Simple Decision-Analytic Functions of the AUC for Ruling Out a Risk Prediction Model and an Added Predictor

When using risk prediction models, an important consideration is weighing performance against the cost (monetary and harms) of ascertaining predictors.

The minimum test tradeoff (MTT) for ruling out a model is the minimum number of all-predictor ascertainments per correct prediction to yield a positive overall expected utility. The MTT for ruling out an added predictor is the minimum number of added-predictor ascertainments per correct prediction to yield a positive overall expected utility.

An approximation to the MTT for ruling out a model is 1/[P (H(AUC_model)], where H(AUC) = AUC - {½ (1-AUC)}^½, AUC is the area under the receiver operating characteristic (ROC) curve, and P is the probability of the predicted event in the target population. An approximation to the MTT for ruling out an added predictor is 1 /[P {(H(AUC_Model:2) - H(AUC_Model:1 )], where Model 2 includes an added predictor relative to Model 1.

The latter approximation requires the Tangent Condition that the true positive rate at the point on the ROC curve with a slope of 1 is larger for Model 2 than Model 1.

These approximations are suitable for back-of-the-envelope calculations. For example, in a study predicting the risk of invasive breast cancer, Model 2 adds to the predictors in Model 1 a set of 7 single nucleotide polymorphisms (SNPs). Based on the AUCs and the Tangent Condition, an MTT of 7200 was computed, which indicates that 7200 sets of SNPs are needed for every correct prediction of breast cancer to yield a positive overall expected utility. If ascertaining the SNPs costs $500, this MTT suggests that SNP ascertainment is not likely worthwhile for this risk prediction.