Time‐Dependent ROC Curves for Censored Survival Data and a Diagnostic Marker

Summary. ROC curves are a popular method for displaying sensitivity and specificity of a continuous marker, X , for a binary disease variable, D. However, many disease outcomes are time dependent, D ( t , and ROC curves that vary as a function of time may be mire appropriate. A common examples of a time‐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 discrimination potential of a marker X , measured at baseline ( t = 0), by calculating ROC Curves for cumulative disease or death incidence by time t , which we denote as ROC( t ). A typical complexity with survival data is that observations may be censored. Two ROC curve estimators are proposed that can accommodate censored data. A simple estimator is based on using the Kaplan‐Meier estimated for each possible subset X > c . However, this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in X . An alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of ( X, T ), where T represents survival time (Akritas, M. J., 1994, Annals of Statistics 22 , 1299–1327). We present an example where ROC( t ) is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the ROC( t ) curve displays the impact of modifying eligibility criteria for sample size and power in HIV prevention trials.