A Screening Test That "Finds the Most Decay": Sensitivity, Specificity, and Two Districts

District A (prevalence 12%, n = 10,000)

• Diseased: 1,200. Non-diseased: 8,800.
• True positives: 1,200 × 0.85 = 1,020. False negatives: 180.
• False positives: 8,800 × 0.10 = 880. True negatives: 7,920.
• Total positives = 1,020 + 880 = 1,900.
• PPV = 1,020 / 1,900 = 53.7%
• NPV = 7,920 / 8,100 = 97.8%

District B (prevalence 2%, n = 10,000)

• Diseased: 200. Non-diseased: 9,800.
• True positives: 200 × 0.85 = 170. False negatives: 30.
• False positives: 9,800 × 0.10 = 980. True negatives: 8,820.
• Total positives = 170 + 980 = 1,150.
• PPV = 170 / 1,150 = 14.8%
• NPV = 8,820 / 8,850 = 99.7%

Why the difference Sensitivity and specificity are properties of the test. Predictive values are properties of the test applied in a population, and they move with prevalence. As prevalence falls, the pool of healthy children who can generate false positives grows much faster than the pool of diseased children who can generate true positives, so a fixed 10% false-positive rate overwhelms the shrinking number of true cases. In District A, about 1 in 2 positive screens is real; in District B, about 1 in 7 is real - even though it is literally the same two-minute exam.

Also note the negative side: the test is very good at ruling disease out in both districts, and it misses 180 cases in District A. A screen is a triage tool, not a diagnosis - every positive must be confirmed by a dentist.

Capacity math for the actual 9,000 children Using the District A prevalence of 12%, the number of positives is 1,710 - the program's 1,500-referral budget is already exceeded by universal screening in District A alone, before District B is touched. Adding District B's universal screening would add roughly 1,035 more positives (1,150 × 0.9) for about 153 true cases.

Cost per true case (per 10,000 children, $45 per false-positive referral)

• District A: 880 × $45 = $39,600, or about $38.80 per true case found (1,020 cases).
• District B: 980 × $45 = $44,100, or about $259 per true case found (170 cases).

Recommendation

1. Universal screening in District A. PPV of ~54% means the referral stream is mostly real disease, and the budget is still the binding constraint - so pair the screen with a treatment pathway (school-linked clinic, Medicaid navigator, portable/restorative services) or the screening simply documents disease it cannot treat.
2. Do not run a universal two-minute visual screen in District B. At a PPV near 15%, you generate roughly 5–6 false-positive referrals for every true case, at ~6.7× the cost per case found, and you consume family time and clinic slots that District A needs.
3. In District B, use risk-based or targeted screening instead: screen only children with a risk marker (previous caries, no dental home, free/reduced lunch, visible plaque on a brief index), or offer a school sealant/fluoride varnish program with a clinical exam built in, so the same professional contact both detects and prevents rather than generating referrals.
4. The board's phrase "finds the most kids" conflates two things. Universal screening in District B finds only about 33 more true cases than no screening at all in per-1,000 terms while burying them in ~980 false positives; targeted screening finds fewer but almost all of them are real.

What to say out loud at the meeting "The test doesn't change between districts; the consequences of a positive do." A screening program without a funded confirmatory exam and treatment pathway is not a screening program - it is a list of names.

District B, specificity 95%, prevalence 2%, n = 10,000

• Diseased: 200; true positives 170; false negatives 30.
• Non-diseased: 9,800; false positives 9,800 × 0.05 = 490; true negatives 9,310.
• Positives total = 170 + 490 = 660.
• PPV = 170 / 660 = 25.8% (up from 14.8%)
• NPV = 9,310 / 9,350 = 99.6%

Interpretation

• A five-point gain in specificity nearly halved the false-positive burden (980 → 490) and raised PPV by 11 points, without touching sensitivity. In low-prevalence settings, specificity is the lever that matters, because false positives are the dominant error.
• Cost per true case in District B falls from ~$259 to about $130 (490 × $45 / 170). Still 3–4× worse than District A, so retraining improves but does not rescue universal screening in a low-prevalence district.

The trade-off the board is really asking about There is no free improvement: maneuvers that raise sensitivity (shorter exam, examiner instructed "when in doubt call it caries," including non-cavitated lesions) typically lower specificity and flood the referral pipeline; maneuvers that raise specificity (stricter case definition, calibration, two examiners, drying teeth, good light) cost time and money per child and will miss a few more true cases.

The defensible framing

• Choose the operating point based on the downstream cost of each error, not on an abstract accuracy number. Here a false positive costs $45 and a parent's day; a false negative costs a child's untreated infection. That asymmetry argues for keeping sensitivity high and reducing false positives by calibration and case definition rather than by making the test harder to pass.
• Best value in District B is still not a better test - it is not screening universally. Investing in the treatment pathway is more cost-effective than investing in the test.