Multivariate character process models for the analysis of two or more correlated function-valued traits

Various methods, including random regression, structured antedependence models, and character process models, have been proposed for the genetic analysis of longitudinal data and other function-valued traits. For univariate problems, the character process models have been shown to perform well in comparison to alternative methods. The aim of this article is to present an extension of these models to the simultaneous analysis of two or more correlated function-valued traits. Analytical forms for stationary and nonstationary cross-covariance functions are studied. Comparisons with the other approaches are presented in a simulation study and in an example of a bivariate analysis of genetic covariance in age-specific fecundity and mortality in Drosophila. As in the univariate case, bivariate character process models with an exponential correlation were found to be quite close to first-order structured antedependence models. The simulation study showed that the choice of the most appropriate methodology is highly dependent on the covariance structure of the data. The bivariate character process approach proved to be able to deal with quite complex nonstationary and nonsymmetric cross-correlation structures and was found to be the most appropriate for the real data example of the fruit fly Drosophila melanogaster.     

Dynamic conditionally linear mixed models for longitudinal data

We develop a new class of models, dynamic conditionally linear mixed models, for longitudinal data by decomposing the within-subject covariance matrix using a special Cholesky decomposition. Here 'dynamic' means using past responses as covariates and 'conditional linearity' means that parameters entering the model linearly may be random, but nonlinear parameters are nonrandom. This setup offers several advantages and is surprisingly similar to models obtained from the first-order linearization method applied to nonlinear mixed models. First, it allows for flexible and computationally tractable models that include a wide array of covariance structures; these structures may depend on covariates and hence may differ across subjects. This class of models includes, e.g., all standard linear mixed models, antedependence models, and Vonesh-Carter models. Second, it guarantees the fitted marginal covariance matrix of the data is positive definite. We develop methods for Bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited.     

A Bayesian antedependence model for whole genome prediction

Hierarchical mixed effects models have been demonstrated to be powerful for predicting genomic merit of livestock and plants, on the basis of high-density single-nucleotide polymorphism (SNP) marker panels, and their use is being increasingly advocated for genomic predictions in human health. Two particularly popular approaches, labeled BayesA and BayesB, are based on specifying all SNP-associated effects to be independent of each other. BayesB extends BayesA by allowing a large proportion of SNP markers to be associated with null effects. We further extend these two models to specify SNP effects as being spatially correlated due to the chromosomally proximal effects of causal variants. These two models, that we respectively dub as ante-BayesA and ante-BayesB, are based on a first-order nonstationary antedependence specification between SNP effects. In a simulation study involving 20 replicate data sets, each analyzed at six different SNP marker densities with average LD levels ranging from r(2) = 0.15 to 0.31, the antedependence methods had significantly (P < 0.01) higher accuracies than their corresponding classical counterparts at higher LD levels (r(2) > 0. 24) with differences exceeding 3%. A cross-validation study was also conducted on the heterogeneous stock mice data resource (http://mus.well.ox.ac.uk/mouse/HS/) using 6-week body weights as the phenotype. The antedependence methods increased cross-validation prediction accuracies by up to 3.6% compared to their classical counterparts (P < 0.001). Finally, we applied our method to other benchmark data sets and demonstrated that the antedependence methods were more accurate than their classical counterparts for genomic predictions, even for individuals several generations beyond the training data.     

Marginalized models for moderate to long series of longitudinal binary response data

Marginalized models (Heagerty, 1999, Biometrics 55, 688-698) permit likelihood-based inference when interest lies in marginal regression models for longitudinal binary response data. Two such models are the marginalized transition and marginalized latent variable models. The former captures within-subject serial dependence among repeated measurements with transition model terms while the latter assumes exchangeable or nondiminishing response dependence using random intercepts. In this article, we extend the class of marginalized models by proposing a single unifying model that describes both serial and long-range dependence. This model will be particularly useful in longitudinal analyses with a moderate to large number of repeated measurements per subject, where both serial and exchangeable forms of response correlation can be identified. We describe maximum likelihood and Bayesian approaches toward parameter estimation and inference, and we study the large sample operating characteristics under two types of dependence model misspecification. Data from the Madras Longitudinal Schizophrenia Study (Thara et al., 1994, Acta Psychiatrica Scandinavica 90, 329-336) are analyzed.     

Structured antedependence models for genetic analysis of repeated measures on multiple quantitative traits

Simultaneous analysis of correlated traits that change with time is an important issue in genetic analyses. Several methodologies have already been proposed for the genetic analysis of longitudinal data on single traits, in particular random regression and character process models. Although the latter proved, in most cases, to compare favourably to alternative approaches for analysis of single function-valued traits, they do not allow a straightforward extension to the multivariate case. In this paper, another methodology (structured antedependence models) is proposed, and methods are derived for the genetic analysis of two or more correlated function-valued traits. Multivariate analyses are presented of fertility and mortality in Drosophila and of milk, fat and protein yields in dairy cattle. These models offer a substantial flexibility for the correlation structure, even in the case of complex non-stationary patterns, and perform better than multivariate random regression models, with fewer parameters.     

On analyzing circadian rhythms data using nonlinear mixed models with harmonic terms

Wang, Ke, and Brown (2003, Biometrics59, 804-812) developed a smoothing-based approach for modeling circadian rhythms with random effects. Their approach is flexible in that fixed and random covariates can affect both the amplitude and phase shift of a nonparametrically smoothed periodic function. In motivating their approach, Wang et al. stated that a simple sinusoidal function is too restrictive. In addition, they stated that "although adding harmonics can improve the fit, it is difficult to decide how many harmonics to include in the model, and the results are difficult to interpret." We disagree with the notion that harmonic models cannot be a useful tool in modeling longitudinal circadian rhythm data. In this note, we show how nonlinear mixed models with harmonic terms allow for a simple and flexible alternative to Wang et al.'s approach. We show how to choose the number of harmonics using penalized likelihood to flexibly model circadian rhythms and to estimate the effect of covariates on the rhythms. We fit harmonic models to the cortisol circadian rhythm data presented by Wang et al. to illustrate our approach. Furthermore, we evaluate the properties of our procedure with a small simulation study. The proposed parametric approach provides an alternative to Wang et al.'s semiparametric approach and has the added advantage of being easy to implement in most statistical software packages.     

A case study on the choice,interpretation and checking of multilevel models for longitudinal binary outcomes

Recent advances in statistical software have led to the rapid diffusion of new methods for modelling longitudinal data. Multilevel (also known as hierarchical or random effects) models for binary outcomes have generally been based on a logistic-normal specification, by analogy with earlier work for normally distributed data. The appropriate application and interpretation of these models remains somewhat unclear, especially when compared with the computationally more straightforward semiparametric or 'marginal' modelling (GEE) approaches. In this paper we pose two interrelated questions. First, what limits should be placed on the interpretation of the coefficients and inferences derived from random-effect models involving binary outcomes? Second, what diagnostic checks are appropriate for evaluating whether such random-effect models provide adequate fits to the data? We address these questions by means of an extended case study using data on adolescent smoking from a large cohort study. Bayesian estimation methods are used to fit a discrete-mixture alternative to the standard logistic-normal model, and posterior predictive checking is used to assess model fit. Surprising parallels in the parameter estimates from the logistic-normal and mixture models are described and used to question the interpretability of the so-called 'subject-specific' regression coefficients from the standard multilevel approach. Posterior predictive checks suggest a serious lack of fit of both multilevel models. The results do not provide final answers to the two questions posed, but we expect that lessons learned from the case study will provide general guidance for further investigation of these important issues.     

Mixed effects models with bivariate and univariate association parameters for longitudinal bivariate binary response data

When two binary responses are measured for each study subject across time, it may be of interest to model how the bivariate associations and marginal univariate risks involving the two responses change across time. To achieve such a goal, marginal models with bivariate log odds ratio and univariate logit components are extended to include random effects for all components. Specifically, separate normal random effects are specified on the log odds ratio scale for bivariate responses and on the logit scale for univariate responses. Assuming conditional independence given the random effects facilitates the modeling of bivariate associations across time with missing at random incomplete data. We fit the model to a dataset for which such structures are feasible: a longitudinal randomized trial of a cardiovascular educational program where the responses of interest are change in hypertension and hypercholestemia status. The proposed model is compared to a naive bivariate model that assumes independence between time points and univariate mixed effects logit models.     

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.     

Variable Selection for Semiparametric Mixed Models in Longitudinal Studies

Summary .  We propose a double-penalized likelihood approach for simultaneous model selection and estimation in semiparametric mixed models for longitudinal data. Two types of penalties are jointly imposed on the ordinary log-likelihood: the roughness penalty on the nonparametric baseline function and a nonconcave shrinkage penalty on linear coefficients to achieve model sparsity. Compared to existing estimation equation based approaches, our procedure provides valid inference for data with missing at random, and will be more efficient if the specified model is correct. Another advantage of the new procedure is its easy computation for both regression components and variance parameters. We show that the double-penalized problem can be conveniently reformulated into a linear mixed model framework, so that existing software can be directly used to implement our method. For the purpose of model inference, we derive both frequentist and Bayesian variance estimation for estimated parametric and nonparametric components. Simulation is used to evaluate and compare the performance of our method to the existing ones. We then apply the new method to a real data set from a lactation study.     

The PX-EM algorithm for fast stable fitting of Henderson's mixed model

This paper presents procedures for implementing the PX-EM algorithm of Liu, Rubin and Wu to compute REML estimates of variance covariance components in Henderson''s linear mixed models. The class of models considered encompasses several correlated random factors having the same vector length e.g., as in random regression models for longitudinal data analysis and in sire-maternal grandsire models for genetic evaluation. Numerical examples are presented to illustrate the procedures. Much better results in terms of convergence characteristics (number of iterations and time required for convergence) are obtained for PX-EM relative to the basic EM algorithm in the random regression.     

Modelling animal growth in random environments: An application using nonparametric estimation

We study a stochastic differential equation growth model to describe individual growth in random environments. In particular, in this paper, we discuss the estimation of the drift and the diffusion coefficients using nonparametric methods for the case of nonequidistant data for several trajectories. We illustrate the methodology by using bovine growth data. Our goal is to assess: (i) if the parametric models (with specific functional forms for the drift and the diffusion coefficients) previously used by us to describe the evolution of bovine weight were adequate choices; (ii) whether some alternative specific parameterized functional forms of these coefficients might be suggested for further parametric analysis of this data.     

A general class of pattern mixture models for nonignorable dropout with many possible dropout times

Summary .   In this article we consider the problem of fitting pattern mixture models to longitudinal data when there are many unique dropout times. We propose a marginally specified latent class pattern mixture model. The marginal mean is assumed to follow a generalized linear model, whereas the mean conditional on the latent class and random effects is specified separately. Because the dimension of the parameter vector of interest (the marginal regression coefficients) does not depend on the assumed number of latent classes, we propose to treat the number of latent classes as a random variable. We specify a prior distribution for the number of classes, and calculate (approximate) posterior model probabilities. In order to avoid the complications with implementing a fully Bayesian model, we propose a simple approximation to these posterior probabilities. The ideas are illustrated using data from a longitudinal study of depression in HIV-infected women.     

A General Framework for Longitudinal Data through Marked Point Processes

This paper reviews a general framework for the modelling of longitudinal data with random measurement times based on marked point processes and presents a worked example. We construct a quite general regression models for longitudinal data, which may in particular include censoring that only depend on the past and outside random variation, and dependencies between measurement times and measurements. The modelling also generalises statistical counting process models. We review a non-parametric Nadarya-Watson kernel estimator of the regression function, and a parametric analysis that is based on a conditional least squares (CLS) criterion. The parametric analysis presented, is a conditional version of the generalised estimation equations of LIANG and ZEGER (1986). We conclude that the usual nonparametric and parametric regression modelling can be applied to this general set-up, with some modifications. The presented framework provides an easily implemented and powerful tool for model building for repeated measurements.     

A non-stationary model for functional mapping of complex traits

SUMMARY: Understanding the genetic control of growth is fundamental to agricultural, evolutionary and biomedical genetic research. In this article, we present a statistical model for mapping quantitative trait loci (QTL) that are responsible for genetic differences in growth trajectories during ontogenetic development. This model is derived within the maximum likelihood context, implemented with the expectation-maximization algorithm. We incorporate mathematical aspects of growth processes to model the mean vector and structured antedependence models to approximate time-dependent covariance matrices for longitudinal traits. Our model has been employed to map QTL that affect body mass growth trajectories in both male and female mice of an F2 population derived from the Large and Small mouse strains. The results from this model are compared with those from the autoregressive-based functional mapping approach. Based on results from computer simulation studies, we suggest that these two models are alternative to one another and should be used simultaneously for the same dataset.     

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 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.     

Analysis of Covariance Models for Data From Observational Field Studies

Abstract We outline the features of a general class of statistical models (i.e., analysis of covariance [ANCOVA] models) that has proven to be effective for the analysis of data from observational studies. In observational studies, treatments are assigned by Nature in a decidedly nonrandom manner; consequently, many of the crucial assumptions and safeguards of the classic experimental design either fail or are absent. Hence, inferences (causal or associative) are more difficult to justify. Typically, investigators can expect the primary factors of interest, which are usually called environmental exposures rather than treatments, to be involved in complex interactions with each other and with other factors, and these factors will be confounded with still other factors. We provide examples illustrating the application of ANCOVA models to adjust for confounding factors and complex interactions, thereby providing relatively clean estimates of association between exposure and response. We summarize information on available software and supporting literature for implementing ANCOVA models for the analysis of cross-sectional and longitudinal observational field data. We conclude with a brief discussion of critical model fitting issues, including proper specification of the functional form of continuous covariates and problems associated with overfitted models and misspecified models that lack important covariates.     

Sample size determination for hierarchical longitudinal designs with differential attrition rates

Summary .   We consider the problem of sample size determination for three-level mixed-effects linear regression models for the analysis of clustered longitudinal data. Three-level designs are used in many areas, but in particular, multicenter randomized longitudinal clinical trials in medical or health-related research. In this case, level 1 represents measurement occasion, level 2 represents subject, and level 3 represents center. The model we consider involves random effects of the time trends at both the subject level and the center level. In the most common case, we have two random effects (constant and a single trend), at both subject and center levels. The approach presented here is general with respect to sampling proportions, number of groups, and attrition rates over time. In addition, we also develop a cost model, as an aid in selecting the most parsimonious of several possible competing models (i.e., different combinations of centers, subjects within centers, and measurement occasions). We derive sample size requirements (i.e., power characteristics) for a test of treatment-by-time interaction(s) for designs based on either subject-level or cluster-level randomization. The general methodology is illustrated using two characteristic examples.     

Joint modelling of longitudinal and time-to-event data with application to predicting abdominal aortic aneurysm growth and rupture

Shared random effects joint models are becoming increasingly popular for investigating the relationship between longitudinal and time‐to‐event data. Although appealing, such complex models are computationally intensive, and quick, approximate methods may provide a reasonable alternative. In this paper, we first compare the shared random effects model with two approximate approaches: a naïve proportional hazards model with time‐dependent covariate and a two‐stage joint model, which uses plug‐in estimates of the fitted values from a longitudinal analysis as covariates in a survival model. We show that the approximate approaches should be avoided since they can severely underestimate any association between the current underlying longitudinal value and the event hazard. We present classical and Bayesian implementations of the shared random effects model and highlight the advantages of the latter for making predictions. We then apply the models described to a study of abdominal aortic aneurysms (AAA) to investigate the association between AAA diameter and the hazard of AAA rupture. Out‐of‐sample predictions of future AAA growth and hazard of rupture are derived from Bayesian posterior predictive distributions, which are easily calculated within an MCMC framework. Finally, using a multivariate survival sub‐model we show that underlying diameter rather than the rate of growth is the most important predictor of AAA rupture.