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.     

Biometrical modeling of twin and family data using standard mixed model software

Summary .   Biometrical genetic modeling of twin or other family data can be used to decompose the variance of an observed response or 'phenotype' into genetic and environmental components. Convenient parameterizations requiring few random effects are proposed, which allow such models to be estimated using widely available software for linear mixed models (continuous phenotypes) or generalized linear mixed models (categorical phenotypes). We illustrate the proposed approach by modeling family data on the continuous phenotype birth weight and twin data on the dichotomous phenotype depression. The example data sets and commands for Stata and R/S-PLUS are available at the Biometrics website.     

Analysis of a Generalized Linear Mixed Model: A Case Study and Simulation Results

A class of generalized linear mixed models can be obtained by introducing random effects in the linear predictor of a generalized linear model, e.g. a split plot model for binary data or count data. Maximum likelihood estimation, for normally distributed random effects, involves high-dimensional numerical integration, with severe limitations on the number and structure of the additional random effects. An alternative estimation procedure based on an extension of the iterative re-weighted least squares procedure for generalized linear models will be illustrated on a practical data set involving carcass classification of cattle. The data is analysed as overdispersed binomial proportions with fixed and random effects and associated components of variance on the logit scale. Estimates are obtained with standard software for normal data mixed models. Numerical restrictions pertain to the size of matrices to be inverted. This can be dealt with by absorption techniques familiar from e.g. mixed models in animal breeding. The final model fitted to the classification data includes four components of variance and a multiplicative overdispersion factor. Basically the estimation procedure is a combination of iterated least squares procedures and no full distributional assumptions are needed. A simulation study based on the classification data is presented. This includes a study of procedures for constructing confidence intervals and significance tests for fixed effects and components of variance. The simulation results increase confidence in the usefulness of the estimation procedure.     

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.     

MIXED MODEL APPROACHES FOR ESTIMATING GENETIC VARIANCES AND COVARIANCES

The limitations of methods for analysis of variance(ANOVA)in estimating genetic variances are discussed. Among the three methods(maximum likelihood ML, restricted maximum likelihood REML, and minimum norm quadratic unbiased estimation MINQUE)for mixed linear models, MINQUE method is presented with formulae for estimating variance components and covariances components and for predicting genetic effects. Several genetic models, which cannot be appropriately analyzed by ANOVA methods, are introduced in forms of mixed linear models. Genetic models with independent random effects can be analyzed by MINQUE(1)method whieh is a MINQUE method with all prior values setting 1. MINQUE(1)method can give unbiased estimation for variance components and covariance components, and linear unbiased prediction (LUP) for genetic effects. There are more complicate genetic models for plant seeds which involve correlated random effects. MINQUE(0/1)method, which is a MINQUE method with all prior covariances setting 0 and all prior variances setting 1, is suitable for estimating variance and covariance components in these models. Mixed model approaches have advantage over ANOVA methods for the capacity of analyzing unbalanced data and complicated models. Some problems about estimation and hypothesis test by MINQUE method are discussed.     

Bayesian multivariate logistic regression

Bayesian analyses of multivariate binary or categorical outcomes typically rely on probit or mixed effects logistic regression models that do not have a marginal logistic structure for the individual outcomes. In addition, difficulties arise when simple noninformative priors are chosen for the covariance parameters. Motivated by these problems, we propose a new type of multivariate logistic distribution that can be used to construct a likelihood for multivariate logistic regression analysis of binary and categorical data. The model for individual outcomes has a marginal logistic structure, simplifying interpretation. We follow a Bayesian approach to estimation and inference, developing an efficient data augmentation algorithm for posterior computation. The method is illustrated with application to a neurotoxicology study.     

Estimation of non-linear growth models by linearization: a simulation study using a Gompertz function

A method based on Taylor series expansion for estimation of location parameters and variance components of non-linear mixed effects models was considered. An attractive property of the method is the opportunity for an easily implemented algorithm. Estimation of non-linear mixed effects models can be done by common methods for linear mixed effects models, and thus existing programs can be used after small modifications. The applicability of this algorithm in animal breeding was studied with simulation using a Gompertz function growth model in pigs. Two growth data sets were analyzed: a full set containing observations from the entire growing period, and a truncated time trajectory set containing animals slaughtered prematurely, which is common in pig breeding. The results from the 50 simulation replicates with full data set indicate that the linearization approach was capable of estimating the original parameters satisfactorily. However, estimation of the parameters related to adult weight becomes unstable in the case of a truncated data set.     

Robust Linear Mixed Models with Normal/Independent Distributions and Bayesian MCMC Implementation

Linear mixed effects models have been widely used in analysis of data where responses are clustered around some random effects, so it is not reasonable to assume independence between observations in the same cluster. In most biological applications, it is assumed that the distributions of the random effects and of the residuals are Gaussian. This makes inferences vulnerable to the presence of outliers. Here, linear mixed effects models with normal/independent residual distributions for robust inferences are described. Specific distributions examined include univariate and multivariate versions of the Student‐ t, the slash and the contaminated normal. A Bayesian framework is adopted and Markov chain Monte Carlo is used to carry out the posterior analysis. The procedures are illustrated using birth weight data on rats in a toxicological experiment. Results from the Gaussian and robust models are contrasted, and it is shown how the implementation can be used for outlier detection. The thick‐tailed distributions provide an appealing robust alternative to the Gaussian process in linear mixed models, and they are easily implemented using data augmentation and MCMC techniques.     

Functional mixed effects models

In this article, a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced. The linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects, which are modeled as realizations of zero-mean stochastic processes. The fixed functional effects and the random functional effects are modeled in the same functional space, which guarantee the population-average and subject-specific curves have the same smoothness property. These models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures, can include continuous covariates as well as dummy factors in both the fixed or random design matrices, and include the nested curves models as special cases. Two estimation procedures are proposed. The first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software. The second procedure is a sequential estimation procedure using Kalman filtering. This algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets. A generalized maximum likelihood (GML) ratio test is proposed for inference and model selection. An application to comparison of cortisol profiles is used as an illustration.     

Inclusion of Dominance Effects in the Multivariate GBLUP Model

New proposals for models and applications of prediction processes with data on molecular markers may help reduce the financial costs of and identify superior genotypes in maize breeding programs. Studies evaluating Genomic Best Linear Unbiased Prediction (GBLUP) models including dominance effects have not been performed in the univariate and multivariate context in the data analysis of this crop. A single cross hybrid construction procedure was performed in this study using phenotypic data and actual molecular markers of 4,091 maize lines from the public database Panzea. A total of 400 simple hybrids resulting from this process were analyzed using the univariate and multivariate GBLUP model considering only additive effects additive plus dominance effects. Historic heritability scenarios of five traits and other genetic architecture settings were used to compare models, evaluating the predictive ability and estimation of variance components. Marginal differences were detected between the multivariate and univariate models. The main explanation for the small discrepancy between models is the low- to moderate-magnitude correlations between the traits studied and moderate heritabilities. These conditions do not favor the advantages of multivariate analysis. The inclusion of dominance effects in the models was an efficient strategy to improve the predictive ability and estimation quality of variance components.     

Factor analysis models for structuring covariance matrices of additive genetic effects: a Bayesian implementation

Multivariate linear models are increasingly important in quantitative genetics. In high dimensional specifications, factor analysis (FA) may provide an avenue for structuring (co)variance matrices, thus reducing the number of parameters needed for describing (co)dispersion. We describe how FA can be used to model genetic effects in the context of a multivariate linear mixed model. An orthogonal common factor structure is used to model genetic effects under Gaussian assumption, so that the marginal likelihood is multivariate normal with a structured genetic (co)variance matrix. Under standard prior assumptions, all fully conditional distributions have closed form, and samples from the joint posterior distribution can be obtained via Gibbs sampling. The model and the algorithm developed for its Bayesian implementation were used to describe five repeated records of milk yield in dairy cattle, and a one common FA model was compared with a standard multiple trait model. The Bayesian Information Criterion favored the FA model.     

The genetical analysis of covariance structure.

The analysis of covariance structures (J?reskog, 1973) is adapted to the simultaneous maximum likelihood estimation of genetical and environmental factor loadings and specific variances. The goodness of fit is tested by chi square and standard errors of parameter estimates can be obtained. Any linear model used in univariate genetical analyses can be extended to the multivariate case. Most biological hypotheses about the relationships between variables can be specified by a variety of factor models. Individual parameters can be given fixed values or set to zero and hypotheses concerning the congruence of genetical and environmental correlations can be tested. The method is illustrated with published twin data on cognitive abilities.     

Spatial models for spatial statistics: some unification

A general statistical framework is proposed for comparing linear models of spatial process and pattern. A spatial linear model for nested analysis of variance can be based on either fixed effects or random effects. Greig-Smith (1952) originally used a fixed effects model, but there are also examples of random effects models in the soil science literature. Assuming intrinsic stationarity for a linear model, the expectations of a spatial nested ANOVA and two term local variance (TTLV, Hill 1973) are functions of the variogram, and several examples are given. Paired quadrat variance (PQV, Ludwig & Goodall 1978) is a variogram estimator which can be used to approximate TTLV, and we provide an example from ecological data. Both nested ANOVA and TTLV can be seen as weighted lag-1 variogram estimators that are functions of support, rather than distance. We show that there are two unbiased estimators for the variogram under aggregation, and computer simulation shows that the estimator with smaller variance depends on the process autocorrelation.     

Continuous time Markov models for binary longitudinal data

Longitudinal data usually consist of a number of short time series. A group of subjects or groups of subjects are followed over time and observations are often taken at unequally spaced time points, and may be at different times for different subjects. When the errors and random effects are Gaussian, the likelihood of these unbalanced linear mixed models can be directly calculated, and nonlinear optimization used to obtain maximum likelihood estimates of the fixed regression coefficients and parameters in the variance components. For binary longitudinal data, a two state, non-homogeneous continuous time Markov process approach is used to model serial correlation within subjects. Formulating the model as a continuous time Markov process allows the observations to be equally or unequally spaced. Fixed and time varying covariates can be included in the model, and the continuous time model allows the estimation of the odds ratio for an exposure variable based on the steady state distribution. Exact likelihoods can be calculated. The initial probability distribution on the first observation on each subject is estimated using logistic regression that can involve covariates, and this estimation is embedded in the overall estimation. These models are applied to an intervention study designed to reduce children's sun exposure.     

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.     

Multivariate multilevel nonlinear mixed effects models for timber yield predictions

Nonlinear mixed effects models have become important tools for growth and yield modeling in forestry. To date, applications have concentrated on modeling single growth variables such as tree height or bole volume. Here, we propose multivariate multilevel nonlinear mixed effects models for describing several plot-level timber quantity characteristics simultaneously. We describe how such models can be used to produce future predictions of timber volume (yield). The class of models and methods of estimation and prediction are developed and then illustrated on data from a University of Georgia study of the effects of various site preparation methods on the growth of slash pine (Pinus elliottii Engelm.).     

Mixed effects models with censored data with application to HIV RNA levels

Mixed effects models are often used for estimating fixed effects and variance components in longitudinal studies of continuous data. When the outcome being modelled is a laboratory measurement, however, it may be subject to lower and upper detection limits (i.e., censoring). In this paper, the usual EM estimation procedure for mixed effects models is modified to account for left and/or right censoring.     

Semiparametric models for missing covariate and response data in regression models

We consider a class of semiparametric models for the covariate distribution and missing data mechanism for missing covariate and/or response data for general classes of regression models including generalized linear models and generalized linear mixed models. Ignorable and nonignorable missing covariate and/or response data are considered. The proposed semiparametric model can be viewed as a sensitivity analysis for model misspecification of the missing covariate distribution and/or missing data mechanism. The semiparametric model consists of a generalized additive model (GAM) for the covariate distribution and/or missing data mechanism. Penalized regression splines are used to express the GAMs as a generalized linear mixed effects model, in which the variance of the corresponding random effects provides an intuitive index for choosing between the semiparametric and parametric model. Maximum likelihood estimates are then obtained via the EM algorithm. Simulations are given to demonstrate the methodology, and a real data set from a melanoma cancer clinical trial is analyzed using the proposed methods.     

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.     

A Simple Illustration of the Failure of PQL,IRREML and APHL as Approximate ML Methods for Mixed Models for Binary Data

Evaluation of the likelihood in mixed models for non-normal data, e.g. dependent binary data, involves high dimensional integration, which offers severe numerical problems. Penalized quasi-likelihood, iterative re-weighted restricted maximum likelihood and adjusted profile h-likelihood estimation are methods which avoid numerical integration. They will be derived by approximation of the maximum likelihood equations. For binary data, these estimation procedures may yield seriously biased estimates for components of variance, intra-class correlation or heritability. An analytical evaluation of a simple example illustrates how very critical the approximations can be for the performance of the variance component estimators.