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# Specificity (tests)

The specificity is a statistical measure of how well a binary classification test correctly identifies the negative cases, or those cases that do not meet the condition under study. For example, given a medical test that determines if a person has a certain disease, the specificity of the test to the disease is the probability that the test indicates `negative' if the person does not have the disease.

That is, the specificity is the proportion of true negatives of all negative cases in the population. It is a parameter of the test.

High specificity is important when the treatment or diagnosis is harmful to the patient mentally and/or physically.

## 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

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

A specificity of 100% means that the test recognizes all healthy people as healthy. The maximum is trivially achieved by a test that claims everybody healthy regardless of the true condition. Therefore, the specificity alone does not tell us how well the test recognizes positive cases. We also need to know the sensitivity of the test to the class, or equivalently, the specificities to the other classes.

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

Specificity is sometimes confused with the precision or the positive predictive value, both of which refer to the fraction of returned positives that are true positives. The distinction is critical when the classes are different sizes. A test with very high specificity can have very low precision if there are far more true negatives than true positives, and vice versa.