Time-dependent AUC with right-censored data: a survey study <s> Time-dependent AUC and Competing risks <s> Methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed. When a test is based on an observed variable that lies on a continuous or graded scale, an assessment of the overall value of the test can be made through the use of a receiver operating characteristic (ROC) curve. The curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates. When two or more empirical curves are constructed based on tests performed on the same individuals, statistical analysis on differences between curves must take into account the correlated nature of the data. This paper presents a nonparametric approach to the analysis of areas under correlated ROC curves, by using the theory on generalized U-statistics to generate an estimated covariance matrix. <s> BIB001 </s> Time-dependent AUC with right-censored data: a survey study <s> Time-dependent AUC and Competing risks <s> ROC curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker, X, for a binary disease variable, D. However, many disease outcomes are time dependent, D(t), and ROC curves that vary as a function of time may be more appropriate. A common example of a time-dependent variable is vital status, where D(t) = 1 if a patient has died prior to time t and zero otherwise. We propose summarizing the discrimination potential of a marker X, measured at baseline (t = 0), by calculating ROC curves for cumulative disease or death incidence by time t, which we denote as ROC(t). A typical complexity with survival data is that observations may be censored. Two ROC curve estimators are proposed that can accommodate censored data. A simple estimator is based on using the Kaplan-Meier estimator for each possible subset X > c. However, this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in X. An alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of (X, T), where T represents survival time (Akritas, M. J., 1994, Annals of Statistics 22, 1299-1327). We present an example where ROC(t) is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the ROC(t) curve displays the impact of modifying eligibility criteria for sample size and power in HIV prevention trials. <s> BIB002 </s> Time-dependent AUC with right-censored data: a survey study <s> Time-dependent AUC and Competing risks <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> BIB003 </s> Time-dependent AUC with right-censored data: a survey study <s> Time-dependent AUC and Competing risks <s> Competing risks arise naturally in time-to-event studies. In this article, we propose time-dependent accuracy measures for a marker when we have censored survival times and competing risks. Time-dependent versions of sensitivity or true positive (TP) fraction naturally correspond to consideration of either cumulative (or prevalent) cases that accrue over a fixed time period, or alternatively to incident cases that are observed among event-free subjects at any select time. Time-dependent (dynamic) specificity (1-false positive (FP)) can be based on the marker distribution among event-free subjects. We extend these definitions to incorporate cause of failure for competing risks outcomes. The proposed estimation for cause-specific cumulative TP/dynamic FP is based on the nearest neighbor estimation of bivariate distribution function of the marker and the event time. On the other hand, incident TP/dynamic FP can be estimated using a possibly nonproportional hazards Cox model for the cause-specific hazards and riskset reweighting of the marker distribution. The proposed methods extend the time-dependent predictive accuracy measures of Heagerty, Lumley, and Pepe (2000, Biometrics 56, 337-344) and Heagerty and Zheng (2005, Biometrics 61, 92-105). <s> BIB004 </s> Time-dependent AUC with right-censored data: a survey study <s> Time-dependent AUC and Competing risks <s> The medical decision-making community has an extensive literature on the use of receiver operating characteristic (ROC) graphs for diagnostic testing. Heagertybiet al. have recently developed this ROC curve theory within the context of survival data (Biometrics 2000; 56:337-344). The time-dependent ROC method allows evaluating the accuracy of a marker to predict a time-dependent failure, whereas the classic methodology focuses on diagnosis. One limitation to this approach, however, is to analyse a single failure. In many medical situations, a marker can be useful to predict different competitive failures. For example in kidney transplantation, the terminal evolution can be a return to dialysis or the death of the patient. With this application in mind, our paper proposes an extension of the time-dependent ROC method for analysing the accuracy of a marker to predict two competitive events. <s> BIB005 </s> Time-dependent AUC with right-censored data: a survey study <s> Time-dependent AUC and Competing risks <s> Time-dependent covariates are frequently encountered in regression analysis for event history data and competing risks. They are often essential predictors, which cannot be substituted by time-fixed covariates. This study briefly recalls the different types of time-dependent covariates, as classified by Kalbfleisch and Prentice [The Statistical Analysis of Failure Time Data, Wiley, New York, 2002] with the intent of clarifying their role and emphasizing the limitations in standard survival models and in the competing risks setting. If random (internal) time-dependent covariates are to be included in the modeling process, then it is still possible to estimate cause-specific hazards but prediction of the cumulative incidences and survival probabilities based on these is no longer feasible. This article aims at providing some possible strategies for dealing with these prediction problems. In a multi-state framework, a first approach uses internal covariates to define additional (intermediate) transient states in the competing risks model. Another approach is to apply the landmark analysis as described by van Houwelingen [Scandinavian Journal of Statistics 2007, 34, 70-85] in order to study cumulative incidences at different subintervals of the entire study period. The final strategy is to extend the competing risks model by considering all the possible combinations between internal covariate levels and cause-specific events as final states. In all of those proposals, it is possible to estimate the changes/differences of the cumulative risks associated with simple internal covariates. An illustrative example based on bone marrow transplant data is presented in order to compare the different methods. <s> BIB006 </s> Time-dependent AUC with right-censored data: a survey study <s> Time-dependent AUC and Competing risks <s> To develop more targeted intervention strategies, an important research goal is to identify markers predictive of clinical events. A crucial step toward this goal is to characterize the clinical performance of a marker for predicting different types of events. In this article, we present statistical methods for evaluating the performance of a prognostic marker in predicting multiple competing events. To capture the potential time-varying predictive performance of the marker and incorporate competing risks, we define time- and cause-specific accuracy summaries by stratifying cases based on causes of failure. Such definition would allow one to evaluate the predictive accuracy of a marker for each type of event and compare its predictiveness across event types. Extending the nonparametric crude cause-specific receiver operating characteristics curve estimators by Saha and Heagerty (2010), we develop inference procedures for a range of cause-specific accuracy summaries. To estimate the accuracy measures and assess how covariates may affect the accuracy of a marker under the competing risk setting, we consider two forms of semiparametric models through the cause-specific hazard framework. These approaches enable a flexible modeling of the relationships between the marker and failure times for each cause, while efficiently accommodating additional covariates. We investigate the asymptotic property of the proposed accuracy estimators and demonstrate the finite sample performance of these estimators through simulation studies. The proposed procedures are illustrated with data from a prostate cancer prognostic study. <s> BIB007 </s> Time-dependent AUC with right-censored data: a survey study <s> Time-dependent AUC and Competing risks <s> This paper describes how population cancer registry data from the Surveillance, Epidemiology, and End Results program of the National Cancer Institute can be used to develop a prognostic model to predict the absolute risk of mortality from cancer and from other causes for an individual with specific covariates. It incorporates previously developed methods for competing risk modeling along with an imputation method to address missing cause of death information. We illustrate these approaches with colorectal cancer and evaluate the model discriminatory and calibration accuracy by time-dependent area under the receiver operating characteristic curve and calibration plot. <s> BIB008
We now consider the setting where a subject might experience multiple type of failures: in this section, we review extensions of time-dependent AUCs to competing risks. For example, we may want to assess the discrimination of a given score on death from prostate cancer with death from other causes acting as a competing event. For the sake of simplicity, we will assume there are only two competing events, and we let δ i = j denote that subject i experienced the competing event of type j ( j = 1, 2, with j = 1 for the event of interest). The observed data consists of a failure time and a failure type (Z i , δ i ) with δ i = 0 denoting a censored observation. In their review paper BIB003 sketched most potential extensions and introduced eventspecific sensitivity and specificity. They also highlighted that the crucial point was to determine whether patients experimenting a competing event should be treated as a control when evaluating the discrimination of the marker under study with respect to the event of interest. More precisely, two settings can be considered. First, if marker X is potentially discriminatory for both the event of interest and the competing event, then both event specific AUCs should be considered simultaneously BIB004 BIB005 . For illustration, in the cumulative/dynamic setting, cases at time t can be stratified according to the event type, Case 1 = {i : T i ≤ t, δ i = 1} and Case 2 = {i : T i ≤ t, δ i = 2}, and controls at time t are the event-free group at time t, Control={i : T i > t}. Following these lines Saha and Heargerty BIB004 proposed event specific versions of (2) Estimation follows from BIB002 , using the conditional cumulative incidence associated to competing event j, IP(T ≤ t, δ = j|X), instead of the conditional survival function of IP(T ≤ t|X). In the context of renal transplantation, Foucher et al. BIB005 considered a slight modification of definitions (5), where controls can also be "event specific". In addition, an extension of the incident/dynamic AUC to the competing events setting was proposed by Saha and Heargerty BIB004 . The other option is to consider both event-free patients and patients with the competing event BIB007 BIB008 as controls. For instance, dynamic controls at time t can be defined as the group {i : T i > t} ∪ {i : T i ≤ i, δ i = 2}. This leads to the estimation of only one ROC curve, for the event of interest. In BIB007 , Zheng et al. based their approach on initial estimates of the conditional cumulative incidence function for the event of interest. Their initial method provides consistent estimators if the proportional hazard assumption holds for each cause specific hazard. To relax this assumption a smooth estimator was also proposed. Another approach was described in BIB008 , which follows the lines of DeLong et al. BIB001 . However, the suitability of this method to deal with censored data is not established. We shall add that, as pointed out in BIB004 BIB007 , employing a direct regression model for the conditional cumulative incidence would lead to a simpler estimation of the cumulative/dynamic AUC and a less convoluted interpretation of the marker effect. However, the extension to the setting of a longitudinal marker BIB006 as well as the evaluation of a risk score (which is usually built with a cause-specific hazard approach) would not be straightforward.