Changing T will change both FNR and FPR in a fashion determined by the shape and the overlap of Distdis and Distnodis. To minimize the overall burden of false-positive and false-negative results combined (with respect to changing T), one can differentiate the expression with respect to T and set the result to zero. That yields
Rearranging, we get:
This can be shown to equal:
Another way of thinking about the cutoff criterion is to understand that it is the point t at which Distdis(t) × (p) × Bfn = Distnodis(t) × (1 − p) × Bfp. That is an equivalent formulation of the same equation because dTPR/dT is simply probability density distribution Distdis, and dFPR/dT is simply probability density distribution Distnodis.
Now, if we plot TPR (vertical axis) against FPR (horizontal axis), we have the receiver operating characteristic (ROC) curve of the test. The slope of that curve at any point is simply dTPR/dFPR. Hence, the optimal operating point is the value of T where the slope of the curve (or its tangent) is numerically equivalent to
Some authors use the term cost of false positive (C) in place of burden of false positive and the term benefit of true positive (B) in place of burden of false negative, all being greater than zero. In that case, the optimal operating point is the value of T where the slope of the ROC curve (or its tangent) is numerically equivalent to
The true and false-positive rates (from which one constructs an ROC curve) are the areas under the tails of the corresponding probability density distributions