The maximum Youden index (Y) for a ROC plot should identify the optimum cut-off point for a test. The Youden Index (described above) is based on combining both the sensitivity and specificity for a test.

Y = (sensitivity S) + (specificity E) - 1

Requirements:

(1) The ROC plot of S vs (1 - E) should be a smooth, continuous, upward sloping curve.

(2) Availability of a computer program to approximate the ROC plot as a second order equation, with S as Y-axis and (1 - E) the X-axis.

(3) For the computer program to provide a reasonable second order equation, it may be necessary to exclude value pairs at very low and very high specificities (i.e., modeling is done on the mid-range of the plot).

The second order equation for sensitivity is:

S =

= (A * ((1 - E)^2)) + (B * (1 - E)) + C

Youden index Y =

= (A * ((1 - E)^2)) + (B * (1 - E)) + C + E - 1 =

= (A * (E^2)) + ((1 - B - 2A) * E) + (A + B + C - 1)

The maximum Youden index will occur where the first derivative of the equation equals 0.

first derivative of Youden index =

= (2A * E) + (1 - B - 2A)

specificity for the maximum Youden index (when the first derivative equals 0) =

= (2A + B - 1) / (2A)

The sensitivity for the maximum Youden index can be estimated from the plot or calculated from the second order equation.