Linear and nonlinear mixed-effects models for censored HIV viral loads using normal/independent distributions

HIV RNA viral load measures are often subjected to some upper and lower detection limits depending on the quantification assays. Hence, the responses are either left or right censored. Linear (and nonlinear) mixed-effects models (with modifications to accommodate censoring) are routinely used to analyze this type of data and are based on normality assumptions for the random terms. However, those analyses might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear (and nonlinear) models replacing the Gaussian assumptions for the random terms with normal/independent (NI) distributions. The NI is an attractive class of symmetric heavy-tailed densities that includes the normal, Student's-t, slash, and the contaminated normal distributions as special cases. The marginal likelihood is tractable (using approximations for nonlinear models) and can be used to develop Bayesian case-deletion influence diagnostics based on the Kullback-Leibler divergence. The newly developed procedures are illustrated with two HIV AIDS studies on viral loads that were initially analyzed using normal (censored) mixed-effects models, as well as simulations.     

A simple imputation method for longitudinal studies with non-ignorable non-responses

Missing data are a common problem in longitudinal studies in the health sciences. Motivated by data from the Muscatine Coronary Risk Factor (MCRF) study, a longitudinal study of obesity, we propose a simple imputation method for handling non-ignorable non-responses (i.e., when non-response is related to the specific values that should have been obtained) in longitudinal studies with either discrete or continuous outcomes. In the proposed approach, two regression models are specified; one for the marginal mean of the response, the other for the conditional mean of the response given non-response patterns. Statistical inference for the model parameters is based on the generalized estimating equations (GEE) approach. An appealing feature of the proposed method is that it can be readily implemented using existing, widely-available statistical software. The method is illustrated using longitudinal data on obesity from the MCRF study.     

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.     

Efficient estimation for patient-specific rates of disease progression using nonnormal linear mixed models

Summary .  This article presents a new class of nonnormal linear mixed models that provide an efficient estimation of subject-specific disease progression in the analysis of longitudinal data from the Modification of Diet in Renal Disease (MDRD) trial. This new analysis addresses the previously reported finding that the distribution of the random effect characterizing disease progression is negatively skewed. We assume a log-gamma distribution for the random effects and provide the maximum likelihood inference for the proposed nonnormal linear mixed model. We derive the predictive distribution of patient-specific disease progression rates, which demonstrates rather different individual progression profiles from those obtained from the normal linear mixed model analysis. To validate the adequacy of the log-gamma assumption versus the usual normality assumption for the random effects, we propose a lack-of-fit test that clearly indicates a better fit for the log-gamma modeling in the analysis of the MDRD data. The full maximum likelihood inference is also advantageous in dealing with the missing at random (MAR) type of dropouts encountered in the MDRD data.     

A mixed-effects regression model for longitudinal multivariate ordinal data

A mixed-effects item response theory model that allows for three-level multivariate ordinal outcomes and accommodates multiple random subject effects is proposed for analysis of multivariate ordinal outcomes in longitudinal studies. This model allows for the estimation of different item factor loadings (item discrimination parameters) for the multiple outcomes. The covariates in the model do not have to follow the proportional odds assumption and can be at any level. Assuming either a probit or logistic response function, maximum marginal likelihood estimation is proposed utilizing multidimensional Gauss-Hermite quadrature for integration of the random effects. An iterative Fisher scoring solution, which provides standard errors for all model parameters, is used. An analysis of a longitudinal substance use data set, where four items of substance use behavior (cigarette use, alcohol use, marijuana use, and getting drunk or high) are repeatedly measured over time, is used to illustrate application of the proposed model.     

Bayesian meta-analysis for longitudinal data models using multivariate mixture priors

We propose a class of longitudinal data models with random effects that generalizes currently used models in two important ways. First, the random-effects model is a flexible mixture of multivariate normals, accommodating population heterogeneity, outliers, and nonlinearity in the regression on subject-specific covariates. Second, the model includes a hierarchical extension to allow for meta-analysis over related studies. The random-effects distributions are decomposed into one part that is common across all related studies (common measure), and one part that is specific to each study and that captures the variability intrinsic between patients within the same study. Both the common measure and the study-specific measures are parameterized as mixture-of-normals models. We carry out inference using reversible jump posterior simulation to allow a random number of terms in the mixtures. The sampler takes advantage of the small number of entertained models. The motivating application is the analysis of two studies carried out by the Cancer and Leukemia Group B (CALGB). In both studies, we record for each patient white blood cell counts (WBC) over time to characterize the toxic effects of treatment. The WBCs are modeled through a nonlinear hierarchical model that gathers the information from both studies.     

Robust joint modeling of longitudinal measurements and time to event data using normal/independent distributions: A Bayesian approach

Joint modeling of longitudinal data and survival data has been used widely for analyzing AIDS clinical trials, where a biological marker such as CD4 count measurement can be an important predictor of survival. In most of these studies, a normal distribution is used for modeling longitudinal responses, which leads to vulnerable inference in the presence of outliers in longitudinal measurements. Powerful distributions for robust analysis are normal/independent distributions, which include univariate and multivariate versions of the Student's t, the slash and the contaminated normal distributions in addition to the normal. In this paper, a linear‐mixed effects model with normal/independent distribution for both random effects and residuals and Cox's model for survival time are used. For estimation, a Bayesian approach using Markov Chain Monte Carlo is adopted. Some simulation studies are performed for illustration of the proposed method. Also, the method is illustrated on a real AIDS data set and the best model is selected using some criteria.     

Marginalized transition shared random effects models for longitudinal binary data with nonignorable dropout

In longitudinal studies investigators frequently have to assess and address potential biases introduced by missing data. New methods are proposed for modeling longitudinal categorical data with nonignorable dropout using marginalized transition models and shared random effects models. Random effects are introduced for both serial dependence of outcomes and nonignorable missingness. Fisher‐scoring and Quasi–Newton algorithms are developed for parameter estimation. Methods are illustrated with a real dataset.     

Structural inference in transition measurement error models for longitudinal data

We propose a new class of models, transition measurement error models, to study the effects of covariates and the past responses on the current response in longitudinal studies when one of the covariates is measured with error. We show that the response variable conditional on the error-prone covariate follows a complex transition mixed effects model. The naive model obtained by ignoring the measurement error correctly specifies the transition part of the model, but misspecifies the covariate effect structure and ignores the random effects. We next study the asymptotic bias in naive estimator obtained by ignoring the measurement error for both continuous and discrete outcomes. We show that the naive estimator of the regression coefficient of the error-prone covariate is attenuated, while the naive estimators of the regression coefficients of the past responses are generally inflated. We then develop a structural modeling approach for parameter estimation using the maximum likelihood estimation method. In view of the multidimensional integration required by full maximum likelihood estimation, an EM algorithm is developed to calculate maximum likelihood estimators, in which Monte Carlo simulations are used to evaluate the conditional expectations in the E-step. We evaluate the performance of the proposed method through a simulation study and apply it to a longitudinal social support study for elderly women with heart disease. An additional simulation study shows that the Bayesian information criterion (BIC) performs well in choosing the correct transition orders of the models.     

Latent-Model Robustness in Joint Models for a Primary Endpoint and a Longitudinal Process

Summary .  Joint modeling of a primary response and a longitudinal process via shared random effects is widely used in many areas of application. Likelihood-based inference on joint models requires model specification of the random effects. Inappropriate model specification of random effects can compromise inference. We present methods to diagnose random effect model misspecification of the type that leads to biased inference on joint models. The methods are illustrated via application to simulated data, and by application to data from a study of bone mineral density in perimenopausal women and data from an HIV clinical trial.     

Pairwise fitting of mixed models for the joint modeling of multivariate longitudinal profiles

A mixed model is a flexible tool for joint modeling purposes, especially when the gathered data are unbalanced. However, computational problems due to the dimension of the joint covariance matrix of the random effects arise as soon as the number of outcomes and/or the number of used random effects per outcome increases. We propose a pairwise approach in which all possible bivariate models are fitted, and where inference follows from pseudo-likelihood arguments. The approach is applicable for linear, generalized linear, and nonlinear mixed models, or for combinations of these. The methodology will be illustrated for linear mixed models in the analysis of 22-dimensional, highly unbalanced, longitudinal profiles of hearing thresholds.     

Multivariate t linear mixed models for irregularly observed multiple repeated measures with missing outcomes

Missing outcomes or irregularly timed multivariate longitudinal data frequently occur in clinical trials or biomedical studies. The multivariate t linear mixed model (MtLMM) has been shown to be a robust approach to modeling multioutcome continuous repeated measures in the presence of outliers or heavy‐tailed noises. This paper presents a framework for fitting the MtLMM with an arbitrary missing data pattern embodied within multiple outcome variables recorded at irregular occasions. To address the serial correlation among the within‐subject errors, a damped exponential correlation structure is considered in the model. Under the missing at random mechanism, an efficient alternating expectation‐conditional maximization (AECM) algorithm is used to carry out estimation of parameters and imputation of missing values. The techniques for the estimation of random effects and the prediction of future responses are also investigated. Applications to an HIV‐AIDS study and a pregnancy study involving analysis of multivariate longitudinal data with missing outcomes as well as a simulation study have highlighted the superiority of MtLMMs on the provision of more adequate estimation, imputation and prediction performances.     

A semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data

Joint models for a time-to-event (e.g., survival) and a longitudinal response have generated considerable recent interest. The longitudinal data are assumed to follow a mixed effects model, and a proportional hazards model depending on the longitudinal random effects and other covariates is assumed for the survival endpoint. Interest may focus on inference on the longitudinal data process, which is informatively censored, or on the hazard relationship. Several methods for fitting such models have been proposed, most requiring a parametric distributional assumption (normality) on the random effects. A natural concern is sensitivity to violation of this assumption; moreover, a restrictive distributional assumption may obscure key features in the data. We investigate these issues through our proposal of a likelihood-based approach that requires only the assumption that the random effects have a smooth density. Implementation via the EM algorithm is described, and performance and the benefits for uncovering noteworthy features are illustrated by application to data from an HIV clinical trial and by simulation.     

A general approach for two-stage analysis of multilevel clustered non-Gaussian data

In this article, we propose a two-stage approach to modeling multilevel clustered non-Gaussian data with sufficiently large numbers of continuous measures per cluster. Such data are common in biological and medical studies utilizing monitoring or image-processing equipment. We consider a general class of hierarchical models that generalizes the model in the global two-stage (GTS) method for nonlinear mixed effects models by using any square-root-n-consistent and asymptotically normal estimators from stage 1 as pseudodata in the stage 2 model, and by extending the stage 2 model to accommodate random effects from multiple levels of clustering. The second-stage model is a standard linear mixed effects model with normal random effects, but the cluster-specific distributions, conditional on random effects, can be non-Gaussian. This methodology provides a flexible framework for modeling not only a location parameter but also other characteristics of conditional distributions that may be of specific interest. For estimation of the population parameters, we propose a conditional restricted maximum likelihood (CREML) approach and establish the asymptotic properties of the CREML estimators. The proposed general approach is illustrated using quartiles as cluster-specific parameters estimated in the first stage, and applied to the data example from a collagen fibril development study. We demonstrate using simulations that in samples with small numbers of independent clusters, the CREML estimators may perform better than conditional maximum likelihood estimators, which are a direct extension of the estimators from the GTS method.     

A penalized spline approach to functional mixed effects model analysis

In this article, we propose penalized spline (P-spline)-based methods for functional mixed effects models with varying coefficients. We decompose longitudinal outcomes as a sum of several terms: a population mean function, covariates with time-varying coefficients, functional subject-specific random effects, and residual measurement error processes. Using P-splines, we propose nonparametric estimation of the population mean function, varying coefficient, random subject-specific curves, and the associated covariance function that represents between-subject variation and the variance function of the residual measurement errors which represents within-subject variation. Proposed methods offer flexible estimation of both the population- and subject-level curves. In addition, decomposing variability of the outcomes as a between- and within-subject source is useful in identifying the dominant variance component therefore optimally model a covariance function. We use a likelihood-based method to select multiple smoothing parameters. Furthermore, we study the asymptotics of the baseline P-spline estimator with longitudinal data. We conduct simulation studies to investigate performance of the proposed methods. The benefit of the between- and within-subject covariance decomposition is illustrated through an analysis of Berkeley growth data, where we identified clearly distinct patterns of the between- and within-subject covariance functions of children's heights. We also apply the proposed methods to estimate the effect of antihypertensive treatment from the Framingham Heart Study data.     

Bayesian Inference in Semiparametric Mixed Models for Longitudinal Data

Summary .  We consider Bayesian inference in semiparametric mixed models (SPMMs) for longitudinal data. SPMMs are a class of models that use a nonparametric function to model a time effect, a parametric function to model other covariate effects, and parametric or nonparametric random effects to account for the within-subject correlation. We model the nonparametric function using a Bayesian formulation of a cubic smoothing spline, and the random effect distribution using a normal distribution and alternatively a nonparametric Dirichlet process (DP) prior. When the random effect distribution is assumed to be normal, we propose a uniform shrinkage prior (USP) for the variance components and the smoothing parameter. When the random effect distribution is modeled nonparametrically, we use a DP prior with a normal base measure and propose a USP for the hyperparameters of the DP base measure. We argue that the commonly assumed DP prior implies a nonzero mean of the random effect distribution, even when a base measure with mean zero is specified. This implies weak identifiability for the fixed effects, and can therefore lead to biased estimators and poor inference for the regression coefficients and the spline estimator of the nonparametric function. We propose an adjustment using a postprocessing technique. We show that under mild conditions the posterior is proper under the proposed USP, a flat prior for the fixed effect parameters, and an improper prior for the residual variance. We illustrate the proposed approach using a longitudinal hormone dataset, and carry out extensive simulation studies to compare its finite sample performance with existing methods.     

Inference in spline-based models for multiple time-to-event data, with applications to a breast cancer prevention trial

As part of the National Surgical Adjuvant Breast and Bowel Project, a controlled clinical trial known as the Breast Cancer Prevention Trial (BCPT) was conducted to assess the effectiveness of tamoxifen as a preventive agent for breast cancer. In addition to the incidence of breast cancer, data were collected on several other, possibly adverse, outcomes, such as invasive endometrial cancer, ischemic heart disease, transient ischemic attack, deep vein thrombosis and/or pulmonary embolism. In this article, we present results from an illustrative analysis of the BCPT data, based on a new modeling technique, to assess the effectiveness of the drug tamoxifen as a preventive agent for breast cancer. We extended the flexible model of Gray (1994, Spline-based test in survival analysis, Biometrics 50, 640-652) to allow inference on multiple time-to-event outcomes in the style of the marginal modeling setup of Wei, Lin, and Weissfeld (1989, Regression analysis of multivariate incomplete failure time data by modeling marginal distributions, Journal of the American Statistical Association 84, 1065-1073). This proposed model makes inference possible for multiple time-to-event data while allowing for greater flexibility in modeling the effects of prognostic factors with nonlinear exposure-response relationships. Results from simulation studies on the small-sample properties of the asymptotic tests will also be presented.     

A Bayesian approach for joint modeling of cluster size and subunit-specific outcomes

In applications that involve clustered data, such as longitudinal studies and developmental toxicity experiments, the number of subunits within a cluster is often correlated with outcomes measured on the individual subunits. Analyses that ignore this dependency can produce biased inferences. This article proposes a Bayesian framework for jointly modeling cluster size and multiple categorical and continuous outcomes measured on each subunit. We use a continuation ratio probit model for the cluster size and underlying normal regression models for each of the subunit-specific outcomes. Dependency between cluster size and the different outcomes is accommodated through a latent variable structure. The form of the model facilitates posterior computation via a simple and computationally efficient Gibbs sampler. The approach is illustrated with an application to developmental toxicity data, and other applications, to joint modeling of longitudinal and event time data, are discussed.     

Hierarchical Bayesian modeling of heterogeneous cluster‐ and subject‐level associations between continuous and binary outcomes in dairy production

The augmentation of categorical outcomes with underlying Gaussian variables in bivariate generalized mixed effects models has facilitated the joint modeling of continuous and binary response variables. These models typically assume that random effects and residual effects (co)variances are homogeneous across all clusters and subjects, respectively. Motivated by conflicting evidence about the association between performance outcomes in dairy production systems, we consider the situation where these (co)variance parameters may themselves be functions of systematic and/or random effects. We present a hierarchical Bayesian extension of bivariate generalized linear models whereby functions of the (co)variance matrices are specified as linear combinations of fixed and random effects following a square‐root‐free Cholesky reparameterization that ensures necessary positive semidefinite constraints. We test the proposed model by simulation and apply it to the analysis of a dairy cattle data set in which the random herd‐level and residual cow‐level effects (co)variances between a continuous production trait and binary reproduction trait are modeled as functions of fixed management effects and random cluster effects.     

Diagnosis of random-effect model misspecification in generalized linear mixed models for binary response

Summary .  Generalized linear mixed models (GLMMs) are widely used in the analysis of clustered data. However, the validity of likelihood-based inference in such analyses can be greatly affected by the assumed model for the random effects. We propose a diagnostic method for random-effect model misspecification in GLMMs for clustered binary response. We provide a theoretical justification of the proposed method and investigate its finite sample performance via simulation. The proposed method is applied to data from a longitudinal respiratory infection study.