A diagnostic 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

## Calculating Positive Predictive Value (PPV) **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 in the population. ### Formula $$PPV = \frac{TP}{TP + FP} = \frac{\text{Sensitivity} \times \text{Prevalence}}{\text{Sensitivity} \times \text{Prevalence} + (1 - \text{Specificity}) \times (1 - \text{Prevalence})}$$ ### Step-by-Step Calculation Given: - Sensitivity = 90% = 0.90 - Specificity = 85% = 0.85 - Prevalence = 10% = 0.10 Substituting into the formula: $$PPV = \frac{0.90 \times 0.10}{(0.90 \times 0.10) + (1 - 0.85) \times (1 - 0.10)}$$ $$PPV = \frac{0.09}{0.09 + 0.15 \times 0.90}$$ $$PPV = \frac{0.09}{0.09 + 0.135}$$ $$PPV = \frac{0.09}{0.225} = 0.40 \approx 41\%$$ ### Clinical Interpretation **High-Yield:** Even with a 90% sensitive and 85% specific test, when disease prevalence is low (10%), a positive test result means there is only a **41% chance** the patient actually has the disease. This illustrates why positive predictive value drops dramatically in low-prevalence populations. **Clinical Pearl:** This is why screening tests in low-prevalence populations generate many false positives, leading to unnecessary anxiety and further investigations. ### Key Insight PPV is inversely related to prevalence — the lower the disease prevalence, the lower the PPV for the same test. Conversely, in high-prevalence populations, the same test will have a much higher PPV.