A screening test has a sensitivity of 90% and specificity of 85%. In a population where the disease prevalence is 10%, what is the positive predictive value (PPV) of this test?

Explanation

## Calculation of Positive Predictive Value **Key Point:** PPV is the probability that a person with a positive test result actually has the disease. It depends on both test characteristics (sensitivity, specificity) AND disease prevalence. ### 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 = 10% = 0.10 **Numerator (True Positives):** $$0.90 \times 0.10 = 0.09$$ **Denominator (True Positives + False Positives):** $$0.09 + (1 - 0.85) \times (1 - 0.10)$$ $$= 0.09 + 0.15 \times 0.90$$ $$= 0.09 + 0.135 = 0.225$$ **PPV:** $$PPV = \frac{0.09}{0.225} = 0.40 = 40\%$$ **High-Yield:** PPV is **inversely related to disease prevalence**. In low-prevalence populations, even a good test will have a low PPV because false positives outnumber true positives. This is why screening tests perform better in high-prevalence or high-risk populations. **Clinical Pearl:** A 90% sensitive, 85% specific test in a 10% prevalence population gives only ~41% PPV — meaning 59% of positive results are false positives. This explains why screening asymptomatic populations often requires confirmatory testing.