Two-Stage Screening for a Rare Disorder
A regional programme is considering adding a one-off blood test to detect a rare metabolic disorder. Prevalence is 1 in 2,000. The test has sensitivity 95% and specificity 98%. The first test costs $40. Any positive first test is repeated with the same test, costing $120, and only two positives are referred for confirmatory genetic testing. The programme has 100,000 eligible people and a budget of $6 million. The test is independent between repeats. A patient advocacy group argues that a 95% sensitive test should be offered to everyone because it will find almost all cases. A finance officer argues that the programme is too expensive per case found. You are asked to model the outcomes before the board meeting.
1. Calculate the positive predictive value of a single positive result and of two consecutive positive results. Explain the difference with numbers.
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Prevalence = 1/2000 = 0.0005. For one positive: true-positive probability = 0.95 × 0.0005 = 0.000475. False-positive probability = 0.02 × 0.9995 = 0.01999. Total positive probability = 0.020465. PPV = 0.000475 / 0.020465 = 0.0232, or 2.3%. In 100,000 people, about 47.5 true cases and 1,999 false positives would test positive, so most positives are false.
For two consecutive positives: true-positive probability = 0.95² × 0.0005 = 0.00045125. False-positive probability = 0.02² × 0.9995 = 0.0003998. Total = 0.00085105. PPV = 0.00045125 / 0.00085105 = 0.530, or 53.0%. Repeating the test removes many false positives because a false positive must occur twice, while most true cases remain positive. In 100,000 people, about 45.1 true cases and 40.0 false positives would be referred.
2. Estimate the total testing cost, the number of people referred, and the number of true cases detected. Calculate the cost per true case detected.
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First tests: 100,000 × $40 = $4,000,000. First positives: 100,000 × 0.020465 = 2,046.5, so about 2,047 repeats. Repeat cost: 2,047 × $120 = $245,640. Two-stage testing total = $4,245,640, before confirmatory genetic testing.
People referred after two positives: 100,000 × 0.00085105 = 85.1, so about 85 people. True cases detected: 100,000 × 0.00045125 = 45.1, so about 45. False positives referred: about 40. Cost per true case detected = $4,245,640 / 45.1 = approximately $94,100. If only the first test is counted, the figure is $4,000,000 / 47.5 = approximately $84,200, but that ignores the repeat stage.
The finance officer is right that the cost per case is high, but the advocacy group is also right that most true cases are detected. The disagreement is about the denominator: the programme buys case detection plus a large number of confirmatory investigations.
3. Recommend whether to adopt the programme, and propose one design change that would improve cost per case without reducing sensitivity.
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A conditional recommendation is appropriate. The two-stage strategy has a 53% PPV, so about 40 of 85 referrals would be false positives, each needing genetic confirmation. That may be acceptable if confirmation is cheap and safe, but the budget of $6 million is nearly exhausted before confirmatory testing. The programme should not be sold as a low-cost case-finding tool at the current prevalence.
The strongest design change is to restrict screening to a high-prevalence subgroup, such as people with a family history or a compatible biochemical marker. If prevalence rises to 1 in 100, the single-test PPV becomes 0.95 × 0.01 / (0.95 × 0.01 + 0.02 × 0.99) = 32.4%, and the two-stage PPV becomes 0.9025 × 0.01 / (0.9025 × 0.01 + 0.0004 × 0.99) = 95.8%. Sensitivity is unchanged, but far fewer unaffected people are tested, so cost per true case falls sharply. The board should also price confirmatory testing before approval.