Showing papers on "Mixed model published in 1991"

A Unified Approach to Mixed Linear Models

Abstract: The mixed model equations as presented by C. R. Henderson offers the base for a methodology that provides flexibility of fitting models with various fixed and random elements with the possible assumption of correlation among random effects. The advantage of teaching analysis of variance applications from this methodology is presented. Particular emphasis is placed upon the relationship between choice of estimable function and inference space.

Some New Estimators for Small-Area Means With Application to the Assessment of Farmland Values

Abstract: Regression models that account for main state effects and nested county effects are considered for the assessment of farmland values. Empirical predictors obtained by replacing the unknown variances in the formulas of the optimal predictors by maximum likelihood estimates are presented. The computations are carried out by simple iterations between two SAS procedures. Estimators for the prediction variances are derived, and a modification to secure the robustness of the predictors is proposed. The procedure is applied to data on nonirrigated cropland in the Corn Belt states and is shown to yield predictors with considerably lower prediction mean squared errors than the survey estimators and other regression-type estimators.

Multiplicative effects in mixed model analysis of variance

Abstract: SUMMARY A class of two-way mixed analysis of variance models is proposed, in which the fixed and random effects enter multiplicatively. Equations are developed for iterative computation of maximum likelihood estimates via a scoring algorithm. Parameter estimation and hypothesis testing are illustrated on a set of plant genetics data.

Mixed model segregation analysis of LDL-C concentration with genotype–covariate interaction

Abstract: Mixed model complex segregation analyses have in the past ignored the possibility of genotype-covariate interaction. Only in the nonmixed model with polygenic heritability equal to zero have segregation analyses been performed that allowed for genotype specific regression of the phenotype on covariates. We present an extension of Hasstedt's [1982] mixed model likelihood approximation which does allow for genotype-covariate interaction in the mixed model. Following description of this approximation, we validate the likelihood calculation by a Monte Carlo procedure based on the actual pedigree and missing data structure used in a complex segregation analysis of low density plus very low density lipoprotein cholesterol (LDL-C + VLDL-C) in baboons. The observed averages of the bootstrap parameter estimates adequately recover the generating values, which included parameters specifying genotype-covariate interaction. We then applied both a traditional complex segregation analysis and an analysis with genotype-covariate interaction to test for the presence of a major locus affecting LDL-C levels in baboons. The model including genotype-covariate interaction was significantly different from the model without interactions, and strongly supported the hypothesis that there is a segregating Mendelian locus as opposed to a random environmental factor. This major locus accounts for approximately 46% of the variance in LDL-C levels, as compared to 40% explained by a locus with no genotype-covariate interaction.

Restricted BLUP for mixed linear models

Abstract: A new estimation procedure for mixed regression models is introduced. It is a development of Henderson's best linear unbiased prediction procedure which uses the joint distribution of the observed dependent random variables and the unknown realisations of the random components of the model. It is proposed to replace the likelihood of the observations given the random components by the asymptotic likelihood of the maximum likelihood estimators and the prior distribution of the random components by a restricted prior distribution which is consistent with the usual restrictions placed on the random components when they are considered conditionally fixed.

On nonlinear random effects models for repeated measurements

Abstract: Linear random effects models for longitudinal data discussed by Laird and Ware (1982), Jennrich and Schluchter (1986), Lange and Laird (1989), and others are extended in a straight forward manner to nonlinear random effects models. This results in a simple computational approach which accommodates patterned covariance matrices and data insufficient for fitting each subject separately. The technique is demonstrated with an interesting medical data set, and a short, simple SAS PROC IML program based on the EM algorithm is presented.

An approach to comparing treatments based on repeated measures

Abstract: SUMMARY A mixed effects model is assumed for comparing treatments based on repeated measurements. The covariance structure of the data is interpreted as some systematic inhomogeneity of individual profiles along the time axis, rather than as serial correlation. Then a method of comparing treatment effects is proposed as well as that of testing the homogeneity of individual profiles. A follow-up analysis of residuals for the resulting model is also mentioned.

The use of fixed effect models and mixed models to estimate single gene associated effects on polygenic traits

Abstract: Single gene associated effects on polygenic traits may often be confounded with the effects of a non-random genetic relationship between individuals sharing a particular allele of the investigated gene. Two different statistical models are suggested to separate the single gene associated effects from the remaining additive genotype: a fixed effect model with ancestor variables and a mixed model with random effects of the additive genotypes of the individual animals (individual animal model). The use of the models is illustrated by an example from an experiment with the chicken major histocompatibility complex (MHC) gene region.

Exact confidence sets for variance components in unbalanced mixed linear models

Abstract: We present a general procedure for obtaining an exact confidence set for the variance components in a mixed linear model. The procedure can be viewed as a generalization of the ANOVA method used with balanced models. Our procedure uses, as pivotal quantities, quadratic forms that are distributed independently as chi-squared variables. These quadratic forms are constructed with reference to spaces that are orthogonal with respect to the covariance matrix of the observation vector, which is a function of the variance components. For balanced models, these pivotal quantities simplify to multiples of the sums of squares used in the ANOVA method. An exact confidence set for the vector of ratios of the effect variances to the error variance is also presented, based on the same collection of quadratic forms. Computing formulas for calculating approximations to these confidence sets are presented, and the results of their application to several two-way data sets are given.

Least Squares and Minimum MSE Estimators of Variance Components in Mixed Linear Models

Abstract: The paper deals with the quadratic invariant estimators of the linear functions of variance components in mixed linear model. The estimator with locally minimal mean square error with respect to a parameter ϑ is derived. Under the condition of normality of the vector Y the theoretical values of MSE of several types of estimators are compared in two different mixed models; under a different types of distributions a simulation study is carried out for the behaviour of derived estimators.

A Mixed Model for Loss Ratio Analysis

Abstract: The model introduced may be treated as a mixed two-way analysis of variance with fixed company effects and random time effects. Further, the risk volumes are integrated into the model in such a way that the unexplained variance is inversely proportional to the risk volume of each company. The proposed model is used to analyze loss ratio data from the general insurance market in Kuwait. The maximum likelihood estimates of the structural parameters are obtained. These estimates are then used to compute the loss ratios and solvency margins for the four domestic insurance companies.

Computing initial estimates with mixed effects models: A general method of moments

Abstract: SUMMARY Use of the maximum likelihood method for parameter estimation in a general mixed effects model requires initial estimates for all the model parameters. When the model is complicated, it may not be easy to obtain such estimates, especially estimates of the variances and covariances. A computational strategy is given for obtaining initial estimates. It is based on a general formulation of the method of moments for mixed effects models which allows the variances and covariances to be estimated for fixed values of the regression parameters. A number of numerical algorithms has been proposed for obtaining maximum, or restricted maximum, likelihood estimates of the parameters of a linear mixed effects model (Hemmerle & Hartley, 1973; Jennrich & Sampson, 1976; Corbeil & Searle, 1976; Laird & Ware, 1982). The maximum likelihood method can also be used with a model in which some fixed effects enter nonlinearly, while all random effects enter linearly; see ? 2. With any of the algorithms it is important to have good initial estimates for all the model parameters. One computational strategy sometimes used for finding initial estimates is to first maximize the full likelihood or log likelihood function, referred to here as the objective function, over a mesh, or grid, of points covering the parameter space and then to use the mesh point corresponding to the maximum as the initial estimate. However, with a complicated mixed effects model the dimension of the parameter space can be large enough to make this strategy impractical. Also, this strategy implies that practical bounds can be placed on all variances and covariances of the model, and though this may be possible, it may not be easy to do, especially when the model is complicated. This paper presents a modification of this strategy. It makes use of a general formulation of the method of moments for mixed effects models which is applicable to obtaining estimates of the variances and covariances of such models for fixed values of the regression parameters. Thus initial estimates of the variances-covariances are obtained without recourse to a grid search over at least those dimensions of the parameter space corresponding to the variances and covariances. The method of moments formulation permits estimates to be obtained under various constraints which might be imposed on the variances and covariances. It is also applicable to multivariate mixed effects models, and

Empirical Bayes Analysis for Systems of Mixed Models with Linked Autocorrelated Random Effects

Abstract: Empirical Bayes concepts are implemented in a simultaneous analysis of a system of mixed linear models having linked and serially correlated random effects. Emphasis is placed on the estimation of the random effects and exploration of the relationships between them. Application is made to the investigation of several series of laboratory assay data that were observed during overlapping time intervals and were therefore subjected to common systematic errors, or “daily effects.” The motivation for this work was the need to investigate methods of adjustment for such daily effects, and to estimate the degree to which concurrently run series are impacted in common. Attention is given to the construction of confidence intervals for daily effects. Tractable methods are proposed that yield approximately correct coverage for large samples. Although derived within a Bayes-empirical Bayes framework, these intervals are somewhat similar to intervals constructed by the method of Kackar and Harville. Implement...

Improved estimation of variance components in balanced hierarchical mixed models

Abstract: The problem of simultaneous estimation of variance components is considered for a balanced hierarchical mixed model under a sum of squared error loss. A new class of estimators is suggested which dominate the usual sensible estimators. These estimators shrink towards the geometric mean of the component mean squares that appear in the ANOVA table. Numerical results are tabled to exhibit the improvement in risk under a simple model.

GENETICS AND BREEDING Derivative-Free Restricted Maximum Likelihood Estimation in Animal Models with a Sparse Matrix Solver'

Abstract: Estimation of (co)variance components by derivative-free REML requires repeated evaluation of the log-likelihood function of the data. Gaussian elimination of the augmented mixed model coefficient matrix is often used to evaluate the likelihood function, but it can be costly for animal models with large coefficient matrices. This study investigated the use of a direct sparse matrix solver to obtain the log-likelihood function. The sparse matrix package SPARSPAK was used to reorder the mixed model equations once and then repeatedly to solve the equations by Cholesky factorization to generate the tenns required to calculate the likelihood. The animal model used for comparison contained 19 fixed levels, 470 maternal permanent environmental effects, and 1586 direct and 1586 maternal genetic effects, resulting in a coefficient matrix of order 3661 with .3% nonzero elements after including numerator relationships. Compared with estimation via Gaussian elimination of the unordered system, utilization of SPARSPAK required 605 and 240 times less central processing unit time on mainframes and personal computers, respectively. The SPARSPAK package also required less memory and provided solutions for all effects in the model.