Sensitivity, specificity and predictive values
Deriving sensitivity, specificity, PPV and NPV from one 2x2 table, why predictive values chase prevalence, and when to pick a sensitive versus a specific test.
One 2x2 table gives every value
| Disease present | Disease absent | |
|---|---|---|
| Test positive | True positive (a) | False positive (b) |
| Test negative | False negative (c) | True negative (d) |
Read down the columns for sensitivity and specificity; read across the rows for predictive values. That one instruction stops most mistakes.
- Sensitivity = a / (a + c) — of the diseased, how many test positive (detects disease)
- Specificity = d / (b + d) — of the healthy, how many test negative (excludes disease)
- PPV = a / (a + b) — of positives, how many are truly diseased
- NPV = d / (c + d) — of negatives, how many are truly healthy
Prevalence is the hidden variable
Sensitivity and specificity are intrinsic to the test — they do not move with prevalence. PPV and NPV do. As prevalence rises, PPV rises and NPV falls; in a low-prevalence population even an excellent test throws mostly false positives, so its PPV collapses. This is why mass screening of rare disease generates so much needless follow-up.
| Setting | Prevalence | PPV | NPV |
|---|---|---|---|
| Screening general population | Low | Low | High |
| Symptomatic clinic / high-risk | High | High | Lower |
Choosing a sensitive vs a specific test — SnNout / SpPin
| Want to... | Choose | Mnemonic |
|---|---|---|
| Rule out disease (screening, dangerous-to-miss) | High sensitivity | SnNout — a Snsitive test, when Negative, rules out |
| Rule in disease (confirm before risky treatment) | High specificity | SpPin — a Specific test, when Positive, rules in |
Use a sensitive test first (screening, e.g. ELISA for HIV) to catch every case, then a specific test (confirmation, e.g. Western blot) to weed out false positives. A highly sensitive test has few false negatives; a highly specific test has few false positives.
Anchor: Sensitivity/specificity belong to the test; PPV/NPV belong to the population. Screen with a sensitive test to rule out, confirm with a specific test to rule in.
Sources
- Park's Textbook of Preventive and Social Medicine, 27e
- Gordis Epidemiology, 6e