi:10.1523/JNEUROSCI.3585-09.2010. PMC 2824341. PMID 20089911.
^ Macmillan, Neil A.; Creelman, C. Douglas (15 September 2004). Detection Theory: A User's Guide. Psychology Press. p. 7. ISBN 978-1-4106-1114-7.
^ Lin, Jennifer S.; Piper, Margaret A.; Perdue, Leslie A.; Rutter, Carolyn M.; Webber, Elizabeth M.; O'Connor, Elizabeth; Smith, Ning; Whitlock, Evelyn P. (21 June 2016). "Screening for Colorectal Cancer". JAMA. 315 (23): 2576–2594. doi:10.1001/jama.2016.3332. ISSN 0098-7484. PMID 27305422.
^ Bénard, Florence; Barkun, Alan N.; Martel, Myriam; Renteln, Daniel von (7 January 2018). "Systematic review of colorectal cancer screening guidelines for average-risk adults: Summarizing the current global recommendations". World Journal of Gastroenterology. 24 (1): 124–138. doi:10.3748/wjg.v24.i1.124. PMC 5757117. PMID 29358889.
^ Online calculator of confidence intervals for predictive parameters

External links

Vassar College's Sensitivity/Specificity Calculator

Commercial

NCSS (statistical software) includes sensitivity and specificity analysis.

Cite error: There are <ref group=lower-alpha> tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).

[Fawcelt2006-1] Fawcelt, Tom (2006). "An Introduction to ROC Analysis". Pattern Recognition Letters. 27 (8): 861–874. doi:10.1016/j.patrec.2005.10.010.

[cebm-2] "SpPins and SnNouts". Centre for Evidence Based Medicine (CEBM). Retrieved 26 December 2013.

[3] Mangrulkar, Rajesh. "Diagnostic Reasoning I and II". Retrieved 24 January 2012.

[4] Michigan State University Evidence Based Medicine resource

[5] Emory University Medical School Evidence Based Medicine course

[pmid8028462-6] Baron, JA (Apr–Jun 1994). "Too bad it isn't true...". Medical decision making : an international journal of the Society for Medical Decision Making. 14 (2): 107. doi:10.1177/0272989X9401400202. PMID 8028462.

[pmid8028470-7] Boyko, EJ (Apr–Jun 1994). "Ruling out or ruling in disease with the most sensitive or specific diagnostic test: short cut or wrong turn?". Medical decision making : an international journal of the Society for Medical Decision Making. 14 (2): 175–179. doi:10.1177/0272989X9401400210. PMID 8028470.

[pmid15271832-8] Pewsner, D; Battaglia, M; Minder, C; Marx, A; Bucher, HC; Egger, M (Jul 24, 2004). "Ruling a diagnosis in or out with "SpPIn" and "SnNOut": a note of caution". BMJ (Clinical research ed.). 329 (7459): 209–13. doi:10.1136/bmj.329.7459.209. PMC 487735. PMID 15271832.

[pmid20089911-9] Gale, SD; Perkel, DJ (Jan 20, 2010). "A basal ganglia pathway drives selective auditory responses in songbird dopaminergic neurons via disinhibition". The Journal of neuroscience : the official journal of the Society for Neuroscience. 30 (3): 1027–1037. doi:10.1523/JNEUROSCI.3585-09.2010. PMC 2824341. PMID 20089911.

[MacmillanCreelman2004-10] Macmillan, Neil A.; Creelman, C. Douglas (15 September 2004). Detection Theory: A User's Guide. Psychology Press. p. 7. ISBN 978-1-4106-1114-7.

[11] Lin, Jennifer S.; Piper, Margaret A.; Perdue, Leslie A.; Rutter, Carolyn M.; Webber, Elizabeth M.; O'Connor, Elizabeth; Smith, Ning; Whitlock, Evelyn P. (21 June 2016). "Screening for Colorectal Cancer". JAMA. 315 (23): 2576–2594. doi:10.1001/jama.2016.3332. ISSN 0098-7484. PMID 27305422.

[12] Bénard, Florence; Barkun, Alan N.; Martel, Myriam; Renteln, Daniel von (7 January 2018). "Systematic review of colorectal cancer screening guidelines for average-risk adults: Summarizing the current global recommendations". World Journal of Gastroenterology. 24 (1): 124–138. doi:10.3748/wjg.v24.i1.124. PMC 5757117. PMID 29358889.

[14] Online calculator of confidence intervals for predictive parameters

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