Improved precision in the analysis of randomized trials with survival outcomes,without assuming proportional hazards

We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the Kaplan–Meier and inverse probability weighted estimators, under identifiability conditions; (iii) it remains consistent under violations of independent censoring (unlike the Kaplan–Meier estimator) when either the censoring or survival distributions, conditional on covariates, are estimated consistently; and (iv) it achieves the nonparametric efficiency bound when both of these distributions are consistently estimated. We illustrate the performance of our method using simulations based on resampling data from a completed, phase 3 randomized clinical trial of a new surgical treatment for stroke; the proposed estimator achieves a 12% gain in relative efficiency compared to the Kaplan–Meier estimator. The proposed estimator has potential advantages over existing approaches for randomized trials with time-to-event outcomes, since existing methods either rely on model assumptions that are untenable in many applications, or lack some of the efficiency and consistency properties (i)–(iv). We focus on estimation of the restricted mean survival time, but our methods may be adapted to estimate any treatment effect measure defined as a smooth contrast between the survival curves for each study arm. We provide R code to implement the estimator.

     

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.     

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.     

Sample size calculations for single-arm survival studies using transformations of the Kaplan–Meier estimator

In single-arm clinical trials with survival outcomes, the Kaplan–Meier estimator and its confidence interval are widely used to assess survival probability and median survival time. Since the asymptotic normality of the Kaplan–Meier estimator is a common result, the sample size calculation methods have not been studied in depth. An existing sample size calculation method is founded on the asymptotic normality of the Kaplan–Meier estimator using the log transformation. However, the small sample properties of the log transformed estimator are quite poor in small sample sizes (which are typical situations in single-arm trials), and the existing method uses an inappropriate standard normal approximation to calculate sample sizes. These issues can seriously influence the accuracy of results. In this paper, we propose alternative methods to determine sample sizes based on a valid standard normal approximation with several transformations that may give an accurate normal approximation even with small sample sizes. In numerical evaluations via simulations, some of the proposed methods provided more accurate results, and the empirical power of the proposed method with the arcsine square-root transformation tended to be closer to a prescribed power than the other transformations. These results were supported when methods were applied to data from three clinical trials.     

The Product-Limit Estimate as an Inverse-Probability-Weighted Average

Abstract

For randomly censored data, (Satten, G. A., Datta S. (2001 Satten, G. A. and Datta, S. 2001. The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. Ass., 55: 207–210. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. Ass. 55:207–210) showed that the Kaplan–Meier estimator (product-limit estimator (PLE)) can be expressed as an inverse-probability-weighted average. In this article, we consider the other two PLEs: the truncation PLE and the censoring-truncation PLE. For the data subject to left-truncation or both left-truncation and right-censoring, it is shown that these two PLEs can be expressed as inverse-probability-weighted averages.     

On Nelson–Aalen type estimation in the partial Koziol–Green model

In this paper, a Nelson–Aalen (NA) type estimator is derived and its sample properties are compared with the partial Abdushukurov–Cheng–Lin (PACL), generalized maximum likelihood (GMLE), and Kaplan–Meier (KM) estimators under the partial Koziol–Green model. These comparisons are made through Monto Carlo simulations under various sample sizes. The results indicate that the NA estimator always performs better than the KM estimator and is competitive with other estimators. Moreover, the PACL, GMLE, and NA estimators are shown to be asymptotically equivalent.     

The central limit theorem under semiparametric random censorship models

We study integrals for arbitrary Borel-measurable functions with respect to a semiparametric estimator of the distribution function in the random censorship model. Based on a representation of these integrals, which is similar to the one given by Stute for Kaplan–Meier integrals, a central limit theorem is established which generalizes a corresponding result of the Cheng and Lin estimator. It is shown that the semiparametric integral estimator is at least as efficient as the corresponding Kaplan–Meier integral estimator in terms of asymptotic variance if the correct semiparametric model is used. Furthermore, a necessary and sufficient condition for a strict gain in efficiency is stated. Finally, this asymptotic result is confirmed in a small simulation study under moderate sample sizes.     

Coverage probabilities of nonparametric simultaneous confidence bands for a survival function

Simultaneous confidence bands provide a useful adjunct to the popular Kaplan–Meier product limit estimator for a survival function, particularly when results are displayed graphically. They allow an assessment of the magnitude of sampling errors and provide a graphical view of a formal goodness-of-fit test. In this paper we evaluate a modified version of Nair's (1981) simultaneous confidence bands. The modification is based on a logistic transformation of the Kaplan–Meier estimator. We show that the modified bands have some important practical advantages.     

Bias and Variance of the Nonparametric MLE Under Length-Biased Censored Sampling: A Simulation Study

Abstract

In some applications, the available data suffer from several sampling problems related to loss of information. This typically happens in Survival Analysis, where models for truncation, censorship, and biasing have been proposed and widely investigated. In this work, we analyze by simulations the (finite sample) bias and variance of the nonparametric MLE under length-biasing and right-censorship, recently introduced by de Uńa-Álvarez [de Uńa-Álvarez, J. (2002a). Product-limit estimation for length-biased censored data. Test 11:109–125]. Comparison with the time-honoured Kaplan–Meier estimate for censored data is included.     

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.     

An Alternative to Efron's Redistribution-of-Mass Construction of the Kaplan—Meier Estimator

Kaplan and Meier (1958) derived the nonparametric maximum likelihood estimator of the survival function for the case in which some survival times are right-censored. Efron (1967) proposed a redistribution-of-mass construction of the Kaplan—Meier estimator that emphasized and illustrated the contribution of the censored observations. This article presents an alternative construction that, unlike Efron's method, redistributes the mass initially associated with each censored observation directly to the uncensored observations. The proposed construction avoids distributing a given mass more than once and provides additional insight into the nature of the Kaplan—Meier estimator.     

Smooth Estimation of the Reliability Function

Problems with censored data arise quite frequently in reliability applications. Estimation of the reliability function is usually of concern. Reliability function estimators proposed by Kaplan and Meier (1958), Breslow (1972), are generally used when dealing with censored data. These estimators have the known properties of being asymptotically unbiased, uniformly strongly consistent, and weakly convergent to the same Gaussian process, when properly normalized. We study the properties of the smoothed Kaplan-Meier estimator with a suitable kernel function in this paper. The smooth estimator is compared with the Kaplan-Meier and Breslow estimators for large sample sizes giving an exact expression for an appropriately normalized difference of the mean square error (MSE) of the two estimators. This quantifies the deficiency of the Kaplan-Meier estimator in comparison to the smoothed version. We also obtain a non-asymptotic bound on an expected ₁-type error under weak conditions. Some simulations are carried out to examine the performance of the suggested method.     

The Jackknife estimate of covariance of two Kaplan–Meier integrals with covariables

In this paper the Jackknife estimate of covariance of two Kaplan–Meier integrals with covariates is introduced. Its strong consistency is established under mild conditions. Several applications of the estimator are discussed.     

Generalized Concept of Relative Risk and Wider Applications of the Proportional Hazards Model and the Kaplan–Meier Estimator

The concepts of relative risk and hazard ratio are generalized for ordinary ordinal and continuous response variables, respectively. Under the generalized concepts, the Cox proportional hazards model with the Breslow's and Efron's methods can be regarded as generalizations of the Mantel–Haenszel estimator for dealing with broader types of covariates and responses. When ordinal responses can be regarded as discretized observations of a hypothetical continuous variable, the estimated relative risks from the Cox model reflect the associations between the responses and covariates. Examples are given to illustrate the generalized concepts and wider applications of the Cox model and the Kaplan–Meier estimator.     

Estimation of the bivariate distribution function for censored gap times

In many medical studies, patients may experience several events during follow-up. The times between consecutive events (gap times) are often of interest and lead to problems that have received much attention recently. In this work, we consider the estimation of the bivariate distribution function for censored gap times. Some related problems such as the estimation of the marginal distribution of the second gap time and the conditional distribution are also discussed. In this article, we introduce a nonparametric estimator of the bivariate distribution function based on Bayes’ theorem and Kaplan–Meier survival function and explore the behavior of the four estimators through simulations. Real data illustration is included.     

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.     

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.     

On the asymptotic normality of the kernel estimators of the density function and its derivatives under censoring

In this paper, we study asymptotic normality of the kernel estimators of the density function and its derivatives as well as the mode in the randomly right censorship model. The mode estimator is defined as the random variable that maximizes the kernel density estimator. Our results are stated under some suitable conditions upon the kernel function, the smoothing parameter and both distributions functions that appear in this model. Here, the Kaplan–Meier estimator of the distribution function is used to build the estimates. We carry out a simulation study which shows how good the normality works.     

Berry–Esseen type bound of conditional mode estimation under truncation and strong mixing assumptions

ABSTRACT

In order to investigate the convergence rate of the asymptotic normality for the estimator of the conditional mode function for the left-truncation model, we derive a Berry–Esseen type bound of the estimator when the lifetime observations with multivariate covariates form a stationary α-mixing sequence. The finite sample performance of the estimator of the conditional mode function is explored through simulations.