Showing papers on "Mixed model published in 1993"

Generalized linear mixed models a pseudo-likelihood approach

Abstract: A useful extension of the generalized linear model involves the addition of random effects andlor correlated errors. A pseudo-likelihood estimation procedure is developed to fit this class of mixed models based on an approximate marginal model for the mean response. The procedure is implemented via iterated fitting of a weighted Gaussian linear mixed model to a modified dependent variable. The approach allows for flexible specification of covariance structures for both the random effects and the correlated errors. An estimate of an additional dispersion parameter for underlying exponential family distributions is optionally automatic. The method allows for subject-specific and population-averaged inference, and the Salamander data example from McCullagh and Nelder (1989) is used to illustrate both.

Covariance structure selection in general mixed models

Abstract: This article describes a unified approach to variance modeling and inference in the context of a general form of the normal-theory linear mixed model. The primary variance modeling objects are parameterized covari-ance structures, examples being diagonal, compound-symmetry, unstructured, timeseries, and spatial. These structures can enter in two different places in the general mixed model, and the combination of one or both of these places with the variety of structures provides a rich class of variance models. The approach is likelihood-based, and involves the use of both maximum likelihood and restricted maximum likelihood. Two examples provide illustration.

Laplace's approximation for nonlinear mixed models.

Abstract: SUMMARY An approximation to Laplace's method for integrals is applied to marginal distributions of data arising from models in which both fixed and random effects enter nonlinearly. The approach provides alternative derivations of some recent algorithms for fitting such models, and it has direct ties with Gaussian restricted maximum likelihood and the accompanying mixed model equations.

The nonlinear mixed effects model with a smooth random effects density

Abstract: SUMMARY The fixed parameters of the nonlinear mixed effects model and the density of the random effects are estimated jointly by maximum likelihood. The density of the random effects is assumed to be smooth but is otherwise unrestricted. The method uses a series expansion that follows from the smoothness assumption to represent the density and quadrature to compute the likelihood. Standard algorithms are used for optimization. Empirical Bayes estimates of random coefficients are obtained by computing posterior modes. The method is applied to data from pharmacokinetics, and properties of the method are investigated by application to simulated data.

REML estimation for survival models with frailty.

Abstract: A method of estimation for generalised mixed models is applied to the estimation of regression parameters in proportional hazards models for failure times when there are repeated observations of failure on each subject. The subject effect is incorporated into the model as a random frailty term. Best linear unbiased predictors are used as an initial step in the computation of maximum likelihood and residual maximum likelihood estimates.

Selection Index and Introduction to Mixed Model Methods

Abstract: The principles covered in Selection Index and Introduction to Mixed Model Methods are developed using easy-to-follow rules for expected values and definitions of variances and covariances. The principles are reenforced with practice problems and exams. Indexes in retrospect as well as restricted indexes are developed. Selection for embedded traits such as maternal, grandmaternal, or paternal effects and for categorically measured traits are discussed. The introduction to mixed models applies expected values to create rules to set up least squares and mixed model equations. The similarity between selection index and mixed model methods is demonstrated with simple examples. The third part of the book introduces basic methods used to estimate genetic parameters such as heritabilities, repeatabilities, and genetic correlations needed for both selection index and mixed model methods. This is an excellent text for advanced undergraduates, graduate students, and researchers who use selection index methods.

Random Factors in ANOVA

Abstract: Fixed and Random Factors Designs With Random Factors Statistical Analysis Statistical Assumptions Interpretation Statistical Power Computer-Assisted Analysis of Designs With Random Factors Conclusion

REML Estimation with Exact Covariance in the Logistic Mixed Model

Abstract: Residual maximum likelihood (REML) estimation is adapted to certain logistic mixed models for which representation of the unconditional mean as a linear function of the fixed effects is possible. Only the first two moments of the unconditional distribution need be evaluated, and except for the form of the covariance, the maximization algorithm carries over directly from linear models. The exact unconditional covariance is computed from the logistic-normal mixture for input into the algorithm. Residual log-likelihood plots provide a means of inference for the dispersion components. The Taylor series approximation to this covariance, besides exhibiting insufficient accuracy, fails to be positive definite for large values of the dispersion components. As a consequence, REML loglikelihood plots based on this approximate covariance attribute misleadingly high precision to the dispersion component estimates. The method is presented in the context of a salamander mating experiment in which random effects corresponding to male and female animals occur in a crossed design. Analyses of these data by several methods are compared.

Heterogeneity in Mantel-Haenszel-type models

Abstract: SUMMARY The Mantel-Haenszel problem involves inferences about a common odds ratio in a set of 2 x 2 tables. Although it is a fairly standard practice to test whether the odds ratios are indeed constant, there is remarkably little methodology available for proceeding when there is evidence of some heterogeneity. Our interest is in models where the log odds ratios Ok, for tables k = 1, 2, .. ., K, are thought of as a sample from a population with mean 0 and standard deviation a, and inferences are desired regarding the parameters (0, a). By Mantel-Haenszel-type models we mean to include the generalization involving pairs of Poisson observations rather than 2 x 2 tables, where the Ok are logarithms of ratios of the Poisson means within pairs. Direct computation of the likelihood function for (0, a) in these settings involves numerical integration, and the main point here is a simple approximation to this likelihood. The approximation is based on Laplace's method, and is very accurate for practical applications. Inference regarding the parameter 0 of primary interest may be made from the profile likelihood function. An alternative approach to likelihood methods, based on approximations to the marginal means and variances, is also considered. The methods explored here can be readily generalized to settings where the parameters Ok depend on covariables as well.

REML estimation in unbalanced multivariate variance components models using an EM algorithm

Abstract: Estimation of covariance matrices when data are missing is a difficult problem. In the multivariate mixed model setting, problems can arise either because the data are unbalanced or because the response vectors are incomplete. Jennrich and Schluchter (1986, Biometrics 42, 805-820) and Laird, Lange, and Stram (1987, Journal of the American Statistical Association 82, 97-105) have proposed EM algorithms for estimating the parameters associated with longitudinal data models. In this paper a new EM algorithm is proposed for the general multivariate mixed model. The algorithm is derivativefree and based on an algorithm for balanced data proposed by Calvin and Dykstra (1991, Annals Qf Statistics 19, 850-869). Once the algorithm is described it is used to produce estimates of the variance components matrices associated with a completely random two-factor nested model for a bivariate response vector. The data for this example come from Calvin and Sedransk (1991, Journal of the American Statistical Association 86, 36-48), who analyze a study of the quality of care received by cancer patients receiving radiation therapy.

Marginal maximum likelihood estimation of variance components in Poisson mixed models using Laplacian integration

Abstract: more commonly used expectation-maximization type algorithm for MML. Numerically, however, a different sequence of iterates is obtained, although the same variance component estimates should result. The Laplacian algorithm is precisely DFREML (derivative free REML) optimization when applied to normally distributed data, and could then be termed DFMML (derivative-free marginal maximum likelihood). Because DFMML is based on

The performance of some real-time statistical flood forecasting models seen through multicriteria analysis

Abstract: In the present work, seven statistical-routing models have been developed and applied to the Medjerdah River (Tunisia) for forecasting the extreme flood events at Jendouba for different forecasting horizons. The performance of the models are characterized by statistical measures of precision, peak error and peak delay between the measured and forecast flow and their variations. Due to the important number of criteria, a multi-criteria analysis is used to rank the models according to four forecasting horizons. The mixed models seem to be the best ones for short (2–4 h) as well as long (6–8 h) forecasting horizons.

Technical note: computing tests of fixed effects in a restricted class of mixed models.

Abstract: Inferences about fixed effects in mixed linear models are important in a variety of animal science studies. The statistical theory for making such inferences is well known, and if the variance components are known up to a proportional- ity constant, then optimal exact tests can be per- formed. Computing the test statistics, however, can still be problematic when the random effects have many levels. In practice, approximate tests that are easily computed but less efficient are usually em- ployed. This article describes reduction in error sum of squares procedures for performing the exact test and for computing associated confidence intervals. By taking advantage of iterative algorithms for solving Henderson's mixed-model equations, the tests can be performed without inverting the covariance matrix or computing a generalized inverse of the mixed-model coefficient matrix. The procedures are illustrated on an animal model that has three random effects, two with 1,372 levels and one with 450 levels.

Segregation analysis under an alternative formulation for the mixed model.

Abstract: Different approaches for segregation analysis that have been proposed in the past are compared to a new approach by Femando et al. [submitted], the finite polygenic mixed model, analyzing a real and a simulated data set

A test of the missing data mechanism for repeated measures data

Abstract: The occurrence of missing data is an often unavoidable consequence of repeated measures studies. Fortunately, multivariate general linear models such as growth curve models and linear mixed models with random effects have been well developed to analyze incomplete normally-distributed repeated measures data. Most statistical methods have assumed that the missing data occur at random. This assumption may include two types of missing data mechanism: missing completely at random (MCAR) and missing at random (MAR) in the sense of Rubin (1976). In this paper, we develop a test procedure for distinguishing these two types of missing data mechanism for incomplete normally-distributed repeated measures data. The proposed test is similar in spiril to the test of Park and Davis (1992). We derive the test for incomplete normally-distribrlted repeated measures data using linear mixed models. while Park and Davis (1992) cleirved thr test for incomplete repeatctl categorical data in the framework of Grizzle Starmer. and...

Choosing and modeling your mixed linear model

Abstract: The literature has recently seen much debate as to what is the most appropriate way to specify the mixed linear model. Three different models are currently in wide use. Two of the models are formulated in terms of constants and random variables while the third specifies the mean and variance-covariance structure of the data as the model. This paper will relate the models for a general design. The mean and variance-covariance formulation will be used to unify the models, incorporate randomization restrictions, motivate when a factor should be called fixed or random, incorporate the inference space into the analysis of the problem and incorporate confounding factors into the design. The most common mixed model discussed in design texts will be shown to have some limitations in the model formulation stage of an analysis.

A simple method for fitting a linear model involving variance components

Abstract: This paper describes combined maximum likelihood estimation of treatment effects and variance components in a linear mixed model. Estimates are obtained by a simple modification of iterative weighted least squares. The method is illustrated using plating efficiencies of Solanum and Lycopersicon species.

Simple estimations of the variance components and the fixed and random effects in mixed, three-stage, hierarchal models

Abstract: Kansas State University SIMPLE ESTIMATIONS OF THE VARIANCE COMPONENTS AND THE FIXED AND RANDOM EFFECTS IN MIXED, THREE-STAGE, HIERARCHAL MODELS C. Philip Cox Iowa State University, Department of Statistics 312 Snedecor Hall, Ames, IA 50011-1210 FRR (Fixed, Random, Random) hierarchal models in which the first-stage" elements are fixed and the second and third-stage elements are random, are used in analyses of comparative experiments and, extensively, in animal breeding contexts where, in the latter, estimates of the second-stage elements and of combinations of them with first-stage elements, are of practical interest. The two procedures, i) empirical BLUP (Best Linear Unbiased Prediction) and ii) a Bayesian approach, used when the ratio of the within-second-stages and the within-third-stages variances is unknown are 'computationally intensive'. When the ratio of the secondto the third-stage variances is large, an alternative and computationally simpler procedure considered here is applicable. This approach provides estimates, including those of the variance components, which jointly maximize likelihood. Another simple method is proposed for further investigation. In all the procedures, estimates of the second-stage elements are obtained by centering or shrinkage translations from the observed means. It is shown that the validity of these adjustments is critically dependent on the distributional assumption made for the second-stage elements. The adjustments will not be centering unless the distribution is Gaussian, in particular, or 'centri-modal' in general.

The estimation of fixed effects in a mixed linear model

Abstract: The estimation of fixed effects is considered for small, unbalanced, mixed linear models. The two-stage estimator, in which the variance components are first estimated by ML or REML, is compared to the intra-block (IB) estimator, the ordinary least squares (OLS) estimator (ignoring the random effects) and the Gauss-Markov (GM) estimator. Comparison is made, based on 100 simulated data sets each, for 6 designs (3 BIBD's and 3 unbalanced designs). In comparing loss of information, relative to the GM lower bound, the two-stage procedures (using either ML or REML) are recommended for all but the smallest and least balanced design. The study also compared estimates of the variance of the two-stage estimators, using either the GM lower bound or the KackarHarville (KH) approximation. Estimators of the variance using REML estimates of the variance components are recommended, since estimators using ML estimates were seriously biased downward for all designs considered.

GENETICS AND BREEDING Solutions to a System of Equations Involving a First-Order Autoregressive Process

Abstract: Animal breeders have often assumed zero covbriances when modeling random effects of biological processes. This assumption changed, at least for genetic effects, with Henderson's rules for the calculation of the inverse of the numerator relationship matrix (5). However, most other random effects are still assumed to ~e uncorrel~ted because alternative assump­ tIons are neIther obvious nor easy to impleA first-order autoregressive process is proposed for modeling certain random effects in instances for which a more general. model would involve too many (co)~anance components. This process ~eqUlres only two parameters, and its Inverse-needed for its inclusion in mixed model methodology-is tridiag­ onal and easy to obtain. The (co)variance matrix for effects that follow a first-order autoregressive structure is introduced and the rules for obtaining its inverse ar~ derived and outlined in an algorithm (an example is also given). Incorporation of such a structure into the mixed model equations is also discussed, and an itera­ tive procedure for obtaining solutions to this potentially large system of equations is outlined.

Optimal Repeated Measurements Designs Under Non-Additive Mixed Effects Models

Abstract: The robustness of some optimality results on repeated measurements designs is studied when tbe effect due to the units is random and an interaction due to the direct and residual effect of treatments is taken into account.