Several authors proposed estimation methods for the time-dependent ROC curve and AUC. Heagerty et al (2000) introduced two estimation methods for time-dependent sensitivity and specicity. The first method uses the empirical distribution function to estimate the marker distribution and the Kapan-Meier estimator of the survival function for subgroups with marker values Z > c and Z c.
However, this method has two main limitations; (i) it is valid only if the censoring time and the marker are independent; (ii) the sensitivity and specicity are not necessarily monotone in c and also notbounded in 0, 1. To overcome these limitations, they proposed the second method called the nearest neighbor estimation (NNE) method. This method uses nonparametric kernel methods to estimate the bivariate distribution of the marker and survival time which was rst introduced by Akritas (1994).The limitation of this method is that there is no proposed rule to nd the optimal bandwidth, although the results are sensitive to the choice this parameter (Li et al., 2016).Chambless and Diao (2006) proposed an alternative method to the Kaplan-Meier approach to the survival function estimation.
This method use a recursive calculation of the ordered times of the event. This method results in a sensitivity which is monotone and bounded in 0, 1. An additional nice property of this method is that, unlike NNE, it does not involve any smoothing parameter. Chambless and Diao (2006) also proposed another method using model based approach. This method is based on the Baye’s theorem to rewrite sensitivity and specicity in terms of conditional survival function in order to estimate the conditional probabilities. It assumes conditional independence between censoring time and the event time given the marker. The main disadvantage of this method is that, it is not invariant to an increasing transformation of the marker.
Song and Zhou (2008) extended the Chambless and Diao (2006) model based approach to account for the covariate eect and to estimate covariate-specic ROC curve. Uno et al. (2007) and Hung and Chiang (2010) independently proposed inverse probability of censoring weighting (IPCW) method to correct Kaplan-Meier estimate of the sensitivity by adding weights to the observations in the sub-sample of uncensored individual before time t.