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Difficulty: HardDiagnostic Testing Performance

A 58-year-old man comes to the clinic for evaluation of exertional chest pain. He describes the pain as a retrosternal tightness that occurs after walking uphill and is relieved by 5 minutes of rest. He has a medical history of hypertension and type 2 diabetes mellitus. Based on his age, sex, and symptom characteristics, his pretest probability of having hemodynamically significant coronary artery disease is estimated to be 25%. The physician orders a treadmill exercise stress echocardiogram. In this hospital's cardiology laboratory, this specific test has a reported sensitivity of 75% and a specificity of 85%. The patient completes the test, and the cardiologist reports the findings as positive for stress-induced ischemia. Based on these results, which of the following is the best estimate of this patient's post-test probability of having significant coronary artery disease?

  1. A
    19%
  2. B
    23%
  3. 63%Answer
  4. D
    75%
  5. E
    83%

Answer

63%
The post-test probability of a disease after a positive test result is equivalent to the test's positive predictive value (PPV) in that specific patient population. To find it, one must integrate the pretest probability (25%25\%) with the test's diagnostic performance properties (sensitivity 75%75\%, specificity 85%85\%). This is most efficiently done using likelihood ratios. First, convert pretest probability to odds: 0.25/0.75=1/30.25 / 0.75 = 1/3. Next, calculate the positive likelihood ratio (LR+LR+): Sensitivity/(1−Specificity)=0.75/0.15=5\text{Sensitivity} / (1 - \text{Specificity}) = 0.75 / 0.15 = 5. Then, multiply the pretest odds by the LR+LR+ to obtain the post-test odds: (1/3)×5=5/3(1/3) \times 5 = 5/3. Finally, convert the post-test odds back to a probability: (5/3)/(8/3)=5/8=62.5%(5/3) / (8/3) = 5/8 = 62.5\%, which rounds to 63%63\%.

Step-by-Step Solution

1
Convert the given pretest probability to pretest odds.
Pretest odds = 0.251−0.25=0.250.75=13\frac{0.25}{1 - 0.25} = \frac{0.25}{0.75} = \frac{1}{3}
Likelihood ratios must be multiplied by odds, not probabilities, to determine post-test odds.
2
Calculate the positive likelihood ratio (LR+LR+) using the given sensitivity and specificity.
LR+=Sensitivity1−Specificity=0.751−0.85=0.750.15=5LR+ = \frac{\text{Sensitivity}}{1 - \text{Specificity}} = \frac{0.75}{1 - 0.85} = \frac{0.75}{0.15} = 5
The LR+LR+ quantifies how much more likely a positive test is to occur in a patient with the disease compared to a patient without the disease.
3
Calculate the post-test odds by multiplying the pretest odds by the LR+LR+.
Post-test odds = 13×5=53\frac{1}{3} \times 5 = \frac{5}{3}
This is the fundamental application of Bayes' theorem in clinical practice.
4
Convert the post-test odds back into a post-test probability.
Post-test probability = Odds1+Odds=5/31+5/3=5/38/3=58=0.625\frac{\text{Odds}}{1 + \text{Odds}} = \frac{5/3}{1 + 5/3} = \frac{5/3}{8/3} = \frac{5}{8} = 0.625 (62.5%62.5\%)
Clinical decisions and risk communication are typically framed in probabilities rather than odds.

Key Concept

The post-test probability of a disease after a positive test is clinically equivalent to the Positive Predictive Value (PPV) and depends entirely on the disease prevalence (pretest probability), test sensitivity, and test specificity.

Alternative Method

You can also solve this using a hypothetical 2x2 contingency table. Assume a cohort of 1000 patients with this exact presentation. Since pretest probability is 25%, 250 have the disease and 750 do not. Of the 250 with disease, the test (75% sensitive) correctly identifies 187.5 (True Positives). Of the 750 without disease, the test (85% specific) correctly identifies 637.5 as negative, leaving 112.5 as False Positives. The total number of positive tests is 187.5 + 112.5 = 300. The post-test probability (PPV) is True Positives / Total Positives = 187.5 / 300 = 0.625 or 62.5%.
Estimated Time:1m 30s
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Diagnostic Testing Performance

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