Semiparametric estimation of time-dependent ROC curves for longitudinal marker data

One approach to evaluating the strength of association between a longitudinal marker process and a key clinical event time is through predictive regression methods such as a time-dependent covariate hazard model. For example, a Cox model with time-varying covariates specifies the instantaneous risk of the event as a function of the time-varying marker and additional covariates. In this manuscript we explore a second complementary approach which characterizes the distribution of the marker as a function of both the measurement time and the ultimate event time. Our goal is to extend the standard diagnostic accuracy concepts of sensitivity and specificity so as to recognize explicitly both the timing of the marker measurement and the timing of disease. The accuracy of a longitudinal marker can be fully characterized using time-dependent receiver operating characteristic (ROC) curves. We detail a semiparametric estimation method for time-dependent ROC curves that adopts a regression quantile approach for longitudinal data introduced by Heagerty and Pepe (1999, Applied Statistics, 48, 533-551). We extend the work of Heagerty and Pepe (1999, Applied Statistics, 48, 533-551) by developing asymptotic distribution theory for the ROC estimators where the distributional shape for the marker is allowed to depend on covariates. To illustrate our method, we analyze pulmonary function measurements among cystic fibrosis subjects and estimate ROC curves that assess how well the pulmonary function measurement can distinguish subjects that progress to death from subjects that remain alive. Comparing the results from our semiparametric analysis to a fully parametric method discussed by Etzioni et al. (1999, Medical Decision Making, 19, 242-251) suggests that the ability to relax distributional assumptions may be important in practice.