A public health task force is evaluating the performance of a newly developed rapid antige… — USMLE Step 3

A public health task force is evaluating the performance of a newly developed rapid antigen test for diagnosing a regional tick-borne infection. During the validation trial, the test is administered to a cohort of $500$ patients who presented with suspected symptoms in an endemic region. A gold standard serological assay, performed simultaneously, confirms that $100$ patients in the cohort truly have the infection. The rapid antigen test yields a positive result in $80$ of the confirmed infected patients and a negative result in $360$ of the uninfected patients. A local primary care physician utilizes this rapid antigen test on a patient presenting with identical symptoms in the same endemic area. The test returns a positive result. Based on the validation trial data, which of the following best represents the approximate probability that this patient actually has the tick-borne infection?

67%Answer
B
80%
C
88%
D
90%
E
95%

Answer

The correct probability is 67%, which represents the positive predictive value calculated from the provided study data.

The question asks for the probability that a patient with a positive test actually has the disease, which is the positive predictive value (PPV) of the test. PPV is calculated as the number of true positive (TP) results divided by the total number of positive results (TP + false positives, FP). Based on the study cohort of

500

patients,

100

patients have the confirmed infection. The test correctly identifies

80

of these patients (TP =

80

). The remaining

400

patients do not have the infection, and the test correctly identifies

360

of them as negative (true negatives, TN =

360

). Consequently,

40

uninfected patients falsely test positive (FP =

400 - 360 = 40

). The PPV is therefore calculated as

80 / (80 + 40) = 80 / 120 = 66.7\%

, rounding to approximately

67\%

Step-by-Step Solution

Determine the total number of True Positive (TP) and False Positive (FP) results from the cohort data.

TP =

80

(since

80

100

infected patients test positive). FP =

40

(since

360

of the

400

uninfected patients test negative, leaving

40

false positives).

Calculating the post-test probability of disease following a positive result requires the ratio of true positives to all positive test results.

Calculate the Positive Predictive Value (PPV).

PPV =

\frac{TP}{TP + FP} = \frac{80}{80 + 40} = \frac{80}{120} = 66.7\%

PPV defines the probability that a patient with a positive test result actually has the disease.

Key Concept

Positive predictive value (PPV) is the probability that a subject with a positive test result truly has the disease, defined mathematically as

\frac{TP}{TP + FP}

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