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.     

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.     

Multiple imputation methods for handling incomplete longitudinal and clustered data where the target analysis is a linear mixed effects model

Multiple imputation (MI) is increasingly popular for handling multivariate missing data. Two general approaches are available in standard computer packages: MI based on the posterior distribution of incomplete variables under a multivariate (joint) model, and fully conditional specification (FCS), which imputes missing values using univariate conditional distributions for each incomplete variable given all the others, cycling iteratively through the univariate imputation models. In the context of longitudinal or clustered data, it is not clear whether these approaches result in consistent estimates of regression coefficient and variance component parameters when the analysis model of interest is a linear mixed effects model (LMM) that includes both random intercepts and slopes with either covariates or both covariates and outcome contain missing information. In the current paper, we compared the performance of seven different MI methods for handling missing values in longitudinal and clustered data in the context of fitting LMMs with both random intercepts and slopes. We study the theoretical compatibility between specific imputation models fitted under each of these approaches and the LMM, and also conduct simulation studies in both the longitudinal and clustered data settings. Simulations were motivated by analyses of the association between body mass index (BMI) and quality of life (QoL) in the Longitudinal Study of Australian Children (LSAC). Our findings showed that the relative performance of MI methods vary according to whether the incomplete covariate has fixed or random effects and whether there is missingnesss in the outcome variable. We showed that compatible imputation and analysis models resulted in consistent estimation of both regression parameters and variance components via simulation. We illustrate our findings with the analysis of LSAC data.     

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.     

A comparison of strategies for Markov chain Monte Carlo computation in quantitative genetics

In quantitative genetics, Markov chain Monte Carlo (MCMC) methods are indispensable for statistical inference in non-standard models like generalized linear models with genetic random effects or models with genetically structured variance heterogeneity. A particular challenge for MCMC applications in quantitative genetics is to obtain efficient updates of the high-dimensional vectors of genetic random effects and the associated covariance parameters. We discuss various strategies to approach this problem including reparameterization, Langevin-Hastings updates, and updates based on normal approximations. The methods are compared in applications to Bayesian inference for three data sets using a model with genetically structured variance heterogeneity.     

Semiparametric estimation of treatment effect in a pretest-posttest study

Inference on treatment effects in a pretest-posttest study is a routine objective in medicine, public health, and other fields. A number of approaches have been advocated. We take a semiparametric perspective, making no assumptions about the distributions of baseline and posttest responses. By representing the situation in terms of counterfactual random variables, we exploit recent developments in the literature on missing data and causal inference, to derive the class of all consistent treatment effect estimators, identify the most efficient such estimator, and outline strategies for implementation of estimators that may improve on popular methods. We demonstrate the methods and their properties via simulation and by application to a data set from an HIV clinical trial.     

Generalized hierarchical multivariate CAR models for areal data

In the fields of medicine and public health, a common application of areal data models is the study of geographical patterns of disease. When we have several measurements recorded at each spatial location (for example, information on p>/= 2 diseases from the same population groups or regions), we need to consider multivariate areal data models in order to handle the dependence among the multivariate components as well as the spatial dependence between sites. In this article, we propose a flexible new class of generalized multivariate conditionally autoregressive (GMCAR) models for areal data, and show how it enriches the MCAR class. Our approach differs from earlier ones in that it directly specifies the joint distribution for a multivariate Markov random field (MRF) through the specification of simpler conditional and marginal models. This in turn leads to a significant reduction in the computational burden in hierarchical spatial random effect modeling, where posterior summaries are computed using Markov chain Monte Carlo (MCMC). We compare our approach with existing MCAR models in the literature via simulation, using average mean square error (AMSE) and a convenient hierarchical model selection criterion, the deviance information criterion (DIC; Spiegelhalter et al., 2002, Journal of the Royal Statistical Society, Series B64, 583-639). Finally, we offer a real-data application of our proposed GMCAR approach that models lung and esophagus cancer death rates during 1991-1998 in Minnesota counties.     

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.     

Modeling disease incidence data with spatial and spatio temporal dirichlet process mixtures

Disease incidence or mortality data are typically available as rates or counts for specified regions, collected over time. We propose Bayesian nonparametric spatial modeling approaches to analyze such data. We develop a hierarchical specification using spatial random effects modeled with a Dirichlet process prior. The Dirichlet process is centered around a multivariate normal distribution. This latter distribution arises from a log-Gaussian process model that provides a latent incidence rate surface, followed by block averaging to the areal units determined by the regions in the study. With regard to the resulting posterior predictive inference, the modeling approach is shown to be equivalent to an approach based on block averaging of a spatial Dirichlet process to obtain a prior probability model for the finite dimensional distribution of the spatial random effects. We introduce a dynamic formulation for the spatial random effects to extend the model to spatio-temporal settings. Posterior inference is implemented through Gibbs sampling. We illustrate the methodology with simulated data as well as with a data set on lung cancer incidences for all 88 counties in the state of Ohio over an observation period of 21 years.     

Hierarchical Bayesian modeling of spatially correlated health service outcome and utilization rates

We present Bayesian hierarchical spatial models for spatially correlated small-area health service outcome and utilization rates, with a particular emphasis on the estimation of both measured and unmeasured or unknown covariate effects. This Bayesian hierarchical model framework enables simultaneous modeling of fixed covariate effects and random residual effects. The random effects are modeled via Bayesian prior specifications reflecting spatial heterogeneity globally and relative homogeneity among neighboring areas. The model inference is implemented using Markov chain Monte Carlo methods. Specifically, a hybrid Markov chain Monte Carlo algorithm (Neal, 1995, Bayesian Learning for Neural Networks; Gustafson, MacNab, and Wen, 2003, Statistics and Computing, to appear) is used for posterior sampling of the random effects. To illustrate relevant problems, methods, and techniques, we present an analysis of regional variation in intraventricular hemorrhage incidence rates among neonatal intensive care unit patients across Canada.     

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.     

Inference on the strength of balancing selection for epistatically interacting loci

Existing inference methods for estimating the strength of balancing selection in multi-locus genotypes rely on the assumption that there are no epistatic interactions between loci. Complex systems in which balancing selection is prevalent, such as sets of human immune system genes, are known to contain components that interact epistatically. Therefore, current methods may not produce reliable inference on the strength of selection at these loci. In this paper, we address this problem by presenting statistical methods that can account for epistatic interactions in making inference about balancing selection. A theoretical result due to Fearnhead (2006) is used to build a multi-locus Wright-Fisher model of balancing selection, allowing for epistatic interactions among loci. Antagonistic and synergistic types of interactions are examined. The joint posterior distribution of the selection and mutation parameters is sampled by Markov chain Monte Carlo methods, and the plausibility of models is assessed via Bayes factors. As a component of the inference process, an algorithm to generate multi-locus allele frequencies under balancing selection models with epistasis is also presented. Recent evidence on interactions among a set of human immune system genes is introduced as a motivating biological system for the epistatic model, and data on these genes are used to demonstrate the methods.     

Modeling individual effects in the Cormack-Jolly-Seber model: a state-space formulation

Summary .   In population and evolutionary biology, there exists considerable interest in individual heterogeneity in parameters of demographic models for open populations. However, flexible and practical solutions to the development of such models have proven to be elusive. In this article, I provide a state-space formulation of open population capture–recapture models with individual effects. The state-space formulation provides a generic and flexible framework for modeling and inference in models with individual effects, and it yields a practical means of estimation in these complex problems via contemporary methods of Markov chain Monte Carlo. A straightforward implementation can be achieved in the software package WinBUGS . I provide an analysis of a simple model with constant parameter detection and survival probability parameters. A second example is based on data from a 7-year study of European dippers, in which a model with year and individual effects is fitted.     

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.     

Structured additive regression for categorical space-time data: a mixed model approach

Motivated by a space-time study on forest health with damage state of trees as the response, we propose a general class of structured additive regression models for categorical responses, allowing for a flexible semiparametric predictor. Nonlinear effects of continuous covariates, time trends, and interactions between continuous covariates are modeled by penalized splines. Spatial effects can be estimated based on Markov random fields, Gaussian random fields, or two-dimensional penalized splines. We present our approach from a Bayesian perspective, with inference based on a categorical linear mixed model representation. The resulting empirical Bayes method is closely related to penalized likelihood estimation in a frequentist setting. Variance components, corresponding to inverse smoothing parameters, are estimated using (approximate) restricted maximum likelihood. In simulation studies we investigate the performance of different choices for the spatial effect, compare the empirical Bayes approach to competing methodology, and study the bias of mixed model estimates. As an application we analyze data from the forest health survey.     

A characterization of missingness at random in a generalized shared‐parameter joint modeling framework for longitudinal and time‐to‐event data,and sensitivity analysis

We consider a conceptual correspondence between the missing data setting, and joint modeling of longitudinal and time‐to‐event outcomes. Based on this, we formulate an extended shared random effects joint model. Based on this, we provide a characterization of missing at random, which is in line with that in the missing data setting. The ideas are illustrated using data from a study on liver cirrhosis, contrasting the new framework with conventional joint models.     

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.     

QUANTITATIVE GENETIC MODELING AND INFERENCE IN THE PRESENCE OF NONIGNORABLE MISSING DATA

Natural selection is typically exerted at some specific life stages. If natural selection takes place before a trait can be measured, using conventional models can cause wrong inference about population parameters. When the missing data process relates to the trait of interest, a valid inference requires explicit modeling of the missing process. We propose a joint modeling approach, a shared parameter model, to account for nonrandom missing data. It consists of an animal model for the phenotypic data and a logistic model for the missing process, linked by the additive genetic effects. A Bayesian approach is taken and inference is made using integrated nested Laplace approximations. From a simulation study we find that wrongly assuming that missing data are missing at random can result in severely biased estimates of additive genetic variance. Using real data from a wild population of Swiss barn owls Tyto alba, our model indicates that the missing individuals would display large black spots; and we conclude that genes affecting this trait are already under selection before it is expressed. Our model is a tool to correctly estimate the magnitude of both natural selection and additive genetic variance.     

An application of a mixed-effects location scale model for analysis of Ecological Momentary Assessment (EMA) data

Summary .   For longitudinal data, mixed models include random subject effects to indicate how subjects influence their responses over repeated assessments. The error variance and the variance of the random effects are usually considered to be homogeneous. These variance terms characterize the within-subjects (i.e., error variance) and between-subjects (i.e., random-effects variance) variation in the data. In studies using ecological momentary assessment (EMA), up to 30 or 40 observations are often obtained for each subject, and interest frequently centers around changes in the variances, both within and between subjects. In this article, we focus on an adolescent smoking study using EMA where interest is on characterizing changes in mood variation. We describe how covariates can influence the mood variances, and also extend the standard mixed model by adding a subject-level random effect to the within-subject variance specification. This permits subjects to have influence on the mean, or location, and variability, or (square of the) scale, of their mood responses. Additionally, we allow the location and scale random effects to be correlated. These mixed-effects location scale models have useful applications in many research areas where interest centers on the joint modeling of the mean and variance structure.     

Permutation inference methods for multivariate meta-analysis

Multivariate meta-analysis is gaining prominence in evidence synthesis research because it enables simultaneous synthesis of multiple correlated outcome data, and random-effects models have generally been used for addressing between-studies heterogeneities. However, coverage probabilities of confidence regions or intervals for standard inference methods for random-effects models (eg, restricted maximum likelihood estimation) cannot retain their nominal confidence levels in general, especially when the number of synthesized studies is small because their validities depend on large sample approximations. In this article, we provide permutation-based inference methods that enable exact joint inferences for average outcome measures without large sample approximations. We also provide accurate marginal inference methods under general settings of multivariate meta-analyses. We propose effective approaches for permutation inferences using optimal weighting based on the efficient score statistic. The effectiveness of the proposed methods is illustrated via applications to bivariate meta-analyses of diagnostic accuracy studies for airway eosinophilia in asthma and a network meta-analysis for antihypertensive drugs on incident diabetes, as well as through simulation experiments. In numerical evaluations performed via simulations, our methods generally provided accurate confidence regions or intervals under a broad range of settings, whereas the current standard inference methods exhibited serious undercoverage properties.