Time-dependent ROC curves for censored survival data and a diagnostic marker

ROC curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker, X, for a binary disease variable, D. How ever, many disease outcomes are time dependent, D(t), and ROC curves that v ary as a function of time may be more appropriate. A common example of a ti me-dependent variable is vital status, where D(t) = 1 if a patient has died prior to time t and zero otherwise. We propose summarizing the discriminat ion potential of a marker X, measured at baseline (t = 0), by calculating R OC curves for cumulative disease or death incidence by time t, which we den ote as ROC(t). A typical complexity with survival data is that observations may be censored. Two ROC curve estimators are proposed that can accommodat e censored data. A simple estimator is based on using the Kaplan-Meier esti mator for each possible subset X > c. However, this estimator does not guar antee the necessary condition that sensitivity and specificity are monotone in X. An alternative estimator that does guarantee monotonicity is based o n a nearest neighbor estimator for the bivariate distribution function of ( X, T), where T represents survival time (Akritas, M. J., 1994, Annals of St atistics 22, 1299-1327). We present an example where ROC(t) is used to comp are a standard and a modified flow cytometry measurement for predicting sur vival after detection of breast cancer and an example where the ROC(t) curv e displays the impact of modifying eligibility criteria for sample size and power in HIV prevention trials.