A partially parametric estimator of survival in the presence of randomly censored data

Many biological or medical experiments have as their goal to estimate the survival function of a specified population of subjects when the time to the specified event may be censored due to loss to follow-up, the occurrence of another event that precludes the occurrence of the event of interest, or the study being terminated before the event of interest occurs. This paper suggests an improvement of the Kaplan-Meier product-limit estimator when the censoring mechanism is random. The proposed estimator treats the uncensored observations nonparametrically and uses a parametric model only for the censored observations. One version of this proposed estimator always has a smaller bias and mean squared error than the product-limit estimator. An example estimating the survival function of patients enrolled in the Ohio State University Bone Marrow Transplant Program is presented.     

Time-dependent ROC curves for censored survival data and a diagnostic marker

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.     

A comparison of several methods of estimating the survival function when there is extreme right censoring

When there is extreme censoring on the right, the Kaplan-Meier product-limit estimator is known to be a biased estimator of the survival function. Several modifications of the Kaplan-Meier estimator are examined and compared with respect to bias and mean squared error.     

What price Kaplan-Meier?

The asymptotic efficiency of the Kaplan-Meier product-limit estimator, relative to the maximum likelihood estimator of a parametric survival function, is examined under a random-censoring model.     

A regression-based method for estimating mean treatment cost in the presence of right-censoring

In many clinical trials and evaluations using medical care administrative databases it is of interest to estimate not only the survival time of a given treatment modality but also the total associated cost. The most widely used estimator for data subject to censoring is the Kaplan-Meier (KM) or product-limit (PL) estimator. The optimality properties of this estimator applied to time-to-event data (consistency, etc.) under the assumptions of random censorship have been established. However, whenever the relationship between cost and survival time includes an error term to account for random differences among patients' costs, the dependency between cumulative treatment cost at the time of censoring and at the survival time results in KM giving biased estimates. A similar phenomenon has previously been noted in the context of estimating quality-adjusted survival time. We propose an estimator for mean cost which exploits the underlying relationship between total treatment cost and survival time. The proposed method utilizes either parametric or nonparametric regression to estimate this relationship and is consistent when this relationship is consistently estimated. We then present simulation results which illustrate the gain in finite-sample efficiency when compared with another recently proposed estimator. The methods are then applied to the estimation of mean cost for two studies where right-censoring was present. The first is the heart failure clinical trial Studies of Left Ventricular Dysfunction (SOLVD). The second is a Health Maintenance Organization (HMO) database study of the cost of ulcer treatment.     

Graphical exploration of covariate effects on survival data through nonparametric quantile curves

Kaplan-Meier curves provide an effective means of presenting the distributional pattern in a sample of survival data. However, in order to assess the effect of a covariate, a standard scatterplot is often difficult to interpret because of the presence of censored observations. Several authors have proposed a running median as an effective way of indicating the effect of a covariate. This article proposes a form of kernel estimation, employing double smoothing, that can be applied in a simple and efficient manner to construct an estimator of a percentile of the survival distribution as a function of one or two covariates. Permutations and bootstrap samples can be used to construct reference bands that help identify whether particular features of the estimates indicate real features of the underlying curve or whether this may be due simply to random variation. The techniques are illustrated on data from a study of kidney transplant patients.     

Bayes Estimator of Survival Probability from Randomly Censored Observations with Weibull Distribution of Time to Death

This paper deals with Bayes estimation of survival probability when the data are randomly censored. Such a situation arises in case of a clinical trial which extends for a limited period T. A fixed number of patients (n) are observed whose times to death have identical Weibull distribution with parameters β and θ. The maximum times of observation for different patients are also independent uniform variables as the patients arrive randomly throughout the trial. For the joint prior distribution of (β, θ) as suggested by Sinha and Kale (1980, page 137) Bayes estimator of survival probability at time t (0<t<T) has been obtained. Considering squared error loss function it is the mean of the survival probability with respect to the posterior distribution of (β, θ). This estimator is then compared with the maximum likelihood estimator, by simulation, for various values of β, θ and censoring percentage. The proposed estimator is found to be better under certain conditions.     

A group sequential test for survival trials: an alternative to rank-based procedures

A method of interim monitoring is described for survival trials in which the proportional hazards assumption may not hold. This method extends the test statistics based on the cumulative weighted difference in the Kaplan-Meier estimates (Pepe and Fleming, 1989, Biometrics 45, 497-507) to the sequential setting. Therefore, it provides a useful alternative to the group sequential linear rank tests. With an appropriate weight function, the test statistic itself provides an estimator for the cumulative weighted difference in survival probabilities, which is an interpretable measure for the treatment difference, especially when the proportional hazards model fails. The method is illustrated based on the design of a real trial. The operating characteristics are studied through a small simulation.     

Inference for the proportional hazards model with misclassified discrete-valued covariates

We consider the Cox proportional hazards model with discrete-valued covariates subject to misclassification. We present a simple estimator of the regression parameter vector for this model. The estimator is based on a weighted least squares analysis of weighted-averaged transformed Kaplan-Meier curves for the different possible configurations of the observed covariate vector. Optimal weighting of the transformed Kaplan-Meier curves is described. The method is designed for the case in which the misclassification rates are known or are estimated from an external validation study. A hybrid estimator for situations with an internal validation study is also described. When there is no misclassification, the regression coefficient vector is small in magnitude, and the censoring distribution does not depend on the covariates, our estimator has the same asymptotic covariance matrix as the Cox partial likelihood estimator. We present results of a finite-sample simulation study under Weibull survival in the setting of a single binary covariate with known misclassification rates. In this simulation study, our estimator performed as well as or, in a few cases, better than the full Weibull maximum likelihood estimator. We illustrate the method on data from a study of the relationship between trans-unsaturated dietary fat consumption and cardiovascular disease incidence.     

Estimation of regression parameters and the hazard function in transformed linear survival models

An estimator of the regression parameters in a semiparametric transformed linear survival model is examined. This estimator consists of a single Newton-like update of the solution to a rank-based estimating equation from an initial consistent estimator. An automated penalized likelihood algorithm is proposed for estimating the optimal weight function for the estimating equations and the error hazard function that is needed in the variance estimator. In simulations, the estimated optimal weights are found to give reasonably efficient estimators of the regression parameters, and the variance estimators are found to perform well. The methodology is applied to an analysis of prognostic factors in non-Hodgkin's lymphoma.     

Semiparametric analysis of survival data with left truncation and dependent right censoring

Studies of chronic life-threatening diseases often involve both mortality and morbidity. In observational studies, the data may also be subject to administrative left truncation and right censoring. Because mortality and morbidity may be correlated and mortality may censor morbidity, the Lynden-Bell estimator for left-truncated and right-censored data may be biased for estimating the marginal survival function of the non-terminal event. We propose a semiparametric estimator for this survival function based on a joint model for the two time-to-event variables, which utilizes the gamma frailty specification in the region of the observable data. First, we develop a novel estimator for the gamma frailty parameter under left truncation. Using this estimator, we then derive a closed-form estimator for the marginal distribution of the non-terminal event. The large sample properties of the estimators are established via asymptotic theory. The methodology performs well with moderate sample sizes, both in simulations and in an analysis of data from a diabetes registry.     

Semiparametric analysis of tumor incidence rates in survival/sacrifice experiments

This paper addresses the problem of comparing treatment groups with respect to the rate of tumor development for animals in a survival experiment with some serial sacrifices. The analysis specifies a parametric model for the tumor incidence function, but places no parametric restrictions on the death rates. The procedure is feasible with as few as two sacrifice times and requires no individual data on cause of death. Other diseases need not act independently of the tumor of interest, nor are any restrictions imposed on tumor lethality or the relationship between the onset and death times for tumor-bearing animals. The proposed methods are illustrated with some survival/sacrifice data.     

Nonparametric analysis of recurrent events and death

This article is concerned with the analysis of recurrent events in the presence of a terminal event such as death. We consider the mean frequency function, defined as the marginal mean of the cumulative number of recurrent events over time. A simple nonparametric estimator for this quantity is presented. It is shown that the estimator, properly normalized, converges weakly to a zero-mean Gaussian process with an easily estimable covariance function. Nonparametric statistics for comparing two mean frequency functions and for combining data on recurrent events and death are also developed. The asymptotic null distributions of these statistics, together with consistent variance estimators, are derived. The small-sample properties of the proposed estimators and test statistics are examined through simulation studies. An application to a cancer clinical trial is provided.     

Nonparametric estimation of the concordance correlation coefficient under univariate censoring

Assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects. Lin's (1989, Biometrics 45, 255-268) concordance correlation coefficient (CCC) has become a popular measure of agreement for correlated continuous outcomes. However, commonly used estimation methods for the CCC do not accommodate censored observations and are, therefore, not applicable for survival outcomes. In this article, we estimate the CCC nonparametrically through the bivariate survival function. The proposed estimator of the CCC is proven to be strongly consistent and asymptotically normal, with a consistent bootstrap variance estimator. Furthermore, we propose a time-dependent agreement coefficient as an extension of Lin's (1989) CCC for measuring the agreement between survival times among subjects who survive beyond a specified time point. A nonparametric estimator is developed for the time-dependent agreement coefficient as well. It has the same asymptotic properties as the estimator of the CCC. Simulation studies are conducted to evaluate the performance of the proposed estimators. A real data example from a prostate cancer study is used to illustrate the method.     

Dimension reduction in survival regressions with censored data via an imputed spline approach

Dimension reduction methods have been proposed for regression analysis with predictors of high dimension, but have not received much attention on the problems with censored data. In this article, we present an iterative imputed spline approach based on principal Hessian directions (PHD) for censored survival data in order to reduce the dimension of predictors without requiring a prespecified parametric model. Our proposal is to replace the right-censored survival time with its conditional expectation for adjusting the censoring effect by using the Kaplan-Meier estimator and an adaptive polynomial spline regression in the residual imputation. A sparse estimation strategy is incorporated in our approach to enhance the interpretation of variable selection. This approach can be implemented in not only PHD, but also other methods developed for estimating the central mean subspace. Simulation studies with right-censored data are conducted for the imputed spline approach to PHD (IS-PHD) in comparison with two methods of sliced inverse regression, minimum average variance estimation, and naive PHD in ignorance of censoring. The results demonstrate that the proposed IS-PHD method is particularly useful for survival time responses approximating symmetric or bending structures. Illustrative applications to two real data sets are also presented.     

Measuring agreement of multivariate discrete survival times using a modified weighted kappa coefficient

Summary .  Assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects. We present a modified weighted kappa coefficient to measure agreement between bivariate discrete survival times. The proposed kappa coefficient accommodates censoring by redistributing the mass of censored observations within the grid where the unobserved events may potentially happen. A generalized modified weighted kappa is proposed for multivariate discrete survival times. We estimate the modified kappa coefficients nonparametrically through a multivariate survival function estimator. The asymptotic properties of the kappa estimators are established and the performance of the estimators are examined through simulation studies of bivariate and trivariate survival times. We illustrate the application of the modified kappa coefficient in the presence of censored observations with data from a prostate cancer study.     

Survival estimation using splines

A nonparametric maximum likelihood procedure is given for estimating the survivor function from right-censored data. It approximates the hazard rate by a simple function such as a spline, with different approximations yielding different estimators. A special case is that proposed by Nelson (1969, Journal of Quality Technology 1, 27-52) and Altshuler (1970, Mathematical Biosciences 6, 1-11). The estimators are uniformly consistent and have the same asymptotic weak convergence properties as the Kaplan-Meier (1958, Journal of the American Statistical Association 53, 457-481) estimator. However, in small and in heavily censored samples, the simplest spline estimators have uniformly smaller mean squared error than do the Kaplan-Meier and Nelson-Altshuler estimators. The procedure is extended to estimate the baseline hazard rate and regression coefficients in the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model and is illustrated using experimental carcinogenesis data.     

Regularized estimation in the accelerated failure time model with high-dimensional covariates

We consider two regularization approaches, the LASSO and the threshold-gradient-directed regularization, for estimation and variable selection in the accelerated failure time model with multiple covariates based on Stute's weighted least squares method. The Stute estimator uses Kaplan-Meier weights to account for censoring in the least squares criterion. The weighted least squares objective function makes the adaptation of this approach to multiple covariate settings computationally feasible. We use V-fold cross-validation and a modified Akaike's Information Criterion for tuning parameter selection, and a bootstrap approach for variance estimation. The proposed method is evaluated using simulations and demonstrated on a real data example.     

The effect of additional follow-up and accrual on survival time data

A method called partial completion is proposed for predicting the gain in precision of the Kaplan-Meier survival curve associated with additional follow-up and accrual. This is accomplished by using the initial data to predict the numbers of patients who would be at risk at the observed death times by the end of the proposed second follow-up period. A consistency result ensures that the predictors will be accurate in large samples while simulation results suggest that the predictors are accurate with moderate sample sizes. The procedures are applied to a bone marrow transplant study and the Channing House data set.     

A statistical measure of tissue heterogeneity with application to 3D PET sarcoma data

In vivo measurement of local tissue characteristics by modern bioimaging techniques such as positron emission tomography (PET) provides the opportunity to analyze quantitatively the role that tissue heterogeneity may play in understanding biological function. This paper develops a statistical measure of the heterogeneity of a tissue characteristic that is based on the deviation of the distribution of the tissue characteristic from a unimodal elliptically contoured spatial pattern. An efficient algorithm is developed for computation of the measure based on volumetric region of interest data. The technique is illustrated by application to data from PET imaging studies of fluorodeoxyglucose utilization in human sarcomas. A set of 74 sarcoma patients (with five-year follow-up survival information) were evaluated for heterogeneity as well as a number of other potential prognostic indicators of survival. A Cox proportional hazards analysis of these data shows that the degree of heterogeneity of the sarcoma is the major risk factor associated with patient death. Some theory is developed to analyze the asymptotic statistical behavior of the heterogeneity estimator. In the context of data arising from Poisson deconvolution (PET being the prime example), the heterogeneity estimator, which is a non-linear functional of the PET image data, is consistent and converges at a rate that is parametric in the injected dose.