On the Kaplan–Meier estimator based on ranked set samples

When quantification of all sampling units is expensive but a set of units can be ranked, without formal measurement, ranked set sampling (RSS) is a cost-efficient alternate to simple random sampling (SRS). In this paper, we study the Kaplan–Meier estimator of survival probability based on RSS under random censoring time setup, and propose nonparametric estimators of the population mean. We present a simulation study to compare the performance of the suggested estimators. It turns out that RSS design can yield a substantial improvement in efficiency over the SRS design. Additionally, we apply the proposed methods to a real data set from an environmental study.     

Landmark estimation of survival and treatment effects in observational studies

Clinical studies aimed at identifying effective treatments to reduce the risk of disease or death often require long term follow-up of participants in order to observe a sufficient number of events to precisely estimate the treatment effect. In such studies, observing the outcome of interest during follow-up may be difficult and high rates of censoring may be observed which often leads to reduced power when applying straightforward statistical methods developed for time-to-event data. Alternative methods have been proposed to take advantage of auxiliary information that may potentially improve efficiency when estimating marginal survival and improve power when testing for a treatment effect. Recently, Parast et al. (J Am Stat Assoc 109(505):384–394, 2014) proposed a landmark estimation procedure for the estimation of survival and treatment effects in a randomized clinical trial setting and demonstrated that significant gains in efficiency and power could be obtained by incorporating intermediate event information as well as baseline covariates. However, the procedure requires the assumption that the potential outcomes for each individual under treatment and control are independent of treatment group assignment which is unlikely to hold in an observational study setting. In this paper we develop the landmark estimation procedure for use in an observational setting. In particular, we incorporate inverse probability of treatment weights (IPTW) in the landmark estimation procedure to account for selection bias on observed baseline (pretreatment) covariates. We demonstrate that consistent estimates of survival and treatment effects can be obtained by using IPTW and that there is improved efficiency by using auxiliary intermediate event and baseline information. We compare our proposed estimates to those obtained using the Kaplan–Meier estimator, the original landmark estimation procedure, and the IPTW Kaplan–Meier estimator. We illustrate our resulting reduction in bias and gains in efficiency through a simulation study and apply our procedure to an AIDS dataset to examine the effect of previous antiretroviral therapy on survival.     

Using categorical markers as auxiliary variables in log‐rank tests and hazard ratio estimation

Markers, which are prognostic longitudinal variables, can be used to replace some of the information lost due to right censoring. They may also be used to remove or reduce bias due to informative censoring. In this paper, the authors propose novel methods for using markers to increase the efficiency of log‐rank tests and hazard ratio estimation, as well as parametric estimation. They propose a «plug‐in» methodology that consists of writing the test statistic or estimate of interest as a functional of Kaplan–Meier estimators. The latter are then replaced by an efficient estimator of the survival curve that incorporates information from markers. Using simulations, the authors show that the resulting estimators and tests can be up to 30% more efficient than the usual procedures, provided that the marker is highly prognostic and that the frequency of censoring is high.     

ON THE ESTIMATION OF SURVIVAL FUNCTION UNDER RANDOM CENSORSHIP

ABSTRACT

The paper deals with an improvement of the well-known Kaplan–Meier estimator of survival function when the censoring mechanism is random and independent of the failure times. Small sample size properties of the new estimator, as well as the original Kaplan–Meier estimator are inspected by means of Monte Carlo simulations. It follows from the simulations that the proposed estimator prevails with respect to some basic statistical characteristics.     

A Comparison of Survival Function Estimators in the Koziol–Green Model

In this paper, three competing survival function estimators are compared under the assumptions of the so-called Koziol– Green model, which is a simple model of informative random censoring. It is shown that the model specific estimators of Ebrahimi and Abdushukurov, Cheng, and Lin are asymptotically equivalent. Further, exact expressions for the (noncentral) moments of these estimators are given, and their biases are analytically compared with the bias of the familiar Kaplan–Meier estimator. Finally, MSE comparisons of the three estimators are given for some selected rates of censoring.     

Product-limit Estimators and Cox Regression with Missing Censoring Information

The Kaplan–Meier estimator of a survival function requires that the censoring indicator is always observed. A method of survival function estimation is developed when the censoring indicators are missing completely at random (MCAR). The resulting estimator is a smooth functional of the Nelson–Aalen estimators of certain cumulative transition intensities. The asymptotic properties of this estimator are derived. A simulation study shows that the proposed estimator has greater efficiency than competing MCAR-based estimators. The approach is extended to the Cox model setting for the estimation of a conditional survival function given a covariate.     

Nonparametric estimation of quantile functions for randomly right censored data

In this paper we compare four nonparametric quantile function estimators for randomly right censored data: the Kaplan–Meier estimator, the linearly interpolated Kaplan–Meier estimator, the kernel-type survival function estimator, and the Bézier curve smoothing estimator. Also, we compare several kinds of confidence intervals of quantiles for four nonparametric quantile function estimators.     

Estimating and comparing adverse event probabilities in the presence of varying follow-up times and competing events

Safety analyses of adverse events (AEs) are important in assessing benefit–risk of therapies but are often rather simplistic compared to efficacy analyses. AE probabilities are typically estimated by incidence proportions, sometimes incidence densities or Kaplan–Meier estimation are proposed. These analyses either do not account for censoring, rely on a too restrictive parametric model, or ignore competing events. With the non-parametric Aalen-Johansen estimator as the “gold standard”, that is, reference estimator, potential sources of bias are investigated in an example from oncology and in simulations, for both one-sample and two-sample scenarios. The Aalen-Johansen estimator serves as a reference, because it is the proper non-parametric generalization of the Kaplan–Meier estimator to multiple outcomes. Because of potential large variances at the end of follow-up, comparisons also consider further quantiles of the observed times. To date, consequences for safety comparisons have hardly been investigated, the impact of using different estimators for group comparisons being unclear. For example, the ratio of two both underestimating or overestimating estimators may not be comparable to the ratio of the reference, and our investigation also considers the ratio of AE probabilities. We find that ignoring competing events is more of a problem than falsely assuming constant hazards by the use of the incidence density and that the choice of the AE probability estimator is crucial for group comparisons.     

Survival estimation through the cumulative hazard function with monotone natural cubic splines

In this paper we explore the estimation of survival probabilities via a smoothed version of the survival function, in the presence of censoring. We investigate the fit of a natural cubic spline on the cumulative hazard function under appropriate constraints. Under the proposed technique the problem reduces to a restricted least squares one, leading to convex optimization. The approach taken in this paper is evaluated and compared via simulations to other known methods such as the Kaplan Meier and the logspline estimator. Our approach is easily extended to address estimation of survival probabilities in the presence of covariates when the proportional hazards model assumption holds. In this case the method is compared to a restricted cubic spline approach that involves maximum likelihood. The proposed approach can be also adjusted to accommodate left censoring.     

An inverse-probability-weighted approach to estimation of the bivariate survival function under left-truncation and right-censoring

In this note, we consider estimating the bivariate survival function when both survival times are subject to random left truncation and one of the survival times is subject to random right censoring. Motivated by Satten and Datta [2001. The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. 55, 207–210], we propose an inverse-probability-weighted (IPW) estimator. It involves simultaneous estimation of the bivariate survival function of the truncation variables and that of the censoring variable and the truncation variable of the uncensored components. We prove that (i) when there is no censoring, the IPW estimator reduces to NPMLE of van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131] and (ii) when there is random left truncation and right censoring on only one of the components and the other component is always observed, the IPW estimator reduces to the estimator of Gijbels and Gürler [1998. Covariance function of a bivariate distribution function estimator for left truncated and right censored data. Statist. Sin. 1219–1232]. Based on Theorem 3.1 of van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131, 1996b. Efficient estimation of the bivariate censoring model and repairing NPMLE. Ann. Statist. 24, 596–627], we prove that the IPW estimator is consistent under certain conditions. Finally, we examine the finite sample performance of the IPW estimator in some simulation studies. For the special case that censoring time is independent of truncation time, a simulation study is conducted to compare the performances of the IPW estimator against that of the estimator proposed by van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131, 1996b. Efficient estimation of the bivariate censoring model and repairing NPMLE. Ann. Statist. 24, 596–627]. For the special case (i), a simulation study is conducted to compare the performances of the IPW estimator against that of the estimator proposed by Huang et al. (2001. Nonnparametric estimation of marginal distributions under bivariate truncation with application to testing for age-of-onset application. Statist. Sin. 11, 1047–1068).     

On estimation of frequency data with censored observations

The problem of the estimation of mean frequency of events in the presence of censoring is important in assessing the efficacy, safety and cost of therapies. The mean frequency is typically estimated by dividing the total number of events by the total number of patients under study. This method, referred to in this paper as the ‘naïve estimator’, ignores the censoring. Other approaches available for this problem require many assumptions that are rarely acceptable. These include the assumption of independence, constant hazard rate over time and other similar distributional assumptions. In this paper a simple non‐parametric estimator based on the sum of the products of Kaplan–Meier estimators is proposed as an estimator of mean frequency, and its approximate variance and standard error are derived. An illustration is provided to show the derivation of the proposed estimator. Although the clinical trial setting is used in this paper, the problem has applications in other areas where survival analysis is used and recurrent events are studied. Copyright © 2003 John Wiley & Sons, Ltd.     

Life table estimator revisited

We explore the standard life table (actuarial) estimator for grouped right-censored survival data and its extensions in order to consider its relationship with the Kaplan–Meier estimator, and to investigate the critical properties of the extended life table estimators (ELTEs). We discuss certain conditions for the ELTE to be consistent and develop a characterization of the standard life table estimator using the consistency property under any choice of at least two observation times of a finite interval. We also perform a comparative analysis of the ELTEs with the corresponding maximum likelihood estimators for grouped right-censored survival data.     

Estimation of bivariate and marginal distributions with censored data

Summary. Consider a pair of random variables, both subject to random right censoring. New estimators for the bivariate and marginal distributions of these variables are proposed. The estimators of the marginal distributions are not the marginals of the corresponding estimator of the bivariate distribution. Both estimators require estimation of the conditional distribution when the conditioning variable is subject to censoring. Such a method of estimation is proposed. The weak convergence of the estimators proposed is obtained. A small simulation study suggests that the estimators of the marginal and bivariate distributions perform well relatively to respectively the Kaplan–Meier estimator for the marginal distribution and the estimators of Pruitt and van der Laan for the bivariate distribution. The use of the estimators in practice is illustrated by the analysis of a data set.     

Multistate Survival Models as Transient Electrical Networks

In multistate survival analysis, the sojourn of a patient through various clinical states is shown to correspond to the diffusion of 1 C of electrical charge through an electrical network. The essential comparison has differentials of probability for the patient to correspond to differentials of charge, and it equates clinical states to electrical nodes. Indeed, if the death state of the patient corresponds to the sink node of the circuit, then the transient current that would be seen on an oscilloscope as the sink output is a plot of the probability density for the survival time of the patient. This electrical circuit analogy is further explored by considering the simplest possible survival model with two clinical states, alive and dead (sink), that incorporates censoring and truncation. The sink output seen on an oscilloscope is a plot of the Kaplan–Meier mass function. Thus, the Kaplan–Meier estimator finds motivation from the dynamics of current flow, as a fundamental physical law, rather than as a nonparametric maximum likelihood estimate (MLE). Generalization to competing risks settings with multiple death states (sinks) leads to cause‐specific Kaplan–Meier submass functions as outputs at sink nodes. With covariates present, the electrical analogy provides for an intuitive understanding of partial likelihood and various baseline hazard estimates often used with the proportional hazards model.     

Nonparametric Estimation for Random Censored Data Based on Ranking Set Sampling

In the case where the population distribution is unknown, the Kaplan–Meier estimator of the reliability function based on a ranked set sample with random right-censored data is first proposed. It is shown to be a unique self-consistent estimator. Then, the censored RSS estimator of the population mean is constructed. A simulation study is conducted to compare the performance of the proposed estimators with the corresponding estimators based on a simple random sample. It is shown that the ranked set sampling has higher efficiency. Finally, the proposed method is applied to a renal carcinoma study.     

A Novel Method to Calculate Mean Survival Time for Time-to-Event Data

In the analysis of time-to-event data, restricted mean survival time has been well investigated in the literature and provided by many commercial software packages, while calculating mean survival time remains as a challenge due to censoring or insufficient follow-up time. Several researchers have proposed a hybrid estimator of mean survival based on the Kaplan–Meier curve with an extrapolated tail. However, this approach often leads to biased estimate due to poor estimate of the parameters in the extrapolated “tail” and the large variability associated with the tail of the Kaplan–Meier curve due to small set of patients at risk. Two key challenges in this approach are (1) where the extrapolation should start and (2) how to estimate the parameters for the extrapolated tail. The authors propose a novel approach to calculate mean survival time to address these two challenges. In the proposed approach, an algorithm is used to search if there are any time points where the hazard rates change significantly. The survival function is estimated by the Kaplan–Meier method prior to the last change point and approximated by an exponential function beyond the last change point. The parameter in the exponential function is estimated locally. Mean survival time is derived based on this survival function. The simulation and case studies demonstrated the superiority of the proposed approach.     

Testing Censoring Point Independence

Identification in censored regression analysis and hazard models of duration outcomes relies on the condition that censoring points are conditionally independent of latent outcomes, an assumption which may be questionable in many settings. This article proposes a test for this assumption based on a Cramer–von-Mises-like test statistic comparing two different nonparametric estimators for the latent outcome cdf: the Kaplan–Meier estimator, and the empirical cdf conditional on the censoring point exceeding (for right-censored data) the cdf evaluation point. The test is consistent and has power against a wide variety of alternatives. Applying the test to unemployment duration data from the NLSY, the SIPP, and the PSID suggests the assumption is frequently suspect.     

Strong consistency of presmoothed Kaplan–Meier integrals when covariables are present

It is known that the Kaplan–Meier estimation may be improved via presmoothing methods. In this article, we introduce an extended presmoothed Kaplan–Meier estimator in the presence of covariates. The main result is the strong consistency of general empirical integrals based on such an estimator. As applications, one can obtain a consis-tent multivariate empirical distribution under censoring, and also can obtain a consistent estimation of regression parameters. We illustrate the new estimation methods through simulations and real data analysis.     

Estimating the survival function based on the semi-Markov model for dependent censoring

In this paper, we study a nonparametric maximum likelihood estimator (NPMLE) of the survival function based on a semi-Markov model under dependent censoring. We show that the NPMLE is asymptotically normal and achieves asymptotic nonparametric efficiency. We also provide a uniformly consistent estimator of the corresponding asymptotic covariance function based on an information operator. The finite-sample performance of the proposed NPMLE is examined with simulation studies, which show that the NPMLE has smaller mean squared error than the existing estimators and its corresponding pointwise confidence intervals have reasonable coverages. A real example is also presented.