A screening test, base rates and a misleading leaflet
A town clinic offers a screening test for a condition that affects 2% of the 5,000 people who take the test. The test gives a positive result for 90% of people who have the condition. It also gives a positive result for 5% of people who do not have the condition. A leaflet produced by the clinic says: 'The test is 90% accurate, so if your result is positive there is a 90% chance that you have the condition.'
1. Construct a two-way table (or tree diagram) showing how many of the 5,000 people fall into each of the four possible groups.
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Of 5,000 people, 2% have the condition and 98% do not:
- Have the condition: 0.02 × 5,000 = 100
- Do not have it: 0.98 × 5,000 = 4,900
Apply the test rates:
- Of the 100 with the condition, 90% test positive: 90 true positives and 10 false negatives.
- Of the 4,900 without it, 5% test positive: 0.05 × 4,900 = 245 false positives and 4,655 true negatives.
| Test positive | Test negative | Total | |
|---|---|---|---|
| Has condition | 90 | 10 | 100 |
| No condition | 245 | 4,655 | 4,900 |
| Total | 335 | 4,665 | 5,000 |
Check: 90 + 10 + 245 + 4,655 = 5,000.
2. Find the probability that a person who tests positive actually has the condition, and comment on the leaflet's claim.
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There are 335 positive results in total, of which only 90 come from people who really have the condition:
P(has condition | tests positive) = 90 / 335 ≈ 0.269, about 27%.
Comment on the leaflet: the claim of 90% is wrong. 90% is the probability of a positive test given that you have the condition - the opposite conditional probability to the one a patient cares about. The two are very different whenever the condition is rare.
Reason: only 100 people out of 5,000 have the condition, so even a small 5% false-positive rate applied to the 4,900 healthy people produces 245 false positives - far more than the 90 genuine cases. Over two-thirds of the positive results are therefore false alarms. The leaflet's reasoning ignores the base rate (the prevalence of 2%).
3. The clinic wants the probability that a positive result is genuine to be above 50%. Using the figures, suggest a change that would achieve this and show that it works.
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Option 1 - make the test more specific (fewer false positives). Keeping 90 true positives, we need 90 / (90 + F) > 0.5, so F < 90. The false-positive rate must satisfy 0.05 × 4,900 = 245 being reduced to below 90, i.e. a false-positive rate below 90 / 4,900 ≈ 1.8%. At a 1.8% rate there would be 88 false positives, giving 90 / 178 ≈ 0.51.
Option 2 - test only people who already have symptoms or risk factors, so the prevalence is higher. Solving 0.9p / (0.9p + 0.05(1 − p)) > 0.5 gives p > 0.0526, so the condition needs to be present in more than about 5.3% of those tested - more than two and a half times the current 2%.
Judgement: Option 2 is usually cheaper and more practical for a clinic, since it means changing who is tested rather than developing a new test; Option 1 is more reliable but may not be achievable. Either change raises the chance that a positive result is genuine, and the leaflet should in any case quote the 27% figure rather than 90%.