A principled approach to setting optimal diagnostic thresholds: where ROC and indifference curves meet

Making clinical decisions on the basis of diagnostic tests is an essential feature of medical practice and the choice of the decision threshold is therefore crucial. A test's optimal diagnostic threshold is the threshold that maximizes expected utility. It is given by the product of the prior odds of a disease and a measure of the importance of the diagnostic test's sensitivity relative to its specificity. Choosing this threshold is the same as choosing the point on the Receiver Operating Characteristic (ROC) curve whose slope equals this product. We contend that a test's likelihood ratio is the canonical decision variable and contrast diagnostic thresholds based on likelihood ratio with two popular rules of thumb for choosing a threshold. The two rules are appealing because they have clear graphical interpretations, but they yield optimal thresholds only in special cases. The optimal rule can be given similar appeal by presenting indifference curves, each of which shows a set of equally good combinations of sensitivity and specificity. The indifference curve is tangent to the ROC curve at the optimal threshold. Whereas ROC curves show what is feasible, indifference curves show what is desirable. Together they show what should be chosen.