An Additive–Multiplicative Restricted Mean Residual Life Model

The mean residual life measures the expected remaining life of a subject who has survived up to a particular time. When survival time distribution is highly skewed or heavy tailed, the restricted mean residual life must be considered. In this paper, we propose an additive–multiplicative restricted mean residual life model to study the association between the restricted mean residual life function and potential regression covariates in the presence of right censoring. This model extends the proportional mean residual life model using an additive model as its covariate dependent baseline. For the suggested model, some covariate effects are allowed to be time‐varying. To estimate the model parameters, martingale estimating equations are developed, and the large sample properties of the resulting estimators are established. In addition, to assess the adequacy of the model, we investigate a goodness of fit test that is asymptotically justified. The proposed methodology is evaluated via simulation studies and further applied to a kidney cancer data set collected from a clinical trial.     

Semiparametric median residual life model and inference

For randomly censored data, the authors propose a general class of semiparametric median residual life models. They incorporate covariates in a generalized linear form while leaving the baseline median residual life function completely unspecified. Despite the non‐identifiability of the survival function for a given median residual life function, a simple and natural procedure is proposed to estimate the regression parameters and the baseline median residual life function. The authors derive the asymptotic properties for the estimators, and demonstrate the numerical performance of the proposed method through simulation studies. The median residual life model can be easily generalized to model other quantiles, and the estimation method can also be applied to the mean residual life model. The Canadian Journal of Statistics 38: 665–679; 2010 © 2010 Statistical Society of Canada     

Simultaneous Modelling of Survival and Longitudinal Data with an Application to Repeated Quality of Life Measures

In biomedical studies, interest often focuses on the relationship between patients characteristics or some risk factors and both quality of life and survival time of subjects under study. In this paper, we propose a simultaneous modelling of both quality of life and survival time using the observed covariates. Moreover, random effects are introduced into the simultaneous models to account for dependence between quality of life and survival time due to unobserved factors. EM algorithms are used to derive the point estimates for the parameters in the proposed model and profile likelihood function is used to estimate their variances. The asymptotic properties are established for our proposed estimators. Finally, simulation studies are conducted to examine the finite-sample properties of the proposed estimators and a liver transplantation data set is analyzed to illustrate our approaches.     

A class of mean residual life regression models with censored survival data

When describing a failure time distribution, the mean residual life is sometimes preferred to the survival or hazard rate. Regression analysis making use of the mean residual life function has recently drawn a great deal of attention. In this paper, a class of mean residual life regression models are proposed for censored data, and estimation procedures and a goodness-of-fit test are developed. Both asymptotic and finite sample properties of the proposed estimators are established, and the proposed methods are applied to a cancer data set from a clinic trial.     

Modeling restricted mean survival time under general censoring mechanisms

Restricted mean survival time (RMST) is often of great clinical interest in practice. Several existing methods involve explicitly projecting out patient-specific survival curves using parameters estimated through Cox regression. However, it would often be preferable to directly model the restricted mean for convenience and to yield more directly interpretable covariate effects. We propose generalized estimating equation methods to model RMST as a function of baseline covariates. The proposed methods avoid potentially problematic distributional assumptions pertaining to restricted survival time. Unlike existing methods, we allow censoring to depend on both baseline and time-dependent factors. Large sample properties of the proposed estimators are derived and simulation studies are conducted to assess their finite sample performance. We apply the proposed methods to model RMST in the absence of liver transplantation among end-stage liver disease patients. This analysis requires accommodation for dependent censoring since pre-transplant mortality is dependently censored by the receipt of a liver transplant.     

Efficient estimation for time series following generalized linear models

In this paper, we consider James–Stein shrinkage and pretest estimation methods for time series following generalized linear models when it is conjectured that some of the regression parameters may be restricted to a subspace. Efficient estimation strategies are developed when there are many covariates in the model and some of them are not statistically significant. Statistical properties of the pretest and shrinkage estimation methods including asymptotic distributional bias and risk are developed. We investigate the relative performances of shrinkage and pretest estimators with respect to the unrestricted maximum partial likelihood estimator (MPLE). We show that the shrinkage estimators have a lower relative mean squared error as compared to the unrestricted MPLE when the number of significant covariates exceeds two. Monte Carlo simulation experiments were conducted for different combinations of inactive covariates and the performance of each estimator was evaluated in terms of its mean squared error. The practical benefits of the proposed methods are illustrated using two real data sets.     

Residual analysis for parametric models fit to censored data

The focus of this paper is on residual analysis for the lognormal and extreme value or Weibull models, although the proposed methods can be applied to any parametric model. Residuals developed by Barlow and Prentice (1988) for the Cox proportional hazards model are extended to the parametric model setting. Three different residuals are proposed based on this approach with two residuals measuring the impact of survival time and one measuring the impact of the covariates included in the model. In addition, a residual derived from the deviations equality presented in Efron and Johnstone (1990) and the residual proposed by Joergensen (1984) for censored data models are discussed.     

Semiparametric Proportional Mean Residual Life Model With Censoring Indicators Missing at Random

For right-censored survival data, the information that whether the observed time is survival or censoring time is frequently lost. This is the case for the competing risk data. In this article, we consider statistical inference for the right-censored survival data with censoring indicators missing at random under the proportional mean residual life model. Simple and augmented inverse probability weighted estimating equation approaches are developed, in which the nonmissingness probability and some unknown conditional expectations are estimated by the kernel smoothing technique. The asymptotic properties of all the proposed estimators are established, while extensive simulation studies demonstrate that our proposed methods perform well under the moderate sample size. At last, the proposed method is applied to a data set from a stage II breast cancer trial.     

Nonparametric Shrinkage Estimation for Aalen's Additive Hazards Model

Aalen's nonparametric additive model in which the regression coefficients are assumed to be unspecified functions of time is a flexible alternative to Cox's proportional hazards model when the proportionality assumption is in doubt. In this paper, we incorporate a general linear hypothesis into the estimation of the time‐varying regression coefficients. We combine unrestricted least squares estimators and estimators that are restricted by the linear hypothesis and produce James‐Stein‐type shrinkage estimators of the regression coefficients. We develop the asymptotic joint distribution of such restricted and unrestricted estimators and use this to study the relative performance of the proposed estimators via their integrated asymptotic distributional risks. We conduct Monte Carlo simulations to examine the relative performance of the estimators in terms of their integrated mean square errors. We also compare the performance of the proposed estimators with a recently devised LASSO estimator as well as with ridge‐type estimators both via simulations and data on the survival of primary billiary cirhosis patients.     

Joint modelling of longitudinal binary data and survival data

The medical costs in an ageing society substantially increase when the incidences of chronic diseases, disabilities and inability to live independently are high. Healthy lifestyles not only affect elderly individuals but also influence the entire community. When assessing treatment efficacy, survival and quality of life should be considered simultaneously. This paper proposes the joint likelihood approach for modelling survival and longitudinal binary covariates simultaneously. Because some unobservable information is present in the model, the Monte Carlo EM algorithm and Metropolis-Hastings algorithm are used to find the estimators. Monte Carlo simulations are performed to evaluate the performance of the proposed model based on the accuracy and precision of the estimates. Real data are used to demonstrate the feasibility of the proposed model.     

Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

Locally Efficient Semiparametric Estimators for Proportional Hazards Models with Measurement Error

We propose a new class of semiparametric estimators for proportional hazards models in the presence of measurement error in the covariates, where the baseline hazard function, the hazard function for the censoring time, and the distribution of the true covariates are considered as unknown infinite dimensional parameters. We estimate the model components by solving estimating equations based on the semiparametric efficient scores under a sequence of restricted models where the logarithm of the hazard functions are approximated by reduced rank regression splines. The proposed estimators are locally efficient in the sense that the estimators are semiparametrically efficient if the distribution of the error‐prone covariates is specified correctly and are still consistent and asymptotically normal if the distribution is misspecified. Our simulation studies show that the proposed estimators have smaller biases and variances than competing methods. We further illustrate the new method with a real application in an HIV clinical trial.     

Regression analysis of restricted mean survival time based on pseudo-observations for competing risks data

Competing risks data often occur in many medical follow-up studies. When the survival time is the outcome variable, the restricted mean survival time has heuristic and clinically meaningful interpretation. In this article, we propose a class of regression models for the restricted mean survival time in the competing risks setting. We adopt a technique of pseudo-observations to develop estimating equation approaches for the model parameters and establish asymptotic properties of the resulting estimators. The finite-sample behavior of the proposed method is evaluated through simulation studies, and an application to the Women’s Interagency HIV Study is provided.     

Additive Transformation Models for Recurrent Events

In this article, we propose a class of additive transformation models for recurrent event data, which includes the additive rates model as a special case. The new models offer great flexibility in formulating the effects of covariates on the mean function of recurrent events. Estimating equation approaches are developed for the model parameters, and asymptotic properties of the resulting estimators are established. In addition, a model checking procedure is presented to assess the adequacy of the model. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a bladder cancer study is presented.     

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.

     

Proportional Hazards Model for Multivariate Failure Time Data

Multivariate failure time data also referred to as correlated or clustered failure time data, often arise in survival studies when each study subject may experience multiple events. Statistical analysis of such data needs to account for intracluster dependence. In this article, we consider a bivariate proportional hazards model using vector hazard rate, in which the covariates under study have different effect on two components of the vector hazard rate function. Estimation of the parameters as well as base line hazard function are discussed. Properties of the estimators are investigated. We illustrated the method using two real life data. A simulation study is reported to assess the performance of the estimator.     

Nonparametric estimation of the conditional mean residual life function with censored data

The conditional mean residual life (MRL) function is the expected remaining lifetime of a system given survival past a particular time point and the values of a set of predictor variables. This function is a valuable tool in reliability and actuarial studies when the right tail of the distribution is of interest, and can be more informative than the survivor function. In this paper, we identify theoretical limitations of some semi-parametric conditional MRL models, and propose two nonparametric methods of estimating the conditional MRL function. Asymptotic properties such as consistency and normality of our proposed estimators are established. We investigate via simulation study the empirical properties of the proposed estimators, including bootstrap pointwise confidence intervals. Using Monte Carlo simulations we compare the proposed nonparametric estimators to two popular semi-parametric methods of analysis, for varying types of data. The proposed estimators are demonstrated on the Veteran’s Administration lung cancer trial.     

Nonparametric inference on mean residual life function with length-biased right-censored data

Abstract

In survival or reliability studies, the mean residual life (MRL) function is an important characteristic in understanding the survival or ageing process. In this article, we consider the problem of nonparametric MRL function estimation with length-biased right-censored data. Two nonparametric estimators of the MRL are proposed and their weak convergence is presented. In order to evaluate the performance of these estimators, small Monte Carlo simulations are carried out. Results show that the proposed estimators work well especially when the sample size is small and their calculations are simple. Finally, a real data example is provided.     

A class of accelerated means regression models for recurrent event data

In this article, we propose a general class of accelerated means regression models for recurrent event data. The class includes the proportional means model, the accelerated failure time model and the accelerated rates model as special cases. The new model offers great flexibility in formulating the effects of covariates on the mean functions of counting processes while leaving the stochastic structure completely unspecified. For the inference on the model parameters, estimating equation approaches are developed and both large and final sample properties of the proposed estimators are established. In addition, some graphical and numerical procedures are presented for model checking. An illustration with multiple-infection data from a clinic study on chronic granulomatous disease is also provided.     

Reducing bias in parameter estimates from stepwise regression in proportional hazards regression with right-censored data

When variable selection with stepwise regression and model fitting are conducted on the same data set, competition for inclusion in the model induces a selection bias in coefficient estimators away from zero. In proportional hazards regression with right-censored data, selection bias inflates the absolute value of parameter estimate of selected parameters, while the omission of other variables may shrink coefficients toward zero. This paper explores the extent of the bias in parameter estimates from stepwise proportional hazards regression and proposes a bootstrap method, similar to those proposed by Miller (Subset Selection in Regression, 2nd edn. Chapman & Hall/CRC, 2002) for linear regression, to correct for selection bias. We also use bootstrap methods to estimate the standard error of the adjusted estimators. Simulation results show that substantial biases could be present in uncorrected stepwise estimators and, for binary covariates, could exceed 250% of the true parameter value. The simulations also show that the conditional mean of the proposed bootstrap bias-corrected parameter estimator, given that a variable is selected, is moved closer to the unconditional mean of the standard partial likelihood estimator in the chosen model, and to the population value of the parameter. We also explore the effect of the adjustment on estimates of log relative risk, given the values of the covariates in a selected model. The proposed method is illustrated with data sets in primary biliary cirrhosis and in multiple myeloma from the Eastern Cooperative Oncology Group.