“If a test with perfect sensitivity to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs?”

## Using PPV and Bayes Formula

There are several ways to solve this problem. Shown below is how calculate the answer using PPV and Bayes formula.

## Visualizing PPV

While understanding the math is important, you can develop a more intuitive understanding by visualizing the population (see figure below). If a draw a sample of 1000 patients from the population (all of the dots), only one will have the disease (red dot). Since the test has a 100% sensitivity, it will be a true positive. The remaining 999 patients do not have the disease, but 50 of them will test positive (blue dots), and the rest will test negative (black dots).

Notice that the false positive blue dots far out number the true positives. If this hypothetical example, those patients would be subjected to unnecessary further workup or treatment.

The solution to the posed question then is easy to see. If your patient has a positive test (which 51 out of 1000 would), only 1 actually have the disease. 1/51 is then calculated at 1.96%. PPV calculation done by visualization without any formula memorization.

References

- N Engl J Med 1978;299(18):999. Interpretation by physicians of clinical laboratory results.
- JAMA Internal Medicine 2014;174(6):991. Medicine’s Uncomfortable Relationship With Math: Calculating Positive Predictive Value.

dave plummerNovember 30, 2014 at 5:24 pmI love the “dot-based” graphic explanation of this solution.

MathematicianFebruary 13, 2018 at 4:28 amTo be absolutly precise, it is P(B)=1/1000+5%*0.999 in Bayes Formula!