Estimation of dependence between paired correlated failure times in the presence of covariate measurement error

Summary. In many biomedical studies, covariates are subject to measurement error. Although it is well known that the regression coefficients estimators can be substantially biased if the measurement error is not accommodated, there has been little study of the effect of covariate measurement error on the estimation of the dependence between bivariate failure times. We show that the dependence parameter estimator in the Clayton–Oakes model can be considerably biased if the measurement error in the covariate is not accommodated. In contrast with the typical bias towards the null for marginal regression coefficients, the dependence parameter can be biased in either direction. We introduce a bias reduction technique for the bivariate survival function in copula models while assuming an additive measurement error model and replicated measurement for the covariates, and we study the large and small sample properties of the dependence parameter estimator proposed.     

A risk set calibration method for failure time regression by using a covariate reliability sample

Regression parameter estimation in the Cox failure time model is considered when regression variables are subject to measurement error. Assuming that repeat regression vector measurements adhere to a classical measurement model, we can consider an ordinary regression calibration approach in which the unobserved covariates are replaced by an estimate of their conditional expectation given available covariate measurements. However, since the rate of withdrawal from the risk set across the time axis, due to failure or censoring, will typically depend on covariates, we may improve the regression parameter estimator by recalibrating within each risk set. The asymptotic and small sample properties of such a risk set regression calibration estimator are studied. A simple estimator based on a least squares calibration in each risk set appears able to eliminate much of the bias that attends the ordinary regression calibration estimator under extreme measurement error circumstances. Corresponding asymptotic distribution theory is developed, small sample properties are studied using computer simulations and an illustration is provided.     

Errors-in-Variables Regression Using Stein Estimates

A method is proposed for estimating regression parameters from data containing covariate measurement errors by using Stein estimates of the unobserved true covariates. The method produces consistent estimates for the slope parameter in the classical linear errors-in-variables model and applies to a broad range of nonlinear regression problems, provided the measurement error is Gaussian with known variance. Simulations are used to examine the performance of the estimates in a nonlinear regression problem and to compare them with the usual naive ones obtained by ignoring error and with other estimates proposed recently in the literature.     

Proportional Rate Model with Incomplete Covariate and Complete Auxiliary Information

This paper deals with the analysis of proportional rate model for recurrent event data when covariates are subject to missing. The true covariate is measured only on a randomly chosen validation set, whereas auxiliary information is available for all cohort subjects. To further utilize the auxiliary information to improve study efficiency, we propose an estimated estimating equation for the regression parameters. The resulting estimators are shown to be consistent and asymptotically normal. Both graphical and numerical techniques for checking the adequacy of the model are presented. Simulations are conducted to evaluate the finite sample performance of the proposed estimators. Illustration with a real medical study is provided.     

Cox regression in cohort studies with validation sampling

An estimation procedure is proposed for the Cox model in cohort studies with validation sampling, where crude covariate information is observed for the full cohort and true covariate information is collected on a validation set sampled randomly from the full cohort. The method proposed makes use of the partial information from data that are available on the entire cohort by fitting a working Cox model relating crude covariates to the failure time. The resulting estimator is consistent regardless of the specification of the working model and is asymptotically more efficient than the validation-set-only estimator. Approximate asymptotic relative efficiencies with respect to some alternative methods are derived under a simple scenario and further studied numerically. The finite sample performance is investigated and compared with alternative methods via simulation studies. A similar procedure also works for the case where the validation set is a stratified random sample from the cohort.     

On maximum likelihood estimation in parametric regression with missing covariates

We consider parametric regression problems with some covariates missing at random. It is shown that the regression parameter remains identifiable under natural conditions. When the always observed covariates are discrete, we propose a semiparametric maximum likelihood method, which does not require parametric specification of the missing data mechanism or the covariate distribution. The global maximum likelihood estimator (MLE), which maximizes the likelihood over the whole parameter set, is shown to exist under simple conditions. For ease of computation, we also consider a restricted MLE which maximizes the likelihood over covariate distributions supported by the observed values. Under regularity conditions, the two MLEs are asymptotically equivalent and strongly consistent for a class of topologies on the parameter set.     

The proportional hazards model with covariate measurement error

The proportional hazards regression model is commonly used to evaluate the relationship between survival and covariates. Covariates are frequently measured with error. Substituting mismeasured values for the true covariates leads to biased estimation. Hu et al. (Biometrics 88 (1998) 447) have proposed to base estimation in the proportional hazards model with covariate measurement error on a joint likelihood for survival and the covariate variable. Nonparametric maximum likelihood estimation (NPMLE) was used and simulations were conducted to assess the asymptotic validity of this approach. In this paper, we derive a rigorous proof of asymptotic normality of the NPML estimators.     

Double sampling for exact values in the normal discriminant model with application to binary regression

Increasing attention is being given to problems involving binary outcomes with covariates subject to measurement error. Here, we consider the two group normal discriminant model where a subset of the continuous variates are subject to error and will typically be replaced by a vector of surrogates, perhaps of different dimension. Correcting for the measurement error is made possible by a double sampling scheme in which the surrogates are collected on all units and true values are obtained on a random subset of units. Such a scheme allows us to consider a rich set of measurement error models which extend the traditional additive error model. Maximum likelihood estimators and their asymptotic properties are derived under a variety of models for the relationship between true values and the surrogates. Specific attention is given to the coefficients in the resulting logistic regression model. Optimal allocations are derived which minimize the variance of the estimated slope subject to cost constraints for the case where there is a univariate covariate but a possibly multivariate surrogate.     

SIMEX method for censored quantile regression with measurement error

Censored quantile regression serves as an important supplement to the Cox proportional hazards model in survival analysis. In addition to being exposed to censoring, some covariates may subject to measurement error. This leads to substantially biased estimate without taking this error into account. The SIMulation-EXtrapolation (SIMEX) method is an effective tool to handle the measurement error issue. We extend the SIMEX approach to the censored quantile regression with covariate measurement error. The algorithm is assessed via extensive simulations. A lung cancer study is analyzed to verify the validation of the proposed method.     

A comparison of regression calibration approaches for designs with internal validation data

We compare the asymptotic relative efficiency of several regression calibration methods of correcting for measurement error in studies with internal validation data, when a single covariate is measured with error. The estimators we consider are appropriate in main study/hybrid validation study designs, where the latter study includes internal validation and may include external validation data. Although all of the methods we consider produce consistent estimates, the method proposed by Spiegelman et al. (Statistics in Medicine, 20 (2001) 139) has an asymptotically smaller variance than the other methods. The methods for measurement error correction are illustrated using a study of the effect of in utero lead exposure on infant birth weight.     

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.     

SMALL AREA ESTIMATION USING SURVEY WEIGHTS WITH FUNCTIONAL MEASUREMENT ERROR IN THE COVARIATE

Nested error linear regression models using survey weights have been studied in small area estimation to obtain efficient model‐based and design‐consistent estimators of small area means. The covariates in these nested error linear regression models are not subject to measurement errors. In practical applications, however, there are many situations in which the covariates are subject to measurement errors. In this paper, we develop a nested error linear regression model with an area‐level covariate subject to functional measurement error. In particular, we propose a pseudo‐empirical Bayes (PEB) predictor to estimate small area means. This predictor borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. We also employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator.     

GMM nonparametric correction methods for logistic regression with error‐contaminated covariates and partially observed instrumental variables

We consider logistic regression with covariate measurement error. Most existing approaches require certain replicates of the error‐contaminated covariates, which may not be available in the data. We propose generalized method of moments (GMM) nonparametric correction approaches that use instrumental variables observed in a calibration subsample. The instrumental variable is related to the underlying true covariates through a general nonparametric model, and the probability of being in the calibration subsample may depend on the observed variables. We first take a simple approach adopting the inverse selection probability weighting technique using the calibration subsample. We then improve the approach based on the GMM using the whole sample. The asymptotic properties are derived, and the finite sample performance is evaluated through simulation studies and an application to a real data set.     

Generalized linear models with covariate measurement error and unknown link function

Generalized linear models (GLMs) with error-in-covariates are useful in epidemiological research due to the ubiquity of non-normal response variables and inaccurate measurements. The link function in GLMs is chosen by the user depending on the type of response variable, frequently the canonical link function. When covariates are measured with error, incorrect inference can be made, compounded by incorrect choice of link function. In this article we propose three flexible approaches for handling error-in-covariates and estimating an unknown link simultaneously. The first approach uses a fully Bayesian (FB) hierarchical framework, treating the unobserved covariate as a latent variable to be integrated over. The second and third are approximate Bayesian approach which use a Laplace approximation to marginalize the variables measured with error out of the likelihood. Our simulation results show support that the FB approach is often a better choice than the approximate Bayesian approaches for adjusting for measurement error, particularly when the measurement error distribution is misspecified. These approaches are demonstrated on an application with binary response.     

基于贝叶斯推断的HIV非线性混合效应联合模型研究

 在纵向数据研究中,混合效应模型的随机误差通常采用正态性假设。而诸如病毒载量和CD4细胞数目等病毒性数据通常呈现偏斜性,因此正态性假设可能影响模型结果甚至导致错误的结论。在HIV动力学研究中,病毒响应值往往与协变量相关,且协变量的测量值通常存在误差,为此论文中联立协变量过程建立具有偏正态分布的非线性混合效应联合模型,并用贝叶斯推断方法估计模型的参数。由于协变量能够解释个体内的部分变化,因此协变量过程的模型选择对病毒载量的拟合效果有重要的影响。该文提出了一次移动平均模型作为协变量过程的改进模型,比较后发现当协变量采用移动平均模型时,病毒载量模型的拟合效果更好。该结果对协变量模型的研究具有重要的指导意义。     

The Extensively Corrected Score for Measurement Error Models

In measurement error problems, two major and consistent estimation methods are the conditional score and the corrected score. They are functional methods that require no parametric assumptions on mismeasured covariates. The conditional score requires that a suitable sufficient statistic for the mismeasured covariate can be found, while the corrected score requires that the object score function can be estimated without bias. These assumptions limit their ranges of applications. The extensively corrected score proposed here is an extension of the corrected score. It yields consistent estimations in many cases when neither the conditional score nor the corrected score is feasible. We demonstrate its constructions in generalized linear models and the Cox proportional hazards model, assess its performances by simulation studies and illustrate its implementations by two real examples.     

Cox regression with missing covariate data using a modified partial likelihood method

Missing covariate values is a common problem in survival analysis. In this paper we propose a novel method for the Cox regression model that is close to maximum likelihood but avoids the use of the EM-algorithm. It exploits that the observed hazard function is multiplicative in the baseline hazard function with the idea being to profile out this function before carrying out the estimation of the parameter of interest. In this step one uses a Breslow type estimator to estimate the cumulative baseline hazard function. We focus on the situation where the observed covariates are categorical which allows us to calculate estimators without having to assume anything about the distribution of the covariates. We show that the proposed estimator is consistent and asymptotically normal, and derive a consistent estimator of the variance–covariance matrix that does not involve any choice of a perturbation parameter. Moderate sample size performance of the estimators is investigated via simulation and by application to a real data example.     

A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

t-Type corrected-loss estimation for error-in-variable model

In this article, we consider a linear model in which the covariates are measured with errors. We propose a t-type corrected-loss estimation of the covariate effect, when the measurement error follows the Laplace distribution. The proposed estimator is asymptotically normal. In practical studies, some outliers that diminish the robustness of the estimation occur. Simulation studies show that the estimators are resistant to vertical outliers and an application of 6-minute walk test is presented to show that the proposed method performs well.     

Incomplete covariates in the Cox model with applications to biological marker data

A common occurrence in clinical trials with a survival end point is missing covariate data. With ignorably missing covariate data, Lipsitz and Ibrahim proposed a set of estimating equations to estimate the parameters of Cox's proportional hazards model. They proposed to obtain parameter estimates via a Monte Carlo EM algorithm. We extend those results to non-ignorably missing covariate data. We present a clinical trials example with three partially observed laboratory markers which are used as covariates to predict survival.