Nonparametric and semiparametric optimal transformations of markers

The receiver operating characteristic (ROC) curve of a likelihood-ratio function has been shown to be the highest among all transformations of continuous markers. For any sampling scheme with the same likelihoods, the induced conditional probability is derived to have the same ROC curve and is found to be more useful for inference purposes. To compromise the difficult task of high-dimensionality in fully nonparametric models and the risk of model misspecification in fully parametric ones, an appealing single-index model is also adopted in our optimization problem. Based on a nonparametric estimator of the area under the ROC curve (AUC), we develop its related inferences and provide some simple and easily checked conditions for the validity of asymptotic results. Since the optimal marker is estimated by using a semiparametric or nonparametric model, conventional theoretical approaches might be inappropriate to some circumstances. The applicability of our procedures are further demonstrated through extensive numerical experiments and data from the studies of Pima-Indian diabetes and liver disorders.