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Difficulty: HardDiagnostic Testing Performance

A public health task force is evaluating a new rapid serologic assay for a tick-borne illness endemic to a specific region. During initial clinical validation at a tertiary care center, the assay was tested on a cohort of 2,0002,000 individuals, exactly half of whom had PCR-confirmed infection. In this validation cohort, the assay correctly identified 900900 of the infected individuals and yielded a positive result in 2525 of the uninfected individuals.

The task force now plans to deploy this assay to a rural outpatient clinic where epidemiological data indicates that 10%10\% of the tested patients actually have the infection.

If a patient at this rural clinic tests positive using the rapid assay, what is the probability (expressed as a percentage) that they truly have the infection?

Answer: 80 %

Answer

80
The positive predictive value (PPV) represents the probability that a patient with a positive test result actually has the disease. Because predictive values are highly dependent on disease prevalence, the PPV from the validation cohort (50%50\% prevalence) cannot be generalized to the rural clinic (10%10\% prevalence). However, the intrinsic properties of the test—sensitivity (90%90\%) and specificity (97.5%97.5\%)—remain constant. By applying these constant values to a hypothetical population representing the rural clinic (e.g., 10,00010,000 patients, yielding 1,0001,000 infected and 9,0009,000 uninfected), we determine the new proportion of true positives (900900) to all positive test results (900+225=1125900 + 225 = 1125). Calculating 900/1125900 / 1125 yields exactly 80%80\%.

Step-by-Step Solution

1
Calculate the test's sensitivity and specificity using data from the validation cohort (prevalence = 50%).
Sensitivity = 90% (900 TP / 1000 infected). Specificity = 97.5% (975 TN / 1000 uninfected, since FP = 25).
Intrinsic test properties remain constant regardless of disease prevalence and are required for downstream calculations.
2
Determine the number of True Positives (TP) and False Positives (FP) for the rural clinic population (prevalence = 10%). Assume a cohort of 10,000 patients for simpler math.
Infected = 1,000; Uninfected = 9,000. TP = 1,000 × 0.90 = 900. FP = 9,000 × 0.025 = 225.
Because PPV is prevalence-dependent, we must calculate the absolute numbers of individuals testing positive in this specific clinical setting.
3
Calculate the Positive Predictive Value (PPV) using the newly calculated TP and FP values.
PPV = 900 / (900 + 225) = 900 / 1125 = 0.80, or 80%.
PPV represents the probability that a positive test result indicates a true infection.

Key Concept

The relationship between disease prevalence and predictive values; while sensitivity and specificity are fixed properties of a test, Positive Predictive Value (PPV) decreases significantly as disease prevalence decreases.
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Diagnostic Testing Performance

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