Showing papers on "Mixed model published in 1996"
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01 Mar 1996
TL;DR: A genetic evaluation with different sources of records and the best linear unbiased prediction of breeding value - univariate models with one random effect, non-additive animal models and dominance relationship matrix animal model for rapid inversion of the dominance matrix epistatis.
Abstract: Part 1 Genetic evaluation with different sources of records: the basic model breeding value prediction from animal own performance breeding value prediction from progeny records breeding value prediction from pedigree breeding value prediction for one trait from another selection index. Part 2 Genetic relationship between relatives: the numerator relationship matrix decomposing the relationship matrix computing inverse of the relationship matrix inverse of the relationship matrix for sizes and maternal grandsires. Part 3 Best linear unbiased prediction of breeding value - univariate models with one random effect: brief theoretical background a model for an animal evaluation (animal model) a sire model reduced animal model animal model with groups. Part 4 Best linear unbiased prediction of breeding value - models with environmental effects: repeatability model models with common environmental effects. Part 5 Best linear unbiased prediction of breeding value - multivariate models: equal design matrices and no missing records canonical transformation equal design matrices with missing records Cholesky transformation unequal design matrices different traits measured on relatives. Part 6 Maternal trait models - animal and reduced animal models: animal model for a maternal trait reduced animal model with maternal effects multivariate maternal animal model. Part 7 Non-additive animal models: dominance relationship matrix animal model with dominance effects method for rapid inversion of the dominance matrix epistatis. Part 8 Solving linear equations: direct inversion iterating on the mixed model equations iterating on the data.
881 citations
TL;DR: In this paper, the authors investigated the impact of the normality assumption for random effects on their estimates in the linear mixed-effects model and showed that if the distribution of random effects is a finite mixture of normal distributions, then the random effects may be badly estimated if normality is assumed.
Abstract: This article investigates the impact of the normality assumption for random effects on their estimates in the linear mixed-effects model. It shows that if the distribution of random effects is a finite mixture of normal distributions, then the random effects may be badly estimated if normality is assumed, and the current methods for inspecting the appropriateness of the model assumptions are not sound. Further, it is argued that a better way to detect the components of the mixture is to build this assumption in the model and then “compare” the fitted model with the Gaussian model. All of this is illustrated on two practical examples.
566 citations
TL;DR: MIXREG is a program that provides estimates for a mixed-effects regression model (MRM) for normally-distributed response data including autocorrelated errors, utilizing both the EM algorithm and a Fisher-scoring solution.
249 citations
TL;DR: In this paper, the authors show that the restricted maximum likelihood (REML) estimates of dispersion parameters (variance components) in a general (non-normal) mixed model are defined as solutions of the REML equations, and give a necessary and sufficient condition for asymptotic normality of Gaussian maximum likelihood estimates in non-normal cases.
Abstract: The restricted maximum likelihood (REML) estimates of dispersion parameters (variance components) in a general (non-normal) mixed model are defined as solutions of the REML equations. In this paper, we show the REML estimates are consistent if the model is asymptotically identifiable and infinitely informative under the (location) invariant class, and are asymptotically normal (A.N.) if in addition the model is asymptotically nondegenerate. The result does not require normality or boundedness of the rank p of design matrix of fixed effects. Moreover, we give a necessary and sufficient condition for asymptotic normality of Gaussian maximum likelihood estimates (MLE) in non-normal cases. As an application, we show for all unconfounded balanced mixed models of the analysis of variance the REML (ANOVA) estimates are consistent; and are also A.N. provided the models are nondegenerate; the MLE are consistent (A.N.) if and only if certain constraints on p are satisfied.
199 citations
TL;DR: The maximum likelihood estimator is shown to converge to the subset characterized by the same density function, and connection is made to the bootstrap method proposed by Aitkin and co-workers and McLachlan for testing the number of components in a finite mixture and deriving confidence regions in a infinite mixture.
Abstract: Statistical inference using the likelihood ratio statistic for the number of components in a mixture model is complicated when the true number of components is less than that of the proposed model since this represents a non-regular problem: the true parameter is on the boundary of the parameter space and in some cases the true parameter is in a non-identifiable subset of the parameter space. The maximum likelihood estimator is shown to converge to the subset characterized by the same density function, and connection is made to the bootstrap method proposed by Aitkin and co-workers and McLachlan for testing the number of components in a finite mixture and deriving confidence regions in a finite mixture.
143 citations
TL;DR: In this paper, three alternative estimation procedures based on the EM algorithm are considered, two of them make use of numerical integration techniques (Gauss-Hermite or Monte Carlo), and the third one is a EM type algorithm based on posterior modes.
137 citations
TL;DR: The proposed model is applied to birth defects data, where continuous data on the size of infants who were exposed to anticonvulsant medications in utero are compared to controls.
Abstract: We discuss latent variable models that allow for fixed effect covariates, as well as covariates affecting the latent variable directly. Restricted maximum likelihood and maximum likelihood are used to estimate model parameters. A generalized likelihood ratio test can be used to test significance of the covariates effecting the latent outcomes. Special cases of the proposed model correspond to factor analysis, mixed models, random effects models, and simultaneous equations. The model is applied to birth defects data, where continuous data on the size of infants who were exposed to anticonvulsant medications in utero are compared to controls.
105 citations
TL;DR: Nonlinear mixed-effects models are used to describe each person's set of scores from a longitudinal design or repeated measures study by a function that includes an overall group effect plus an effect for the individual.
Abstract: Nonlinear mixed-effects models are used to describe each person's set of scores from a longitudinal design or repeated measures study by a function that includes an overall group effect plus an effect for the individual. The model is ideal for many kinds of behavioral data. Some characteristics of mixed models are reviewed in this article and illustrated by a series of examples.
98 citations
TL;DR: Laplace's method is adapted for analytic approximation within the E-step and retains much of the conceptual simplicity of the conventional EM algorithm, although the usual convergence properties are not guaranteed.
Abstract: Application of the EM algorithm for estimation in the generalized mixed model has been largely unsuccessful because the E-step cannot be determined in most instances. The E-step computes the conditional expectation of the complete data log-likelihood and when the random effect distribution is normal, this expectation remains an intractable integral. The problem can be approached by numerical or analytic approximations; however, the computational burden imposed by numerical integration methods and the absence of an accurate analytic approximation have limited the use of the EM algorithm. In this paper, Laplace's method is adapted for analytic approximation within the E-step. The proposed algorithm is computationally straightforward and retains much of the conceptual simplicity of the conventional EM algorithm, although the usual convergence properties are not guaranteed. The proposed algorithm accommodates multiple random factors and random effect distributions besides the normal, e.g., the log-gamma distribution. Parameter estimates obtained for several data sets and through simulation show that this modified EM algorithm compares favorably with other generalized mixed model methods.
62 citations
TL;DR: To identify the testing problem underlying the different rank procedures, the relations between the hypotheses in the general model and in the standard linear model are investigated in the chapter.
Abstract: Publisher Summary This chapter focuses on pure rank statistics in factorial designs. Pure rank statistics are invariant under any strict monotone transformation of the data and are robust against outliers. In addition, they are applicable to ordinal data such as scores in psychological tests, grading scales to describe the degree of the damage of plants or trees in ecological or environmental studies. The classical models of analysis of variance are generalized (ANOVA) in such a way that not only the assumption of normality of the error terms is relaxed but also the structure of the designs is introduced in a broader framework. In addition, the concept of treatment effects is redefined within this framework. To identify the testing problem underlying the different rank procedures, the relations between the hypotheses in the general model and in the standard linear model are investigated in the chapter. The chapter concludes with a discussion of random-factor model along with the rank procedures for heteroscedastic mixed models.
TL;DR: A MINQUE(1) procedure, which is minimum norm quadratic unbiased estimation (MINQUE) method with 1 for all the prior values, is suggested for estimating variance and covariance components in a bio-model for diallel crosses and a procedure of adjusted unbiased prediction (AUP) is proposed for predicting random genetic effects in the bio- model.
Abstract: A MINQUE(1) procedure, which is minimum norm quadratic unbiased estimation (MINQUE) method with 1 for all the prior values, is suggested for estimating variance and covariance components in a bio-model for diallel crosses. Unbiasedness and efficiency of estimation were compared for MINQUE(1), restricted maximum likelihood (REML) and MINQUE theta which has parameter values for the prior values. MINQUE(1) is almost as efficient as MINQUE theta for unbiased estimation of genetic variance and covariance components. The bio-model is efficient and robust for estimating variance and covariance components for maternal and paternal effects as well as for nuclear effects. A procedure of adjusted unbiased prediction (AUP) is proposed for predicting random genetic effects in the bio-model. The jack-knife procedure is suggested for estimation of sampling variances of estimated variance and covariance components and of predicted genetic effects. Worked examples are given for estimation of variance and covariance components and for prediction of genetic merits.
TL;DR: In this paper, the mixed effects model for binary responses was extended to accommodate ordinal responses in general and discrete time survival data with ordinal response in particular, and a Newton-Raphson estimation procedure was proposed without resorting to numerical, approximation-based or Monte Carlo integration techniques.
Abstract: The mixed effects model for binary responses due to Conaway (1990, A Random Effects Model for Binary Data) is extended to accommodate ordinal responses in general and discrete time survival data with ordinal responses in particular. Given a multinomial likelihood, cumulative complementary log-log link function, and log-gamma random effects distribution, the resulting marginal likelihood has a closed form. As a result, a Newton-Raphson estimation procedure is feasible without resorting to numerical, approximation-based, or Monte Carlo integration techniques. The parameters in the model have a proportional hazards interpretation in terms of multivariate discrete time data with ordinal responses. Using data from a psychological example, the proposed method is compared with other mixed effects approaches as well as population-averaged models.
TL;DR: In this article, the effects of measurement error in intra-individual covariates on inference for the nonlinear mixed effects model were investigated for a controlled variable measurement error model, and it was shown that a major consequence may be substantial bias in estimates of parameters characterising intraindividual variation.
Abstract: SUMMARY Little is known about the effects of measurement error in intra-individual covariates on inference for the nonlinear mixed effects model. We investigate this issue for a controlled variable measurement error model, and find that a major consequence may be substantial bias in estimates of parameters characterising intra-individual variation. Estimation of population parameters may also be affected. The dramatic effect of measurement error on estimation of intra-individual variance parameters has implications not only in the repeated measurement context, but also in individual nonlinear regression models.
TL;DR: The authors investigated the efficiency of conditional and semiparametric likelihood methods for estimating the regression parameters of mixed-effects logistic regression models and showed that the asymptotic relative efficiency of the conditional likelihood estimators relative to parametric estimators is a decreasing function of within-cluster covariate correlation.
Abstract: SUMMARY We investigate the efficiency of likelihood methods for estimating the regression parameters of mixed-effects logistic regression models. One approach uses a conditional likelihood which eliminates the random intercept terms. A second uses the likelihood generated from the marginal distribution of the data where the random intercepts are integrated out. Parametric estimates result from assuming a parametric form for the intercept distribution, whereas we obtain semiparametric estimates when the intercept distribution is left unspecified. We present an expression which shows that the asymptotic relative efficiency of conditional likelihood estimators relative to parametric estimators is a decreasing function of within-cluster covariate correlation. Simulation results show the same for the asymptotic relative efficiency of the semiparametric estimator relative to the conditional. For fixed covariate correlation, the asymptotic relative efficiency of the parametric versus the conditional increases as cluster sizes increase. Example data further illustrate our findings.
TL;DR: In this paper, marginal maximum likelihood (MML) estimates of genetic variances obtained under a Poisson "sire" model (PSM) and under an animal model (PAM), accounting for half-sib relationships, were compared in two simulations.
Abstract: Count data models are developed for animal breeding applications to account for more variability than in a Poisson mixed effects model. A gamma distribution is assigned to Poisson parameters, thereby leading to a negative binomial model. The natural log of the expected value of the Poisson parameter is expressed as a linear function of fixed and random polygenic effects. The negative binomial and Poisson mixed models were compared in two simulations. In the first, marginal maximum likelihood (MML) estimates of genetic variances obtained under a Poisson "sire" model (PSM) and under a Poisson "animal" model (PAM), accounting for half-sib relationships, were different, contrary to what occurs in a Gaussian mixed linear model. MML estimates of genetic variance under a negative binomial "sire" model were less biased than estimates under a PSM, and had a slightly smaller mean squared error (MSE). The second simulation compared "animal" models in which the variance of the residuals was larger than the genetic variance. Empirical relative bias and MSE of MML estimates of genetic variance were larger under a PAM that ignored the residuals than under a negative binomial animal model. Differences in performance widened as genetic variance increased. An application to the analysis of number of artificial inseminations until conception in dairy heifers is presented to illustrate potential differences in genetic variance estimates under the two animal models.
TL;DR: In this paper, two approaches for the analysis of growth data by means of longitudinal mixed linear models are discussed. But neither of them is suitable for real data analysis, and neither can be applied to real-world data.
TL;DR: In this paper, the authors present statistical tests for departures from random expectation in spatial memory tasks and derive analytical expressions for the probability mass functions, expectations, and variances associated with each type of null hypothesis.
Abstract: We present statistical tests for departures from random expectation in spatial memory tasks. We consider two common protocols for spatial memory experiments. In the first one, subjects are allowed to search a fixed number of sites. In the second protocol, subjects are allowed to search until they achieve a fixed number of successes. In either of these protocols, the subjects involved may or may not revisit sites that have been previously searched or exploited. This yields four situations to consider: fixed number of sites searched or fixed number of successes, with or without revisits. We derive analytical expressions for the probability mass functions, expectations, and variances associated with each type of null hypothesis. We present three statistical tests of these hypotheses: the Kolmogorov-Smirnov test, the ordinary sign test, and theZ test. We use our results to demonstrate a priori calculation of sample sizes and statistical power and to consider a mixed model of sampling with and without replacement.
TL;DR: Maximum likelihood methodology was applied to determine the mode of inheritance of rare binary traits with data structures typical for swine populations with main emphasis on the detection of major genes acting on a polygenic background.
Abstract: Maximum likelihood methodology was applied to determine the mode of inheritance of rare binary traits with data structures typical for swine populations. The genetic models considered included a monogenic, a digenic, a polygenic, and three mixed polygenic and major gene models. The main emphasis was on the detection of major genes acting on a polygenic background. Deterministic algorithms were employed to integrate and maximize likelihoods. A simulation study was conducted to evaluate model selection and parameter estimation. Three designs were simulated that differed in the number of sires/number of dams within sires (10/10, 30/30, 100/30). Major gene effects of at least one SD of the liability were detected with satisfactory power under the mixed model of inheritance, except for the smallest design. Parameter estimates were empirically unbiased with acceptable standard errors, except for the smallest design, and allowed to distinguish clearly between the genetic models. Distributions of the likelihood ratio statistic were evaluated empirically, because asymptotic theory did not hold. For each simulation model, the Average Information Criterion was computed for all models of analysis. The model with the smallest value was chosen as the best model and was equal to the true model in almost every case studied.
TL;DR: In this article, a method of robust estimation of shift and scale parameters in a mixed unbalanced interlaboratory model is described. But this method is not suitable for highdimensional nonlinear equations and it is ''inconvenient'' for assessing variability of estimators.
Abstract: This paper describes a method of robust estimation of shift and scale parameters in a mixed unbalanced interlaboratory model. Estimators presented result from \easily computable" Fr echet differentiable functionals which enjoy some optimal properties in a small neighborhood of the model. A rigorous treatment of their asymptotic behaviour under departures from the model assumptions and a simulation study are given. 1. Introduction. A method of robust analysis of random effects in mixed models is introduced in the paper by Rocke (1991) [see also Iglewicz (1983)] and a general treatment of robust estimators of variance components is presented by Fellner (1986). Rocke proposes estimators of variance components applying robust scale estimators to residuals and laboratory effects. In the rst step he uses Huber’s method to robustly assess location parameters. A similar point of view is taken by Lischer (1994), who improves breakdown properties of the estimators. He also gives a comprehensive practical motivation for the use of robust methods in interlaboratory experiments. The above-mentioned estimation technique may lead to high-dimensional nonlinear equations and it is \inconvenient" for assessing variability of estimators. A possible improvement may be in the simultaneous estimation of xed effects and components of variation, where problems of a rigorous asymptotic treatment, smoothness of estimators and their optimality become more feasible. In a preliminary study by Bednarski, Zmy slony and Zontek (1992) it is shown that Fr echet differentiability of statistical functionals easily leads to reasonable robust estimators of variance components and treatment fixed effects in a simple interlaboratory model. The basic idea there is to construct a smooth Fisher consistent functional T for the parameter De 1;:::; a; ; ee t at the model
TL;DR: In this paper, the authors present a model that incorporates random trends through time and also allows correlations to exist among observations taken at the same time from the different units, based on intuitively reasonable sums of squares that, under the usual normality assumptions, can be shown to possess desirable distributional properties.
Abstract: The use of longitudinal data is common in environmental and agricultural applications where interest lies in trends in a response variable through time. Various methodologies have been applied in the analysis of such data. Although many methods allow for correlation among the repeated measurements taken on the same experimental unit, nearly all assume independence of those units. Additionally, in mixed model settings, where trends are random, interest has often focused on Best Linear Unbiased Predictors (BLUP's), rather than on variance components, and on techniques for large unbalanced datasets, rather than for relatively small balanced datasets. We present a model that incorporates random trends through time and also allows correlations to exist among observations taken at the same time from the different units. The analysis (for the balanced case) focuses on variance components and an overall fixed trend through time, and is based on intuitively reasonable sums of squares that, under the usual normality assumptions, can be shown to possess desirable distributional properties.
TL;DR: Bootstrapping the log-likelihood differences of non-hierarchical non-linear mixed effects models and constructing a bootstrap confidence interval on the LLDs is proposed for comparing the goodness-of-fit of such models.
TL;DR: In this article, an extension of the iterative re-weighted least squares procedure for generalized linear models is presented on a practical data set involving carcass classification of cattle, which is analyzed as overdispersed binomial proportions with fixed and random effects and associated components of variance.
Abstract: 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.
TL;DR: Results of a simulation study with two methods of analysis of data simulated under the mixed model on a 232‐member pedigree indicate that, unlike PAP, the starting configurations of most parameter estimates do not substantially influence the final parameter estimates in analysis with MIXD.
Abstract: Results of a simulation study with two methods of analysis of data simulated under the mixed model on a 232-member pedigree are presented. The programs Pedigree Analysis Package (PAP), which approximate the likelihoods needed in a complex segregation analysis, and MIXD, which uses Monte Carlo Markov chain (MCMC), to estimate likelihoods were used. PAP obtained unbiased estimates of the major locus genotype means and the gene frequency, but biased estimates of the environmental variance component, and thus the heritability. A substantial fraction of the runs did not converge to an internal set of parameter estimates when analyzed with PAP. MIXD, which uses the Gibbs sampler to perform the MCMC sampling, produced unbiased estimates of all parameters with considerably more accuracy than obtained with PAP, and did not suffer from convergence of estimates to the boundary of the parameter space. The difference in behavior and accuracy of parameter estimates between PAP and MIXD was most apparent for models with either high or low residual additive genetic variance. Thus in situations where accuracy of the model is important, use of MCMC methods may be useful. In situations where less accuracy is needed, approximation methods may be adequate. Practical issues in using MCMC as implemented in MIXD to fit the mixed model are also discussed. Results of the simulations indicate that, unlike PAP, the starting configurations of most parameter estimates do not substantially influence the final parameter estimates in analysis with MIXD.
TL;DR: In this article, a method of estimation for generalised mixed models is applied to the estimation of regression parameters in a proportional hazards model with time dependent frailty, where a parameter representing change over time is introduced and is modelled in turn into a fixed effect, a normally distributed random effect and a longitudinal effect in which the random component relates to the patient characteristics.
Abstract: SUMMARY
A method of estimation for generalised mixed models is applied to the estimation of regression parameters in a proportional hazards model with time dependent frailty. A parameter representing change over time is introduced and is modelled in turn into a fixed effect, a normally distributed random effect and a longitudinal effect in which the random component relates to the patient characteristics. Both maximum likelihood and residual maximum likelihood estimators are given.
TL;DR: In this paper, the authors address the important practical problem of selecting covariates in mixed linear models when the covariance structure is known from the data collection process and there are a possibly large number of covariates available.
TL;DR: In this article, a simulation study is carried out for the analysis of clustered survival data based on the generalised linear mixed model (GLMM) methodology, where data are simulated for a proportional hazards model containing random components corresponding to a cluster effect and a subject effect.
Abstract: A simulation study is carried out for the analysis of clustered survival data based on the generalised linear mixed model (GLMM) methodology. Data are simulated for a proportional hazards model containing random components corresponding to a cluster effect and a subject effect. The performance of approximate maximum likelihood (ML) and residual maximum likelihood (REML) estimation of the regression parameter and variance components are evaluated. For a range of true parameter values,average biases of estimators, standard error of average bias and standard error of estimates over simulations are reported. A discussion of the use of random effects predictions is also given.
TL;DR: In this paper, a method for simultaneously estimating the fixed parameters of a generalized linear mixed-effects model and the random-effects distribution of which no parametric assumption is made is discussed.
Abstract: We discuss a method for simultaneously estimating the fixed parameters of a generalized linear mixed-effects model and the random-effects distribution of which no parametric assumption is made. In addition, classifying subjects into clusters according to the random regression coefficients is a natural by-product of the proposed method. An alternative approach to maximum-likelihood method, maximum-penalized-likelihood method, is used to avoid estimating “too many” clusters. Consistency and asymptotic normality properties of the estimators are presented. We also provide robust variance estimators of the fixed parameters estimators which remain consistent even in presence of misspecification. The methodology is illustrated by an application to a weight loss study.
TL;DR: I use Laguerre polynomials to model growth curves or time--response curves known to approach an asymptote as time approaches infinity and present two examples that involve groups of subjects recovering from surgery.
Abstract: I use Laguerre polynomials to model growth curves or time--response curves known to approach an asymptote as time approaches infinity. An example is with measurements on a variable or variables from subjects recovering from surgery. These variables can often vary in a non-monotonic fashion for which a functional form of the curve is unknown. Using a longitudinal data mixed model, one can include in the model random subject effects, within-subject serial correlation and fixed or time varying covariates. I present two examples that involve groups of subjects recovering from surgery.