Planning the duration of a survival group sequential trial with a fixed follow-up time for all subjects

To explore the operation characteristics of survival group sequential trials with a fixed follow-up period, the accrual time and total trial duration to ensure power and type I error rate requirements are explained and investigated for hazard ratios ranging from 1.3 to 3.0, with slow or high accrual rate, and in the presence or absence of censoring. Impacts of hazard rate, accrual rate, and competitive censoring on accrual time and subsequently on total trial duration are carefully illustrated. Real time for interim analyses, needed number of events, and recruited number of subjects at time of interim analyses are also tabulated.     

Distribution of ranks in a two-sample problem with right censored data

We consider different censoring models for a two-sample and find the joint distribution of the rank vector and number of uncensored observations under each censoring model when the distributions of life times and/or distributions of censoring of censoring variables satisfy the condition for the Lehmann type of alternatives.     

Tobit regression for modeling mean survival time using data subject to multiple sources of censoring

Mean survival time is often of inherent interest in medical and epidemiologic studies. In the presence of censoring and when covariate effects are of interest, Cox regression is the strong default, but mostly due to convenience and familiarity. When survival times are uncensored, covariate effects can be estimated as differences in mean survival through linear regression. Tobit regression can validly be performed through maximum likelihood when the censoring times are fixed (ie, known for each subject, even in cases where the outcome is observed). However, Tobit regression is generally inapplicable when the response is subject to random right censoring. We propose Tobit regression methods based on weighted maximum likelihood which are applicable to survival times subject to both fixed and random censoring times. Under the proposed approach, known right censoring is handled naturally through the Tobit model, with inverse probability of censoring weighting used to overcome random censoring. Essentially, the re‐weighting data are intended to represent those that would have been observed in the absence of random censoring. We develop methods for estimating the Tobit regression parameter, then the population mean survival time. A closed form large‐sample variance estimator is proposed for the regression parameter estimator, with a semiparametric bootstrap standard error estimator derived for the population mean. The proposed methods are easily implementable using standard software. Finite‐sample properties are assessed through simulation. The methods are applied to a large cohort of patients wait‐listed for kidney transplantation.     

Slope estimation of covariates that influence renal outcome following renal transplant adjusting for informative right censoring

A new statistical model is proposed to estimate population and individual slopes that are adjusted for covariates and informative right censoring. Individual slopes are assumed to have a mean that depends on the population slope for the covariates. The number of observations for each individual is modeled as a truncated discrete distribution with mean dependent on the individual subjects’ slopes. Our simulation study results indicated that the associated bias and mean squared errors for the proposed model were comparable to those associated with the model that only adjusts for informative right censoring. The proposed model was illustrated using renal transplant dataset to estimate population slopes for covariates that could impact the outcome of renal function following renal transplantation.     

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.     

Variable screening for survival data in the presence of heterogeneous censoring

Variable screening for censored survival data is most challenging when both survival and censoring times are correlated with an ultrahigh-dimensional vector of covariates. Existing approaches to handling censoring often make use of inverse probability weighting by assuming independent censoring with both survival time and covariates. This is a convenient but rather restrictive assumption which may be unmet in real applications, especially when the censoring mechanism is complex and the number of covariates is large. To accommodate heterogeneous (covariate-dependent) censoring that is often present in high-dimensional survival data, we propose a Gehan-type rank screening method to select features that are relevant to the survival time. The method is invariant to monotone transformations of the response and of the predictors, and works robustly for a general class of survival models. We establish the sure screening property of the proposed methodology. Simulation studies and a lymphoma data analysis demonstrate its favorable performance and practical utility.     

Likelihood-based confidence sets for partially identified parameters

There has been growing interest in partial identification of probability distributions and parameters. This paper considers statistical inference on parameters that are partially identified because data are incompletely observed, due to nonresponse or censoring, for instance. A method based on likelihood ratios is proposed for constructing confidence sets for partially identified parameters. The method can be used to estimate a proportion or a mean in the presence of missing data, without assuming missing-at-random or modeling the missing-data mechanism. It can also be used to estimate a survival probability with censored data without assuming independent censoring or modeling the censoring mechanism. A version of the verification bias problem is studied as well.     

Regression analysis of longitudinal data with time-dependent covariates in the presence of informative observation and censoring times

In longitudinal observational studies, repeated measures are often correlated with observation times as well as censoring time. This article proposes joint modeling and analysis of longitudinal data with time-dependent covariates in the presence of informative observation and censoring times via a latent variable. Estimating equation approaches are developed for parameter estimation and asymptotic properties of the proposed estimators are established. In addition, a generalization of the semiparametric model with time-varying coefficients for the longitudinal response is considered. Furthermore, a lack-of-fit test is provided for assessing the adequacy of the model, and some tests are presented for investigating whether or not covariate effects vary with time. The finite-sample behavior of the proposed methods is examined in simulation studies, and an application to a bladder cancer study is illustrated.     

Failure time studies with intermittent observation and losses to follow-up

In health research interest often lies in modeling a failure time process but in many cohort studies failure status is only determined at scheduled assessment times. While the assessment times may be fixed upon study entry, individuals may become lost to follow-up and miss visits subsequent to the time of loss to follow-up. We consider a three-state model to characterize a joint failure and loss to follow-up process, and use it to investigate the impact of dependent loss to follow-up on standard parametric, nonparametric, and semiparametric analysis. The effect of dependent loss to follow-up is mitigated by fitting the joint model. The performance of standard methods is studied using the asymptotic theory of misspecified models, and the finite sample performance is examined for the standard and joint analyses through simulation studies. An application to data from a youth smoking prevention study is presented for illustration.     

Parameters estimation for weibull distributed lifetimes under progressive censoring with random removeals

This paper considers the estimation problem when lifetimes are Weibull distributed and are collected under a Type-II progressive censoring with random removals, where the number of units removed at each failure time follows a uniform discrete distribution. The expected time of this censoring plan is discussed and compared numerically to that under a Type II censoring without removal. Maximum likelihood estimator of the parameters and their asymptotic variances are derived.     

On small sample properties of the generalized signed rank and generalized sign tests

The generalized signed rank (GSR) and generalized sign (GS) tests were recently proposed for matched pair studies with censored observations (Woolson and Lechenbruch, 1980). The results provided in that paper were asymptotic, and no indicatin of small sample behavior was given. In this paper we report on simulation studied of these statistics for a variety of distributions. We find that the GSR is more powerful than the GS, and that censoring does not affect power greatly. In the original paper, we assumed each member of the pair has the same censoring time. We consider a variant of this in which each member of the pair has a censoring time chosen from a uniform distribution, and the minimum of these times is selected as the censoring time for the pair. It is found that the power of the test is slightly reduced because the number of doubly censored pairs is increased.     

An analytic method for randomized trials with informative censoring: Part II

Consider a randomized trial in which time to the occurrence of a particular disease, say pneumocystic pneumonia in an AIDS trial or breast cancer in a mammographic screening trial, is the failure time of primary interest. Suppose that time to disease is subject to informative censoring by the minimum of time to death, loss to and end of follow-up. In such a trial, the potential censoring time is observed for all study subjects, including failures. In the presence of informative censoring, it is not possible to consistently estimate the effect of treatment on time to disease without imposing additional non-identifiable assumptions. Robins (1995) specified two non-identifiable assumptions that allow one to test for and estimate an effect of treatment on time to disease in the presence of informative censoring. The goal of this paper is to provide a class of consistent and reasonably efficient semiparametric tests and estimators for the treatment effect under these assumptions. The tests in our class, like standard weighted-log-rank tests, are asymptotically distribution-free -level tests under the null hypothesis of no causal effect of treatment on time to disease whenever the censoring and failure distributions are conditionally independent given treatment arm. However, our tests remain asymptotically distribution-free -level tests in the presence of informative censoring provided either of our assumptions are true. In contrast, a weighted log-rank test will be an -level test in the presence of informative censoring only if (1) one of our two non-identifiable assumptions hold, and (2) the distribution of time to censoring is the same in the two treatment arms. We also study the estimation, in the presence of informative censoring, of the effect of treatment on the evolution over time of the mean of repeated measures outcome such as CD4 count.     

Inverse probability of censoring weighting for visual predictive checks of time-to-event models with time-varying covariates

When constructing models to summarize clinical data to be used for simulations, it is good practice to evaluate the models for their capacity to reproduce the data. This can be done by means of Visual Predictive Checks (VPC), which consist of several reproductions of the original study by simulation from the model under evaluation, calculating estimates of interest for each simulated study and comparing the distribution of those estimates with the estimate from the original study. This procedure is a generic method that is straightforward to apply, in general. Here we consider the application of the method to time-to-event data and consider the special case when a time-varying covariate is not known or cannot be approximated after event time. In this case, simulations cannot be conducted beyond the end of the follow-up time (event or censoring time) in the original study. Thus, the simulations must be censored at the end of the follow-up time. Since this censoring is not random, the standard KM estimates from the simulated studies and the resulting VPC will be biased. We propose to use inverse probability of censoring weighting (IPoC) method to correct the KM estimator for the simulated studies and obtain unbiased VPCs. For analyzing the Cantos study, the IPoC weighting as described here proved valuable and enabled the generation of VPCs to qualify PKPD models for simulations. Here, we use a generated data set, which allows illustration of the different situations and evaluation against the known truth.     

THE ASYMPTOTIC EQUIVALENCE OF THE FISHER INFORMATION MATRICES FOR TYPE I AND TYPE II CENSORED DATA FROM LOCATION-SCALE FAMILIES

Type I and Type II censored data arise frequently in controlled laboratory studies concerning time to a particular event (e.g., death of an animal or failure of a physical device). Log-location-scale distributions (e.g., Weibull, lognormal, and loglogistic) are commonly used to model the resulting data. Maximum likelihood (ML) is generally used to obtain parameter estimates when the data are censored. The Fisher information matrix can be used to obtain large-sample approximate variances and covariances of the ML estimates or to estimate these variances and covariances from data. The derivations of the Fisher information matrix proceed differently for Type I (time censoring) and Type II (failure censoring) because the number of failures is random in Type I censoring, but length of the data collection period is random in Type II censoring. Under regularity conditions (met with the above-mentioned log-location-scale distributions), we outline the different derivations and show that the Fisher information matrices for Type I and Type II censoring are asymptotically equivalent.     

Macro-Economic Factors in Credit Risk Calculations: Including Time-Varying Covariates in Mixture Cure Models

The prediction of the time of default in a credit risk setting via survival analysis needs to take a high censoring rate into account. This rate is because default does not occur for the majority of debtors. Mixture cure models allow the part of the loan population that is unsusceptible to default to be modeled, distinct from time of default for the susceptible population. In this article, we extend the mixture cure model to include time-varying covariates. We illustrate the method via simulations and by incorporating macro-economic factors as predictors for an actual bank dataset.     

An analytic method for randomized trials with informative censoring: Part 1

Consider a randomized trial in which time to the occurrence of a particular disease, say pneumocystis pneumonia in an AIDS trial or breast cancer in a mammographic screening trial, is the failure time of primary interest. Suppose that time to disease is subject to informative censoring by the minimum of time to death, loss to and end of follow-up. In such a trial, the censoring time is observed for all study subjects, including failures. In the presence of informative censoring, it is not possible to consistently estimate the effect of treatment on time to disease without imposing additional non-identifiable assumptions. The goals of this paper are to specify two non-identifiable assumptions that allow one to test for and estimate an effect of treatment on time to disease in the presence of informative censoring. In a companion paper (Robins, 1995), we provide consistent and reasonably efficient semiparametric estimators for the treatment effect under these assumptions. In this paper we largely restrict attention to testing. We propose tests that, like standard weighted-log-rank tests, are asymptotically distribution-free -level tests under the null hypothesis of no causal effect of treatment on time to disease whenever the censoring and failure distributions are conditionally independent given treatment arm. However, our tests remain asymptotically distribution-free -level tests in the presence of informative censoring provided either of our assumptions are true. In contrast, a weighted log-rank test will be an -level test in the presence of informative censoring only if (1) one of our two non-identifiable assumptions hold, and (2) the distribution of time to censoring is the same in the two treatment arms. We also extend our methods to studies of the effect of a treatment on the evolution over time of the mean of a repeated measures outcome, such as CD-4 count.     

Slope Estimation for Bivariate Longitudinal Outcomes Adjusting for Informative Right Censoring Using Discrete Survival Model: Application to the Renal Transplant Cohort

Patients undergoing renal transplantation are prone to graft failure which causes lost of follow-up measures on their blood urea nitrogen and serum creatinine levels. These two outcomes are measured repeatedly over time to assess renal function following transplantation. Loss of follow-up on these bivariate measures results in informative right censoring, a common problem in longitudinal data that should be adjusted for so that valid estimates are obtained. In this study, we propose a bivariate model that jointly models these two longitudinal correlated outcomes and generates population and individual slopes adjusting for informative right censoring using a discrete survival approach. The proposed approach is applied to the clinical dataset of patients who had undergone renal transplantation. A simulation study validates the effectiveness of the approach.     

Recurrent events analysis in the presence of time-dependent covariates and dependent censoring

Summary.  Recurrent events models have had considerable attention recently. The majority of approaches show the consistency of parameter estimates under the assumption that censoring is independent of the recurrent events process of interest conditional on the covariates that are included in the model. We provide an overview of available recurrent events analysis methods and present an inverse probability of censoring weighted estimator for the regression parameters in the Andersen–Gill model that is commonly used for recurrent event analysis. This estimator remains consistent under informative censoring if the censoring mechanism is estimated consistently, and it generally improves on the naïve estimator for the Andersen–Gill model in the case of independent censoring. We illustrate the bias of ad hoc estimators in the presence of informative censoring with a simulation study and provide a data analysis of recurrent lung exacerbations in cystic fibrosis patients when some patients are lost to follow-up.     

C55. Quick and accurate rule of thumb for determining the critical value of the coefficient of correlation

Life table analysis techniques in epidemiology depend upon the asymptotic properties of the statistical test methods employed. In some instances, the statistical procedures indicate highly significant results which are, in reality, unjustified. The phenomenon may occur when the asymptotic methods are applied in situations where the cases of interest are few in number. This situation is illustrated by the 20 multiple myeloma deaths observed in the RERF Life Span Study cohort. A permutation test is applied to the life table data, although the test requires the false assumption that the censoring distribution is independent of the radiation dose. A simulation test is developed which does not require equal censoring, which has the same asymptotics as the usual test methods, and which is less likely to overestimate significance in small samples. It is found that both of these small-sample tests provide reasonable numerical solutions. In addition, the simulation test is recommended in general for analyzing life table data with unequal censoring. Finally, by using the small-sample tests, the frequency of death from multiple myeloma is shown to be positively associated with radiation dose (P<0.01).     

Statistical analysis of Weibull distributed lifetime data under Type II progressive censoring with binomial removals

This paper considers the analysis of Weibull distributed lifetime data observed under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial distribution. Maximum likelihood estimators of the parameters and their asymptotic variances are derived. The expected time required to complete the life test under this censoring scheme is investigated.