The Screening Letter
A 500-person company runs a voluntary screening day. The condition being screened for affects about 6% of employees in this workforce. The test detects 90% of people who have the condition, and returns a false positive for 8% of people who do not. HR plans to send everyone with a positive result a letter stating that they 'most likely have' the condition and should begin treatment immediately. A confirmatory test costs R900 and takes two weeks. The occupational health nurse supports the wording because, she says, 'the test is 90% accurate'. The company has budgeted for 40 confirmatory tests this year.
1. Starting from 500 employees, how many positive results would you expect, and how many of those people would actually have the condition? Show your working.
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Work from the whole population down, not from the positives up.
| Has condition | Does not | Total | |
|---|---|---|---|
| Number | 500 x 0.06 = 30 | 500 - 30 = 470 | 500 |
| Test positive | 30 x 0.90 = 27 | 470 x 0.08 = 37.6 ≈ 38 | 65 |
| Test negative | 3 | 432 | 435 |
Positive predictive value = 27 / 65 ≈ 42%.
So roughly 65 letters go out, and about 38 of them go to people who do not have the condition. Fewer than half of the recipients actually have it. Note that rounding 37.6 to 38 barely changes the conclusion (27/64.6 is still about 42%).
2. Evaluate the nurse's statement that 'the test is 90% accurate' and HR's proposed wording. What does the arithmetic imply for the letter?
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The nurse's figure is real but is the wrong figure for this decision.
- 90% is the sensitivity - the share of diseased people the test catches. HR's letter makes a claim about a different group: people who have tested positive. That quantity is the PPV, which here is about 42%.
- Specificity is 92%, so 8% of the 470 healthy employees test positive. Because healthy people vastly outnumber sick people at 6% prevalence, the false positives (38) outnumber the true positives (27).
Implication for the letter: 'most likely have the condition' is false for the majority of recipients - about 58% of them do not have it. Telling someone to 'begin treatment immediately' before confirmation is both inaccurate and potentially harmful (anxiety, unnecessary medication, stigma).
Better wording: 'Your screening result was positive. This does not mean you have the condition; about 4 in 10 positive results in this programme are confirmed. Please book the confirmatory test, which is free and takes two weeks.' That is accurate, actionable, and does not pre-empt the diagnosis.
3. The budget covers 40 confirmatory tests. Does the arithmetic support going ahead with the screening day as planned? Advise the company.
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The budget and the expected result are not compatible. Expected positives ≈ 65 against 40 funded confirmations, a shortfall of about 25 people. Those 25 would be told they may have the condition and then be unable to confirm it - the worst of both worlds.
Options, with the trade-off stated:
- Fund 65-70 confirmations (about R58,500-R63,000). Screening proceeds as planned; roughly 38 of those tests will be spent clearing false positives, which is the unavoidable cost of screening at low prevalence.
- Do not screen yet. Spend nothing and lose nothing; the condition is rare enough that opportunistic detection may be adequate.
- Screen a higher-prevalence subgroup (e.g. only employees over 50 or with a family history). Positives fall, PPV rises, and the 40-test budget may then fit - but coverage and equity suffer and the group left out gets no benefit.
Recommendation: do not run the day on the current budget. Either raise the confirmation budget to match the expected positives, or narrow the screened group. The decision should be driven by the number of positives, not by the number of employees screened.