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.     

A Simple Corrected Score for Logistic Regression with Errors-in-Covariates

We develop a simple corrected score for logistic regression with errors-in-covariates. The new method is an extension of the consistent functional methods proposed by Huang and Wang (2001) and is closely related to the corrected score method by Nakamura (1990 Nakamura, T. (1990). Corrected score function for errors-in-variables models: Methodology and application to generalized linear models. Biometrika. 77:127–137.[Crossref], [Web of Science ®] , [Google Scholar]) and Stefanski (1989 Stefanski, L.A. (1989). Unbiased estimation of a nonlinear function a normal mean with application to measurement error models. Commun. Stat. Ser. A - Theory Methods. 18:4335–4358.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The new method requires that the measurement error distribution is known, but does not require normality. The new method yields a consistent and asymptotically normal estimator under regularity conditions. We examine the finite-sample performance of the new estimator through simulation studies. Finally, we illustrate the new method by applying it to an AIDS study.     

An Exact Corrected Log-Likelihood Function for Cox's Proportional Hazards Model under Measurement Error and Some Extensions

Abstract.  This paper studies Cox's proportional hazards model under covariate measurement error. Nakamura's [ Biometrika 77 (1990) 127] methodology of corrected log-likelihood will be applied to the so-called Breslow likelihood, which is, in the absence of measurement error, equivalent to partial likelihood. For a general error model with possibly heteroscedastic and non-normal additive measurement error, corrected estimators of the regression parameter as well as of the baseline hazard rate are obtained. The estimators proposed by Nakamura [Biometrics 48 (1992) 829], Kong et al. [ Scand. J. Statist. 25 (1998) 573] and Kong & Gu [ Statistica Sinica 9 (1999) 953] are re-established in the special cases considered there. This sheds new light on these estimators and justifies them as exact corrected score estimators. Finally, the method will be extended to some variants of the Cox model.     

A Multiple Imputation Approach to the Analysis of Current Status Data with the Additive Hazards Model

This article discusses regression analysis of current status data, which occur in many fields including cross-sectional studies, demographical investigations, and tumorigenicity experiments (Keiding, 1991 Keiding , N. ( 1991 ). Age-specific incidence and prevalence: a statistical perspective (with discussion) . J. Roy. Statist. Soc. Ser. A 154 : 371 – 412 .[Crossref] , [Google Scholar]; Sun 2006 Sun , J. ( 2006 ). The Statistical Analysis of Interval-Censored Failure Time Data . New York : Springer-Verlag . [Google Scholar]). For the problem, we focus on the situation where the survival time of interest can be described by the additive hazards model and a multiple imputation approach is presented for inference. A major advantage of the approach is its simplicity and it can be easily implemented by using the existing software packages for right-censored failure time data. Extensive simulation studies are conducted and indicate that the approach performs well for practical situations and is comparable to the existing methods. The methodology is applied to a set of current status data arising from a tumorigenicity experiment and the model checking is discussed.     

Empirical Likelihood for the Additive Hazards Model with Current Status Data

We investigate empirical likelihood for the additive hazards model with current status data. An empirical log-likelihood ratio for a vector or subvector of regression parameters is defined and its limiting distribution is shown to be a standard chi-squared distribution. The proposed inference procedure enables us to make empirical likelihood-based inference for the regression parameters. Finite sample performance of the proposed method is assessed in simulation studies to compare with that of a normal approximation method, it shows that the empirical likelihood method provides more accurate inference than the normal approximation method. A real data example is used for illustration.     

A Semiparametric Approach for Accelerated Failure Time Models with Covariates Subject to Measurement Error

There are relatively few discussions about measurement error in the accelerated failure time (AFT) model, particularly for the semiparametric AFT model. In this article, we propose an adjusted estimation procedure for the semiparametric AFT model with covariates subject to measurement error, based on the profile likelihood approach and simulation and exploration (SIMEX) method. The simulation studies show that the proposed semiparametric SIMEX approach performs well. The proposed approach is applied to a coronary heart disease dataset from the Busselton Health study for illustration.     

Regression Analysis of Left-truncated and Case I Interval-censored Data with the Additive Hazards Model

In recent years the analysis of interval-censored failure time data has attracted a great deal of attention and such data arise in many fields including demographical studies, economic and financial studies, epidemiological studies, social sciences, and tumorigenicity experiments. This is especially the case in medical studies such as clinical trials. In this article, we discuss regression analysis of one type of such data, Case I interval-censored data, in the presence of left-truncation. For the problem, the additive hazards model is employed and the maximum likelihood method is applied for estimations of unknown parameters. In particular, we adopt the sieve estimation approach that approximates the baseline cumulative hazard function by linear functions. The resulting estimates of regression parameters are shown to be consistent and efficient and have an asymptotic normal distribution. An illustrative example is provided.     

Cure Rate Model with Measurement Error

This simulation study focuses on the relative small sample properties of some widely applied predictors in regression with AR(1) errors where there errors are allowed to follow normal and non-normal distributions. The conclusions are: all predictors considered are significantly unbiased; the relative performances of predictors, from the efficiency point of view, seemed insensitive to the nature of the error distribution; and the standard errors of predictors computed from the asymptotic formulas are very useful for purposes of inference in small sample and under all assumed distributions.     

Generalized Method of Moments for Additive Hazards Model with Clustered Dental Survival Data

For multivariate survival data, we study the generalized method of moments (GMM) approach to estimation and inference based on the marginal additive hazards model. We propose an efficient iterative algorithm using closed‐form solutions, which dramatically reduces the computational burden. Asymptotic normality of the proposed estimators is established, and the corresponding variance–covariance matrix can be consistently estimated. Inference procedures are derived based on the asymptotic chi‐squared distribution of the GMM objective function. Simulation studies are conducted to empirically examine the finite sample performance of the proposed method, and a real data example from a dental study is used for illustration.     

Weibull Regression for Lifetimes Measured with Error

Models are considered in which true lifetimes are generated by a Weibull regression model and measured lifetimes are determined from the true times by certain measurement error models. Adjusted estimators are obtained under one parametric specification. The bias properties of these estimators and standard estimators are compared both theoretically, using small measurement error asymptotics, and by simulation. The standard estimators of regression coefficients, other than the intercept, are bias-robust. The adjusted estimator of the shape parameter removes the bias of the standard estimator.     

Validation of A Heteroscedastic Hazards Regression Model

A Cox-type regression model accommodating heteroscedasticity, with a power factor of the baseline cumulative hazard, is investigated for analyzing data with crossing hazards behavior. Since the approach of partial likelihood cannot eliminate the baseline hazard, an overidentified estimating equation (OEE) approach is introduced in the estimation procedure. Its by-product, a model checking statistic, is presented to test for the overall adequacy of the heteroscedastic model. Further, under the heteroscedastic model setting, we propose two statistics to test the proportional hazards assumption. Implementation of this model is illustrated in a data analysis of a cancer clinical trial.     

Multiple Imputation for the Cox Proportional Hazards Model with Missing Covariates

We present three multiple imputation estimates for the Cox model with missing covariates. Two of the suggested estimates are asymptotically equivalent to estimates in the literature when the number of multiple imputations approaches infinity. The third estimate can be implemented using standard software that could handle time-varying covariates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Accounting for Response Misclassification and Covariate Measurement Error Using a Random Effects Logit Model

Often in longitudinal data arising out of epidemiologic studies, measurement error in covariates and/or classification errors in binary responses may be present. The goal of the present work is to develop a random effects logistic regression model that corrects for the classification errors in binary responses and/or measurement error in covariates. The analysis is carried out under a Bayesian set up. Simulation study reveals the effect of ignoring measurement error and/or classification errors on the estimates of the regression coefficients.     

Bayesian Model Selection and Averaging in Additive and Proportional Hazards Models

Although Cox proportional hazards regression is the default analysis for time to event data, there is typically uncertainty about whether the effects of a predictor are more appropriately characterized by a multiplicative or additive model. To accommodate this uncertainty, we place a model selection prior on the coefficients in an additive-multiplicative hazards model. This prior assigns positive probability, not only to the model that has both additive and multiplicative effects for each predictor, but also to sub-models corresponding to no association, to only additive effects, and to only proportional effects. The additive component of the model is constrained to ensure non-negative hazards, a condition often violated by current methods. After augmenting the data with Poisson latent variables, the prior is conditionally conjugate, and posterior computation can proceed via an efficient Gibbs sampling algorithm. Simulation study results are presented, and the methodology is illustrated using data from the Framingham heart study.     

Empirical Likelihood Inference for Longitudinal Data with Missing Response Variables and Error-Prone Covariates

We consider statistical inference for longitudinal partially linear models when the response variable is sometimes missing with missingness probability depending on the covariate that is measured with error. The block empirical likelihood procedure is used to estimate the regression coefficients and residual adjusted block empirical likelihood is employed for the baseline function. This leads us to prove a nonparametric version of Wilk's theorem. Compared with methods based on normal approximations, our proposed method does not require a consistent estimators for the asymptotic variance and bias. An application to a longitudinal study is used to illustrate the procedure developed here. A simulation study is also reported.     

The Method to Identify a Biomarker and to Evaluate Its Efficiency for Survival by Using the Joint Model of the Accelerate Failure Time and Longitudinal Data

In this article, we discuss how to identify longitudinal biomarkers in survival analysis under the accelerated failure time model and also discuss the effectiveness of biomarkers under the accelerated failure time model. Two methods proposed by Shcemper et al. are deployed to measure the efficacy of biomarkers. We use simulations to explore how the factors can influence the power of a score test to detect the association of a longitudinal biomarker and the survival time. These factors include the functional form of the random effects from the longitudinal biomarkers, in the different number of individuals, and time points per individual. The simulations are used to explore how the number of individuals, the number of time points per individual influence the effectiveness of the biomarker to predict survival at the given endpoint under the accelerated failure time model. We illustrate our methods using a prothrombin index as a predictor of survival in liver cirrhosis patients.     

Regression Analysis of Current Status Data Under the Additive Hazards Model with Auxiliary Covariates

This paper discusses regression analysis of current status or case I interval‐censored failure time data arising from the additive hazards model. In this situation, some covariates could be missing because of various reasons, but there may exist some auxiliary information about the missing covariates. To address the problem, we propose an estimated partial likelihood approach for estimation of regression parameters, which makes use of the available auxiliary information. The method can be easily implemented, and the asymptotic properties of the resulting estimates are established. To assess the finite sample performance of the proposed method, an extensive simulation study is conducted and indicates that the method works well.     

A corrected likelihood method for the proportional hazards model with covariates subject to measurement error

There has been extensive interest in discussing inference methods for survival data when some covariates are subject to measurement error. It is known that standard inferential procedures produce biased estimation if measurement error is not taken into account. With the Cox proportional hazards model a number of methods have been proposed to correct bias induced by measurement error, where the attention centers on utilizing the partial likelihood function. It is also of interest to understand the impact on estimation of the baseline hazard function in settings with mismeasured covariates. In this paper we employ a weakly parametric form for the baseline hazard function and propose simple unbiased estimating functions for estimation of parameters. The proposed method is easy to implement and it reveals the connection between the naive method ignoring measurement error and the corrected method with measurement error accounted for. Simulation studies are carried out to evaluate the performance of the estimators as well as the impact of ignoring measurement error in covariates. As an illustration we apply the proposed methods to analyze a data set arising from the Busselton Health Study [Knuiman, M.W., Cullent, K.J., Bulsara, M.K., Welborn, T.A., Hobbs, M.S.T., 1994. Mortality trends, 1965 to 1989, in Busselton, the site of repeated health surveys and interventions. Austral. J. Public Health 18, 129–135].     

Comparison of Additive and Multiplicative Bayesian Models for Longitudinal Count Data with Overdispersion Parameters: A Simulation Study

In applied statistical data analysis, overdispersion is a common feature. It can be addressed using both multiplicative and additive random effects. A multiplicative model for count data incorporates a gamma random effect as a multiplicative factor into the mean, whereas an additive model assumes a normally distributed random effect, entered into the linear predictor. Using Bayesian principles, these ideas are applied to longitudinal count data, based on the so-called combined model. The performance of the additive and multiplicative approaches is compared using a simulation study.     

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.