Skew‐normal/independent linear mixed models for censored responses with applications to HIV viral loads

Often in biomedical studies, the routine use of linear mixed‐effects models (based on Gaussian assumptions) can be questionable when the longitudinal responses are skewed in nature. Skew‐normal/elliptical models are widely used in those situations. Often, those skewed responses might also be subjected to some upper and lower quantification limits (QLs; viz., longitudinal viral‐load measures in HIV studies), beyond which they are not measurable. In this paper, we develop a Bayesian analysis of censored linear mixed models replacing the Gaussian assumptions with skew‐normal/independent (SNI) distributions. The SNI is an attractive class of asymmetric heavy‐tailed distributions that includes the skew‐normal, skew‐t, skew‐slash, and skew‐contaminated normal distributions as special cases. The proposed model provides flexibility in capturing the effects of skewness and heavy tail for responses that are either left‐ or right‐censored. For our analysis, we adopt a Bayesian framework and develop a Markov chain Monte Carlo algorithm to carry out the posterior analyses. The marginal likelihood is tractable, and utilized to compute not only some Bayesian model selection measures but also case‐deletion influence diagnostics based on the Kullback–Leibler divergence. The newly developed procedures are illustrated with a simulation study as well as an HIV case study involving analysis of longitudinal viral loads.     

Robust Bayesian hierarchical model using normal/independent distributions

The multilevel item response theory (MLIRT) models have been increasingly used in longitudinal clinical studies that collect multiple outcomes. The MLIRT models account for all the information from multiple longitudinal outcomes of mixed types (e.g., continuous, binary, and ordinal) and can provide valid inference for the overall treatment effects. However, the continuous outcomes and the random effects in the MLIRT models are often assumed to be normally distributed. The normality assumption can sometimes be unrealistic and thus may produce misleading results. The normal/independent (NI) distributions have been increasingly used to handle the outlier and heavy tail problems in order to produce robust inference. In this article, we developed a Bayesian approach that implemented the NI distributions on both continuous outcomes and random effects in the MLIRT models and discussed different strategies of implementing the NI distributions. Extensive simulation studies were conducted to demonstrate the advantage of our proposed models, which provided parameter estimates with smaller bias and more reasonable coverage probabilities. Our proposed models were applied to a motivating Parkinson's disease study, the DATATOP study, to investigate the effect of deprenyl in slowing down the disease progression.     

Mixed-model analysis of a censored normal distribution with reference to animal breeding

A mixed-model procedure for analysis of censored data assuming a multivariate normal distribution is described. A Bayesian framework is adopted which allows for estimation of fixed effects and variance components and prediction of random effects when records are left-censored. The procedure can be extended to right- and two-tailed censoring. The model employed is a generalized linear model, and the estimation equations resemble those arising in analysis of multivariate normal or categorical data with threshold models. Estimates of variance components are obtained using expressions similar to those employed in the EM algorithm for restricted maximum likelihood (REML) estimation under normality.     

Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system

HIV dynamics studies have significantly contributed to the understanding of HIV infection and antiviral treatment strategies. But most studies are limited to short-term viral dynamics due to the difficulty of establishing a relationship of antiviral response with multiple treatment factors such as drug exposure and drug susceptibility during long-term treatment. In this article, a mechanism-based dynamic model is proposed for characterizing long-term viral dynamics with antiretroviral therapy, described by a set of nonlinear differential equations without closed-form solutions. In this model we directly incorporate drug concentration, adherence, and drug susceptibility into a function of treatment efficacy, defined as an inhibition rate of virus replication. We investigate a Bayesian approach under the framework of hierarchical Bayesian (mixed-effects) models for estimating unknown dynamic parameters. In particular, interest focuses on estimating individual dynamic parameters. The proposed methods not only help to alleviate the difficulty in parameter identifiability, but also flexibly deal with sparse and unbalanced longitudinal data from individual subjects. For illustration purposes, we present one simulation example to implement the proposed approach and apply the methodology to a data set from an AIDS clinical trial. The basic concept of the longitudinal HIV dynamic systems and the proposed methodologies are generally applicable to any other biomedical dynamic systems.     

A Bayesian Approach to Joint Mixed‐Effects Models with a Skew‐Normal Distribution and Measurement Errors in Covariates

Summary In recent years, nonlinear mixed‐effects (NLME) models have been proposed for modeling complex longitudinal data. Covariates are usually introduced in the models to partially explain intersubject variations. However, one often assumes that both model random error and random effects are normally distributed, which may not always give reliable results if the data exhibit skewness. Moreover, some covariates such as CD4 cell count may be often measured with substantial errors. In this article, we address these issues simultaneously by jointly modeling the response and covariate processes using a Bayesian approach to NLME models with covariate measurement errors and a skew‐normal distribution. A real data example is offered to illustrate the methodologies by comparing various potential models with different distribution specifications. It is showed that the models with skew‐normality assumption may provide more reasonable results if the data exhibit skewness and the results may be important for HIV/AIDS studies in providing quantitative guidance to better understand the virologic responses to antiretroviral treatment.     

Bayesian nonparametric meta-analysis using Polya tree mixture models

Summary .   A common goal in meta-analysis is estimation of a single effect measure using data from several studies that are each designed to address the same scientific inquiry. Because studies are typically conducted in geographically disperse locations, recent developments in the statistical analysis of meta-analytic data involve the use of random effects models that account for study-to-study variability attributable to differences in environments, demographics, genetics, and other sources that lead to heterogeneity in populations. Stemming from asymptotic theory, study-specific summary statistics are modeled according to normal distributions with means representing latent true effect measures. A parametric approach subsequently models these latent measures using a normal distribution, which is strictly a convenient modeling assumption absent of theoretical justification. To eliminate the influence of overly restrictive parametric models on inferences, we consider a broader class of random effects distributions. We develop a novel hierarchical Bayesian nonparametric Polya tree mixture (PTM) model. We present methodology for testing the PTM versus a normal random effects model. These methods provide researchers a straightforward approach for conducting a sensitivity analysis of the normality assumption for random effects. An application involving meta-analysis of epidemiologic studies designed to characterize the association between alcohol consumption and breast cancer is presented, which together with results from simulated data highlight the performance of PTMs in the presence of nonnormality of effect measures in the source population.     

Bayesian nonparametric hierarchical modeling

In biomedical research, hierarchical models are very widely used to accommodate dependence in multivariate and longitudinal data and for borrowing of information across data from different sources. A primary concern in hierarchical modeling is sensitivity to parametric assumptions, such as linearity and normality of the random effects. Parametric assumptions on latent variable distributions can be challenging to check and are typically unwarranted, given available prior knowledge. This article reviews some recent developments in Bayesian nonparametric methods motivated by complex, multivariate and functional data collected in biomedical studies. The author provides a brief review of flexible parametric approaches relying on finite mixtures and latent class modeling. Dirichlet process mixture models are motivated by the need to generalize these approaches to avoid assuming a fixed finite number of classes. Focusing on an epidemiology application, the author illustrates the practical utility and potential of nonparametric Bayes methods.     

A mixture model for longitudinal data with application to assessment of noncompliance

In clinical trials of a self-administered drug, repeated measures of a laboratory marker, which is affected by study medication and collected in all treatment arms, can provide valuable information on population and individual summaries of compliance. In this paper, we introduce a general finite mixture of nonlinear hierarchical models that allows estimates of component membership probabilities and random effect distributions for longitudinal data arising from multiple subpopulations, such as from noncomplying and complying subgroups in clinical trials. We outline a sampling strategy for fitting these models, which consists of a sequence of Gibbs, Metropolis-Hastings, and reversible jump steps, where the latter is required for switching between component models of different dimensions. Our model is applied to identify noncomplying subjects in the placebo arm of a clinical trial assessing the effectiveness of zidovudine (AZT) in the treatment of patients with HIV, where noncompliance was defined as initiation of AZT during the trial without the investigators' knowledge. We fit a hierarchical nonlinear change-point model for increases in the marker MCV (mean corpuscular volume of erythrocytes) for subjects who noncomply and a constant mean random effects model for those who comply. As part of our fully Bayesian analysis, we assess the sensitivity of conclusions to prior and modeling assumptions and demonstrate how external information and covariates can be incorporated to distinguish subgroups.     

Cumulative disease progression models for cross‐sectional data: A review and comparison

A better understanding of disease progression is beneficial for early diagnosis and appropriate individual therapy. Many different approaches for statistical modelling of cumulative disease progression have been proposed in the literature, including simple path models up to complex restricted Bayesian networks. Important fields of application are diseases such as cancer and HIV. Tumour progression is measured by means of chromosome aberrations, whereas people infected with HIV develop drug resistances because of genetic changes of the HI‐virus. These two very different diseases have typical courses of disease progression, which can be modelled partly by consecutive and partly by independent steps. This paper gives an overview of the different progression models and points out their advantages and drawbacks. Different models are compared via simulations to analyse how they work if some of their assumptions are violated. In a simulation study, we evaluate how models perform in terms of fitting induced multivariate probability distributions and topological relationships. We often find that the true model class used for generating data is outperformed by either a less or a more complex model class. The more flexible conjunctive Bayesian networks can be used to fit oncogenetic trees, whereas mixtures of oncogenetic trees with three tree components can be well fitted by mixture models with only two tree components.     

Bayesian analysis of multivariate mixed models for a prospective cohort study using skew‐elliptical distributions

Classical multivariate mixed models that acknowledge the correlation of patients through the incorporation of normal error terms are widely used in cohort studies. Violation of the normality assumption can make the statistical inference vague. In this paper, we propose a Bayesian parametric approach by relaxing this assumption and substituting some flexible distributions in fitting multivariate mixed models. This strategy allows for the skewness and the heavy tails of error‐term distributions and thus makes inferences robust to the violation. This approach uses flexible skew‐elliptical distributions, including skewed, fat, or thin‐tailed distributions, and imposes the normal model as a special case. We use real data obtained from a prospective cohort study on the low back pain to illustrate the usefulness of our proposed approach.     

Bayesian experimental design for nonlinear mixed-effects models with application to HIV dynamics

Bayesian experimental design is investigated for Bayesian analysis of nonlinear mixed-effects models. Existence of the posterior risk for parameter estimation is shown. When the same prior distribution is used for both design and inference, existence of the preposterior risk for design is also proven. If the prior distribution used in design is different from that used for inference, sufficient conditions are established for existence of the preposterior risk for design. A case study of design for an experiment in population HIV dynamics is provided.     

An Approximate EM Algorithm for Nonlinear Mixed Effects Models

In this article, we construct an approximate EM algorithm to estimate the parameters of a nonlinear mixed effects model. The iterative procedure can be viewed as an iterative method of moments procedure for estimating the variance components and an iterative reweighted least squares estimates for estimating the fixed effects. Therefore, it is valid without the normality assumptions on the random components. A computationally simple method of moments estimates of the model parameters are used as the starting values for our iterative procedure. A simulation study was conducted to compare the performances of the proposed procedure with the procedure proposed by Lindstrom and Bates (1990) for some normal models and nonnormal models.     

Goodness-of-fit in generalized nonlinear mixed-effects models

In recent years, generalized linear and nonlinear mixed-effects models have proved to be powerful tools for the analysis of unbalanced longitudinal data. To date, much of the work has focused on various methods for estimating and comparing the parameters of mixed-effects models. Very little work has been done in the area of model selection and goodness-of-fit, particularly with respect to the assumed variance-covariance structure. In this paper, we present a goodness-of-fit statistic which can be used in a manner similar to the R2 criterion in linear regression for assessing the adequacy of an assumed mean and variance-covariance structure. In addition, we introduce an approximate pseudo-likelihood ratio test for testing the adequacy of the hypothesized convariance structure. These methods are illustrated and compared to the usual normal theory likelihood methods (Akaike's information criterion and the likelihood ratio test) using three examples. Simulation results indicate the pseudo-likelihood ratio test compares favorably with the standard normal theory likelihood ratio test, but both procedures are sensitive to departures from normality.     

A semiparametric Bayesian approach for structural equation models

In the development of structural equation models (SEMs), observed variables are usually assumed to be normally distributed. However, this assumption is likely to be violated in many practical researches. As the non‐normality of observed variables in an SEM can be obtained from either non‐normal latent variables or non‐normal residuals or both, semiparametric modeling with unknown distribution of latent variables or unknown distribution of residuals is needed. In this article, we find that an SEM becomes nonidentifiable when both the latent variable distribution and the residual distribution are unknown. Hence, it is impossible to estimate reliably both the latent variable distribution and the residual distribution without parametric assumptions on one or the other. We also find that the residuals in the measurement equation are more sensitive to the normality assumption than the latent variables, and the negative impact on the estimation of parameters and distributions due to the non‐normality of residuals is more serious. Therefore, when there is no prior knowledge about parametric distributions for either the latent variables or the residuals, we recommend making parametric assumption on latent variables, and modeling residuals nonparametrically. We propose a semiparametric Bayesian approach using the truncated Dirichlet process with a stick breaking prior to tackle the non‐normality of residuals in the measurement equation. Simulation studies and a real data analysis demonstrate our findings, and reveal the empirical performance of the proposed methodology. A free WinBUGS code to perform the analysis is available in Supporting Information.     

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.     

Mixed model analysis of censored longitudinal data with flexible random-effects density

Mixed models are commonly used to represent longitudinal or repeated measures data. An additional complication arises when the response is censored, for example, due to limits of quantification of the assay used. While Gaussian random effects are routinely assumed, little work has characterized the consequences of misspecifying the random-effects distribution nor has a more flexible distribution been studied for censored longitudinal data. We show that, in general, maximum likelihood estimators will not be consistent when the random-effects density is misspecified, and the effect of misspecification is likely to be greatest when the true random-effects density deviates substantially from normality and the number of noncensored observations on each subject is small. We develop a mixed model framework for censored longitudinal data in which the random effects are represented by the flexible seminonparametric density and show how to obtain estimates in SAS procedure NLMIXED. Simulations show that this approach can lead to reduction in bias and increase in efficiency relative to assuming Gaussian random effects. The methods are demonstrated on data from a study of hepatitis C virus.     

Genomic-assisted prediction of genetic value with semiparametric procedures

Semiparametric procedures for prediction of total genetic value for quantitative traits, which make use of phenotypic and genomic data simultaneously, are presented. The methods focus on the treatment of massive information provided by, e.g., single-nucleotide polymorphisms. It is argued that standard parametric methods for quantitative genetic analysis cannot handle the multiplicity of potential interactions arising in models with, e.g., hundreds of thousands of markers, and that most of the assumptions required for an orthogonal decomposition of variance are violated in artificial and natural populations. This makes nonparametric procedures attractive. Kernel regression and reproducing kernel Hilbert spaces regression procedures are embedded into standard mixed-effects linear models, retaining additive genetic effects under multivariate normality for operational reasons. Inferential procedures are presented, and some extensions are suggested. An example is presented, illustrating the potential of the methodology. Implementations can be carried out after modification of standard software developed by animal breeders for likelihood-based or Bayesian analysis.     

Design of long-term HIV dynamic studies using semiparametric mixed-effects models

Studies of HIV dynamics in AIDS research are very important in understanding the pathogenesis of HIV-1 infection and also in assessing the effectiveness of antiviral therapies. There are many AIDS clinical trials on HIV dynamics currently in development worldwide, giving rise to many design issues yet to be addressed. For example, most studies are focused on short-term viral dynamics and the existing models may not be applicable to describe long-term virologic response. In this paper, we use a simulation-based approach to study the designs of long-term viral dynamics under semiparametric nonlinear mixed-effects models. These models not only can preserve the meaningful interpretation of the short-term HIV dynamics, but also characterize the long-term virologic responses to antiretroviral (ARV) treatment. We investigate a number of feasible clinical protocol designs similar to those currently used in AIDS clinical trials. In particular, we evaluate whether earlier samplings can result in more useful information about the viral response trajectory; we also evaluate the effectiveness of two strategies: more frequent samplings per subject with fewer subjects versus fewer samplings per subject with more subjects while keeping the total number of samplings constant. The results of our investigation provide quantitative guidance for designing and selecting ARV therapy.     

Quantitative genetic models for describing simultaneous and recursive relationships between phenotypes

Multivariate models are of great importance in theoretical and applied quantitative genetics. We extend quantitative genetic theory to accommodate situations in which there is linear feedback or recursiveness between the phenotypes involved in a multivariate system, assuming an infinitesimal, additive, model of inheritance. It is shown that structural parameters defining a simultaneous or recursive system have a bearing on the interpretation of quantitative genetic parameter estimates (e.g., heritability, offspring-parent regression, genetic correlation) when such features are ignored. Matrix representations are given for treating a plethora of feedback-recursive situations. The likelihood function is derived, assuming multivariate normality, and results from econometric theory for parameter identification are adapted to a quantitative genetic setting. A Bayesian treatment with a Markov chain Monte Carlo implementation is suggested for inference and developed. When the system is fully recursive, all conditional posterior distributions are in closed form, so Gibbs sampling is straightforward. If there is feedback, a Metropolis step may be embedded for sampling the structural parameters, since their conditional distributions are unknown. Extensions of the model to discrete random variables and to nonlinear relationships between phenotypes are discussed.     

There Is No Need To Be Normal: Generalized Linear Models of Natural Variation

Both ecological field studies and attempts to extrapolate from laboratory experiments to natural populations generally encounter the high degree of natural variability and chaotic behavior that typify natural ecosystems. Regardless of this variability and non-normal distribution, most statistical models of natural systems use normal error which assumes independence between the variance and mean. However, environmental data are often random or clustered and are better described by probability distributions which have more realistic variance to mean relationships. Until recently statistical software packages modeled only with normal error and researchers had to assume approximate normality on the original or transformed scale of measurement and had to live with the consequences of often incorrectly assuming independence between the variance and mean. Recent developments in statistical software allow researchers to use generalized linear models (GLMs) and analysis can now proceed with probability distributions from the exponential family which more realistically describe natural conditions: binomial (even distribution with variance less than mean), Poisson (random distribution with variance equal mean), negative binomial (clustered distribution with variance greater than mean). GLMs fit parameters on the original scale of measurement and eliminate the need for obfuscating transformations, reduce bias for proportions with unequal sample size, and provide realistic estimates of variance which can increase power of tests. Because GLMs permit modeling according to the non-normal behavior of natural systems and obviate the need for normality assumptions, they will likely become a widely used tool for analyzing toxicity data. To demonstrate the broad-scale utility of GLMs, we present several examples where the use of GLMs improved the statistical power of field and laboratory studies to document the rapid ecological recovery of Prince William Sound following the Exxon Valdez oil spill.