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 $X_1, \dots, X_m$ from healthy individuals and n responses $Y_1, \dots, Y_n$ 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 $0 \leq t \leq 1$ 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.