Nonparametric and semiparametric estimation of the receiver operating characteristic curve

The receiver operating characteristic (ROC) curve describes the performance of a diagnostic test used to discriminate between healthy and diseased individuals based on a variable measured on a continuous scale. The data consist of a training set of m responses X₁, , Xₘ from healthy individuals and n responses Y₁, , Yₙ from diseased individuals. The responses are assumed i.i.d. from unknown distributions F and G, respectively. We consider estimation of the ROC curve defined by 1 - G(F⁻¹ (1 - t)) for 0 ≤ t ≤ 1 or, equivalently, the ordinal dominance curve (ODC) given by F(G⁻¹ (t)). First we consider nonparametric estimators based on empirical distribution functions and derive asymptotic properties. Next we consider the so-called semiparametric "binormal" model, in which it is assumed that the distributions F and G are normal after some unknown monotonic transformation of the measurement scale. For this model, we propose a generalized least squares procedure and compare it with the estimation algorithm of Dorfman and Alf, which is based on grouped data. Asymptotic results are obtained; small sample properties are examined via a simulation study. Finally, we describe a minimum distance estimator for the ROC curve, which does not require grouping the data.