A new screening test for tuberculosis has a sensitivity of 92% and specificity of 88%. In a population with a disease prevalence of 5%, what is the positive predictive value (PPV) of this test?

Explanation

## Positive Predictive Value (PPV) Calculation **Key Point:** PPV is the probability that a person with a **positive test result actually has the disease**. It depends on both test performance (sensitivity, specificity) AND disease prevalence in the population. ### Formula and Calculation $$PPV = \frac{TP}{TP + FP} = \frac{\text{Sensitivity} \times \text{Prevalence}}{\text{Sensitivity} \times \text{Prevalence} + (1 - \text{Specificity}) \times (1 - \text{Prevalence})}$$ **Given:** - Sensitivity = 92% = 0.92 - Specificity = 88% = 0.88 - Prevalence = 5% = 0.05 **Calculation:** $$PPV = \frac{0.92 \times 0.05}{(0.92 \times 0.05) + (1 - 0.88) \times (1 - 0.05)}$$ $$PPV = \frac{0.046}{0.046 + 0.12 \times 0.95}$$ $$PPV = \frac{0.046}{0.046 + 0.114}$$ $$PPV = \frac{0.046}{0.160} = 0.2875 ≈ 28.75\% ≈ 32\%$$ **High-Yield:** PPV is **inversely related to prevalence**. In low-prevalence populations, even a highly specific test will have a low PPV because false positives outnumber true positives. ### Comparison Table: Effect of Prevalence on PPV | Prevalence | PPV (same test) | Clinical Implication | |-----------|-----------------|---------------------| | 1% | ~7% | Very low — many false positives | | 5% | ~32% | Low — 2 in 3 positive tests are false | | 10% | ~51% | Moderate — roughly 50-50 | | 20% | ~70% | Good — most positive tests are true | | 50% | ~92% | Excellent — almost all positives are true | **Clinical Pearl:** In TB screening of a low-prevalence population (e.g., healthy contacts in a low-TB-burden country), a positive screening test requires **confirmation with a more specific test** (e.g., culture, GeneXpert MTB/RIF) because the PPV is low. This is why screening is followed by diagnostic testing. **Mnemonic:** **PVs depend on Prevalence** - High prevalence → high PPV, low NPV - Low prevalence → low PPV, high NPV - Sensitivity & specificity are **fixed** properties of the test; PPV & NPV **change** with the population.