A state health department is reviewing the implementation strategy for a novel point-of-ca… — USMLE Step 3

A state health department is reviewing the implementation strategy for a novel point-of-care rapid assay used to detect a re-emerging vector-borne pathogen. Validation studies demonstrate that the assay has a positive likelihood ratio ( $LR+$ ) of 9.0 and a negative likelihood ratio ( $LR-$ ) of 0.1.

The department plans to deploy this assay in two distinct clinical environments:
- Environment X: A routine occupational health screening program for forestry workers in an endemic region. The baseline prevalence of the infection in this cohort is 10%.
- Environment Y: A specialized infectious disease referral clinic evaluating patients who present with classic constitutional symptoms and rash. The pretest probability of the infection in this cohort is 50%.

An asymptomatic forestry worker evaluated in Environment X receives a positive assay result. A symptomatic patient evaluated in Environment Y receives a negative assay result.

Assuming test performance characteristics are constant across populations, which of the following best represents the approximate post-test probability of infection for the forestry worker and the clinic patient, respectively?

Forestry worker: 50%; Clinic patient: 9%Answer
B
Forestry worker: 90%; Clinic patient: 5%
C
Forestry worker: 1%; Clinic patient: 90%
D
Forestry worker: 10%; Clinic patient: 50%
E
Forestry worker: 99%; Clinic patient: 10%

Answer

Forestry worker: 50%; Clinic patient: 9%

The correct answer is derived by properly utilizing Bayes' theorem via the odds form. First, the pre-test probability must be converted to pre-test odds (

Odds = \frac{Probability}{1 - Probability}

). The forestry worker's 10% probability converts to 1:9 odds. Multiplying these odds by the

LR+

of 9.0 yields post-test odds of 1:1, which converts back to a 50% post-test probability (

Probability = \frac{Odds}{1 + Odds}

). For the clinic patient, a 50% pre-test probability is equal to 1:1 odds. Multiplying by the

LR-

of 0.1 yields post-test odds of 0.1 (or 1:10). Converting 1:10 odds back to probability yields

\frac{1}{11}

, which is approximately 9%.

Step-by-Step Solution

Convert the pre-test probability to pre-test odds for both patients.

Forestry worker (10%):

Odds = \frac{0.1}{1 - 0.1} = \frac{1}{9}

. Clinic patient (50%):

Odds = \frac{0.5}{1 - 0.5} = 1

Likelihood ratios must be multiplied by pre-test odds, not pre-test probabilities, to determine the post-test disease likelihood.

Multiply the pre-test odds by the appropriate likelihood ratio based on the test result.

Forestry worker (Positive test):

\frac{1}{9} \times 9.0 = 1

(Post-test odds of 1:1). Clinic patient (Negative test):

1 \times 0.1 = 0.1

(Post-test odds of 1:10).

A positive test utilizes the positive likelihood ratio (

LR+

), while a negative test utilizes the negative likelihood ratio (

LR-

Convert the post-test odds back into post-test probabilities.

Forestry worker:

Probability = \frac{1}{1 + 1} = 50\%

. Clinic patient:

Probability = \frac{0.1}{1 + 0.1} = \frac{1}{11} \approx 9.1\%

Post-test probability is calculated using the formula:

Probability = \frac{Odds}{1 + Odds}

Key Concept

Applying Likelihood Ratios using Bayesian Odds

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