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Difficulty: HardDiagnostic and Screening Test Performance

An infectious disease research group is evaluating a new rapid serological test for identifying acute Zika virus infection in an endemic region. In the initial validation study, 1,0001,000 symptomatic individuals were tested. Among the 200200 patients with PCR-confirmed Zika virus, the rapid test was positive in 180180. Among the 800800 patients without Zika virus, the rapid test was positive in 8080.

The test is subsequently deployed to a different clinic during a localized outbreak, where the prevalence of acute Zika virus among symptomatic patients is 25%25\%. Assuming the test's intrinsic properties remain unchanged, if a symptomatic patient at this new clinic receives a positive result on the rapid test, what is the probability (in percent) that they actually have the infection?

Answer: 75 %

Answer

75
The probability that a patient with a positive test actually has the disease is the Positive Predictive Value (PPV). Unlike sensitivity and specificity, which are intrinsic properties of the test, PPV changes with the prevalence of the disease in the tested population. First, the intrinsic properties are calculated from the validation study: Sensitivity = 180/200=90%180 / 200 = 90\%, and Specificity = (800−80)/800=90%(800 - 80) / 800 = 90\%. When the test is moved to a clinic with a 25%25\% prevalence, a theoretical cohort of 1,0001,000 patients would have 250250 diseased and 750750 healthy individuals. The test would correctly identify 225225 of the diseased patients (250×0.90250 \times 0.90) and incorrectly identify 7575 of the healthy patients (750×0.10750 \times 0.10). The PPV is the ratio of true positives to total positives: 225/(225+75)=225/300=75%225 / (225 + 75) = 225 / 300 = 75\%.

Step-by-Step Solution

1
Determine the test's sensitivity from the initial validation study.
Sensitivity is 90%90\%.
Sensitivity is the proportion of patients with the disease who test positive. Out of 200200 patients with Zika virus, 180180 tested positive. 180200=0.90\frac{180}{200} = 0.90.
2
Determine the test's specificity from the initial validation study.
Specificity is 90%90\%.
Specificity is the proportion of patients without the disease who test negative. Out of 800800 patients without Zika virus, 8080 tested positive (false positives). True negatives = 800−80=720800 - 80 = 720. 720800=0.90\frac{720}{800} = 0.90.
3
Calculate the expected number of true positives in the new clinic.
For every 1,0001,000 patients, there are 225225 true positives.
The new prevalence is 25%25\%, so in a cohort of 1,0001,000 patients, 250250 have the disease. True Positives = 250×0.90=225250 \times 0.90 = 225.
4
Calculate the expected number of false positives in the new clinic.
For every 1,0001,000 patients, there are 7575 false positives.
In the cohort of 1,0001,000 patients, 750750 do not have the disease. The false positive rate is 1−Specificity=1−0.90=0.101 - \text{Specificity} = 1 - 0.90 = 0.10. False Positives = 750×0.10=75750 \times 0.10 = 75.
5
Calculate the Positive Predictive Value (PPV).
The PPV is 75%75\%.
PPV=True PositivesTrue Positives+False Positives=225225+75=225300=0.75\text{PPV} = \frac{\text{True Positives}}{\text{True Positives} + \text{False Positives}} = \frac{225}{225 + 75} = \frac{225}{300} = 0.75.

Key Concept

Positive Predictive Value (PPV) depends heavily on disease prevalence; as prevalence changes, PPV must be recalculated using the test's intrinsic sensitivity and specificity.
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Diagnostic and Screening Test Performance

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