Youden's J statistic

Youden's J statistic (also called Youden's index) is a single statistic that captures the performance of a dichotomous diagnostic test. (Bookmaker) Informedness is its generalization to the multiclass case and estimates the probability of an informed decision.

Definition

Youden's J statistic is

${\displaystyle J={\text{sensitivity}}+{\text{specificity}}-1={\text{recall}}_{1}+{\text{recall}}_{0}-1}$

with the two right-hand quantities being sensitivity and specificity. Thus the expanded formula is:

${\displaystyle J={\frac {\text{true positives}}{{\text{true positives}}+{\text{false negatives}}}}+{\frac {\text{true negatives}}{{\text{true negatives}}+{\text{false positives}}}}-1}$

The index was suggested by W. J. Youden in 1950^[1] as a way of summarising the performance of a diagnostic test; however, the formula was earlier published in Science by C. S. Pierce in 1884.^[2] Its value ranges from -1 through 1 (inclusive),^[1] and has a zero value when a diagnostic test gives the same proportion of positive results for groups with and without the disease, i.e the test is useless. A value of 1 indicates that there are no false positives or false negatives, i.e. the test is perfect. The index gives equal weight to false positive and false negative values, so all tests with the same value of the index give the same proportion of total misclassified results. While it is possible to obtain a value of less than zero from this equation, e.g. Classification yields only False Positives and False Negatives, a value of less than zero just indicates that the positive and negative labels have been switched. After correcting the labels the result will then be in the 0 through 1 range.

Youden's index is often used in conjunction with receiver operating characteristic (ROC) analysis.^[3] The index is defined for all points of an ROC curve, and the maximum value of the index may be used as a criterion for selecting the optimum cut-off point when a diagnostic test gives a numeric rather than a dichotomous result. The index is represented graphically as the height above the chance line, and it is also equivalent to the area under the curve subtended by a single operating point.^[4]

Youden's index is also known as deltaP' ^[5] and generalizes from the dichotomous to the multiclass case as informedness.^[4]

The use of a single index is "not generally to be recommended",^[6] but informedness or Youden's index is the probability of an informed decision (as opposed to a random guess) and takes into account all predictions.^[4]

An unrelated but commonly used combination of basic statistics from