Examkin

Question

Difficulty: Very hardDiagnostic Testing Performance

A 35-year-old man presents to the outpatient clinic with chronic, intermittent joint pain, fatigue, and a mild photosensitive rash. Based on the clinical presentation and demographics, the physician estimates the patient's pre-test probability of a novel systemic autoimmune syndrome to be 10%10\%. To further evaluate, the physician utilizes a newly validated two-step serological testing algorithm. First, a screening enzyme-linked immunosorbent assay (ELISA) is performed. This test has a sensitivity of 96%96\% and a specificity of 84%84\%. The patient's ELISA result is positive. Next, a confirmatory immunoblot assay is performed. This test has a sensitivity of 84%84\% and a specificity of 96%96\%. The patient's immunoblot result is negative. Assuming the performance characteristics of the two assays are statistically independent, which of the following is the patient's post-test probability of having the systemic autoimmune syndrome?

  1. A
    2%2\%
  2. 10%10\%Answer
  3. C
    40%40\%
  4. D
    50%50\%
  5. E
    90%90\%

Answer

The correct answer is 10%10\%.
The correct answer is 10%10\%. To determine the final probability after sequential testing, one must use likelihood ratios (LRLR). The LR+LR^+ of the ELISA is Sensitivity/(1−Specificity)=0.96/(1−0.84)=6Sensitivity / (1 - Specificity) = 0.96 / (1 - 0.84) = 6. The LR−LR^- of the immunoblot is (1−Sensitivity)/Specificity=(1−0.84)/0.96=1/6(1 - Sensitivity) / Specificity = (1 - 0.84) / 0.96 = 1/6. The combined likelihood ratio for these sequential, independent results is 6×(1/6)=16 \times (1/6) = 1. A combined likelihood ratio of 11 means the test results provide no net change to the pre-test odds. Therefore, the post-test probability remains identically at the pre-test probability of 10%10\%.

Step-by-Step Solution

1
Convert the initial pre-test probability to pre-test odds.
Odds = 0.10/(1−0.10)=1/90.10 / (1 - 0.10) = 1/9.
Likelihood ratios must be multiplied by odds, not probabilities.
2
Calculate the positive likelihood ratio (LR+LR^+) for the screening ELISA.
LR+=Sensitivity/(1−Specificity)=0.96/(1−0.84)=0.96/0.16=6LR^+ = Sensitivity / (1 - Specificity) = 0.96 / (1 - 0.84) = 0.96 / 0.16 = 6.
This determines how much the positive ELISA increases the odds of disease.
3
Calculate the negative likelihood ratio (LR−LR^-) for the confirmatory immunoblot.
LR−=(1−Sensitivity)/Specificity=(1−0.84)/0.96=0.16/0.96=1/6LR^- = (1 - Sensitivity) / Specificity = (1 - 0.84) / 0.96 = 0.16 / 0.96 = 1/6.
This determines how much the negative immunoblot decreases the odds of disease.
4
Calculate the final post-test odds and convert back to probability.
Combined LR=6×(1/6)=1LR = 6 \times (1/6) = 1. Final Odds = 1/9×1=1/91/9 \times 1 = 1/9. Final Probability = (1/9)/(1+1/9)=1/10=10%(1/9) / (1 + 1/9) = 1/10 = 10\%.
Multiplying the sequential likelihood ratios provides the net change in odds, which is then converted back to a final probability.

Key Concept

When sequential diagnostic tests are performed independently, their combined effect on diagnostic probability is the product of their respective likelihood ratios. If the LR+LR^+ of a positive test and the LR−LR^- of a negative test are exact reciprocals, the post-test probability reverts back to the original pre-test probability.
Rate this question

Topics

Diagnostic Testing Performance

More questions from this topic