A 50-year-old man from Delhi undergoes screening for coronary artery disease (CAD) with an exercise stress test. The test has 85% sensitivity and 70% specificity. In the general population (CAD prevalence ≈ 10%), a positive test result is obtained. Which likelihood ratio BEST explains why a positive stress test in this low-prevalence population is less conclusive than in a high-risk subset (prevalence ≈ 50%)?

Explanation

## Likelihood Ratios vs. Predictive Values: The Prevalence Paradox **Key Point:** Likelihood ratios (LR+ and LR−) are properties of the test itself and do NOT change with prevalence. However, the clinical interpretation—captured by PPV and NPV—DOES change with prevalence. ### Calculating LR+ for the Stress Test $$LR^+ = \frac{\text{Sensitivity}}{1 - \text{Specificity}} = \frac{0.85}{1 - 0.70} = \frac{0.85}{0.30} ≈ 2.83$$ This LR+ of 2.83 is **invariant**—it applies whether prevalence is 10% or 50%. ### PPV Changes with Prevalence Using Bayes' theorem: $$PPV = \frac{\text{Sensitivity} \times \text{Prevalence}}{(\text{Sensitivity} \times \text{Prevalence}) + ((1 - \text{Specificity}) \times (1 - \text{Prevalence}))}$$ **In low-prevalence population (10%):** $$PPV = \frac{0.85 \times 0.10}{(0.85 \times 0.10) + (0.30 \times 0.90)} = \frac{0.085}{0.085 + 0.27} = \frac{0.085}{0.355} ≈ 23.9\%$$ **In high-prevalence population (50%):** $$PPV = \frac{0.85 \times 0.50}{(0.85 \times 0.50) + (0.30 \times 0.50)} = \frac{0.425}{0.425 + 0.15} = \frac{0.425}{0.575} ≈ 73.9\%$$ **High-Yield:** The same positive test result has a 24% chance of true CAD in the general population but a 74% chance in a high-risk subset—despite the LR+ remaining constant at 2.83. ### Why LR+ Stays Constant LR+ depends only on test characteristics (sensitivity and specificity), not on how common the disease is in the population being tested. It answers: "How much more likely is a positive test in someone WITH disease versus someone WITHOUT disease?" This ratio is intrinsic to the test. **Clinical Pearl:** When counseling a patient with a positive stress test, the clinician must know the pre-test probability (prevalence in that patient's risk category) to interpret the result. The LR+ tells you how much the test shifts your odds, but the starting point (pre-test probability) determines the final answer (post-test probability / PPV). ### Relationship Table | Parameter | Depends on Prevalence? | Depends on Test Characteristics? | |-----------|------------------------|----------------------------------| | Sensitivity | No | Yes | | Specificity | No | Yes | | LR+ | No | Yes | | LR− | No | Yes | | PPV | **Yes** | Yes | | NPV | **Yes** | Yes | **Mnemonic:** **LR is FIXED, PPV is FLEX** - **LR** (Likelihood Ratio) is **FIXED** by test performance alone - **PPV** (Positive Predictive Value) is **FLEX**ible—it bends with prevalence