A 64-year-old man with a 30-pack-year smoking history is evaluated in the clinic for a sol… — USMLE Step 3

A 64-year-old man with a 30-pack-year smoking history is evaluated in the clinic for a solitary pulmonary nodule incidentally found on a chest radiograph. Based on clinical and radiographic features, the physician estimates his pre-test probability of malignancy to be $20\%$ . The physician is considering ordering a novel functional imaging study to further risk-stratify the nodule before deciding on an invasive biopsy.

The medical literature states that the new imaging study has a positive likelihood ratio ( $LR+$ ) of $6.0$ . To safely defer biopsy and continue with serial observation alone if the test is negative, the physician determines that the post-test probability of malignancy must be $4\%$ or less.

Assuming the test is performed, which of the following best represents the patient's post-test probability of malignancy if the result is positive, and what is the maximum acceptable negative likelihood ratio ( $LR-$ ) of the test to achieve the physician's goal for a negative result?

A
Post-test probability if positive: $55\%$ ; Maximum acceptable $LR-$ : $1/5$
B
Post-test probability if positive: $55\%$ ; Maximum acceptable $LR-$ : $1/6$
C
Post-test probability if positive: $60\%$ ; Maximum acceptable $LR-$ : $1/25$
Post-test probability if positive: $60\%$ ; Maximum acceptable $LR-$ : $1/6$ Cevap
E
Post-test probability if positive: $60\%$ ; Maximum acceptable $LR-$ : $1/5$

Cevap

Post-test probability if positive:

60\%

; Maximum acceptable

LR-

1/6

To determine the post-test probability of disease, one must use the odds form of Bayes' theorem (Fagan's theorem). First, convert the pre-test probability (

20\%

) to pre-test odds:

0.20 / (1 - 0.20) = 1/4

. For a positive test result, multiply the pre-test odds by the

LR+

(

6.0

) to get post-test odds of

6/4

, or

3/2

. Converting this back to probability yields

(3/2) / (1 + 3/2) = 3/5

, which is

60\%

. To find the required

LR-

for a negative result, convert the target post-test probability (

4\%

) to target post-test odds:

0.04 / 0.96 = 1/24

. Because Post-test odds = Pre-test odds

\times LR-

, the maximum acceptable

LR-

is the Target post-test odds divided by the Pre-test odds:

(1/24) / (1/4) = 4/24

, which simplifies to

1/6

Adım Adım Çözüm

Convert the pre-test probability to pre-test odds.

Pre-test odds =

0.20 / (1 - 0.20) = 0.20 / 0.80 = 1/4

(or

0.25

Likelihood ratios can only be mathematically applied to odds, not directly to probabilities.

Calculate the post-test odds for a positive test result and convert it back to probability.

Post-test odds = Pre-test odds

\times LR+ = (1/4) \times 6.0 = 6/4 = 3/2

(or

1.5

). Post-test probability = Odds / (1 + Odds) =

(3/2) / (5/2) = 3/5 = 60\%

This determines the patient's likelihood of malignancy if the imaging study is positive.

Convert the target post-test probability for a negative result into target post-test odds.

Target post-test odds =

0.04 / (1 - 0.04) = 0.04 / 0.96 = 1/24

To find the required

LR-

, the target probability goal must first be expressed as odds.

Calculate the maximum acceptable

LR-

required to reach the target post-test odds.

LR- = \text{Target post-test odds} / \text{Pre-test odds} = (1/24) / (1/4) = 4/24 = 1/6

Fagan's theorem states that Post-test odds = Pre-test odds

\times LR

. Rearranging this allows us to solve for the necessary test characteristic.

Anahtar Kavram

Applying Fagan's theorem (Odds form of Bayes' theorem) to calculate post-test probabilities and required likelihood ratios.

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