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Difficulty: MediumDiagnostic and Screening Test Performance

A public health task force is assessing a newly deployed rapid antigen assay for an endemic respiratory virus. Validation studies of the assay demonstrate a sensitivity of 80%80\% and a specificity of 95%95\%. Epidemiological data show a significant seasonal variation in the disease burden among symptomatic patients seeking urgent care: during the peak winter months, the prevalence of the viral infection in this population is 20%20\%, whereas during the summer months, the prevalence drops to 5%5\%.

In early August, a symptomatic patient undergoes the rapid antigen assay at a local urgent care center, and the result is positive. Based on the seasonal prevalence, which of the following best estimates the probability that this patient actually has the viral infection?

  1. A
    5%5\%
  2. 46%46\%Answer
  3. C
    80%80\%
  4. D
    95%95\%
  5. E
    99%99\%

Answer

Approximately 46%46\%
The probability that a patient with a positive test actually has the disease is defined as the Positive Predictive Value (PPV). Unlike sensitivity and specificity, which are intrinsic properties of the test, PPV is highly dependent on the disease prevalence in the tested population. The scenario states the patient is tested in August (summer), so the applicable prevalence is 5%5\%. Using a hypothetical cohort of 10,000 patients, 500 will have the disease (5%5\%) and 9,500 will not. With a sensitivity of 80%80\%, the test correctly identifies 400 of the 500 infected patients as positive (True Positives). With a specificity of 95%95\%, the test incorrectly identifies 5%5\% of the 9,500 uninfected patients as positive, resulting in 475 False Positives. The PPV is the ratio of true positives to all positive test results: 400400+475=400875≈45.7%\frac{400}{400 + 475} = \frac{400}{875} \approx 45.7\%, which is best estimated as 46%46\%.

Step-by-Step Solution

1
Identify the correct pre-test probability (prevalence) for the patient based on the clinical scenario.
The scenario occurs in early August (summer), so the prevalence to use is 5%5\%.
Predictive values are strictly dependent on the prevalence of the disease in the population being tested at that specific time.
2
Calculate the number of True Positives (TP) using a hypothetical cohort (e.g., 10,00010,000 patients).
Patients with disease = 10,000×0.05=50010,000 \times 0.05 = 500. TP = 500×0.80 (Sensitivity)=400500 \times 0.80 \text{ (Sensitivity)} = 400.
True positives are the diseased individuals correctly identified by the test.
3
Calculate the number of False Positives (FP) in the same cohort.
Patients without disease = 10,000×0.95=9,50010,000 \times 0.95 = 9,500. FP = 9,500×0.05 (1 - Specificity)=4759,500 \times 0.05 \text{ (1 - Specificity)} = 475.
False positives are healthy individuals incorrectly identified as positive by the test.
4
Calculate the Positive Predictive Value (PPV).
PPV=400400+475=400875≈45.7%PPV = \frac{400}{400 + 475} = \frac{400}{875} \approx 45.7\%, which rounds to 46%46\%.
PPV is the proportion of positive test results that are true positives.

Key Concept

Positive Predictive Value (PPV) calculation and its direct dependence on disease prevalence.
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Topics

Diagnostic and Screening Test Performance

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