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

Abstract

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 Statistics22, 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.