Time-dependent AUC with right-censored data: a survey study <s> Methods based on IPCW estimators <s> The Kaplan-Meier (product-limit) estimator of the survival function of randomly-censored time-to-event data is a central quantity in survival analysis. It is usually introduced as a nonparametric maximum likelihood estimator, or else as the output of an imputation scheme for censored observations such as redistribute-to-the-right or self-consistency. Following recent work by Robins and Rotnitzky, we show that the Kaplan-Meier estimator can also be represented as a weighted average of identically distributed terms, where the weights are related to the survival function of censoring times. We give two demonstrations of this representation; the first assumes a Kaplan-Meier form for the censoring time survival function, the second estimates the survival functions of failure and censoring times simultaneously and can be developed without prior introduction to the Kaplan-Meier estimator. <s> BIB001 </s> Time-dependent AUC with right-censored data: a survey study <s> Methods based on IPCW estimators <s> Suppose that we are interested in establishing simple but reliable rules for predicting future t-year survivors through censored regression models. In this article we present inference procedures for evaluating such binary classification rules based on various prediction precision measures quantified by the overall misclassification rate, sensitivity and specificity, and positive and negative predictive values. Specifically, under various working models, we derive consistent estimators for the above measures through substitution and cross-validation estimation procedures. Furthermore, we provide large-sample approximations to the distributions of these nonsmooth estimators without assuming that the working model is correctly specified. Confidence intervals, for example, for the difference of the precision measures between two competing rules can then be constructed. All of the proposals are illustrated with real examples, and their finite-sample properties are evaluated through a simulation study. <s> BIB002 </s> Time-dependent AUC with right-censored data: a survey study <s> Methods based on IPCW estimators <s> The performance of clinical tests for disease screening is often evaluated using the area under the receiver-operating characteristic (ROC) curve (AUC). Recent developments have extended the traditional setting to the AUC with binary time-varying failure status. Without considering covariates, our first theme is to propose a simple and easily computed nonparametric estimator for the time-dependent AUC. Moreover, we use generalized linear models with time-varying coefficients to characterize the time-dependent AUC as a function of covariate values. The corresponding estimation procedures are proposed to estimate the parameter functions of interest. The derived limiting Gaussian processes and the estimated asymptotic variances enable us to construct the approximated confidence regions for the AUCs. The finite sample properties of our proposed estimators and inference procedures are examined through extensive simulations. An analysis of the AIDS Clinical Trials Group (ACTG) 175 data is further presented to show the applicability of the proposed methods. The Canadian Journal of Statistics 38:8–26; 2010 © 2009 Statistical Society of Canada <s> BIB003
In BIB003 and BIB002 , the authors independently suggested to use IPCW-type estimates: , whereŜ C (·) is the Kaplan-Meier estimator of the survival function of the censoring time C. The expression of the false positive rate estimator is more compact because weights all equal 1/(nŜ C (t)) under the assumption of independence between C and X, and then vanish. FPR D (c,t) corresponds to 1 minus the empirical distribution function of X among individuals for whom Z i > t. In the absence of censoring before time t, TPR C (c,t) also reduces to the usual empirical version of TPR C (c,t), i.e., 1 minus the empirical distribution function of X among individuals for whom T i ≤ t. It can be shown (see BIB003 BIB001 ) that an estimator of AUC C,D (t) is then given by