Two-Stage Screening in a District Camp
A district health camp screens 10,000 adults for a condition using a rapid test. Past surveys put prevalence at 3%. The rapid test has sensitivity 92% and specificity 88%. A positive rapid test is followed by a confirmatory test that is not part of this calculation. The camp has budget for 1,500 confirmatory tests. The chief medical officer wants to know how many positives the rapid test will produce, what proportion of them are truly affected, and whether the confirmatory budget is sufficient. A volunteer argues that because sensitivity is 92%, a positive result means the person has a 92% chance of having the condition. The officer also wants to know how many affected people the rapid test would miss among the 10,000, and whether lowering the cut-off to raise sensitivity to 96% (but lowering specificity to 80%) would be a good trade-off given the confirmatory-test budget.
1. Using a natural-frequency table for 10,000 people, find the expected number of true positives, false positives, false negatives, and the positive predictive value. Is the volunteer’s 92% claim correct?
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Take 10,000 people. Expected affected = 0.03 × 10,000 = 300; unaffected = 9,700.
True positives = 0.92 × 300 = 276; false negatives = 24. False positives = 0.12 × 9,700 = 1,164; true negatives = 8,536. Total rapid positives = 276 + 1,164 = 1,440. PPV = 276/1,440 ≈ 0.192, i.e. 19.2%.
The volunteer’s 92% is the sensitivity P(positive | affected), not P(affected | positive). Because the condition is rare, most positives are false positives.
2. Is the confirmatory-test budget of 1,500 sufficient? What practical decision follows?
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About 1,440 rapid positives are expected, which is below the 1,500 confirmatory-test budget by about 60 tests. So the budget is sufficient on the stated prevalence, but only narrowly.
If prevalence is higher than 3%, or if the camp attracts a higher-risk group, positives could exceed 1,500. The practical decision is to prioritise confirmatory slots, monitor the daily positive rate, and have a contingency for any excess rather than turning people away.
3. Evaluate the proposed cut-off with sensitivity 96% and specificity 80%. Recompute positives and PPV, compare with the current test, and decide, considering the 1,500-test budget.
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New sensitivity 0.96, new specificity 0.80.
True positives = 0.96 × 300 = 288; false negatives = 12. False positives = 0.20 × 9,700 = 1,940. Total positives = 288 + 1,940 = 2,228. PPV = 288/2,228 ≈ 0.129, or 12.9%.
The new cut-off finds 12 more true cases (288 vs 276) but adds 776 false positives (1,940 vs 1,164) and exceeds the 1,500-test budget by 728. Unless confirmatory capacity is expanded, the current cut-off is the better operational choice. The lower cut-off would help only if the cost of a missed case is very high and extra confirmatory capacity can be funded.