A screening test for tuberculosis has a sensitivity of 95% and a specificity of 90%. In a population where the prevalence of TB is 5%, what is the most common reason for a positive test result being a false positive rather than a true positive?

Explanation

## Understanding Positive Predictive Value (PPV) in Low-Prevalence Settings ### The Concept Positive Predictive Value (PPV) is the probability that a person with a positive test result actually has the disease. PPV is heavily influenced by disease prevalence, not just test characteristics. **Key Point:** PPV = $\frac{TP}{TP+FP} = \frac{Sensitivity \times Prevalence}{Sensitivity \times Prevalence + (1-Specificity) \times (1-Prevalence)}$ ### Calculation for This Scenario Given: - Sensitivity = 95% (0.95) - Specificity = 90% (0.90) - Prevalence = 5% (0.05) $$PPV = \frac{0.95 \times 0.05}{0.95 \times 0.05 + (1-0.90) \times (1-0.05)}$$ $$PPV = \frac{0.0475}{0.0475 + 0.10 \times 0.95} = \frac{0.0475}{0.0475 + 0.095} = \frac{0.0475}{0.1425} ≈ 33.3\%$$ This means only **33% of positive tests are true positives**, while **67% are false positives**. ### Why Low Prevalence Causes High False Positive Rate | Factor | Impact on False Positives | |--------|---------------------------| | Low prevalence | Fewer diseased individuals in population → more false positives relative to true positives | | High specificity | Reduces false positives, but cannot overcome low prevalence effect | | High sensitivity | Detects most true cases, but doesn't affect false positive rate | **High-Yield:** In a population where disease is rare (low prevalence), even a highly specific test will generate many false positives because there are far more non-diseased individuals than diseased ones. A 10% false positive rate applied to 95% of the population (non-diseased) produces more false positives than the 95% sensitivity applied to 5% (diseased). ### Clinical Pearl This is why screening tests work best in high-prevalence populations or high-risk groups. A test with identical sensitivity and specificity will have much higher PPV in a population with 50% prevalence than in one with 5% prevalence. **Mnemonic: "SNOUT and SPIN"** - **SNOUT** = Sensitivity rules OUT (high sensitivity means few false negatives; negative test reliably excludes disease) - **SPIN** = Specificity rules IN (high specificity means few false positives; positive test reliably confirms disease) - However, this applies best when prevalence is moderate to high. Low prevalence breaks SPIN.