Semiparametric Least Squares Based Estimation of the Receiver Operating Characteristic(ROC) Curve

The receiver operating characteristics (ROC) curve is a standard statistical tool to characterize the accuracy of diagnostic tests when test results are continuous. It provides a complete description of test performance and a meaningful way to compare the performances of different tests. The empirical (nonparametric) ROC curve is the most popular estimator of the ROC curve. Semiparametric estimation of the ROC curve assumes the ROC curve takes a parametric form, while leaving the distributional form of the test results unspecified. Various semiparametric methods have been developed in recent years, yet few of them are intuitive or easy to apply in practice. Here we propose a least squares based semiparametric method to estimate the ROC curve. The method is simple yet elegant and can be implemented using standard linear regression algorithms. Covariates can be included in the linear regression. We derive two estimators: an ordinary least squares (OLS) estimator and a generalized least squares (GLS) estimator. Asymptotic theory shows the new estimators are asymptotically unbiased and normally distributed. In large samples, the OLS estimator is more efficient than the nonparametric estimator as well as a related least squares based estimator of categorized data proposed by Hsieh & Turnbull (1996). It performs at least as well as other semiparametric estimators but is much easier to implement. We illustrate the new method on a pancreatic cancer biomarker data set.