# Sensitivity (tests)

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Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]; Assistant Editor(s)-In-Chief: Kristin Feeney, B.S.

## Overview

Sensitivity refers to the statistical measure of how well a binary classification test correctly identifies a condition. In epidemiology, this is referred to as medical screening tests that detect preclinical disease. In quality control, this is referred to as a recall rate, whereby factories decided if a new product is at an acceptable level to be mass-produced and sold for distribution.

## Critical Considerations

• 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.
• The sensitivity is the proportion of true positives of all diseased cases in the population. It is a parameter of the test.
• High sensitivity is required when early diagnosis and treatment is beneficial, and when the disease is infectious.

## Worked Example

Relationships among terms
 Condition(as determined by "Gold standard") True False Testoutcome Positive True Positive False Positive(Type I error, P-value) → Positive predictive value Negative False Negative(Type II error) True Negative → Negative predictive value ↓Sensitivity ↓Specificity
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 endoscopy) True False ? FOBtest Positive TP = 2 FP = 18 = TP / (TP + FP)= 2 / (2 + 18)= 2 / 20 ≡ 10% Negative FN = 1 TN = 182 = TN / (TN + FN)182 / (1 + 182)= 182 / 183 ≡ 99.5% ↓= TP / (TP + FN)= 2 / (2 + 1)= 2 / 3 ≡ 66.67% ↓= TN / (FP + TN)= 182 / (18 + 182)= 182 / 200 ≡ 91%

Related calculations

• False positive rate (α) = FP / (FP + TN) = 18 / (18 + 182) = 9% = 1 - specificity
• False negative rate (β) = FN / (TP + FN) = 1 / (2 + 1) = 33% = 1 - sensitivity
• Power = 1 − β

Hence with large numbers of false positives and few false negatives, a positive FOB screen test is in itself poor at confirming cancer (PPV=10%) and further investigations must be undertaken, it will though pickup 66.7% of all cancers (the sensitivity). However as a screening test, a negative result is very good at reassuring that a patient does not have cancer (NPV=99.5%) and at this initial screen correctly identifies 91% of those who do not have cancer (the specificity).

## Definition

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

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).

## 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:

$\rm precision} \times {\rm recall}) / ({\rm precision} + {\rm recall$

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.