A Two-Step Screening Programme on a Fixed Budget
A district health office is considering a one-time screening programme for a chronic condition in 10,000 adults aged 40–70. The condition is asymptomatic in its early stage. The best local estimate of prevalence is 4%. The first test costs $3 per person, has sensitivity 90% and specificity 92%. People who test positive are offered a confirmatory test costing $20, with sensitivity 95% and specificity 98% among those who test positive on the first test. The office has a budget of $60,000 for testing and cannot fund lifelong treatment for false positives. A patient asks what a positive first test means. The programme manager wants to know whether the budget is sufficient and whether the two-step strategy is worth it.
1. Build the expected 2 × 2 table for 10,000 people for the first test and calculate its positive predictive value. Explain why the PPV is low even though sensitivity and specificity look good.
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Diseased: 4% × 10,000 = 400. Non-diseased: 9,600. True positives: 90% × 400 = 360. False negatives: 40. False positives: 8% × 9,600 = 768. True negatives: 8,832. First-test positives: 360 + 768 = 1,128. PPV = 360/1,128 = 0.319, or about 32%.
The PPV is modest because prevalence is low. Most people tested do not have the condition, so even a specificity of 92% produces many false positives from the large healthy group: 768 false positives versus 360 true positives. Sensitivity and specificity describe the test; PPV describes the patient in front of you and depends on prevalence. A positive first test therefore means roughly a one-in-three chance of disease, not a diagnosis. The negative predictive value is high, but some diseased people (40) are still missed.
2. Apply the confirmatory test to the first-test positives. Calculate the final PPV and the total testing cost. Is the budget sufficient?
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Among the 1,128 first-test positives, 360 truly have the condition and 768 do not. Confirmatory true positives: 95% × 360 = 342. Confirmatory false negatives: 18. Confirmatory false positives: 2% × 768 = 15.36, about 15. Confirmatory true negatives: about 753. Final positives: 342 + 15.4 = 357.4. Final PPV = 342/357.4 = 0.957, or about 96%.
Cost: first test = 10,000 × $3 = $30,000. Confirmatory test = 1,128 × $20 = $22,560. Total = $52,560, below the $60,000 budget, leaving about $7,440. The two-step strategy detects 342 true cases; 58 diseased people are missed overall (40 at the first step, 18 at the second). About 15 people receive a false-positive final result and need careful follow-up. Cost per true case detected is $52,560/342 ≈ $154. The budget is sufficient for testing, but the office must also budget for diagnostic follow-up and treatment, and must not describe the programme as “free” if those costs fall on patients.
3. How would you explain a positive first test to the patient, and what caveats should the programme manager state publicly?
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Explain: “Your first test was positive, but that does not mean you have the condition. About one in three people with this result actually has it. We recommend the confirmatory test before any treatment decision. The confirmatory test is much more specific, and if it is positive the chance of disease is about 96%. If it is negative, we will discuss whether repeat testing is needed.” Avoid saying “you are positive” without the probability, and avoid starting treatment on the first result alone.
Public caveats: the programme is a one-time screening, not a guarantee; it will miss some cases; it will generate anxiety and false positives; confirmatory testing and follow-up must be funded; and people with symptoms should seek care regardless of screening. The manager should also state that the PPV depends on the 4% prevalence estimate, and that the budget covers testing but not lifelong treatment. Reporting should include the number needed to screen, the false-positive burden, and the 58 expected missed cases, not only the 342 detected.