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Difficulty: MediumDiagnostic Testing Performance

A 35-year-old woman presents to the rheumatology clinic with a 4-month history of symmetrical polyarthritis affecting the small joints of her hands and wrists. She also reports significant fatigue. A novel serum biomarker assay for early systemic lupus erythematosus (SLE) has recently been approved for clinical use. The manufacturer reports that the test has a sensitivity of 60%60\% and a specificity of 90%90\% for diagnosing SLE in patients with undifferentiated inflammatory arthritis. Based on the patient's clinical presentation and demographic profile, the physician estimates her pretest probability of having SLE is 10%10\%.

The patient's blood sample is drawn, and the biomarker assay returns a positive result. Which of the following is the best estimate of the patient's posttest probability of having SLE?

  1. A
    6%6\%
  2. B
    20%20\%
  3. 40%40\%Answer
  4. D
    60%60\%
  5. E
    86%86\%

Answer

The correct posttest probability is 40%40\%.
The patient's pretest probability of SLE is estimated at 10%10\%. To find the posttest probability following a positive test result, one must first determine the pretest odds and the positive likelihood ratio (LR+LR+).

1. Pretest odds = Pretest probability1−Pretest probability=0.100.90=19\frac{\text{Pretest probability}}{1 - \text{Pretest probability}} = \frac{0.10}{0.90} = \frac{1}{9}

2. LR+LR+ = Sensitivity1−Specificity=0.601−0.90=0.600.10=6\frac{\text{Sensitivity}}{1 - \text{Specificity}} = \frac{0.60}{1 - 0.90} = \frac{0.60}{0.10} = 6

3. Posttest odds = Pretest odds ×LR+=19×6=69=23\times LR+ = \frac{1}{9} \times 6 = \frac{6}{9} = \frac{2}{3}

4. Posttest probability = Posttest odds1+Posttest odds=231+23=2353=25=40%\frac{\text{Posttest odds}}{1 + \text{Posttest odds}} = \frac{\frac{2}{3}}{1 + \frac{2}{3}} = \frac{\frac{2}{3}}{\frac{5}{3}} = \frac{2}{5} = 40\%

Therefore, the best estimate of the patient's posttest probability is 40%40\%.

Step-by-Step Solution

1
Calculate the pretest odds from the given pretest probability.
Pretest odds = 0.100.90=19\frac{0.10}{0.90} = \frac{1}{9}
Likelihood ratios must be multiplied by odds, not probabilities, to properly update disease estimates using Bayes' theorem.
2
Calculate the positive likelihood ratio (LR+LR+) using the given sensitivity and specificity.
LR+=Sensitivity1−Specificity=0.601−0.90=0.600.10=6LR+ = \frac{\text{Sensitivity}}{1 - \text{Specificity}} = \frac{0.60}{1 - 0.90} = \frac{0.60}{0.10} = 6
The LR+LR+ determines how many times more likely a positive test is to occur in a patient with the disease compared to a patient without the disease.
3
Calculate the posttest odds by multiplying the pretest odds by the LR+LR+.
Posttest odds = 19×6=69=23\frac{1}{9} \times 6 = \frac{6}{9} = \frac{2}{3}
This applies Bayesian updating to find the odds of the patient having the disease given the positive test result.
4
Convert the posttest odds back into a posttest probability.
Posttest probability = Odds1+Odds=231+23=2353=25=40%\frac{\text{Odds}}{1 + \text{Odds}} = \frac{\frac{2}{3}}{1 + \frac{2}{3}} = \frac{\frac{2}{3}}{\frac{5}{3}} = \frac{2}{5} = 40\%
Clinical decisions and standard reporting utilize probabilities (predictive values) rather than odds.

Key Concept

Calculating posttest probability using pretest probability and likelihood ratios (Bayesian updating).

Alternative Method

Instead of using likelihood ratios and odds, you can create a 2x2 contingency table using a hypothetical population of 1,0001,000 patients. With a 10%10\% pretest probability, 100100 patients have SLE and 900900 do not. Applying the 60%60\% sensitivity, 6060 of the 100100 SLE patients test positive (True Positives). Applying the 90%90\% specificity, 810810 of the 900900 non-SLE patients test negative, leaving 9090 who test positive (False Positives). The total number of positive tests is 150150 (6060 true + 9090 false). The posttest probability (positive predictive value) is the true positives divided by total positives: 60150=615=25=40%\frac{60}{150} = \frac{6}{15} = \frac{2}{5} = 40\%.
Estimated Time:1m 30s
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Diagnostic Testing Performance

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