A screening test for tuberculosis has a sensitivity of 90% and a specificity of 85%. In a population where the prevalence of TB is 5%, what is the positive predictive value (PPV) of this test?

Explanation

## Calculating Positive Predictive Value **Key Point:** PPV depends not only on test sensitivity and specificity, but critically on disease prevalence in the population being tested. ### Formula $$PPV = \frac{TP}{TP + FP} = \frac{Sensitivity \times Prevalence}{(Sensitivity \times Prevalence) + ((1 - Specificity) \times (1 - Prevalence))}$$ ### Step-by-Step Calculation Given: - Sensitivity = 90% = 0.90 - Specificity = 85% = 0.85 - Prevalence = 5% = 0.05 **Numerator (True Positives):** $$TP = 0.90 \times 0.05 = 0.045$$ **Denominator (True Positives + False Positives):** $$FP = (1 - 0.85) \times (1 - 0.05) = 0.15 \times 0.95 = 0.1425$$ $$TP + FP = 0.045 + 0.1425 = 0.1875$$ **PPV:** $$PPV = \frac{0.045}{0.1875} = 0.24 = 24\%$$ ### Clinical Interpretation | Metric | Value | Meaning | |--------|-------|----------| | Sensitivity | 90% | Test catches 90% of true TB cases | | Specificity | 85% | Test correctly identifies 85% of non-TB individuals | | Prevalence | 5% | TB is uncommon in this population | | **PPV** | **24%** | Only 24% of positive tests are true TB cases | **High-Yield:** In low-prevalence populations, even a good test produces many false positives, drastically reducing PPV. This is why screening tests are most useful in higher-prevalence or higher-risk populations. **Clinical Pearl:** A positive TB screening test in a low-prevalence population requires confirmatory testing (e.g., culture, GeneXpert MTB/RIF) before initiating treatment, because 76% of positive results are false positives.