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.     

An estimator for the proportional hazards model with multiple longitudinal covariates measured with error

In many longitudinal studies, it is of interest to characterize the relationship between a time-to-event (e.g. survival) and several time-dependent and time-independent covariates. Time-dependent covariates are generally observed intermittently and with error. For a single time-dependent covariate, a popular approach is to assume a joint longitudinal data-survival model, where the time-dependent covariate follows a linear mixed effects model and the hazard of failure depends on random effects and time-independent covariates via a proportional hazards relationship. Regression calibration and likelihood or Bayesian methods have been advocated for implementation; however, generalization to more than one time-dependent covariate may become prohibitive. For a single time-dependent covariate, Tsiatis and Davidian (2001) have proposed an approach that is easily implemented and does not require an assumption on the distribution of the random effects. This technique may be generalized to multiple, possibly correlated, time-dependent covariates, as we demonstrate. We illustrate the approach via simulation and by application to data from an HIV clinical trial.     

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.     

Flexible modeling via a hybrid estimation scheme in generalized mixed models for longitudinal data

To circumvent the computational complexity of likelihood inference in generalized mixed models that assume linear or more general additive regression models of covariate effects, Laplace's approximations to multiple integrals in the likelihood have been commonly used without addressing the issue of adequacy of the approximations for individuals with sparse observations. In this article, we propose a hybrid estimation scheme to address this issue. The likelihoods for subjects with sparse observations use Monte Carlo approximations involving importance sampling, while Laplace's approximation is used for the likelihoods of other subjects that satisfy a certain diagnostic check on the adequacy of Laplace's approximation. Because of its computational tractability, the proposed approach allows flexible modeling of covariate effects by using regression splines and model selection procedures for knot and variable selection. Its computational and statistical advantages are illustrated by simulation and by application to longitudinal data from a fecundity study of fruit flies, for which overdispersion is modeled via a double exponential family.     

Bayesian nonparametric regression analysis of data with random effects covariates from longitudinal measurements

We consider nonparametric regression analysis in a generalized linear model (GLM) framework for data with covariates that are the subject-specific random effects of longitudinal measurements. The usual assumption that the effects of the longitudinal covariate processes are linear in the GLM may be unrealistic and if this happens it can cast doubt on the inference of observed covariate effects. Allowing the regression functions to be unknown, we propose to apply Bayesian nonparametric methods including cubic smoothing splines or P-splines for the possible nonlinearity and use an additive model in this complex setting. To improve computational efficiency, we propose the use of data-augmentation schemes. The approach allows flexible covariance structures for the random effects and within-subject measurement errors of the longitudinal processes. The posterior model space is explored through a Markov chain Monte Carlo (MCMC) sampler. The proposed methods are illustrated and compared to other approaches, the "naive" approach and the regression calibration, via simulations and by an application that investigates the relationship between obesity in adulthood and childhood growth curves.     

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.     

Goodness-of-fit methods for generalized linear mixed models

We develop graphical and numerical methods for checking the adequacy of generalized linear mixed models (GLMMs). These methods are based on the cumulative sums of residuals over covariates or predicted values of the response variable. Under the assumed model, the asymptotic distributions of these stochastic processes can be approximated by certain zero-mean Gaussian processes, whose realizations can be generated through Monte Carlo simulation. Each observed process can then be compared, both visually and analytically, to a number of realizations simulated from the null distribution. These comparisons enable one to assess objectively whether the observed residual patterns reflect model misspecification or random variation. The proposed methods are particularly useful for checking the functional form of a covariate or the link function. Extensive simulation studies show that the proposed goodness-of-fit tests have proper sizes and are sensitive to model misspecification. Applications to two medical studies lead to improved models.     

A note on conditional AIC for linear mixed-effects models

The conventional model selection criterion, the Akaike informationcriterion, AIC, has been applied to choose candidate modelsin mixed-effects models by the consideration of marginal likelihood.Vaida & Blanchard (2005) demonstrated that such a marginalAIC and its small sample correction are inappropriate when theresearch focus is on clusters. Correspondingly, these authorssuggested the use of conditional AIC. Their conditional AICis derived under the assumption that the variance-covariancematrix or scaled variance-covariance matrix of random effectsis known. This note provides a general conditional AIC but withoutthese strong assumptions. Simulation studies show that the proposedmethod is promising.     

Hazard ratio estimation for biomarker-calibrated dietary exposures

Uncertainty concerning the measurement error properties of self-reported diet has important implications for the reliability of nutritional epidemiology reports. Biomarkers based on the urinary recovery of expended nutrients can provide an objective measure of short-term nutrient consumption for certain nutrients and, when applied to a subset of a study cohort, can be used to calibrate corresponding self-report nutrient consumption assessments. A nonstandard measurement error model that makes provision for systematic error and subject-specific error, along with the usual independent random error, is needed for the self-report data. Three estimation procedures for hazard ratio (Cox model) parameters are extended for application to this more complex measurement error structure. These procedures are risk set regression calibration, conditional score, and nonparametric corrected score. An estimator for the cumulative baseline hazard function is also provided. The performance of each method is assessed in a simulation study. The methods are then applied to an example from the Women's Health Initiative Dietary Modification Trial.     

Marginally specified generalized linear mixed models: a robust approach

Longitudinal data modeling is complicated by the necessity to deal appropriately with the correlation between observations made on the same individual. Building on an earlier nonrobust version proposed by Heagerty (1999, Biometrics 55, 688-698), our robust marginally specified generalized linear mixed model (ROBMS-GLMM) provides an effective method for dealing with such data. This model is one of the first to allow both population-averaged and individual-specific inference. As well, it adopts the flexibility and interpretability of generalized linear mixed models for introducing dependence but builds a regression structure for the marginal mean, allowing valid application with time-dependent (exogenous) and time-independent covariates. These new estimators are obtained as solutions of a robustified likelihood equation involving Huber's least favorable distribution and a collection of weights. Huber's least favorable distribution produces estimates that are resistant to certain deviations from the random effects distributional assumptions. Innovative weighting strategies enable the ROBMS-GLMM to perform well when faced with outlying observations both in the response and covariates. We illustrate the methodology with an analysis of a prospective longitudinal study of laryngoscopic endotracheal intubation, a skill that numerous health-care professionals are expected to acquire. The principal goal of our research is to achieve robust inference in longitudinal analyses.     

The effect of miss-specified baseline characteristics on inference for longitudinal trends in linear mixed models

The main advantage of longitudinal studies is that they can distinguish changes over time within individuals (longitudinal effects) from differences among subjects at the start of the study (baseline characteristics, cross-sectional effects). Often, especially in observational studies, longitudinal trends are studied after correction for many potentially important baseline differences between subjects. We show that, in the context of linear mixed models, inference for longitudinal trends is in general biased if a wrong model for the baseline characteristics is used. However, we will argue that this bias is small in most practical situations and completely vanishes in the special case of a growth curve model for complete balanced data. In the latter case, inference for longitudinal trends is completely independent of additional baseline covariates that might have been omitted from the model.     

An estimation method for the semiparametric mixed effects model

A semiparametric mixed effects regression model is proposed for the analysis of clustered or longitudinal data with continuous, ordinal, or binary outcome. The common assumption of Gaussian random effects is relaxed by using a predictive recursion method (Newton and Zhang, 1999) to provide a nonparametric smooth density estimate. A new strategy is introduced to accelerate the algorithm. Parameter estimates are obtained by maximizing the marginal profile likelihood by Powell's conjugate direction search method. Monte Carlo results are presented to show that the method can improve the mean squared error of the fixed effects estimators when the random effects distribution is not Gaussian. The usefulness of visualizing the random effects density itself is illustrated in the analysis of data from the Wisconsin Sleep Survey. The proposed estimation procedure is computationally feasible for quite large data sets.     

Robustified maximum likelihood estimation in generalized partial linear mixed model for longitudinal data

Summary .  In this article, we study the robust estimation of both mean and variance components in generalized partial linear mixed models based on the construction of robustified likelihood function. Under some regularity conditions, the asymptotic properties of the proposed robust estimators are shown. Some simulations are carried out to investigate the performance of the proposed robust estimators. Just as expected, the proposed robust estimators perform better than those resulting from robust estimating equations involving conditional expectation like Sinha (2004, Journal of the American Statistical Association 99, 451–460) and Qin and Zhu (2007, Journal of Multivariate Analysis 98, 1658–1683). In the end, the proposed robust method is illustrated by the analysis of a real data set.     

A longitudinal measurement error model with a semicontinuous covariate

Covariate measurement error in regression is typically assumed to act in an additive or multiplicative manner on the true covariate value. However, such an assumption does not hold for the measurement error of sleep-disordered breathing (SDB) in the Wisconsin Sleep Cohort Study (WSCS). The true covariate is the severity of SDB, and the observed surrogate is the number of breathing pauses per unit time of sleep, which has a nonnegative semicontinuous distribution with a point mass at zero. We propose a latent variable measurement error model for the error structure in this situation and implement it in a linear mixed model. The estimation procedure is similar to regression calibration but involves a distributional assumption for the latent variable. Modeling and model-fitting strategies are explored and illustrated through an example from the WSCS.     

A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution

A popular way to represent clustered binary, count, or other data is via the generalized linear mixed model framework, which accommodates correlation through incorporation of random effects. A standard assumption is that the random effects follow a parametric family such as the normal distribution; however, this may be unrealistic or too restrictive to represent the data. We relax this assumption and require only that the distribution of random effects belong to a class of 'smooth' densities and approximate the density by the seminonparametric (SNP) approach of Gallant and Nychka (1987). This representation allows the density to be skewed, multi-modal, fat- or thin-tailed relative to the normal and includes the normal as a special case. Because an efficient algorithm to sample from an SNP density is available, we propose a Monte Carlo EM algorithm using a rejection sampling scheme to estimate the fixed parameters of the linear predictor, variance components and the SNP density. The approach is illustrated by application to a data set and via simulation.