Showing papers on "Mixed model published in 2000"

Modelling covariance structure in the analysis of repeated measures data.

Abstract: The term 'repeated measures' refers to data with multiple observations on the same sampling unit. In most cases, the multiple observations are taken over time, but they could be over space. It is usually plausible to assume that observations on the same unit are correlated. Hence, statistical analysis of repeated measures data must address the issue of covariation between measures on the same unit. Until recently, analysis techniques available in computer software only offered the user limited and inadequate choices. One choice was to ignore covariance structure and make invalid assumptions. Another was to avoid the covariance structure issue by analysing transformed data or making adjustments to otherwise inadequate analyses. Ignoring covariance structure may result in erroneous inference, and avoiding it may result in inefficient inference. Recently available mixed model methodology permits the covariance structure to be incorporated into the statistical model. The MIXED procedure of the SAS((R)) System provides a rich selection of covariance structures through the RANDOM and REPEATED statements. Modelling the covariance structure is a major hurdle in the use of PROC MIXED. However, once the covariance structure is modelled, inference about fixed effects proceeds essentially as when using PROC GLM. An example from the pharmaceutical industry is used to illustrate how to choose a covariance structure. The example also illustrates the effects of choice of covariance structure on tests and estimates of fixed effects. In many situations, estimates of linear combinations are invariant with respect to covariance structure, yet standard errors of the estimates may still depend on the covariance structure.

Estimation of Disease Rates in Small Areas: A new Mixed Model for Spatial Dependence

Abstract: In this paper, a new model is proposed for spatial dependence that includes separate parameters for overdispersion and the strength of spatial dependence The new dependence structure is incorporated into a generalized linear mixed model useful for the estimation of disease incidence rates in small geographic regions The mixed model allows for log-linear covariate adjustment and local smoothing of rates through estimation of the spatially correlated random effects Computer simulation studies compare the new model with the following sub-models: intrinsic autoregression, an independence model, and a model with no random effects The major finding was that regression coefficient estimates based on fitting intrinsic autoregression to independent data can have very low precision compared with estimates based on the full model Additional simulation studies demonstrate that penalized quasi-likelihood (PQL) estimation generally performs very well although the estimates are slightly biased for very small counts

Maximum Likelihood for Generalized Linear Models with Nested Random Effects via High-Order, Multivariate Laplace Approximation

Abstract: Nested random effects models are often used to represent similar processes occurring in each of many clusters. Suppose that, given cluster-specific random effects b, the data y are distributed according to f(y|b, Θ), while b follows a density p(b|Θ). Likelihood inference requires maximization of ∫ f(y|b, Θ)p(b|Θdb with respect to Θ. Evaluation of this integral often proves difficult, making likelihood inference difficult to obtain. We propose a multivariate Taylor series approximation of the log of the integrand that can be made as accurate as desired if the integrand and all its partial derivatives with respect to b are continuous in the neighborhood of the posterior mode of b|Θ,y. We then apply a Laplace approximation to the integral and maximize the approximate integrated likelihood via Fisher scoring. We develop computational formulas that implement this approach for two-level generalized linear models with canonical link and multivariate normal random effects. A comparison with approximation...

Reference Bayesian Methods for Generalized Linear Mixed Models

Abstract: Bayesian methods furnish an attractive approach to inference in generalized linear mixed models. In the absence of subjective prior information for the random-effect variance components, these analyses are typically conducted using either the standard invariant prior for normal responses or diffuse conjugate priors. Previous work has pointed out serious difficulties with both strategies, and we show here that as in normal mixed models, the standard invariant prior leads to an improper posterior distribution for generalized linear mixed models. This article proposes and investigates two alternative reference (i.e., “objective” or “noninformative”) priors: an approximate uniform shrinkage prior and an approximate Jeffreys's prior. We give conditions for the existence of the posterior distribution under any prior for the variance components in conjunction with a uniform prior for the fixed effects. The approximate uniform shrinkage prior is shown to satisfy these conditions for several families of d...

The Analysis of Variance: Fixed, Random and Mixed Models

Abstract: One-way classification model two-crossed classification with one observation per cell two-way crossed classification with more than one observation per cell three-way and higher order crossed classification two-way nested (hierarchical) classification three-way and higher order nested classifications partially nested classification finite population and other models some simple experimental designs analysis and variance using statistical computing packages.

A Mixed-Model Approach to Mapping Quantitative Trait Loci in Barley on the Basis of Multiple Environment Data

Abstract: In this article, I propose a mixed-model method to detect QTL with significant mean effect across environments and to characterize the stability of effects across multiple environments. I demonstrate the method using the barley dataset by the North American Barley Genome Mapping Project. The analysis raises the need for mixed modeling in two different ways. First, it is reasonable to regard environments as a random sample from a population of target environments. Thus, environmental main effects and QTL-by-environment interaction effects are regarded as random. Second, I expect a genetic correlation among pairs of environments caused by undetected QTL. I show how random QTL-by-environment effects as well as genetic correlations are straightforwardly handled in a mixed-model framework. The main advantage of this method is the ability to assess the stability of QTL effects. Moreover, the method allows valid statistical inferences regarding average QTL effects.

Modeling nonstationary longitudinal data.

Abstract: An important theme of longitudinal data analysis in the past two decades has been the development and use of explicit parametric models for the data's variance-covariance structure. A variety of these models have been proposed, of which most are second-order stationary. A few are flexible enough to accommodate nonstationarity, i.e., nonconstant variances and/or correlations that are not a function solely of elapsed time between measurements. We review five nonstationary models that we regard as most useful: (1) the unstructured covariance model, (2) unstructured antedependence models, (3) structured antedependence models, (4) autoregressive integrated moving average and similar models, and (5) random coefficients models. We evaluate the relative strengths and limitations of each model, emphasizing when it is inappropriate or unlikely to be useful. We present three examples to illustrate the fitting and comparison of the models and to demonstrate that nonstationary longitudinal data can be modeled effectively and, in some cases, quite parsimoniously. In these examples, the antedependence models generally prove to be superior and the random coefficients models prove to be inferior. We conclude that antedependence models should be given much greater consideration than they have historically received.

Generalized linear latent and mixed models

Abstract: What is a latent variable? • Random variable ηj for unit j whose realized values are hidden • (Hierarchical Bayes: ηj exchangeable, prior has free hyperparameters) • Properties must be indirectly inferred based on a statistical model connecting observed variables yj = (y1j , y2j , . . . , ynj) to ηj Basic construction principle of latent variable models: Conditional independence of observed variables given latent variable

Mixed model analysis of quantitative trait loci

Abstract: We develop a mixed model approach of quantitative trait locus (QTL) mapping for a hybrid population derived from the crosses of two or more distinguished outbred populations. Under the mixed model, we treat the mean allelic value of each source population as the fixed effect and the allelic deviations from the mean as random effects so that we can partition the total genetic variance into between- and within-population variances. Statistical inference of the QTL parameters is obtained by using the Bayesian method implemented by Markov chain Monte Carlo (MCMC). This unified QTL mapping algorithm treats the fixed and random model approaches as special cases of the general mixed model methodology. Utility and flexibility of the method are demonstrated by using a set of simulated data.

Asymptotics for Analysis of Variance When the Number of Levels is Large

Abstract: We study asymptotic results for F tests in analysis of variance models as the number of factor levels goes to ∞ but the number of observations for each factor combination is fixed. Asymptotic derivations of the type discussed in this article would be relevant whenever both the numerator and denominator degrees of freedom go to ∞ (at the same rate). We consider null and alternative distributions of F, the usual F statistic, for fixed-effects and random-effects, balanced and unbalanced, one-way and two-way, and normal and nonnormal analysis of variance (ANOVA) models. The results may be most relevant for random-effects and mixed models. For example, we may have an agricultural experiment in which the number of cows is quite large but the number of measurements on each cow is small. The results would also be relevant for fixed-effects models in which there are many factor levels but not many observations for each factor level.

Methods and systems for plant performance analysis

Abstract: Method and systems for assessing wide area performance of a crop variety using a linear mixed model that incorporates geologic statistical components and includes parameters for fixed effects, random effects and covariance (J20) A wide area database is constructed that includes covariate data, spatial coordinates of testing locations of one or more crop varieties, geographic areas in which the testing location reside, and performance trait values of one or more crop varieties (H25, H30) The parameters for the fixed effects, random effects and covariances are estimated by fitting the linear mixed model with data in the wide area database Long term expected performance of the crop variety may be estimated using the parameter estimates Average performance of the crop variety for a given time period may be predicted using the parameter estimates

Variance components analysis for pedigree-based censored survival data using generalized linear mixed models (GLMMs) and Gibbs sampling in BUGS.

Abstract: Complex human diseases are an increasingly important focus of genetic research. Many of the determinants of these diseases are unknown and there is often a strong residual covariance between relatives even when all known genetic and environmental factors have been taken into account. This must be modeled correctly whether scientific interest is focused on fixed effects, as in an association analysis, or on the covariance structure itself. Analysis is straightforward for multivariate normally distributed traits, but difficulties arise with other types of trait. Generalized linear mixed models (GLMMs) offer a potentially unifying approach to analysis for many classes of phenotype including right censored survival times. This includes age-at-onset and age-at-death data and a variety of other censored traits. Markov chain Monte Carlo (MCMC) methods, including Gibbs sampling, provide a convenient framework within which such GLMMs may be fitted. In this paper, we use BUGS ("Bayesian inference using Gibbs sampling": a readily available, generic Gibbs sampler) to fit GLMMs for right-censored survival times in nuclear and extended families. We discuss parameter interpretation and statistical inference, and show how to circumvent a number of important theoretical and practical problems. Using simulated data, we show that model parameters are consistent. We further illustrate our methods using data from an ongoing cohort study. Finally, we propose that the random effects associated with a genetic component of variance (e.g., sigma(2)(A)) in a GLMM may be regarded as an adjusted "phenotype" and used as input to a conventional model-based or model-free linkage analysis. This provides a simple way to conduct a linkage analysis for a trait reflected in a right-censored survival time while comprehensively adjusting for observed confounders at the level of the individual and latent environmental effects shared across families.

A robust mixed linear model analysis for longitudinal data.

Abstract: This paper describes robust procedures for estimating parameters of a mixed effects linear model as applied to longitudinal data. In addition to fixed regression parameters, the model incorporates random subject effects to accommodate between-subjects variability and autocorrelation for within-subject variability. Robust empirical Bayesian estimation of subject effects is briefly discussed. As an illustration, the procedures are applied to data from a multiple sclerosis clinical trial.

Genetic analysis of the age at menopause by using estimating equations and Bayesian random effects models.

Abstract: Multi-wave self-report data on age at menopause in 2182 female twin pairs (1355 monozygotic and 827 dizygotic pairs), were analysed to estimate the genetic, common and unique environmental contribution to variation in age at menopause. Two complementary approaches for analysing correlated time-to-onset twin data are considered: the generalized estimating equations (GEE) method in which one can estimate zygosity-specific dependence simultaneously with regression coefficients that describe the average population response to changing covariates; and a subject-specific Bayesian mixed model in which heterogeneity in regression parameters is explicitly modelled and the different components of variation may be estimated directly. The proportional hazards and Weibull models were utilized, as both produce natural frameworks for estimating relative risks while adjusting for simultaneous effects of other covariates. A simple Markov chain Monte Carlo method for covariate imputation of missing data was used and the actual implementation of the Bayesian model was based on Gibbs sampling using the freeware package BUGS.

Estimation in longitudinal random effects models with measurement error

Abstract: Estimation of the regression parameters and variance components in a longitudinal mixed model with measurement error in a time-varying covariate is considered. The positive bias in variance estimators caused by covariate measure- ment error in a normal linear mixed model has recently been identified and studied (Tosteson, Buonaccorsi and Demidenko (1997)). The methods suggested there for correction of the bias involve convenient adaptations of existing software for a par- ticular model. In this paper, we study alternative methods of estimation which achieve higher efficiencies and extend readily to a more general class of models. Full and pseudo-maximum likelihood estimators under normality are considered as is a pseudo-moment approach relying on initial estimation of nuisance parameters. The latter lead to a "regression calibration" method for estimating the regression parameters, in which a substitution is made for the unknown covariates, followed by a correction for estimation of the variance parameters. It is shown that for some cases this yields the pseudo-maximum likelihood estimates and, in these cases, the resulting estimators are highly efficient relative to the full maximum likelihood estimators. We first consider a model with no additional data, where identifia- bility follows from assumptions about the longitudinal model for the unobserved true covariates, and then describe some extensions to cases where either replicate or validation data is available. We illustrate with an example investigating the relationship between dietary and serum beta-carotene.

EM-REML estimation of covariance parameters in Gaussian mixed models for longitudinal data analysis

Abstract: This paper presents procedures for implementing the EM algorithm to compute REML estimates of variance covariance components in Gaussian mixed models for longitudinal data analysis. The class of models considered includes random coefficient factors, stationary time processes and measurement errors. The EM algorithm allows separation of the computations pertaining to parameters involved in the random coefficient factors from those pertaining to the time processes and errors. The procedures are illustrated with Pothoff and Roy's data example on growth measurements taken on 11 girls and 16 boys at four ages. Several variants and extensions are discussed.

A scaled linear mixed model for multiple outcomes

Abstract: Summary. We propose a scaled linear mixed model to assess the effects of exposure and other covariates on multiple continuous outcomes. The most general form of the model allows a different exposure effect for each outcome. An important special case is a model that represents the exposure effects using a common global measure that can be characterized in terms of effect sizes. Correlations among different outcomes within the same subject are accommodated using random effects. We develop two approaches to model fitting, including the maximum likelihood method and the working parameter method. A key feature of both methods is that they can be easily implemented by repeatedly calling software for fitting standard linear mixed models, e.g., SAS PROC MIXED. Compared to the maximum likelihood method, the working parameter method is easier to implement and yields fully efficient estimators of the parameters of interest. We illustrate the proposed methods by analyzing data from a study of the effects of occupational pesticide exposure on semen quality in a cohort of Chinese men.

Comparing Traditional and Bayesian Analyses of Selection Experiments in Animal Breeding

Abstract: The mixed linear model is commonly used in animal breeding to predict or estimate the breeding values of individual animals. Breeding values are used to select animals for use in breeding subsequent generations. A traditional analysis of data under this model involves estimating the variance components with restricted maximum likelihood. These estimates are then used to find best linear unbiased predictors for the animal breeding values. A Bayesian analysis of data under this model involves treating all of the parameters, including the variance components, as random and finding the joint posterior distribution of all of the parameters given the data. Because the selection decision depends on the values of the variance components, a Bayesian analysis can yield different selection outcomes than the traditional analysis. We demonstrate both types of data analysis on data from an animal breeding experiment and compare the resulting selections.

Bayesian hierarchical models.

Abstract: Publisher Summary This chapter describes the Bayesian hierarchical models. The evaluation of this study, by a Bayesian hierarchical linear model is derived from the data that include the other large clinical trials of thrombolytic therapy and suggests that treatment is also beneficial for patients, arriving much later than six hours after symptom onset. Many random processes, generating data for which statistical analyses are required, involve multiple sources of variation. These sources represent randomness introduced at different levels of a nested data structure. The standard statistical approach, for describing processes with multiple components of variation, uses mixed models of fixed and random effects. The fixed effects are quantities, about which inference is to be made directly, whereas the random effects are quantities, sampled from a population about which inference is desired. Meta-analysis, a technique used to pool data from different studies to estimate some common parameter, such as a treatment effect in medical studies, is a particular example of this type. An important clinical objective for cardiologists is determining the latest time, at which treatment still shows a benefit.

Estimating data transformations in nonlinear mixed effects models.

Abstract: A routine practice in the analysis of repeated measurement data is to represent individual responses by a mixed effects model on some transformed scale. For example, for pharmacokinetic, growth, and other data, both the response and the regression model are typically transformed to achieve approximate within-individual normality and constant variance on the new scale; however, the choice of transformation is often made subjectively or by default, with adoption of a standard choice such as the log. We propose a mixed effects framework based on the transform-both-sides model, where the transformation is represented by a monotone parametric function and is estimated from the data. For this model, we describe a practical fitting strategy based on approximation of the marginal likelihood. Inference is complicated by the fact that estimation of the transformation requires modification of the usual standard errors for estimators of fixed effects; however, we show that, under conditions relevant to common applications, this complication is asymptotically negligible, allowing straightforward implementation via standard software.

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

Abstract: 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.

Exploring spatio-temporal patterns of mortality using mixed effects models.

Abstract: A linear mixed effects (LME) model previously used for a spatial analysis of mortality data for a single time period is extended to include time trends and spatio-temporal interactions. This model includes functions of age and time period that can account for increasing and decreasing death rates over time and age, and a change-point of rates at a predetermined age. A geographic hierarchy is included that provides both regional and small area age-specific rate estimates, stabilizing rates based on small numbers of deaths by sharing information within a region. The proposed log-linear analysis of rates allows the use of commercially available software for parameter estimation, and provides an estimator of overdispersion directly as the residual variance. Because of concerns about the accuracy of small area rate estimates when there are many instances of no observed deaths, we consider potential sources of error, focusing particularly on the similarity of likelihood inferences using the LME model for rates as compared to an exact Poisson-normal mixed effects model for counts. The proposed LME model is applied to breast cancer deaths which occurred among white women during 1979-1996. For this example, application of diagnostics for multiparameter likelihood comparisons suggests a restriction of age to a minimum of either 25 or 35, depending on whether small area rate estimates are required. Investigation into a convergence problem led to the discovery that the changes in breast cancer geographic patterns over time are related more to urbanization than to region, as previously thought. Published in 2000 by John Wiley & Sons, Ltd.

Generalized additive mixed models

Abstract: Following the extension from linear mixed models to additive mixed models, extension from generalized linear mixed models to generalized additive mixed models is made, Algorithms are developed to compute the MLE's of the nonlinear effects and the covariance structures based on the penalized marginal likelihood. Convergence of the algorithms and selection of the smooth param¬eters are discussed.

Issues in use of sas proc.mixed to test the significance of treatment effects in controlled clinical trials

Abstract: A project that originated with the aim of documenting the implications of dropouts for tests of significance based on general linear mixed model procedures resulted in recognition of problems in the use of SAS PROC.MIXED for this purpose. In responding to suggestions and criticisms, we have further analyzed simulated clinical trial data with realistic autoregressive structure, using alternative error model formulations, different approaches to the use of covariates to model dropout patterns, and different ways to include the critical time variable in the mixed model. Results emphasize the sensitivity of the PROC.MIXED tests of significance for GROUP and TIME x GROUP equal slopes hypothesis to less than optimal modeling of the error covariance structure. Even with the authoritatively recommended best available modeling of the error structure, model formulations that made use of the REPEATED statement did not maintain conservative test sizes when covariates were required to model dropout data patterns. Random coefficients models that employed the RANDOM statement did permit appropriate covariate controls, but the tests of significance for treatment effects were lacking in power. After examining a variety of alternative PROC.MIXED model formulations, it is concluded that none provided both Type I error protection and power comparable to that of simple two-stage analysis of covariance (ANCOVA) procedures for confirming the presence of true treatment effects in controlled clinical trials. Other issues examined in this article concern treating baseline scores as both covariate and initial repeated measurement to which a linear means model is fitted, failure to take advantage of the regression of repeated measurements on time in modeling time as an unordered categorical variable, and fitting linear regression models to nonlinear response patterns.

Dynamic Mixed Models for Irregularly Observed Time Series

Abstract: We review the conventional dynamic linear model in state-space form and give a useful generalization that admits fixed covariates to the measurement equation while treating the state vectors as time-varying random effects.

Flexible modelling of the covariance matrix in a linear random effects model

Abstract: A flexible approach is proposed for modelling the covariance matrix of a linear mixed model for longitudinal data. The method combines parametric modelling of the random effects part with flexible modelling of the serial correlation component. The approach is exemplified on weight gain data and on the evolution of height of children in their first year of life of the Jimma Infant Survival Study, an Ethiopian cohort study. The analyses show the usefulness of the approach.

Lines in Random Effects Plots from the Linear Mixed-Effects Model

Abstract: After fitting a linear mixed-effects model to a set of repeated-measures or longitudinal data, it is common practice to plot the estimated random effects. On occasion it may be observed that in these plots a straight line may appear. How did this line arise? What influence does it have on the interpretation of the results from the model? This article demonstrates an artifact that can occur in the plots of random effects. If a cluster has exactly one observation, the plot of any estimated random effect against any other estimated random effect will fall on a straight line. The line in the plot of a pair of estimated random effects may even lie in the opposite direction of the direction suggested by the correlation from the random effects covariance matrix. For clusters with two observations at the same design points of the variables for the random effects, the estimated random effects will lie on a plane. Using an example, we demonstrate the patterns in the plots of the estimated random effects. T...

Biometrical Models for Predicting Future Performance in Plant Breeding.

Abstract: The plant breeding process begins with the selection o f parents and crosses. Promising progeny from these crosses progress through a series o f selection stages that typically culminate in multi-environment trials. I evaluated best linear unbiased predictors (BLUP). other predictors and prediction models at the initial (cross prediction), early replicated testing and late (multi-location) stages o f a sugarcane breeding selection cycle. Model and predictor accuracy was assessed in the first two stages by using cross-validation procedures. I compared statistical models o f progeny test data in their ability to predict the cross performance o f untested sugarcane crosses. Random parental effect predictors and a random cross effect predictors were compared to mid-parent values (MPV) derived from a fixed female-male parental effect model. The cross effect model was evaluated with and without incorporating the genetic relationships among tested crosses into the BLUP derivation. Models with BLUP-based predictors showed smaller mean square prediction error and higher fidelity o f top cross identification than the MPV for all traits evaluated. The MP-BLUP was consistently the best one. Prediction o f per se (genotype) performance is needed during the selection process and requires combining information from different trials. The study investigated three mixed models involving three versions o f BLUPs estimated under different strategies, a fixed least squares genotype means model, and four check-based methods for combining information at early replicated stages. BLUP-based predictors