Time-dependent AUC with right-censored data: a survey study <s> Methods based on primary estimates of IP(T > t|X = x) <s> We consider the problem of estimating the bivariate distribution of the random vector (X, Y) when Y may be subject to random censoring. The censoring variable C is allowed to depend on X but it is assumed that Y and C are conditionally independent given X = x. The estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of Y given X = x over a range of values of x. The weak convergence of the centered estimator multiplied by n 1/2 is obtained, and a closed-form expression for the covariance function of the limiting process is given. It is shown that the proposed estimator is optimal in the Beran sense. This is similar to an optimality property the product-limit estimator enjoys. Using the proposed estimator of the bivariate distribution, an extension of the least squares estimator to censored data polynomial regression is obtained and its asymptotic normality established <s> BIB001 </s> Time-dependent AUC with right-censored data: a survey study <s> Methods based on primary estimates of IP(T > t|X = x) <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> Methods based on primary estimates of IP(T > t|X = x) <s> SUMMARY Absolute risk is the probability that an individual who is free of a given disease at an initial age, a, will develop that disease in the subsequent interval (a, t]. Absolute risk is reduced by mortality from competing risks. Models of absolute risk that depend on covariates have been used to design intervention studies, to counsel patients regarding their risks of disease and to inform clinical decisions, such as whether or not to take tamoxifen to prevent breast cancer. Several general criteria have been used to evaluate models of absolute risk, including how well the model predicts the observed numbers of events in subsets of the population (‘calibration’), and ‘discriminatory power,’ measured by the concordance statistic. In this paper we review some general criteria and develop specific loss function-based criteria for two applications, namely whether or not to screen a population to select subjects for further evaluation or treatment and whether or not to use a preventive intervention that has both beneficial and adverse effects. We find that high discriminatory power is much more crucial in the screening application than in the preventive intervention application. These examples indicate that the usefulness of a general criterion such as concordance depends on the application, and that using specific loss functions can lead to more appropriate assessments. <s> BIB003 </s> Time-dependent AUC with right-censored data: a survey study <s> Methods based on primary estimates of IP(T > t|X = x) <s> 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. <s> BIB004 </s> Time-dependent AUC with right-censored data: a survey study <s> Methods based on primary estimates of IP(T > t|X = x) <s> The predictive capacity of a marker in a population can be described using the population distribution of risk (Huang et al. 2007; Pepe et al. 2008a; Stern 2008). Virtually all standard statistical summaries of predictability and discrimination can be derived from it (Gail and Pfeiffer 2005). The goal of this paper is to develop methods for making inference about risk prediction markers using summary measures derived from the risk distribution. We describe some new clinically motivated summary measures and give new interpretations to some existing statistical measures. Methods for estimating these summary measures are described along with distribution theory that facilitates construction of confidence intervals from data. We show how markers and, more generally, how risk prediction models, can be compared using clinically relevant measures of predictability. The methods are illustrated by application to markers of lung function and nutritional status for predicting subsequent onset of major pulmonary infection in children suffering from cystic fibrosis. Simulation studies show that methods for inference are valid for use in practice. <s> BIB005 </s> Time-dependent AUC with right-censored data: a survey study <s> Methods based on primary estimates of IP(T > t|X = x) <s> Summary Recently meta-analysis has been widely utilized to combine information across multiple studies to evaluate a common effect. Integrating data from similar studies is particularly useful in genomic studies where the individual study sample sizes are not large relative to the number of parameters of interest. In this article, we are interested in developing robust prognostic rules for the prediction of t-year survival based on multiple studies. We propose to construct a composite score for prediction by fitting a stratified semiparametric transformation model that allows the studies to have related but not identical outcomes. To evaluate the accuracy of the resulting score, we provide point and interval estimators for the commonly used accuracy measures including the time-specific receiver operating characteristic curves, and positive and negative predictive values. We apply the proposed procedures to develop prognostic rules for the 5-year survival of breast cancer patients based on five breast cancer genomic studies. <s> BIB006 </s> Time-dependent AUC with right-censored data: a survey study <s> Methods based on primary estimates of IP(T > t|X = x) <s> To assess the classification accuracy of a continuous diagnostic result, the receiver operating characteristic (ROC) curve is commonly used in applications. The partial area under the ROC curve (pAUC) is one of widely accepted summary measures due to its generality and ease of probability interpretation. In the field of life science, a direct extension of the pAUC into the time-to-event setting can be used to measure the usefulness of a biomarker for disease detection over time. Without using a trapezoidal rule, we propose nonparametric estimators, which are easily computed and have closed-form expressions, for the time-dependent pAUC. The asymptotic Gaussian processes of the estimators are established and the estimated variance-covariance functions are provided, which are essential in the construction of confidence intervals. The finite sample performance of the proposed inference procedures are investigated through a series of simulations. Our method is further applied to evaluate the classification ability of CD4 cell counts on patient's survival time in the AIDS Clinical Trials Group (ACTG) 175 study. In addition, the inferences can be generalized to compare the time-dependent pAUCs between patients received the prior antiretroviral therapy and those without it. <s> BIB007 </s> Time-dependent AUC with right-censored data: a survey study <s> Methods based on primary estimates of IP(T > t|X = x) <s> Finding out biomarkers and building risk scores to predict the occurrence of survival outcomes is a major concern of clinical epidemiology, and so is the evaluation of prognostic models. In this paper, we are concerned with the estimation of the time-dependent AUC -- area under the receiver operating curve -- which naturally extends standard AUC to the setting of survival outcomes and enables to evaluate the discriminative power of prognostic models. We establish a simple and useful relation between the predictiveness curve and the time-dependent AUC -- AUC(t). This relation confirms that the predictiveness curve is the key concept for evaluating calibration and discrimination of prognostic models. It also highlights that accurate estimates of the conditional absolute risk function should yield accurate estimates for AUC(t). From this observation, we derive several estimators for AUC(t) relying on distinct estimators of the conditional absolute risk function. An empirical study was conducted to compare our estimators with existing ones and assess the effect of model misspecification -- when estimating the conditional absolute risk function-- on the AUC(t) estimation. We further illustrate the methodology on the Mayo PBC and the VA lung cancer data sets. <s> BIB008
Bayes' theorem yields the following expressions for TPR C (c,t) and FPR D (c,t) From (1), it readily follows that Since IP( . In BIB004 , the authors suggested to use a Cox model to derive estimates S n (t|x), while one of the methods described in Heagerty et al. BIB002 reduces to using the conditional Kaplan-Meier estimator as in BIB001 . Some theoretical results for these two methods can be found in  and BIB006  BIB007 respectively. We shall add that Viallon and Latouche BIB008 related AUC C,D (t) to the quantity IP(T ≤ t|X = G −1 (q)) -a time-dependent version of the predictiveness curve: This confirms that most standard statistical summaries of predictability and discrimination can be derived from the predictiveness curve, as pointed out in BIB003 BIB005 .