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.     

Conditional estimation for generalized linear models when covariates are subject-specific parameters in a mixed model for longitudinal measurements

The relationship between a primary endpoint and features of longitudinal profiles of a continuous response is often of interest, and a relevant framework is that of a generalized linear model with covariates that are subject-specific random effects in a linear mixed model for the longitudinal measurements. Naive implementation by imputing subject-specific effects from individual regression fits yields biased inference, and several methods for reducing this bias have been proposed. These require a parametric (normality) assumption on the random effects, which may be unrealistic. Adapting a strategy of Stefanski and Carroll (1987, Biometrika74, 703-716), we propose estimators for the generalized linear model parameters that require no assumptions on the random effects and yield consistent inference regardless of the true distribution. The methods are illustrated via simulation and by application to a study of bone mineral density in women transitioning to menopause.     

Repeated probit regression when covariates are measured with error

This paper develops a model for repeated binary regression when a covariate is measured with error. The model allows for estimating the effect of the true value of the covariate on a repeated binary response. The choice of a probit link for the effect of the error-free covariate, coupled with normal measurement error for the error-free covariate, results in a probit model after integrating over the measurement error distribution. We propose a two-stage estimation procedure where, in the first stage, a linear mixed model is used to fit the repeated covariate. In the second stage, a model for the correlated binary responses conditional on the linear mixed model estimates is fit to the repeated binary data using generalized estimating equations. The approach is demonstrated using nutrient safety data from the Diet Intervention of School Age Children (DISC) study.     

Logistic regression when covariates are random effects from a non‐linear mixed model

In many studies, the association of longitudinal measurements of a continuous response and a binary outcome are often of interest. A convenient framework for this type of problems is the joint model, which is formulated to investigate the association between a binary outcome and features of longitudinal measurements through a common set of latent random effects. The joint model, which is the focus of this article, is a logistic regression model with covariates defined as the individual‐specific random effects in a non‐linear mixed‐effects model (NLMEM) for the longitudinal measurements. We discuss different estimation procedures, which include two‐stage, best linear unbiased predictors, and various numerical integration techniques. The proposed methods are illustrated using a real data set where the objective is to study the association between longitudinal hormone levels and the pregnancy outcome in a group of young women. The numerical performance of the estimating methods is also evaluated by means of simulation.     

Random effects logistic regression analysis with auxiliary covariates

We study a semiparametric estimation method for the random effects logistic regression when there is auxiliary covariate information about the main exposure variable. We extend the semiparametric estimator of Pepe and Fleming (1991, Journal of the American Statistical Association 86, 108-113) to the random effects model using the best linear unbiased prediction approach of Henderson (1975, Biometrics 31, 423-448). The method can be used to handle the missing covariate or mismeasured covariate data problems in a variety of real applications. Simulation study results show that the proposed method outperforms the existing methods. We analyzed a data set from the Collaborative Perinatal Project using the proposed method and found that the use of DDT increases the risk of preterm births among U.S. children.     

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 of random regression parameters

The problem of the best linear unbiased estimation (BLUE) of random regression parameters is considered. It is proved that increasing informations about the mean value of the parameters both extend the class of estimable linear functionals and improve on the estimation. In all investigated cases the uniqueness of BLUE is proved. In the case of known mean values the BLUE is shown to be numerically equivalent with the MMSEE almost everywhere. A numerical example shows the improvements of BLUE due to increasing informations about the mean values of the parameters.     

Multilevel models for survival analysis with random effects

A method for modeling survival data with multilevel clustering is described. The Cox partial likelihood is incorporated into the generalized linear mixed model (GLMM) methodology. Parameter estimation is achieved by maximizing a log likelihood analogous to the likelihood associated with the best linear unbiased prediction (BLUP) at the initial step of estimation and is extended to obtain residual maximum likelihood (REML) estimators of the variance component. Estimating equations for a three-level hierarchical survival model are developed in detail, and such a model is applied to analyze a set of chronic granulomatous disease (CGD) data on recurrent infections as an illustration with both hospital and patient effects being considered as random. Only the latter gives a significant contribution. A simulation study is carried out to evaluate the performance of the REML estimators. Further extension of the estimation procedure to models with an arbitrary number of levels is also discussed.     

Monte Carlo EM for missing covariates in parametric regression models

We propose a method for estimating parameters for general parametric regression models with an arbitrary number of missing covariates. We allow any pattern of missing data and assume that the missing data mechanism is ignorable throughout. When the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). We extend this method to continuous or mixed categorical and continuous covariates, and for arbitrary parametric regression models, by adapting a Monte Carlo version of the EM algorithm as discussed by Wei and Tanner (1990, Journal of the American Statistical Association 85, 699-704). In addition, we discuss the Gibbs sampler for sampling from the conditional distribution of the missing covariates given the observed data and show that the appropriate complete conditionals are log-concave. The log-concavity property of the conditional distributions will facilitate a straightforward implementation of the Gibbs sampler via the adaptive rejection algorithm of Gilks and Wild (1992, Applied Statistics 41, 337-348). We assume the model for the response given the covariates is an arbitrary parametric regression model, such as a generalized linear model, a parametric survival model, or a nonlinear model. We model the marginal distribution of the covariates as a product of one-dimensional conditional distributions. This allows us a great deal of flexibility in modeling the distribution of the covariates and reduces the number of nuisance parameters that are introduced in the E-step. We present examples involving both simulated and real data.     

Modeling random effects for censored data by a multivariate normal regression model

A normal distribution regression model with a frailty-like factor to account for statistical dependence between the observed survival times is introduced. This model, as opposed to standard hazard-based frailty models, has survival times that, conditional on the shared random effect, have an accelerated failure time representation. The dependence properties of this model are discussed and maximum likelihood estimation of the model's parameters is considered. A number of examples are considered to illustrate the approach. The estimated degree of dependence is comparable to other models, but the present approach has the advantage that the interpretation of the random effect is simpler than in the frailty model.     

A zero‐augmented generalized gamma regression calibration to adjust for covariate measurement error: A case of an episodically consumed dietary intake

Measurement error in exposure variables is a serious impediment in epidemiological studies that relate exposures to health outcomes. In nutritional studies, interest could be in the association between long‐term dietary intake and disease occurrence. Long‐term intake is usually assessed with food frequency questionnaire (FFQ), which is prone to recall bias. Measurement error in FFQ‐reported intakes leads to bias in parameter estimate that quantifies the association. To adjust for bias in the association, a calibration study is required to obtain unbiased intake measurements using a short‐term instrument such as 24‐hour recall (24HR). The 24HR intakes are used as response in regression calibration to adjust for bias in the association. For foods not consumed daily, 24HR‐reported intakes are usually characterized by excess zeroes, right skewness, and heteroscedasticity posing serious challenge in regression calibration modeling. We proposed a zero‐augmented calibration model to adjust for measurement error in reported intake, while handling excess zeroes, skewness, and heteroscedasticity simultaneously without transforming 24HR intake values. We compared the proposed calibration method with the standard method and with methods that ignore measurement error by estimating long‐term intake with 24HR and FFQ‐reported intakes. The comparison was done in real and simulated datasets. With the 24HR, the mean increase in mercury level per ounce fish intake was about 0.4; with the FFQ intake, the increase was about 1.2. With both calibration methods, the mean increase was about 2.0. Similar trend was observed in the simulation study. In conclusion, the proposed calibration method performs at least as good as the standard method.     

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.