Screening Test: PPV, Repeat Testing, and a Limited Confirmation Budget
In this fictional probability and expected-cost exercise, a regional clinic screens 10,000 adults, of whom 2,000 are in a high-risk subgroup with family history. The general-population prevalence is 0.80%; in the high-risk subgroup it is 6.0%. The screening test has sensitivity 92% and specificity 96%. The clinic can run a second, independent confirmatory test that costs 60 currency units per patient; treatment started on a false positive causes 250 units of avoidable harm, while a true case missed because treatment is delayed costs 400 units. The clinic's current pathway treats every positive result immediately, without confirmation, and has budget for at most 500 confirmatory tests per 10,000 people screened. A nurse argues that because the test is '92% accurate', a positive result in general screening is probably real. The clinic wants to know whether to confirm all positives, confirm none, or restrict confirmation to one subgroup. Assume the second test is conditionally independent given true status and has the same sensitivity and specificity as the first.
1. Compute the positive predictive value of a single positive test in (a) the general population and (b) the high-risk subgroup. Support the answer with natural frequencies per 10,000 in each group.
Read the worked answer
Use PPV = sensitivityp / [sensitivityp + (1 - specificity)(1 - p)]. General, p = 0.008: numerator = 0.92(0.008) = 0.00736; false-positive term = 0.04(0.992) = 0.03968; denominator = 0.04704; PPV = 0.00736/0.04704 ≈ 0.156, or 15.6%. Per 10,000: 80 diseased, 73.6 true positives; 9,920 healthy, 396.8 false positives; total positives 470.4, so 73.6/470.4 ≈ 15.6%. High-risk, p = 0.06: numerator = 0.92(0.06) = 0.0552; false-positive term = 0.04(0.94) = 0.0376; denominator = 0.0928; PPV ≈ 0.595, or 59.5%. Per 10,000: 600 diseased, 552 true positives; 9,400 healthy, 376 false positives; total positives 928, so 552/928 ≈ 59.5%.
2. Compute the PPV after two independent positive tests in each group. Does the second test resolve the nurse's concern?
Read the worked answer
For two independent positives, the positive likelihood is sensitivity^2 if diseased and (1 - specificity)^2 if healthy. Thus PPV = 0.92^2 p / [0.92^2 p + 0.04^2(1 - p)]. General, p = 0.008: numerator = 0.8464(0.008) = 0.0067712; healthy term = 0.0016(0.992) = 0.0015872; PPV = 0.0067712/0.0083584 ≈ 0.810, or 81.0%. High-risk, p = 0.06: numerator = 0.8464(0.06) = 0.050784; healthy term = 0.0016(0.94) = 0.001504; PPV = 0.050784/0.052288 ≈ 0.971, or 97.1%. The second test greatly strengthens the general-screening result, but about 19% of general-screening double positives would still be false positives, so a double-positive result is more likely true than false under the model, but is not near-certain. In the high-risk group the second positive is much more decisive.
3. Using the expected-cost figures and the 500-test budget, recommend a confirmation policy for the 8,000 general and 2,000 high-risk patients. Show the expected cost per group for no confirmation and for confirm-all, and state how many confirmatory tests your policy uses.
Read the worked answer
General group, 8,000 people: diseased = 64, healthy = 7,936. Single positives: true = 58.88, false = 317.44, total = 376.32. No confirmation costs 250(317.44) = 79,360. Confirm-all costs 60(376.32) + 250[7,936(0.04^2)] + 400[58.88(0.08)] = 22,579.2 + 3,174.4 + 1,884.2 = 27,637.8. Saving ≈ 51,722, or about 137.4 per confirmatory test. High-risk group, 2,000 people: diseased = 120, healthy = 1,880. Single positives: true = 110.4, false = 75.2, total = 185.6. No confirmation costs 250(75.2) = 18,800. Confirm-all costs 60(185.6) + 250[1,880(0.04^2)] + 400[110.4(0.08)] = 11,136 + 752 + 3,532.8 = 15,420.8. Saving ≈ 3,379.2, or about 18.2 per confirmatory test. Confirmation has a much larger expected saving in the low-prevalence general group because false positives are numerous there. With a 500-test budget, confirm all 376.32 general positives first, using about 376 tests, then use the remaining ~124 tests in the high-risk group. Expected cost ≈ 27,637.8 + [18,800 - 18.2(124)] ≈ 44,181. Confirming all high-risk positives first and only part of the general group would cost more, about 51,600. Under the supplied simplified costs, prioritize general-group positives and allocate remaining tests to high-risk positives. These are incremental pathway costs; common first-test false-negative costs cancel in the comparison. Fractional test counts are expected values, not a guarantee that a realized500-test budget suffices. This exercise cannot establish clinical policy or the net benefit of screening.