Estimation of time-dependent area under the ROC curve for long-term risk prediction.

Sensitivity, specificity, and area under the ROC curve (AUC) are often used to measure the ability of survival models to predict future risk. Estimation of these parameters is complicated by the fact that these parameters are time-dependent and by the fact that censoring affects their estimation just as it affects estimation of survival curves or coefficients of survival regression models. The authors present several estimators that overcome these complications. One approach is a recursive calculation over the ordered times of events, analogous to the Kaplan-Meier approach to survival function estimation. Another is to first apply Bayes' theorem to write the parameters of interest in terms of conditional survival functions that are then estimated by survival analysis methods. Simulation studies demonstrate that the proposed estimators perform well in practical situations, when compared with an estimator (c-statistic, from logistic regression) that ignores time. An illustration with data from a cardiovascular follow-up study is provided.