Rapid Antigen Test Interpretation at Different Prevalences
A community clinic evaluates a new rapid antigen test for a respiratory infection. In a validation study of 10,000 people, the prevalence of active infection is 5.0%. The test has a sensitivity of 92% and a specificity of 88%. The clinic plans to use the test for all symptomatic patients and then confirm positives with a slower PCR test that has sensitivity 98% and specificity 99%. A clinician argues that because sensitivity and specificity are both high, a positive rapid test means the patient almost certainly has the infection. The clinic's laboratory director is worried about false positives and asks for a quantitative justification. The clinic sees about 40 symptomatic patients per day and has PCR capacity for 15 confirmatory tests per day. Assume the rapid test result is independent of the PCR result conditional on infection status.
1. Calculate the positive predictive value and negative predictive value of the rapid test at 5% prevalence and evaluate the clinician's claim.
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Set up a 10,000-person table. Diseased = 0.05 × 10,000 = 500; healthy = 9,500. Sensitivity 92%: true positives = 0.92 × 500 = 460; false negatives = 40. Specificity 88%: true negatives = 0.88 × 9,500 = 8,360; false positives = 0.12 × 9,500 = 1,140.
PPV = TP/(TP + FP) = 460/(460 + 1,140) = 460/1,600 = 0.2875, or 28.8%. NPV = TN/(TN + FN) = 8,360/(8,360 + 40) = 0.995, or 99.5%.
The clinician is wrong: fewer than one in three positive rapid tests is a true positive at this prevalence, so confirmation is essential. The high NPV means a negative test is reassuring.
2. The clinic tests 40 symptomatic patients per day. Estimate the expected number of positive rapid tests and decide whether PCR capacity is sufficient.
Read the worked answer
Expected positives per 40 patients = 40 × P(positive). P(positive) = prevalence × sensitivity + (1 − prevalence) × (1 − specificity) = 0.05 × 0.92 + 0.95 × 0.12 = 0.046 + 0.114 = 0.160.
So 40 × 0.160 = 6.4 positive rapid tests per day on average. PCR capacity is 15 per day, so 15 > 6.4 and capacity is sufficient on average. The clinic should still plan for day-to-day variability, prioritize high-risk patients during surges, and confirm all positives rather than a subset to avoid missing true cases.
3. If the same test were used in an outbreak setting with 20% prevalence, recalculate the PPV and explain the implication for confirmatory testing.
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At 20% prevalence: diseased = 2,000 and healthy = 8,000. TP = 0.92 × 2,000 = 1,840; FP = 0.12 × 8,000 = 960. PPV = 1,840/(1,840 + 960) = 1,840/2,800 = 0.657, or 65.7%.
The same test performs much better when prevalence is higher because the base rate of true positives rises. This is Bayes' rule in action: PPV depends on prevalence, not just sensitivity and specificity. In an outbreak, confirmatory testing is still useful but less urgent; in a low-prevalence screening setting, it is essential. PPVs from different settings cannot be compared without adjusting for prevalence.