Time-dependent AUC with right-censored data: a survey study <s> The incident static AUC: AUC I,S (t) <s> Summary. We introduce a new estimator of the conditional survival function given some subset of the covariate values under a proportional hazards regression. The new estimate does not require estimating the base-line cumulative hazard function. An estimate of the variance is given and is easy to compute, involving only those quantities that are routinely calculated in a Cox model analysis. The asymptotic normality of the new estimate is shown by using a central limit theorem for Kaplan-Meier integrals. We indicate the straightforward extension of the estimation procedure under models with multiplicative relative risks, including non-proportional hazards, and to stratified and frailty models. The estimator is applied to a gastric cancer study where it is of interest to predict patients' survival based only on measurements obtained before surgery, the time at which the most important prognostic variable, stage, becomes known. <s> BIB001 </s> Time-dependent AUC with right-censored data: a survey study <s> The incident static AUC: AUC I,S (t) <s> The predictive accuracy of a survival model can be summarized using extensions of the proportion of variation explained by the model, or R2, commonly used for continuous response models, or using extensions of sensitivity and specificity, which are commonly used for binary response models. In this article we propose new time-dependent accuracy summaries based on time-specific versions of sensitivity and specificity calculated over risk sets. We connect the accuracy summaries to a previously proposed global concordance measure, which is a variant of Kendall's tau. In addition, we show how standard Cox regression output can be used to obtain estimates of time-dependent sensitivity and specificity, and time-dependent receiver operating characteristic (ROC) curves. Semiparametric estimation methods appropriate for both proportional and nonproportional hazards data are introduced, evaluated in simulations, and illustrated using two familiar survival data sets. <s> BIB002 </s> Time-dependent AUC with right-censored data: a survey study <s> The incident static AUC: AUC I,S (t) <s> ∗Harvard University, tcai@hsph.harvard.edu †University of Washington, mspepe@u.washington.edu ‡University of Washington, tlumley@u.washington.edu ∗∗Fred Hutchinson Cancer Research Center, yzheng@fhcrc.org ††University of Vermont, Nancy.Jenny@uvm.edu This working paper is hosted by The Berkeley Electronic Press (bepress) and may not be commercially reproduced without the permission of the copyright holder. <s> BIB003 </s> Time-dependent AUC with right-censored data: a survey study <s> The incident static AUC: AUC I,S (t) <s> Receiver operating characteristic (ROC) curves play a central role in the evaluation of biomarkers and tests for disease diagnosis. Predictors for event time outcomes can also be evaluated with ROC curves, but the time lag between marker measurement and event time must be acknowledged. We discuss different definitions of time-dependent ROC curves in the context of real applications. Several approaches have been proposed for estimation. We contrast retrospective versus prospective methods in regards to assumptions and flexibility, including their capacities to incorporate censored data, competing risks and different sampling schemes. Applications to two datasets are presented. <s> BIB004 </s> Time-dependent AUC with right-censored data: a survey study <s> The incident static AUC: AUC I,S (t) <s> For censored survival outcomes, it can be of great interest to evaluate the predictive power of individual markers or their functions. Compared with alternative evaluation approaches, approaches based on the time-dependent receiver operating characteristics (ROC) rely on much weaker assumptions, can be more robust, and hence are preferred. In this article, we examine evaluation of markers' predictive power using the time-dependent ROC curve and a concordance measure that can be viewed as a weighted area under the time-dependent area under the ROC curve profile. This study significantly advances from existing time-dependent ROC studies by developing nonparametric estimators of the summary indexes and, more importantly, rigorously establishing their asymptotic properties. It reinforces the statistical foundation of the time-dependent ROC-based evaluation approaches for censored survival outcomes. Numerical studies, including simulations and application to an HIV clinical trial, demonstrate the satisfactory finite-sample performance of the proposed approaches. <s> BIB005 </s> Time-dependent AUC with right-censored data: a survey study <s> The incident static AUC: AUC I,S (t) <s> A major biomedical goal associated with evaluating a candidate biomarker or developing a predictive model score for event-time outcomes is to accurately distinguish between incident cases from the controls surviving beyond t throughout the entire study period. Extensions of standard binary classification measures like time-dependent sensitivity, specificity, and receiver operating characteristic (ROC) curves have been developed in this context (Heagerty, P. J., and others, 2000. Time-dependent ROC curves for censored survival data and a diagnostic marker. Biometrics 56, 337-344). We propose a direct, non-parametric method to estimate the time-dependent Area under the curve (AUC) which we refer to as the weighted mean rank (WMR) estimator. The proposed estimator performs well relative to the semi-parametric AUC curve estimator of Heagerty and Zheng (2005. Survival model predictive accuracy and ROC curves. Biometrics 61, 92-105). We establish the asymptotic properties of the proposed estimator and show that the accuracy of markers can be compared very simply using the difference in the WMR statistics. Estimators of pointwise standard errors are provided. <s> BIB006
Using the dynamic definition of controls, the control group varies with time, and so does the x-axis of the corresponding ROC curves: in situations where trends over time are of particular interest, this renders their interpretation more difficult (since such trends may be partly due to changing control groups). Moreover, the group of static controls is interesting in that it tries to mimic the group of individuals who never develop the disease, which can be seen as an ideal control group in some situations. In particular, patients with preclinical diseases are eliminated from the control group as far as possible, if τ is large enough. Regarding cases, the incident definition has several advantages over the dynamic definition BIB004 . The cumulative TPR does not distinguish between events that occur early versus late, and it shows redundant information over time (since early events are also included in the cumulative TPR for late evaluation times). Moreover, as pointed out by BIB003 , the cumulative TPR can be computed from the incident TPR when the distribution of the event time is known. Putting all this together, several authors have proposed estimators of the timedependent ROC curve relying on the incident definition of cases and static definition of controls. Standard numerical integration techniques are then used to compute an estimate for AUC I,S (t) from the estimators of the ROC curve. Incident true positive rates and static false positive rates are defined as Applying Bayes' theorem, they can further be rewritten (see, e.g.,  ) where f (t|x) = ∂ IP(T ≤ t|X = x)/∂t is the conditional density function of T given X = x. Under a standard Cox model of the form λ (t; X) = λ 0 (t) exp(β X) -here λ (t; X) stands for the conditional hazard rate of T given X while λ 0 is the unspecified baseline hazard rate -Song and Zhou  deduced that , where Λ 0 (t) = t −∞ λ 0 (u)du is the cumulative baseline hazard function. Estimation of TPR I (c,t) and FPR S τ (c) can then be achieved by plug-in methods. We shall add that Song and Zhou actually considered a slightly more general set-up where additional covariates can be accounted for. In BIB002 , Heagerty and Zheng adopted a slightly different approach. To estimate TPR I (c,t), they used a (possibly time-varying-coefficients) Cox model of the form λ (t; X) = λ 0 (t) exp(β (t)X) in combination with the fact that the distribution of X· exp(β X) for subjects in the risk set at time t is equal to the conditional distribution of X given T = t (see, e.g., BIB001 ). Setting R(t) = {i : Z i ≥ t}, this leads to As for the estimation of FPR S τ (c), they proposed a model-free approach using the empirical distribution function for marker values among the control set S τ := {i : Z i > τ}. Namely, denoting by n τ the cardinality of S τ , they proposed which is FPR D (c,t) of Section 3.1.2, except τ is used instead of t. Cai et al. BIB003 proposed another approach in the context of longitudinal markers; it will be described in more details in Section 4. In addition, two non parametric approaches were recently proposed (see BIB005 and BIB006 ). Note also that estimators for the time-dependant incident/dynamic AUC, AUC I,D (t), can be obtained by simply replacing τ by t in the definitions of FPR S τ (c) above BIB002 . A global accuracy measure has further been derived from the definition of AUC I,D (t), which is particularly appealing when no a priori time t is identified and/or when trends over time are not of interest BIB002 .