A 28-year-old man presents to an outpatient clinic with a 2-day history of low-grade fever… — USMLE Step 2 CK

A $28$ -year-old man presents to an outpatient clinic with a $2$ -day history of low-grade fever, myalgias, and mild fatigue. A newly FDA-approved point-of-care rapid antigen test for a circulating respiratory virus is available. Regional epidemiologic surveillance indicates that the current prevalence of this viral infection in the community is $10\%$ . The manufacturer's validation studies report that the assay has a sensitivity of $80\%$ and a specificity of $95\%$ . The physician performs the swab, and the test yields a positive result. Based on these statistical parameters, what is the approximate probability that this patient is actually infected with the virus?

A
$10\%$
B
$36\%$
$64\%$ Answer
D
$80\%$
E
$94\%$

Answer

The correct probability that the patient has the infection is

64\%

The question asks for the Positive Predictive Value (PPV) of the test, which is the probability that a patient with a positive test truly has the disease. PPV depends on sensitivity, specificity, and the disease prevalence in the tested population.

Using a hypothetical cohort of

1,000

patients:
- With a prevalence of

10\%

100

patients have the disease and

900

do not.
- The test has

80\%

sensitivity, meaning it will correctly identify

80

true positives out of the

100

diseased patients.
- The test has

95\%

specificity, meaning its false positive rate is

5\%

. Out of the

900

healthy patients,

5\%

will falsely test positive, yielding

45

false positives.
- The total number of positive tests is

80

(true positives) +

45

(false positives) =

125

.

The PPV is the ratio of true positives to total positives:

80 / 125 = 0.64

, or

64\%

. Therefore, the probability that the patient actually has the infection is

64\%

Step-by-Step Solution

Determine the true prevalence and set up a hypothetical population (e.g.,

1,000

people).

With a

10\%

prevalence, there are

100

people with the disease and

900

people without the disease.

Applying percentages to a hypothetical cohort simplifies Bayesian calculations and helps visualize the

2\times2

table.

Calculate the number of True Positives (TP) using the test's sensitivity.

Sensitivity is

80\%

80\%

of the

100

diseased patients will test positive.

TP = 80

Sensitivity measures the proportion of actual positives correctly identified.

Calculate the number of False Positives (FP) using the test's specificity.

Specificity is

95\%

, meaning the false positive rate is

5\%

5\%

of the

900

healthy patients will incorrectly test positive.

FP = 0.05 \times 900 = 45

Specificity measures the true negative rate; its complement (

1 - \text{specificity}

) determines the false positive rate in the healthy population.

Calculate the Positive Predictive Value (PPV).

\text{Total positive results} = TP + FP = 80 + 45 = 125

\text{PPV} = \frac{TP}{\text{Total Positives}} = \frac{80}{125} = 0.64

64\%

PPV answers the clinical question: 'Given a positive test result, what is the probability that the disease is actually present?'

Key Concept

Positive Predictive Value (PPV) depends heavily on disease prevalence and must be calculated using both the test's intrinsic characteristics (sensitivity, specificity) and the pre-test probability.

Alternative Method

This problem can also be solved rapidly using Likelihood Ratios (LR).

\text{LR+} = \frac{\text{Sensitivity}}{1 - \text{Specificity}} = \frac{0.80}{0.05} = 16

. The pre-test odds are calculated from prevalence:

\frac{0.10}{0.90} = \frac{1}{9}

. The post-test odds equal the pre-test odds multiplied by

\text{LR+}

\frac{1}{9} \times 16 = \frac{16}{9}

. Finally, converting post-test odds back to probability yields

\frac{16/9}{1 + 16/9} = \frac{16}{25} = 64\%

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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