A simulation-based marginal method for longitudinal data with dropout and mismeasured covariates

Longitudinal data often contain missing observations and error-prone covariates. Extensive attention has been directed to analysis methods to adjust for the bias induced by missing observations. There is relatively little work on investigating the effects of covariate measurement error on estimation of the response parameters, especially on simultaneously accounting for the biases induced by both missing values and mismeasured covariates. It is not clear what the impact of ignoring measurement error is when analyzing longitudinal data with both missing observations and error-prone covariates. In this article, we study the effects of covariate measurement error on estimation of the response parameters for longitudinal studies. We develop an inference method that adjusts for the biases induced by measurement error as well as by missingness. The proposed method does not require the full specification of the distribution of the response vector but only requires modeling its mean and variance structures. Furthermore, the proposed method employs the so-called functional modeling strategy to handle the covariate process, with the distribution of covariates left unspecified. These features, plus the simplicity of implementation, make the proposed method very attractive. In this paper, we establish the asymptotic properties for the resulting estimators. With the proposed method, we conduct sensitivity analyses on a cohort data set arising from the Framingham Heart Study. Simulation studies are carried out to evaluate the impact of ignoring covariate measurement error and to assess the performance of the proposed method.     

Semiparametric approaches for joint modeling of longitudinal and survival data with time-varying coefficients

Summary .   We study joint modeling of survival and longitudinal data. There are two regression models of interest. The primary model is for survival outcomes, which are assumed to follow a time-varying coefficient proportional hazards model. The second model is for longitudinal data, which are assumed to follow a random effects model. Based on the trajectory of a subject's longitudinal data, some covariates in the survival model are functions of the unobserved random effects. Estimated random effects are generally different from the unobserved random effects and hence this leads to covariate measurement error. To deal with covariate measurement error, we propose a local corrected score estimator and a local conditional score estimator. Both approaches are semiparametric methods in the sense that there is no distributional assumption needed for the underlying true covariates. The estimators are shown to be consistent and asymptotically normal. However, simulation studies indicate that the conditional score estimator outperforms the corrected score estimator for finite samples, especially in the case of relatively large measurement error. The approaches are demonstrated by an application to data from an HIV clinical trial.     

A simulation study of measurement error correction methods in logistic regression

Measurement error models in logistic regression have received considerable theoretical interest over the past 10-15 years. In this paper, we present the results of a simulation study that compares four estimation methods: the so-called regression calibration method, probit maximum likelihood as an approximation to the logistic maximum likelihood, the exact maximum likelihood method based on a logistic model, and the naive estimator, which is the result of simply ignoring the fact that some of the explanatory variables are measured with error. We have compared the behavior of these methods in a simple, additive measurement error model. We show that, in this situation, the regression calibration method is a very good alternative to more mathematically sophisticated methods.     

Estimation in Semiparametric Transition Measurement Error Models for Longitudinal Data

Summary .  We consider semiparametric transition measurement error models for longitudinal data, where one of the covariates is measured with error in transition models, and no distributional assumption is made for the underlying unobserved covariate. An estimating equation approach based on the pseudo conditional score method is proposed. We show the resulting estimators of the regression coefficients are consistent and asymptotically normal. We also discuss the issue of efficiency loss. Simulation studies are conducted to examine the finite-sample performance of our estimators. The longitudinal AIDS Costs and Services Utilization Survey data are analyzed for illustration.     

A Semiparametric Joint Model for Longitudinal and Survival Data with Application to Hemodialysis Study

Summary .  In many longitudinal clinical studies, the level and progression rate of repeatedly measured biomarkers on each subject quantify the severity of the disease and that subject's susceptibility to progression of the disease. It is of scientific and clinical interest to relate such quantities to a later time-to-event clinical endpoint such as patient survival. This is usually done with a shared parameter model. In such models, the longitudinal biomarker data and the survival outcome of each subject are assumed to be conditionally independent given subject-level severity or susceptibility (also called frailty in statistical terms). In this article, we study the case where the conditional distribution of longitudinal data is modeled by a linear mixed-effect model, and the conditional distribution of the survival data is given by a Cox proportional hazard model. We allow unknown regression coefficients and time-dependent covariates in both models. The proposed estimators are maximizers of an exact correction to the joint log likelihood with the frailties eliminated as nuisance parameters, an idea that originated from correction of covariate measurement error in measurement error models. The corrected joint log likelihood is shown to be asymptotically concave and leads to consistent and asymptotically normal estimators. Unlike most published methods for joint modeling, the proposed estimation procedure does not rely on distributional assumptions of the frailties. The proposed method was studied in simulations and applied to a data set from the Hemodialysis Study.     

Bias analysis and the simulation‐extrapolation method for survival data with covariate measurement error under parametric proportional odds models

It has been well known that ignoring measurement error may result in substantially biased estimates in many contexts including linear and nonlinear regressions. For survival data with measurement error in covariates, there has been extensive discussion in the literature with the focus on proportional hazards (PH) models. Recently, research interest has extended to accelerated failure time (AFT) and additive hazards (AH) models. However, the impact of measurement error on other models, such as the proportional odds model, has received relatively little attention, although these models are important alternatives when PH, AFT, or AH models are not appropriate to fit data. In this paper, we investigate this important problem and study the bias induced by the naive approach of ignoring covariate measurement error. To adjust for the induced bias, we describe the simulation‐extrapolation method. The proposed method enjoys a number of appealing features. Its implementation is straightforward and can be accomplished with minor modifications of existing software. More importantly, the proposed method does not require modeling the covariate process, which is quite attractive in practice. As the precise values of error‐prone covariates are often not observable, any modeling assumption on such covariates has the risk of model misspecification, hence yielding invalid inferences if this happens. The proposed method is carefully assessed both theoretically and empirically. Theoretically, we establish the asymptotic normality for resulting estimators. Numerically, simulation studies are carried out to evaluate the performance of the estimators as well as the impact of ignoring measurement error, along with an application to a data set arising from the Busselton Health Study. Sensitivity of the proposed method to misspecification of the error model is studied as well.     

Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information

This article presents a new method for maximum likelihood estimation of logistic regression models with incomplete covariate data where auxiliary information is available. This auxiliary information is extraneous to the regression model of interest but predictive of the covariate with missing data. Ibrahim (1990, Journal of the American Statistical Association 85, 765-769) provides a general method for estimating generalized linear regression models with missing covariates using the EM algorithm that is easily implemented when there is no auxiliary data. Vach (1997, Statistics in Medicine 16, 57-72) describes how the method can be extended when the outcome and auxiliary data are conditionally independent given the covariates in the model. The method allows the incorporation of auxiliary data without making the conditional independence assumption. We suggest tests of conditional independence and compare the performance of several estimators in an example concerning mental health service utilization in children. Using an artificial dataset, we compare the performance of several estimators when auxiliary data are available.     

Maximum likelihood estimation in the additive hazards model

The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As the non-parametric model, additive hazards offer a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper, we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We show that the maximum likelihood estimator may be obtained by separately maximizing the log-likelihood contribution of each event time point, and we show that the maximizing problem is equivalent to fitting a series of Poisson regression models with an identity link under non-negativity constraints. We derive an analytic solution to the maximum likelihood estimator. We contrast the maximum likelihood estimator with the ordinary least-squares estimator in a simulation study and show that the maximum likelihood estimator has smaller mean squared error than the ordinary least-squares estimator. An illustration with data on patients with carcinoma of the oropharynx is provided.     

Structural Nested Mean Models for Assessing Time-Varying Effect Moderation

Summary .  This article considers the problem of assessing causal effect moderation in longitudinal settings in which treatment (or exposure) is time varying and so are the covariates said to moderate its effect.  Intermediate causal effects  that describe time-varying causal effects of treatment conditional on past covariate history are introduced and considered as part of Robins' structural nested mean model. Two estimators of the intermediate causal effects, and their standard errors, are presented and discussed: The first is a proposed two-stage regression estimator. The second is Robins' G-estimator. The results of a small simulation study that begins to shed light on the small versus large sample performance of the estimators, and on the bias–variance trade-off between the two estimators are presented. The methodology is illustrated using longitudinal data from a depression study.     

Estimation in capture-recapture models when covariates are subject to measurement errors

We consider estimation problems in capture-recapture models when the covariates or the auxiliary variables are measured with errors. The naive approach, which ignores measurement errors, is found to be unacceptable in the estimation of both regression parameters and population size: it yields estimators with biases increasing with the magnitude of errors, and flawed confidence intervals. To account for measurement errors, we derive a regression parameter estimator using a regression calibration method. We develop modified estimators of the population size accordingly. A simulation study shows that the resulting estimators are more satisfactory than those from either the naive approach or the simulation extrapolation (SIMEX) method. Data from a bird species Prinia flaviventris in Hong Kong are analyzed with and without the assumption of measurement errors, to demonstrate the effects of errors on estimations.     

Approximate nonparametric corrected-score method for joint modeling of survival and longitudinal data measured with error

We consider the problem of jointly modeling survival time and longitudinal data subject to measurement error. The survival times are modeled through the proportional hazards model and a random effects model is assumed for the longitudinal covariate process. Under this framework, we propose an approximate nonparametric corrected-score estimator for the parameter, which describes the association between the time-to-event and the longitudinal covariate. The term nonparametric refers to the fact that assumptions regarding the distribution of the random effects and that of the measurement error are unnecessary. The finite sample size performance of the approximate nonparametric corrected-score estimator is examined through simulation studies and its asymptotic properties are also developed. Furthermore, the proposed estimator and some existing estimators are applied to real data from an AIDS clinical trial.     

Semiparametric additive time-varying coefficients model for longitudinal data with censored time origin

Statistical analysis of longitudinal data often involves modeling treatment effects on clinically relevant longitudinal biomarkers since an initial event (the time origin). In some studies including preventive HIV vaccine efficacy trials, some participants have biomarkers measured starting at the time origin, whereas others have biomarkers measured starting later with the time origin unknown. The semiparametric additive time-varying coefficient model is investigated where the effects of some covariates vary nonparametrically with time while the effects of others remain constant. Weighted profile least squares estimators coupled with kernel smoothing are developed. The method uses the expectation maximization approach to deal with the censored time origin. The Kaplan–Meier estimator and other failure time regression models such as the Cox model can be utilized to estimate the distribution and the conditional distribution of left censored event time related to the censored time origin. Asymptotic properties of the parametric and nonparametric estimators and consistent asymptotic variance estimators are derived. A two-stage estimation procedure for choosing weight is proposed to improve estimation efficiency. Numerical simulations are conducted to examine finite sample properties of the proposed estimators. The simulation results show that the theory and methods work well. The efficiency gain of the two-stage estimation procedure depends on the distribution of the longitudinal error processes. The method is applied to analyze data from the Merck 023/HVTN 502 Step HIV vaccine study.     

Joint models for a primary endpoint and multiple longitudinal covariate processes

Summary .   Joint models are formulated to investigate the association between a primary endpoint and features of multiple longitudinal processes. In particular, the subject-specific random effects in a multivariate linear random-effects model for multiple longitudinal processes are predictors in a generalized linear model for primary endpoints. Li, Zhang, and Davidian (2004, Biometrics 60 , 1–7) proposed an estimation procedure that makes no distributional assumption on the random effects but assumes independent within-subject measurement errors in the longitudinal covariate process. Based on an asymptotic bias analysis, we found that their estimators can be biased when random effects do not fully explain the within-subject correlations among longitudinal covariate measurements. Specifically, the existing procedure is fairly sensitive to the independent measurement error assumption. To overcome this limitation, we propose new estimation procedures that require neither a distributional or covariance structural assumption on covariate random effects nor an independence assumption on within-subject measurement errors. These new procedures are more flexible, readily cover scenarios that have multivariate longitudinal covariate processes, and can be implemented using available software. Through simulations and an analysis of data from a hypertension study, we evaluate and illustrate the numerical performances of the new estimators.     

Simulation‐selection‐extrapolation: Estimation in high‐dimensional errors‐in‐variables models

Errors‐in‐variables models in high‐dimensional settings pose two challenges in application. First, the number of observed covariates is larger than the sample size, while only a small number of covariates are true predictors under an assumption of model sparsity. Second, the presence of measurement error can result in severely biased parameter estimates, and also affects the ability of penalized methods such as the lasso to recover the true sparsity pattern. A new estimation procedure called SIMulation‐SELection‐EXtrapolation (SIMSELEX) is proposed. This procedure makes double use of lasso methodology. First, the lasso is used to estimate sparse solutions in the simulation step, after which a group lasso is implemented to do variable selection. The SIMSELEX estimator is shown to perform well in variable selection, and has significantly lower estimation error than naive estimators that ignore measurement error. SIMSELEX can be applied in a variety of errors‐in‐variables settings, including linear models, generalized linear models, and Cox survival models. It is furthermore shown in the Supporting Information how SIMSELEX can be applied to spline‐based regression models. A simulation study is conducted to compare the SIMSELEX estimators to existing methods in the linear and logistic model settings, and to evaluate performance compared to naive methods in the Cox and spline models. Finally, the method is used to analyze a microarray dataset that contains gene expression measurements of favorable histology Wilms tumors.     

Missing covariates in longitudinal data with informative dropouts: bias analysis and inference

We consider estimation in generalized linear mixed models (GLMM) for longitudinal data with informative dropouts. At the time a unit drops out, time-varying covariates are often unobserved in addition to the missing outcome. However, existing informative dropout models typically require covariates to be completely observed. This assumption is not realistic in the presence of time-varying covariates. In this article, we first study the asymptotic bias that would result from applying existing methods, where missing time-varying covariates are handled using naive approaches, which include: (1) using only baseline values; (2) carrying forward the last observation; and (3) assuming the missing data are ignorable. Our asymptotic bias analysis shows that these naive approaches yield inconsistent estimators of model parameters. We next propose a selection/transition model that allows covariates to be missing in addition to the outcome variable at the time of dropout. The EM algorithm is used for inference in the proposed model. Data from a longitudinal study of human immunodeficiency virus (HIV)-infected women are used to illustrate the methodology.     

On Method of Moments Estimation in Linear Mixed Effects Models with Measurement Error on Covariates and Response with Application to a Longitudinal Study of Gene-Environment Interaction

We study a linear mixed effects model for longitudinal data, where the response variable and covariates with fixed effects are subject to measurement error. We propose a method of moment estimation that does not require any assumption on the functional forms of the distributions of random effects and other random errors in the model. For a classical measurement error model we apply the instrumental variable approach to ensure identifiability of the parameters. Our methodology, without instrumental variables, can be applied to Berkson measurement errors. Using simulation studies, we investigate the finite sample performances of the estimators and show the impact of measurement error on the covariates and the response on the estimation procedure. The results show that our method performs quite satisfactory, especially for the fixed effects with measurement error (even under misspecification of measurement error model). This method is applied to a real data example of a large birth and child cohort study.     

A Positive Stable Frailty Model for Clustered Failure Time Data with Covariate‐Dependent Frailty

Summary In this article, we propose a positive stable shared frailty Cox model for clustered failure time data where the frailty distribution varies with cluster‐level covariates. The proposed model accounts for covariate‐dependent intracluster correlation and permits both conditional and marginal inferences. We obtain marginal inference directly from a marginal model, then use a stratified Cox‐type pseudo‐partial likelihood approach to estimate the regression coefficient for the frailty parameter. The proposed estimators are consistent and asymptotically normal and a consistent estimator of the covariance matrix is provided. Simulation studies show that the proposed estimation procedure is appropriate for practical use with a realistic number of clusters. Finally, we present an application of the proposed method to kidney transplantation data from the Scientific Registry of Transplant Recipients.     

Family-specific approaches to the analysis of case-control family data

Case-control studies augmented by the values of responses and covariates from family members allow investigators to study the association between the response and genetics and environment by relating differences in the response directly to within-family differences in covariates. However, existing approaches for case-control family data parameterize covariate effects in terms of the marginal probability of response, the same effects that one estimates from standard case-control studies. This article focuses on the estimation of family-specific covariate effects and develops efficient methods to fit family-specific models such as binary mixed-effects models. We also extend the approach to cover any setting where one has a fully specified model for the vector of responses in a family. We illustrate our approach using data from a case-control family study of brain cancer and consider the use of weighted and conditional likelihood methods as alternatives.     

Regression analysis when covariates are regression parameters of a random effects model for observed longitudinal measurements

We consider regression analysis when covariate variables are the underlying regression coefficients of another linear mixed model. A naive approach is to use each subject's repeated measurements, which are assumed to follow a linear mixed model, and obtain subject-specific estimated coefficients to replace the covariate variables. However, directly replacing the unobserved covariates in the primary regression by these estimated coefficients may result in a significantly biased estimator. The aforementioned problem can be evaluated as a generalization of the classical additive error model where repeated measures are considered as replicates. To correct for these biases, we investigate a pseudo-expected estimating equation (EEE) estimator, a regression calibration (RC) estimator, and a refined version of the RC estimator. For linear regression, the first two estimators are identical under certain conditions. However, when the primary regression model is a nonlinear model, the RC estimator is usually biased. We thus consider a refined regression calibration estimator whose performance is close to that of the pseudo-EEE estimator but does not require numerical integration. The RC estimator is also extended to the proportional hazards regression model. In addition to the distribution theory, we evaluate the methods through simulation studies. The methods are applied to analyze a real dataset from a child growth study.     

Extended Mantel-Haenszel estimating procedure for multivariate logistic regression models

A class of estimating functions is proposed for the estimation of multivariate relative risk in stratified case-control studies. It reduces to the well-known Mantel-Haenszel estimator when there is a single binary risk factor. Large-sample properties of the solutions to the proposed estimating equations are established for two distinct situations. Efficiency calculations suggest that the proposed estimators are nearly fully efficient relative to the conditional maximum likelihood estimator for the parameters considered. Application of the proposed method to family data and longitudinal data, where the conditional likelihood approach fails, is discussed. Two examples from case-control studies and one example from a study on familial aggregation are presented.