Rapid Screening Test and Confirmatory Testing Policy
A clinic is evaluating a rapid screening test for a condition that affects 3.5% of its patients. In a validation study, the test has sensitivity 92% and specificity 96%. The clinic plans to test 500 patients in a month. A positive result is followed by a confirmatory test with sensitivity 98% and specificity 99.5%. A negative rapid result is usually not confirmed. The clinic manager is worried about false positives and wants to know how many positives to expect and whether the confirmatory step is worthwhile. The prevalence in the clinic is assumed to match the 3.5% figure. The tests are independent conditional on true status. The clinic has capacity for 40 confirmatory tests per month. The rapid test costs €18 and the confirmatory test €75; the manager also has a budget constraint.
1. Using a hypothetical cohort of 500 patients, calculate expected true positives, false positives, true negatives, and false negatives on the rapid test. Compute the positive predictive value (PPV).
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Prevalence 3.5% gives 500 × 0.035 = 17.5 true cases and 482.5 non-cases. True positives TP = 17.5 × 0.92 = 16.1. False negatives FN = 17.5 × 0.08 = 1.4. False positives FP = 482.5 × (1 − 0.96) = 482.5 × 0.04 = 19.3. True negatives TN = 482.5 × 0.96 = 463.2. Total rapid positives = 16.1 + 19.3 = 35.4. PPV = 16.1 / 35.4 ≈ 0.455, or 45.5%. So fewer than half of rapid positives are true cases.
2. Calculate the PPV after a positive confirmatory test. Interpret the result for the clinic.
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Among rapid positives, the pre-confirm probability is 45.5%. Apply sensitivity 98% and specificity 99.5%: PPV₂ = (0.455 × 0.98) / (0.455 × 0.98 + 0.545 × 0.005) = 0.4459 / (0.4459 + 0.002725) = 0.4459 / 0.448625 ≈ 0.994, or 99.4%. Expected confirmatory positives = 35.4 × 0.4486 ≈ 15.9. The confirmatory step turns a weak positive into a strong one.
3. The manager suggests dropping the confirmatory test to save money. Use the PPV and expected numbers to argue for or against.
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Dropping the confirmatory test would mean about 19.3 false positives for every 16.1 true positives, so roughly 55% of the 35.4 rapid positives would be false. That is unacceptable for a clinic making treatment or isolation decisions. With 40 confirmatory tests available, the expected 35.4 rapid positives fit within capacity. The confirmatory test is worthwhile: it raises predictive value from 45.5% to 99.4%. If resources were tighter, the clinic could prioritise confirmatory testing for higher-risk subgroups, but universal confirmation is justified here.