Positive predictive value

The formula

• PPV = TP / (TP + FP)

TP, the true positives, are the people with the condition whom the test flags; FP, the false positives, are the people without it whom it flags. The denominator is everyone with a positive result, and the PPV is the share of them who really have the condition: the question a person holding a positive result is asking.

A worked example

30 true positives and 10 false positives: PPV = 30 / (30 + 10) = 0.75. Three positive results in four are right. Turned round, a PPV of 0.75 with 10 false positives means 30 true ones, as the card works out.

Why it depends on how common the condition is

Take a test with a sensitivity of 90% and a specificity of 90%. Among 1,000 people of whom 100 have the condition, it flags 90 true cases and 90 of the 900 healthy: PPV = 90 / 180 = 50%. Among 1,000 people of whom 10 have it, it flags 9 true cases and 99 healthy: PPV = 9 / 108 = 8%. The test is the same; the people tested are not, and the PPV follows the prevalence. That is Bayes’ theorem written as counts.

What it is for

Sensitivity and specificity describe the test for the people who make it; the predictive values describe a result for the person who gets it. A table from a study that chose its cases and controls separately gives a PPV for that study’s mix, not for a clinic, which is why the sensitivity and specificity calculator also takes a prevalence and the two rates.

Common mistakes

• Reading it as the sensitivity. Sensitivity reads down a column; PPV reads across a row.
• Carrying a PPV from one population to another. It travels with the prevalence.
• Reading 1 − PPV as the false positive rate. The false positive rate is FP over the healthy, 1 − Sp.

Frequently asked questions