# Sensitivity and specificity

(Redirected from Prevalence Threshold)
Sensitivity and specificity

Sensitivity and specificity are statistical measures of the performance of a binary classification test that are widely used in medicine:

• Sensitivity measures the proportion of true positives that are correctly identified (e.g., the proportion of those who truly have a condition (affected) who are correctly identified as having some condition).
• Specificity measures the proportion of true negatives (e.g. the proportion of those who truly do not have the condition (unaffected) who are correctly identified as not having the condition).

The terms "true positive" and "true negative" do not refer to benefit. They refer to the presence or absence of a condition. If the condition is a disease, "true positive" means "diseased" and "true negative" means "not diseased".

In medical testing medical tests, sensitivity is a measure of the extent to which true positives (people who truly have a condition) are identified by the test. Sensitivity is defined as the true positive "rate" (TP). This is the percentage (or proportion) of people who truly have the condition that are identified by the test as having the condition. People who truly have the condition but have a negative test for the condition are falsely negative. This is called the false negative "rate" (FN). This is the percentage (or proportion) of people who truly have the condition that are identified (falsely) as not having the condition. The true positive "rate" (TP) and the false negative "rate" (FN) equal 1.0. That is, everyone with a positive test is either a true positive (they tested positive and they truly have the condition) or a false negative (they truly have the disease but they tested negative).

In a diagnostic test, specificity is the extent to which true negatives (people who truly DO NOT have a condition) are correctly identified as not having the condition. Specificity is defined as the true negative "rate" (TN). This is the percentage (or proportion) of people who truly DO NOT have the condition that are identified by the test as NOT having the condition. People who have truly DO NOT have the condition but have a positive test for the condition are falsely positive on the test. This is called the false positive rate" (FP). This is the percentage (or proportion) of people who truly DO NOT have the condition who are identified (falsely) as not having the condition. The true negative "rate" (TN) and the false positive "rate" (FP) equal 1.0.

In a "good" diagnostic test (one that attempts to identify with precision people who have the condition), the false positives (FP) should be very low. That is, people who are identified a having a condition should be highly likely to truly have the condition. This is because people who are identified as having a condition (but do not have it, in truth) may be subjected to: more testing (which could be expensive); stigma (e.g. HIV positive test); anxiety (e.g., I'm sick...I might die).

In testing with one test, everyone with a positive test is either a true positive (they tested positive and they truly have the condition) or a false positive (they tested positive but they truly do not have the condition). Everyone with a negative test is either a false negative (they tested negative but they truly have the condition) or a true negative (they tested negative and they truly do not have the condition).

A test done in a person who has symptoms of a condition (the prior probability of disease is high) and has high sensitivity--0.98 or more--is highly likely to identify a true positive--a person who truly has the condition.

A test done in a person who does not have symptoms of a condition (the prior probability of disease is low) will have a high rate of false positives (FP).

For all testing, both diagnostic and screening, there is a trade-off between sensitivity and specificity.

The terms "sensitivity" and "specificity" were introduced by American biostatistician Jacob Yerushalmy in 1947.[1]

## Definitions

In the terminology true/false positive/negative, true or false refers to the assigned classification being correct or incorrect, while positive or negative refers to assignment to the positive or the negative category.

 condition positive (P) the number of real positive cases in the data condition negative (N) the number of real negative cases in the data 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, recall, hit rate, or true positive rate (TPR) ${\displaystyle \mathrm {TPR} ={\frac {\mathrm {TP} }{\mathrm {P} }}={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FN} }}=1-\mathrm {FNR} }$ specificity, selectivity or true negative rate (TNR) ${\displaystyle \mathrm {TNR} ={\frac {\mathrm {TN} }{\mathrm {N} }}={\frac {\mathrm {TN} }{\mathrm {TN} +\mathrm {FP} }}=1-\mathrm {FPR} }$ precision or positive predictive value (PPV) ${\displaystyle \mathrm {PPV} ={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FP} }}=1-\mathrm {FDR} }$ negative predictive value (NPV) ${\displaystyle \mathrm {NPV} ={\frac {\mathrm {TN} }{\mathrm {TN} +\mathrm {FN} }}=1-\mathrm {FOR} }$ miss rate or false negative rate (FNR) ${\displaystyle \mathrm {FNR} ={\frac {\mathrm {FN} }{\mathrm {P} }}={\frac {\mathrm {FN} }{\mathrm {FN} +\mathrm {TP} }}=1-\mathrm {TPR} }$ fall-out or false positive rate (FPR) ${\displaystyle \mathrm {FPR} ={\frac {\mathrm {FP} }{\mathrm {N} }}={\frac {\mathrm {FP} }{\mathrm {FP} +\mathrm {TN} }}=1-\mathrm {TNR} }$ false discovery rate (FDR) ${\displaystyle \mathrm {FDR} ={\frac {\mathrm {FP} }{\mathrm {FP} +\mathrm {TP} }}=1-\mathrm {PPV} }$ false omission rate (FOR) ${\displaystyle \mathrm {FOR} ={\frac {\mathrm {FN} }{\mathrm {FN} +\mathrm {TN} }}=1-\mathrm {NPV} }$ Prevalence Threshold (PT) ${\displaystyle PT={\frac {{\sqrt {TPR(-TNR+1)}}+TNR-1}{(TPR+TNR-1)}}}$ Threat score (TS) or critical success index (CSI) ${\displaystyle \mathrm {TS} ={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FN} +\mathrm {FP} }}}$ accuracy (ACC) ${\displaystyle \mathrm {ACC} ={\frac {\mathrm {TP} +\mathrm {TN} }{\mathrm {P} +\mathrm {N} }}={\frac {\mathrm {TP} +\mathrm {TN} }{\mathrm {TP} +\mathrm {TN} +\mathrm {FP} +\mathrm {FN} }}}$ balanced accuracy (BA) ${\displaystyle \mathrm {BA} ={\frac {TPR+TNR}{2}}}$ F1 score is the harmonic mean of precision and sensitivity ${\displaystyle \mathrm {F} _{1}=2\cdot {\frac {\mathrm {PPV} \cdot \mathrm {TPR} }{\mathrm {PPV} +\mathrm {TPR} }}={\frac {2\mathrm {TP} }{2\mathrm {TP} +\mathrm {FP} +\mathrm {FN} }}}$ Matthews correlation coefficient (MCC) ${\displaystyle \mathrm {MCC} ={\frac {\mathrm {TP} \times \mathrm {TN} -\mathrm {FP} \times \mathrm {FN} }{\sqrt {(\mathrm {TP} +\mathrm {FP} )(\mathrm {TP} +\mathrm {FN} )(\mathrm {TN} +\mathrm {FP} )(\mathrm {TN} +\mathrm {FN} )}}}}$ Fowlkes–Mallows index (FM) ${\displaystyle \mathrm {FM} ={\sqrt {{\frac {TP}{TP+FP}}\cdot {\frac {TP}{TP+FN}}}}={\sqrt {PPV\cdot TPR}}}$ informedness or bookmaker informedness (BM) ${\displaystyle \mathrm {BM} =\mathrm {TPR} +\mathrm {TNR} -1}$ markedness (MK) or deltaP ${\displaystyle \mathrm {MK} =\mathrm {PPV} +\mathrm {NPV} -1}$ Sources: Fawcett (2006),[2] Powers (2011),[3] Ting (2011),[4], CAWCR[5] D. Chicco & G. Jurman (2020),[6] Tharwat (2018).[7]

### Application to screening study

Imagine a study evaluating a 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 (classifying the person as having the disease) or negative (classifying the person as not having 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 identified 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

### Confusion matrix

Consider a group with P positive instances and N negative instances of some condition. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix, as follows:

 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 conditionpositive True positive Positive predictive value (PPV), Precision = Σ True positive/Σ Predicted condition positive False discovery rate (FDR) = Σ False positive/Σ Predicted condition positive Predicted conditionnegative True negative False omission rate (FOR) = Σ False negative/Σ Predicted condition negative Negative predictive value (NPV) = Σ True negative/Σ Predicted condition negative True positive rate (TPR), Recall, Sensitivity, 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− F1 score = 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

Consider the example of a medical test for diagnosing a condition. Sensitivity refers to the test's ability to correctly detect ill patients who do have the condition.[8] In the example of a medical test used to identify a condition, the sensitivity (sometimes also named the detection rate in a clinical setting) of the test is the proportion of people who test positive for the disease among those who have the disease. Mathematically, this can be expressed as:

{\displaystyle {\begin{aligned}{\text{sensitivity}}&={\frac {\text{number of true positives}}{{\text{number of true positives}}+{\text{number of false negatives}}}}\\[8pt]&={\frac {\text{number of true positives}}{\text{total number of sick individuals in population}}}\\[8pt]&={\text{probability of a positive test given that the patient has the disease}}\end{aligned}}}

A negative result in a test with high sensitivity is useful for ruling out disease.[8] 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. However, a positive result in a test with high sensitivity is not necessarily useful for ruling in disease. Suppose a 'bogus' test kit is designed to always give a positive reading. When used on diseased patients, all patients test positive, giving the test 100% sensitivity. However, sensitivity 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 detecting 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).

## Specificity

Consider the example of a medical test for diagnosing a disease. Specificity relates to the test's ability to correctly reject healthy patients without a condition. Specificity of a test is the proportion of who truly do not have the condition who test negative for the condition. Mathematically, this can also be written as:

{\displaystyle {\begin{aligned}{\text{specificity}}&={\frac {\text{number of true negatives}}{{\text{number of true negatives}}+{\text{number of false positives}}}}\\[8pt]&={\frac {\text{number of true negatives}}{\text{total number of well individuals in population}}}\\[8pt]&={\text{probability of a negative test given that the patient is well}}\end{aligned}}}

A positive result in a test with high specificity is useful for ruling in disease. The test rarely gives positive results in healthy patients. A positive result signifies a high probability of the presence of disease.[9]

A test with a higher specificity has a lower type I error rate.

## Medical examples

In medical diagnosis, 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 completely 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.[10] 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. These concepts are illustrated graphically in this applet Bayesian clinical diagnostic model which show the positive and negative predictive values as a function of the prevalence, the sensitivity and specificity.

### Prevalence threshold

The relationship between a screening tests' positive predictive value, and its target prevalence, is proportional - though not linear in all but a special case. In consequence, there is a point of local extrema and maximum curvature defined only as a function of the sensitivity and specificity beyond which the rate of change of a test's positive predictive value drops at a differential pace relative to the disease prevalence. Using differential equations, this point was first defined by Balayla et al. [11] and is termed the prevalence threshold (${\displaystyle \phi _{e}}$). The equation for the prevalence threshold is given by the following formula, where a = sensitivity and b = specificity:

${\displaystyle \phi _{e}={\frac {{\sqrt {a(-b+1)}}+b-1}{(a+b-1)}}}$

Where this point lies in the screening curve has critical implications for clinicians and the interpretation of positive screening tests in real time.

### 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.[12][13] This has led to the widely used mnemonics SPPIN and SNNOUT, 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.[14][15][16]

The tradeoff between specificity and sensitivity is explored in ROC analysis as a trade off between TPR and FPR (that is, recall and fallout).[17] Giving them equal weight optimizes informedness = specificity+sensitivity-1 = TPR-FPR, the magnitude of which gives the probability of an informed decision between the two classes (>0 represents appropriate use of information, 0 represents chance-level performance, <0 represents perverse use of information).[18]

### 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 ${\displaystyle \mu _{S}}$ and ${\displaystyle \sigma _{S}}$, and ${\displaystyle \mu _{N}}$ and ${\displaystyle \sigma _{N}}$, respectively, d' is defined as:

${\displaystyle d'={\frac {\mu _{S}-\mu _{N}}{\sqrt {{\frac {1}{2}}(\sigma _{S}^{2}+\sigma _{N}^{2})}}}}$ [19]

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),[20]

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%
 Patients with bowel cancer(as confirmed on endoscopy) Condition positive Condition negative Prevalence= (TP+FN)/Total_Population= (20+10)/2030≈1.48% Accuracy (ACC) = (TP+TN)/Total_Population= (20+1820)/2030≈90.64% Fecaloccultbloodscreentestoutcome Testoutcomepositive True positive(TP) = 20(2030 x 1.48% x 67%) False positive(FP) = 180(2030 x (100 - 1.48%) x (100 - 91%)) Positive predictive value (PPV), Precision= TP / (TP + FP)= 20 / (20 + 180)= 10% False discovery rate (FDR)= FP/(TP+FP)= 180/(20+180)= 90.0% Testoutcomenegative False negative(FN) = 10(2030 x 1.48% x (100 - 67%)) True negative(TN) = 1820(2030 x (100 -1.48%) x 91%) False omission rate (FOR)= FN / (FN + TN)= 10 / (10 + 1820)≈ 0.55% Negative predictive value (NPV)= TN / (FN + TN)= 1820 / (10 + 1820)≈ 99.45% TPR, Recall, Sensitivity= TP / (TP + FN)= 20 / (20 + 10)≈ 66.7% False positive rate (FPR),Fall-out, probability of false alarm = FP/(FP+TN)= 180/(180+1820)=9.0% Positive likelihood ratio (LR+) = TPR/FPR= (20/30)/(180/2000)≈7.41 Diagnostic odds ratio (DOR) = LR+/LR−≈20.2 F1 score = 2 · Precision · Recall/Precision + Recall≈0.174 False negative rate (FNR), Miss rate = FN/(TP+FN)= 10/(20+10) ≈ 33.3% Specificity, Selectivity, True negative rate (TNR)= TN / (FP + TN)= 1820 / (180 + 1820)= 91% Negative likelihood ratio (LR−) = FNR/TNR= (10/30)/(1820/2000)≈0.366

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
• Likelihood ratio positive = sensitivity / (1 − specificity) = 0.67 / (1 − 0.91) = 7.4
• Likelihood ratio negative = (1 − sensitivity) / specificity = (1 − 0.67) / 0.91 = 0.37
• Prevalence threshold = ${\displaystyle PT={\frac {{\sqrt {TPR(-TNR+1)}}+TNR-1}{(TPR+TNR-1)}}}$ = 0.19 => 19.1%

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 ruling out cancer as the cause of gastrointestinal symptoms or 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%).[23]

## Terminology in information retrieval

In information retrieval, the positive predictive value is called precision, and sensitivity is called recall. Unlike the Specificity vs Sensitivity tradeoff, these measures are both independent of the number of true negatives, which is generally unknown and much larger than the actual numbers of relevant and retrieved documents. This assumption of very large numbers of true negatives versus positives is rare in other applications.[18]

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

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

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.

## Notes

1. ^ There are advantages and disadvantages for all medical screening tests. Clinical practice guidelines, such as those for colorectal cancer screening, describe these risks and benefits.[21][22]

## References

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23. ^