Sensitivity (tests)

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Sensitivity is a statistical measure of how well a binary classification test correctly identifies a condition, whether this be medical screening tests picking up on a disease or quality control in factories deciding if a new product is good enough to be sold.

The results of the screening test are compared to some absolute (Gold standard); for example, for a medical test to determine if a person has a certain disease, the sensitivity to the disease is the probability that if the person has the disease, the test will be positive.

		Condition (e.g. Disease) As determined by "Gold" standard
		True	False
Test outcome	Positive	True Positive	False Positive	→ Positive Predictive Value
Test outcome	Negative	False Negative	True Negative	→ Negative Predictive Value
		↓ Sensitivity	↓ Specificity

That is, the sensitivity is the proportion of true positives of all positive cases in the population. It is a parameter of the test.

${\rm sensitivity}=\frac{\rm number\ of\ True\ Positives}{{\rm number\ of\ True\ Positives}+{\rm number\ of\ False\ Negatives}}.$

A sensitivity of 100% means that the test recognizes all sick people as such.

Sensitivity alone does not tell us how well the test predicts other classes (that is, about the negative cases). In the binary classification, as illustrated above, this is the corresponding specificity test, or equivalently, the sensitivity for the other classes.

Sensitivity is not the same as the 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, the options are to exclude indeterminate samples from analyses (but the number of exclusions 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 may therefore underestimate it).

A worked example: the Fecal occult blood (FOB) screen test is used in 203 people to look for bowel cancer:

		Patients with Bowel cancer (as confirmed on endocopy)
		True	False
FOB test	Positive	2	18
FOB test	Negative	1	182

Sensitivity is: 2 / ( 2 + 1 ) = 2 / 3 = 0.66 = 66%, meaning that two thirds of cases are identified.

Specificity is: 182 / (18 + 182 ) = 182 / 200 = 0.91 = 91%

Positive Predictive Value is: 2 / ( 2 + 18 ) = 2 / 20 = 0.1 = 10%

Negative Predictive Value is: 182 / ( 1 + 182 ) = 182 / 183 = 0.9945 = 99.45%, meaning a negative test result is likely to correctly exclude cancer.

[edit] Terminology in information retrieval

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

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

$F = 2 \times {\rm precision} \times {\rm recall} / ({\rm precision} + {\rm recall}).$

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.