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Difficulty: HardDiagnostic Testing Performance

A 55-year-old man presents to the clinic reporting episodes of substernal chest pressure radiating to his left arm, typically occurring when he walks briskly up a hill and relieved by rest. He has a history of hypertension and hyperlipidemia. Based on his age, sex, and symptom characteristics, his pretest probability for hemodynamically significant coronary artery disease (CAD) is estimated to be 50%.

The physician initially orders a standard exercise treadmill test. The test literature states this modality has a sensitivity of 70% and a specificity of 80% for significant CAD. The patient completes the test and shows 1.5 mm of ST-segment depression, which is interpreted as a positive result.

To further stratify his risk before considering invasive angiography, the physician immediately orders a dobutamine stress echocardiogram. This test is reported to have a sensitivity of 85% and a specificity of 90%. The stress echocardiogram result is interpreted as negative for inducible ischemia.

Assuming the two diagnostic tests are statistically independent, what is the most accurate estimate of this patient's final post-test probability of having significant CAD?

  1. A
    14%
  2. 37%Answer
  3. C
    58%
  4. D
    78%
  5. E
    95%

Answer

The most accurate estimate of the patient's final post-test probability is 37%.
To determine the final post-test probability after sequential testing, one must use likelihood ratios. First, convert the 50% pretest probability to odds: 0.5/(1−0.5)=10.5 / (1 - 0.5) = 1. Next, calculate the LR+LR^+ for the positive treadmill test: 0.70/(1−0.80)=3.50.70 / (1 - 0.80) = 3.5. Multiply the pretest odds by this LR+LR^+ to get the intermediate odds: 1×3.5=3.51 \times 3.5 = 3.5. These intermediate odds serve as the new pretest odds for the second test. Then, calculate the LR−LR^- for the negative stress echocardiogram: (1−0.85)/0.90=0.15/0.90=1/6(1 - 0.85) / 0.90 = 0.15 / 0.90 = 1/6. Multiply the intermediate odds by this LR−LR^- to get the final odds: 3.5×(1/6)=7/123.5 \times (1/6) = 7/12. Finally, convert the final odds back to probability: (7/12)/(1+7/12)=7/19≈36.8%(7/12) / (1 + 7/12) = 7/19 \approx 36.8\%, which rounds to 37%.

Step-by-Step Solution

1
Convert the initial pretest probability to pretest odds.
Pretest odds = 0.501−0.50=1\frac{0.50}{1 - 0.50} = 1.
Likelihood ratios must be multiplied by odds, not probabilities.
2
Calculate the Positive Likelihood Ratio (LR+LR^+) for the exercise treadmill test and update the odds.
LR+=Sensitivity1−Specificity=0.700.20=3.5LR^+ = \frac{\text{Sensitivity}}{1 - \text{Specificity}} = \frac{0.70}{0.20} = 3.5. Intermediate post-test odds = 1×3.5=3.51 \times 3.5 = 3.5.
The first test is positive, so its LR+LR^+ is used to calculate the new post-test odds.
3
Calculate the Negative Likelihood Ratio (LR−LR^-) for the stress echocardiogram and update the odds again.
LR−=1−SensitivitySpecificity=1−0.850.90=0.150.90=16LR^- = \frac{1 - \text{Sensitivity}}{\text{Specificity}} = \frac{1 - 0.85}{0.90} = \frac{0.15}{0.90} = \frac{1}{6}. Final post-test odds = 3.5×16=3.56=7123.5 \times \frac{1}{6} = \frac{3.5}{6} = \frac{7}{12}.
The second test is negative, so its LR−LR^- is applied sequentially to the odds generated after the first test.
4
Convert the final post-test odds back to a post-test probability.
Probability = Odds1+Odds=7121+712=7121912=719≈0.368\frac{\text{Odds}}{1 + \text{Odds}} = \frac{\frac{7}{12}}{1 + \frac{7}{12}} = \frac{\frac{7}{12}}{\frac{19}{12}} = \frac{7}{19} \approx 0.368 (37%).
The clinical question asks for the final probability, requiring conversion from the calculated odds.

Key Concept

In sequential diagnostic testing, the post-test odds of the first test become the pretest odds for the second test. Probability must be converted to odds to use likelihood ratios (LR+LR^+ and LR−LR^-), and then converted back to probability at the end.
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Diagnostic Testing Performance

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