Receiver operating characteristic studies and measurement errors

A receiver operating characteristic (ROC) curve expresses the probability of a true positive (PTP) as a function of the probability of a false positive (PFP) for all possible values of the cutoff between cases and controls. Theta, the area under ROC curve, is a measure of the diagnostic ability of the separator variable. The usual nonparametric estimate of theta is shown to be based when the separator is measured with error. An expression for the largest-order term of the bias is found. The observed values and the measurement error variance are used to form a kernel estimate of the underlying distribution. These kernel estimates are used to estimate the bias. Monte Carlo simulation indicates that, for several families of distributions, the bias-corrected estimators have smaller bias and comparable MSE to the usual estimator. An application to the data of Clayton, Moncrieff, and Roberts (1967, British Medical Journal 3, 133-136) illustrates the technique.