Youden's J Index

W. J. Youden proposed this in 1950 in a short paper in Cancer, asking for a single figure that would let two diagnostic tests be ranked. His answer was disarmingly simple: add sensitivity and specificity and subtract one. A useless test — one whose positive rate is the same in sick and healthy people — scores 0, and a perfect test scores 1.

Geometrically, J is the vertical distance from a point on the ROC curve up to the diagonal line of no information, and that reading explains what it is used for. Choosing the cut-off that maximises J picks the point on the curve furthest from useless, which is the most common automated rule for selecting a threshold from validation data.

The rule embeds an assumption that is almost never true, and it should be stated rather than absorbed. Adding sensitivity and specificity with equal weight asserts that a missed case and a false alarm cost the same. In screening for a treatable cancer they emphatically do not — a missed case may cost a life while a false alarm costs an anxious fortnight and a biopsy — and in that setting the J-maximising cut-off is the wrong cut-off. It is a decision that ought to be made with an explicit loss ratio, and J is what you get when you decline to name one.

Two smaller cautions. A negative J is arithmetically possible and means the test performs worse than chance; in practice it almost always means the positive and negative labels are reversed somewhere in the data rather than that a genuinely perverse test has been found. And because J collapses two numbers into one, very different tests share a score: 0.85 is both a 90/95 test and a 95/90 test, which behave nothing alike in a low-prevalence population. J ranks; it does not describe.