Sensitivity and Specificity

We sometimes use tests to try to detect diseases, setting a cutoff value on the test to indicate whether or not the individual has the disease. Of course, our tests are not perfect, so test results may not match reality/truth. As shown in the table below, a person can either truly have (+) or not have (-) the disease, and the test can either say the person has (+) or do not have (-) the disease. We use the terms positive predictive power, negative predictive power, sensitivity, and specificity to describe how well the test decision matches reality in a population of individuals.

Has Disease No disease
Test=true True positive False positive
Test = healthy False Negative True Negative

Positive Predictive Power (PPP) = True Positive divided by [True Positive + False Positive]

  • The reference value (i.e., the demoninator) for PPP is TOTAL WHO TEST POSITIVE FOR DISEASE.

Negative Predictive Power (NPP) = True Negative divided by [True Negative plus False Negative]

  • The reference value for NPP is TOTAL WHO TEST NEGATIVE FOR DISEASE.

Sensitivity – ability to pick out individuals with disease; the True Positive rate of a test

Sensitivity = True Positive / [True positive + False Negative]

  • The reference value for sensitivity is TOTAL WHO REALLY HAVE DISEASE.

Specificity – ability to pick out individuals without disease; the True Negative rate of a test

Specificity = True Negative / [True Negative + False Positive]

  • The reference value for specificity is TOTAL WHO REALLY DO NOT HAVE DISEASE.

Base Rate Summary

The base rates of the disease in the general population affect predictive power (Bayesian analysis).

Positive predictive power (PPP) – probability that an individual who receives an abnormal test score actually has the disorder of interest

  • PPP is determined by the test’s sensitivity and specificity in the context of base rate info
  • Even w/ 90% sensitivity and specificity, if base rate is relatively low (condition is rare), the majority of individuals who exhibit that sign or test score will not have the condition.
    • In unreferred population of 1,000 children and 4% base rate for ADHD, 40 children are expected to have ADHD.
    • Using a test w/ 90% sensitivity and specificity, only 27% of children who receive an abnormal score on the test can be expected to actually have disorder. That’s more than a random guess (which would yield only 4% success in identifying kids with the disorder), but it’s still not great.
ADHD present No Adhd
Abnormal score 36 96 PPP= 36/(26+96) = 27%
Normal score 4 864 NPP= 864/{864+4) = 99%

Sensitivity: 36/(36+4) = 90% Specificity: 864/(864+96) = 90%

  • As illustrated in the chart below, the best Positive Predictive Power comes with high prevalence and high specificity.
50% 43% 10% 1% 0.1% 0.04%
90% 91 88 53 9 1 0.4
95% 95 94 69 17 2 0.8
99% 99 99 92 50 9 4
99.9% 99.9 99.9 99 91 50 29