Interpreting a rapid HIV test in a low-prevalence donor pool
A district hospital's blood transfusion service screens every donated unit with a rapid HIV assay that, under local field conditions, has a sensitivity of 98.0% and a specificity of 97.0%. The service collects 12,000 units a year from voluntary donors, and the estimated HIV prevalence in this donor pool is 0.12%. Every reactive rapid result is sent for confirmatory testing, which costs Rs 4,500, while each rapid test costs Rs 250. At a quality meeting the HIV counsellor argues that because the assay is "98% accurate", roughly 98 out of every 100 reactive donors must truly be infected, and that routine confirmation is an unnecessary expense that frightens donors. The laboratory officer disagrees but cannot produce the figures. The medical director asks for the expected numbers over one year.
1. Using a hypothetical 12,000 donors, calculate the expected true positives, false positives, PPV and NPV at this prevalence.
Read the worked answer
Set up the table. Infected = 12,000 × 0.0012 = 14.4; not infected = 11,985.6.
Test results.
- True positives = 14.4 × 0.98 = 14.11
- False negatives = 14.4 × 0.02 = 0.29
- False positives = 11,985.6 × 0.03 = 359.57
- True negatives = 11,985.6 × 0.97 = 11,626.0
Predictive values.
- PPV = 14.11 ÷ (14.11 + 359.57) = 14.11 ÷ 373.68 = 3.8%
- NPV = 11,626 ÷ (11,626 + 0.29) = 99.998%
Reading. About 374 donors a year will be reactive, but only about 14 will truly be infected: roughly 26 false positives for every true positive. A reactive result in this setting means "probably not infected, confirm urgently", not "infected". The counsellor has confused sensitivity with predictive value; sensitivity describes test performance among the infected, not the probability of infection among the reactive.
2. Cost the annual screening and confirmation programme, and estimate the cost per infected donor detected. What does this imply for policy?
Read the worked answer
Costs. Rapid tests: 12,000 × Rs 250 = Rs 3,000,000. Confirmatory tests: 373.7 reactive results, rounded to 374 × Rs 4,500 = Rs 1,683,000. Total ≈ Rs 4.68 million per year.
Yield. About 14.1 infected donors detected, so the cost per infected donor identified is roughly 4,680,000 ÷ 14.1 ≈ Rs 332,000. That looks expensive per case - but it is the wrong denominator. The purpose of screening is to stop transmission through transfusion, and each intercepted unit prevents a lifelong infection in a recipient.
Policy implications.
- Confirmation is not optional; without it, 360 donors a year would be told they are HIV-positive wrongly.
- The confirmatory cost is 36% of the programme, so improving the rapid assay's specificity is where quality investment pays.
- Counselling capacity must be planned around ~374 reactive notifications, not 14.
- A second, different rapid assay as a tie-breaker would cut confirmations substantially at modest cost.
3. What specificity would be required for the PPV to reach 50% at 0.12% prevalence? Compare with the PPV at 5% prevalence using the current test.
Read the worked answer
Algebra. PPV = se·p / [se·p + (1 − sp)(1 − p)]. Setting PPV = 0.5 means the true positives equal the false positives:
se·p = (1 − sp)(1 − p) 0.98 × 0.0012 = (1 − sp)(0.9988) 1 − sp = 0.001176 ÷ 0.9988 = 0.001177 sp ≈ 99.88%
So at this prevalence a PPV of only 50% demands a specificity of about 99.9% - far beyond any field rapid test. No single assay can carry the diagnosis here; a two-step algorithm is mathematically necessary.
At 5% prevalence (e.g. a high-risk clinic). PPV = (0.98 × 0.05) ÷ [(0.98 × 0.05) + (0.03 × 0.95)] = 0.049 ÷ (0.049 + 0.0285) = 63.2%
The same test, same sensitivity, same specificity, but the PPV rises from 3.8% to 63%. Predictive value belongs to the test and the population, which is why donor screening and diagnostic testing of symptomatic patients cannot share the same interpretation rules.