Henderson's Mixed Model Equations Research Articles

A new REML (parameter expanded) EM algorithm for linear mixed models

SummaryLinear mixed models are regularly applied to animal and plant breeding data to evaluate genetic potential. Residual maximum likelihood (REML) is the preferred method for estimating variance parameters associated with this type of model. Typically an iterative algorithm is required for the estimation of variance parameters. Two algorithms which can be used for this purpose are the expectation‐maximisation (EM) algorithm and the parameter expanded EM (PX‐EM) algorithm. Both, particularly the EM algorithm, can be slow to converge when compared to a Newton‐Raphson type scheme such as the average information (AI) algorithm. The EM and PX‐EM algorithms require specification of the complete data, including the incomplete and missing data. We consider a new incomplete data specification based on a conditional derivation of REML. We illustrate the use of the resulting new algorithm through two examples: a sire model for lamb weight data and a balanced incomplete block soybean variety trial. In the cases where the AI algorithm failed, a REML PX‐EM based on the new incomplete data specification converged in 28% to 30% fewer iterations than the alternative REML PX‐EM specification. For the soybean example a REML EM algorithm using the new specification converged in fewer iterations than the current standard specification of a REML PX‐EM algorithm. The new specification integrates linear mixed models, Henderson's mixed model equations, REML and the REML EM algorithm into a cohesive framework.

An evaluation of ridge estimator in linear mixed models: an example from kidney failure data

ABSTRACTThis paper is concerned with the ridge estimation of fixed and random effects in the context of Henderson's mixed model equations in the linear mixed model. For this purpose, a penalized likelihood method is proposed. A linear combination of ridge estimator for fixed and random effects is compared to a linear combination of best linear unbiased estimator for fixed and random effects under the mean-square error (MSE) matrix criterion. Additionally, for choosing the biasing parameter, a method of MSE under the ridge estimator is given. A real data analysis is provided to illustrate the theoretical results and a simulation study is conducted to characterize the performance of ridge and best linear unbiased estimators approach in the linear mixed model.

Genetic evaluation for three-way crossbreeding.

BackgroundCommercial pig producers generally use a terminal crossbreeding system with three breeds. Many pig breeding organisations have started to use genomic selection for which genetic evaluation is often done by applying single-step methods for which the pedigree-based additive genetic relationship matrix is replaced by a combined relationship matrix based on both marker genotypes and pedigree. Genomic selection is implemented for purebreds, but it also offers opportunities for incorporating information from crossbreds and selecting for crossbred performance. However, models for genetic evaluation for the three-way crossbreeding system have not been developed.ResultsFour-variate models for three-way terminal crossbreeding are presented in which the first three variables contain the records for the three pure breeds and the fourth variable contains the records for the three-way crossbreds. For purebred animals, the models provide breeding values for both purebred and crossbred performances. Heterogeneity of genetic architecture between breeds and genotype by environment interactions are modelled through genetic correlations between these breeding values. Specification of the additive genetic relationships is essential for these models and can be defined either within populations or across populations. Based on these two types of additive genetic relationships, both pedigree-based, marker-based and combined relationships based on both pedigree and marker information are presented. All these models for three-way crossbreeding can be formulated using Kronecker matrix products and therefore fitted using Henderson’s mixed model equations and standard animal breeding software.ConclusionsModels for genetic evaluation in the three-way crossbreeding system are presented. They provide estimated breeding values for both purebred and crossbred performances, and can use pedigree-based or marker-based relationships, or combined relationships based on both pedigree and marker information. This provides a framework that allows information from three-way crossbred animals to be incorporated into a genetic evaluation system.

Exploring the potential of hierarchical generalized linear models in animal breeding and genetics

Exploring the potential of hierarchical generalized linear models in animal breeding and genetics

Technical Note: Computing Strategies in Genome-Wide Selection

Genome-wide genetic evaluation might involve the computation of BLUP-like estimations, potentially including thousands of covariates (i.e., single-nucleotide polymorphism markers) for each record. This implies dense Henderson's mixed-model equations and considerable computing resources in time and storage, even for a few thousand records. Possible computing options include the type of storage and the solving algorithm. This work evaluated several computing options, including half-stored Cholesky decomposition, Gauss-Seidel, and 3 matrix-free strategies: Gauss-Seidel, Gauss-Seidel with residuals update, and preconditioned conjugate gradients. Matrix-free Gauss-Seidel with residuals update adjusts the residuals after computing the solution for each effect. This avoids adjusting the left-hand side of the equations by all other effects at every step of the algorithm and saves considerable computing time. Any Gauss-Seidel algorithm can easily be extended for variance component estimation by Markov chain-Monte Carlo. Let m and n be the number of records and markers, respectively. Computing time for Cholesky decomposition is proportional to n3. Computing times per round are proportional to mn2 in matrix-free Gauss-Seidel, to n2 for half-stored Gauss-Seidel, and to n and m for the rest of the algorithms. Algorithms were tested on a real mouse data set, which included 1,928 records and 10,946 single-nucleotide polymorphism markers. Computing times were in the order of a few minutes for Gauss-Seidel with residuals update and preconditioned conjugate gradients, more than 1h for half-stored Gauss-Seidel, 2h for Cholesky decomposition, and 4 d for matrix-free Gauss-Seidel. Preconditioned conjugate gradients was the fastest. Gauss-Seidel with residuals update would be the method of choice for variance component estimation as well as solving.

Mixture model equations for marker‐assisted genetic evaluation

Marker-assisted genetic evaluation needs to infer genotypes at quantitative trait loci (QTL) based on the information of linked markers. As the inference usually provides the probability distribution of QTL genotypes rather than a specific genotype, marker-assisted genetic evaluation is characterized by the mixture model because of the uncertainty of QTL genotypes. It is, therefore, necessary to develop a statistical procedure useful for mixture model analyses. In this study, a set of mixture model equations was derived based on the normal mixture model and the EM algorithm for evaluating linear models with uncertain independent variables. The derived equations can be seen as an extension of Henderson's mixed model equations to mixture models and provide a general framework to deal with the issues of uncertain incidence matrices in linear models. The mixture model equations were applied to marker-assisted genetic evaluation with different parameterizations of QTL effects. A sire-QTL-effect model and a founder-QTL-effect model were used to illustrate the application of the mixture model equations. The potential advantages of the mixture model equations for marker-assisted genetic evaluation were discussed. The mixed-effect mixture model equations are flexible in modelling QTL effects and show desirable properties in estimating QTL effects, compared with Henderson's mixed model equations.

The effect of using approximate gametic variance covariance matrices on marker assisted selection by BLUP.

Under additive inheritance, the Henderson mixed model equations (HMME) provide an efficient approach to obtaining genetic evaluations by marker assisted best linear unbiased prediction (MABLUP) given pedigree relationships, trait and marker data. For large pedigrees with many missing markers, however, it is not feasible to calculate the exact gametic variance covariance matrix required to construct HMME. The objective of this study was to investigate the consequences of using approximate gametic variance covariance matrices on response to selection by MABLUP. Two methods were used to generate approximate variance covariance matrices. The first method (Method A) completely discards the marker information for individuals with an unknown linkage phase between two flanking markers. The second method (Method B) makes use of the marker information at only the most polymorphic marker locus for individuals with an unknown linkage phase. Data sets were simulated with and without missing marker data for flanking markers with 2, 4, 6, 8 or 12 alleles. Several missing marker data patterns were considered. The genetic variability explained by marked quantitative trait loci (MQTL) was modeled with one or two MQTL of equal effect. Response to selection by MABLUP using Method A or Method B were compared with that obtained by MABLUP using the exact genetic variance covariance matrix, which was estimated using 15 000 samples from the conditional distribution of genotypic values given the observed marker data. For the simulated conditions, the superiority of MABLUP over BLUP based only on pedigree relationships and trait data varied between 0.1% and 13.5% for Method A, between 1.7% and 23.8% for Method B, and between 7.6% and 28.9% for the exact method. The relative performance of the methods under investigation was not affected by the number of MQTL in the model.

The effect of using approximate gametic variance covariance matrices on marker assisted selection by BLUP

Under additive inheritance, the Henderson mixed model equations (HMME) provide an efficient approach to obtaining genetic evaluations by marker assisted best linear unbiased prediction (MABLUP) given pedigree relationships, trait and marker data. For large pedigrees with many missing markers, however, it is not feasible to calculate the exact gametic variance covariance matrix required to construct HMME. The objective of this study was to investigate the consequences of using approximate gametic variance covariance matrices on response to selection by MABLUP. Two methods were used to generate approximate variance covariance matrices. The first method (Method A) completely discards the marker information for individuals with an unknown linkage phase between two flanking markers. The second method (Method B) makes use of the marker information at only the most polymorphic marker locus for individuals with an unknown linkage phase. Data sets were simulated with and without missing marker data for flanking markers with 2, 4, 6, 8 or 12 alleles. Several missing marker data patterns were considered. The genetic variability explained by marked quantitative trait loci (MQTL) was modeled with one or two MQTL of equal effect. Response to selection by MABLUP using Method A or Method B were compared with that obtained by MABLUP using the exact genetic variance covariance matrix, which was estimated using 15 000 samples from the conditional distribution of genotypic values given the observed marker data. For the simulated conditions, the superiority of MABLUP over BLUP based only on pedigree relationships and trait data varied between 0.1% and 13.5% for Method A, between 1.7% and 23.8% for Method B, and between 7.6% and 28.9% for the exact method. The relative performance of the methods under investigation was not affected by the number of MQTL in the model.

Dos desvios das médias dos grupos contemporâneos aos modelos animais

O objetivo deste trabalho foi apresentar a reparametrização de um modelo de desvios das médias dos grupos contemporâneos até um modelo animal não-restrito (equações dos modelos mistos de Lush - EMML). Aplicando restrições sobre as EMML, foi obtido o modelo animal segundo Henderson (equações dos modelos mistos de Henderson - EMMH). Para obter as EMMH, restrições do tipo <img border=0 width=32 height=32 id="_x0000_i1026" src="../../../../img/revistas/rbz/v28n3/a07fr1.gif">foram impostas e estas geraram a necessidade de pressuposições fortes, como ausência de seleção, para que as propriedades das EMMH pudessem ser provadas. Para diferentes situações examinadas, as restrições foram confirmadas. O somatório destas restrições pode ser erroneamente interpretado como propriedade das soluções de um modelo animal de que os animais-base tenham a soma dos seus valores genéticos igual a zero.

A quasi-score approach to the analysis of ordered categorical data via a mixed heteroskedastic threshold model

This article presents an extension of the methodology developed by Gilmour et al. (19), for ordered categorical data, taking into account the hetero- geneity of residual variances of latent variables. Heterogeneity of residual variances is described via a structural linear model on log-variances. This method involves ' two main steps: i) a 'marginalization' with respect to the random effects leading to quasi-score estimators; ii) an approximation of the variance-covariance matrix of the observations which leads to an analogue of the Henderson mixed model equations for continuous Gaussian data. This methodology is illustrated by a numerical example of footshape in sheep. © Inra/Elsevier, Paris

AN ALTERNATIVE FOR MIXED MODEL ANALYSES OF LARGE, MESSY DATA SETS (MTDFREML)

Portable Fortran based programs (MTDFREML) were developed using a derivative-free algorithm to obtain REML estimates of (co)variance components. Computations are based on Henderson's mixed model equations for multiple-trait models with missing observations on some traits and incorporation of relationships among relatives. Many fixed and random factors are allowed with number of levels dependent on computer memory. Data sets with more than 40,000 genetic effects have been analyzed. Options allow solving MME at convergence. Constraints are automatically imposed. Expectations, standard errors of contrasts of solutions for fixed effects and prediction error variances of solutions for random effects can be obtained. Dimensions can be changed to match data with computer capability. A Fortran compiler is necessary. No fee is charged but the University of Waterloo must certify a license has been obtained for sparse matrix subroutines (SPARSPAK) used in the program. As an example, birth weights of 4891 progeny of 389 sires nested within 12 breeds and of 2893 dams nested within 3 breeds of dam were analyzed to estimate components of variance due to sires and dams and to estimate differences among breeds of sires. For MTDFREML the analysis was trivial but for PROC MIXED the analysis was impossible unless dams were dropped from the model.

Genetic evaluation by BLUP in two-breed terminal crossbreeding systems under dominance.

To obtain estimates of breeding values by BLUP using Henderson's mixed-model equations, it is necessary to invert the covariance matrix for each random effect in the model. In a model in which the genotypic value is included as a random effect (genotypic model), it is necessary to invert the genotypic covariance matrix. Under additive inheritance, the inverse of the genotypic covariance matrix can be computed efficiently. Under dominance inheritance, however, an efficient method to invert the genotypic covariance matrix has not yet been developed, especially for crossbred populations. Thus, the use of a genotypic model for BLUP is not suitable for genetic evaluation in large, crossbred populations. We present an equivalent model in which the genotypic effect is partitioned into additive and dominance effects. With this equivalent model, methods used for within-breed genetic evaluation by BLUP can be used for a two-breed terminal cross under dominance.

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

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 proportionality constant, then optimal exact tests can be performed. 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 employed. 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.

Effects of Selection on Variances and Covariances of Simulated First and Second Lactations

When variances and covariances are estimated for a number of traits of individuals, frequently each animal does not have records on all traits as a result of selection for one or more traits. Usual estimators of components of variances and covariances are biased by selection. Components of variance for sires and errors for production in first and second lactations and for sire and error covariances between those lactations were estimated from unselected and selected simulated data by Maximum Likelihood and Henderson's Method 1. Estimates by Maximum Likelihood were computed by an iterative algorithm based on Henderson's mixed model equations. Estimates by Method 1 from unselected data were unbiased. Estimates by Method 1 of variances of second records and covariances between records from data with 50% selection were biased. There were only slight differences between the parameters and estimates by Maximum Likelihood from either unselected or selected data. These results suggest the usefulness of Maximum Likelihood to estimate variances and covariances when selection within fixed subclasses exists.

Some Useful Representations for Constrained Mixed-Model Estimation

Abstract Mixed-model computations can be accomplished efficiently by using the Henderson mixed-model equations. An augmented version of the equations is developed for the case where there are constraints on the fixed effects. Equations comparable to the conjugate normal equations are also developed. These equations are useful for direct computation and, in the case of patterned designs, lend themselves well to algebraic simplification.

Some Useful Representations for Constrained Mixed-Model Estimation

Abstract Mixed-model computations can be accomplished efficiently by using the Henderson mixed-model equations. An augmented version of the equations is developed for the case where there are constraints on the fixed effects. Equations comparable to the conjugate normal equations are also developed. These equations are useful for direct computation and, in the case of patterned designs, lend themselves well to algebraic simplification.