Screening arithmetic without the confusion
Sensitivity, specificity and predictive values from one 2x2 table, plus why prevalence ruins a good test.
One table, every answer
| 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 single instruction prevents most errors.
- Sensitivity = a / (a + c) — of those with disease, how many the test catches
- Specificity = d / (b + d) — of those without disease, how many the test clears
- Positive predictive value = a / (a + b) — of those testing positive, how many truly have it
- Negative predictive value = d / (c + d)
The property that matters most
Sensitivity and specificity are properties of the test. They do not change with prevalence. Predictive values are properties of the test in a population, and they move sharply with prevalence.
Take a test with 99% sensitivity and 99% specificity applied to 10,000 people:
| Prevalence | True positives | False positives | PPV |
|---|---|---|---|
| 10% | 990 | 90 | 92% |
| 1% | 99 | 99 | 50% |
| 0.1% | ~10 | ~100 | 9% |
The test never changed. Screening a low-prevalence population with an excellent test still produces mostly false alarms — which is the entire argument against indiscriminate mass screening.
Choosing a cut-off
Moving the cut-off trades one error for the other:
- Lower the threshold → more positives → sensitivity ↑, specificity ↓
- Raise the threshold → fewer positives → specificity ↑, sensitivity ↓
Use a sensitive test when missing the disease is dangerous and follow-up is cheap (screening). Use a specific test when a false positive is harmful (confirmation, before starting toxic treatment). This is why a screening test is followed by a confirmatory one — the classic HIV ELISA then Western blot / NAT sequence.
SnNout: a Snsitive test, when Negative, rules out. SpPin: a Specific test, when Positive, rules in.
Two more that get asked
- Likelihood ratio positive = sensitivity / (1 − specificity). Unlike predictive values, it is prevalence-independent and can be applied to any individual patient's pre-test odds.
- Lead-time bias and length-time bias make screened patients appear to survive longer without anyone living a day longer. Any question that says "screening improved 5-year survival" is inviting you to name one of these.
Anchor
Columns give you test properties, rows give you patient answers, and prevalence only ever touches the rows.
Questions this note answers
- EpidemiologyA screening test is applied to 1000 people, of whom 100 truly have the disease. The test is positive in 90 of the diseased and in 180 of the non-diseased. What is the sensitivity o…
Sources
- Park's Textbook of Preventive and Social Medicine, 27e