Rapid Screening at MediCorp: What Does One Positive Result Mean?
MediCorp screens all 10,000 job applicants for a treatable condition using a rapid test. The condition affects 2% of applicants. The test has sensitivity 95% (positive in 95% of people who have the condition) and specificity 90% (negative in 90% of people who do not). Applicants with a positive result are currently excluded from hiring pending a confirmatory laboratory test, which takes three weeks and costs €300. HR proposes skipping the laboratory test to speed up hiring and instead excluding anyone with a positive rapid result. The occupational health physician objects, noting that the rapid test was validated on a hospital population with a much higher prevalence. The company has budgeted €60,000 for confirmatory testing this year and expects 10,000 applicants. A second, independent rapid test is available at €20 per person with the same sensitivity and specificity; its errors are independent of the first test's errors.
1. Among applicants with one positive rapid result, what proportion actually have the condition? Give the expected counts on 10,000 applicants.
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Expected counts (prevalence 2%)
- Have condition: 200. Test positive (95%): 190 true positives; 10 false negatives.
- Do not have condition: 9,800. Test positive (10% false-positive rate): 980 false positives; 8,820 true negatives.
Total positives = 190 + 980 = 1,170.
Positive predictive value 190 / 1,170 = 16.2%.
So about five in six positive results are false. The base rate dominates: because the condition is rare, the 10% false-positive rate applied to 9,800 healthy applicants produces far more positives than the 95% sensitivity applied to 200 affected applicants. The physician's point about prevalence is decisive: in a high-prevalence hospital population the same test would have a much higher predictive value, but that figure cannot be transferred to a 2% population.
2. If a second, independent rapid test is given only to those who test positive the first time, what is the probability of the condition after two positives, and what would this strategy cost compared with laboratory confirmation of every positive?
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Two independent positives
- True positives both times: 0.95 × 0.95 × 200 = 180.5.
- False positives both times: 0.10 × 0.10 × 9,800 = 98.
- Total double positives ≈ 278.5; predictive value = 180.5/278.5 = 64.8%.
Costs
- Second test on 1,170 first positives: 1,170 × €20 = €23,400.
- Laboratory confirmation of all 1,170 positives: 1,170 × €300 = €351,000 - nearly six times the €60,000 budget, which covers only 200 confirmations.
- Two-test triage then laboratory: €23,400 + 278.5 × €300 ≈ €23,400 + €83,550 = €106,950.
Even the triage route exceeds the budget, though it cuts laboratory volume by about 76%. The programme as designed is unfunded, and the budget must be raised, the policy changed, or screening dropped.
3. Evaluate HR's proposal to exclude anyone with a single positive rapid result, using the numbers above.
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Under the proposal, of 1,170 excluded applicants, 980 (83.8%) do not have the condition. The policy would also hire 10 people who do have it, since 5% of the 200 affected applicants test negative.
So the proposal is both unfair and ineffective: it imposes a serious consequence on a large majority of healthy people while still missing some affected ones. The three-week laboratory delay is inconvenient, but removing it by acting on a 16% predictive value is not a defensible trade.
Better options, in order of strength:
- Do not use the rapid test as a hiring gate at this prevalence; screen for the condition through a clinical pathway instead.
- If screening continues, use two independent rapid tests (64.8% predictive value) and confirm only double positives by laboratory test.
- Fund the laboratory budget to match the policy: at €300 per confirmation, 1,170 positives require €351,000, not €60,000.
HR's error is treating a test result as a diagnosis; the correct decision rule must incorporate the 2% base rate.