Heterogeneous variances in Gaussian linear mixed models

This paper reviews some problems encountered in estimating heterogeneous variances in Gaussian linear mixed models. The one-way and multiple classification cases are considered. EM-REML algorithms and Bayesian procedures are derived. A structural mixed linear model on log-variance components is also presented, which allows identification of meaningful sources of variation of heterogeneous residual and genetic components of variance and assessment of their magnitude and mode of action.Cet article fait le point sur un certain nombre de problèmes qui surviennent lors de l’estimation de variances hétérogènes dans des modèles linéaires mixtes gaussiens. On considère le cas d’un ou plusieurs facteurs d’hétéroscédasticité. On développe des algorithmes EM-REML et bayésiens. On propose également un modèle linéaire mixte structurel des logarithmes des variances qui permet de mettre en évidence des sources significatives de variation des variances résiduelles et génétiques et d’appréhender leur importance et leur mode d’action

Bayesian analysis of calving ease scores and birth weights

International audienc

Inferences on homogeneity of between-family components of variance and covariance among environments in balanced cross-classified designs

Estimation and testing of homogeneity of between-family components of variance and covariance among environments are investigated for balanced cross-classified designs. The variance-covariance structure of the residuals is assumed to be diagonal and heteroskedastic. The testing procedure for homogeneity of family components is based on the ratio of maximized log-restricted likelihoods for the reduced (hypothesis of homogeneity) and saturated models. An expectation-maximization (EM) algorithm is proposed for calculating restricted maximum likelihood (REML) estimates of the residual and between-family components of variance and covariance. The EM formulae to implement this are iterative and use the classical analysis of variance (ANOVA) statistics, ie the between- and within-family sums of squares and cross-products. They can be applied both to the saturated and reduced models and guarantee the solutions to be in the parameter space. Procedures presented in this paper are illustrated with the analysis of 5 vegetative and reproductive traits recorded in an experiment on 20 full-sib families of black medic (Medicago lupulina L) tested in 3 environments. Application to pure maximum likelihood procedures, extension to unbalanced designs and comparison with approaches relying on alternative models are also discussed.Cet article étudie les problèmes d’estimation et de test d’homogénéité des composantes familiales de variance et de covariance entre milieux dans des dispositifs factoriels équilibrés. La structure des variances et des covariances résiduelles est supposée diagonale et hétéroscédastique. La procédure de test d’homogénéité des composantes familiales repose sur le rapport des vraisemblances restreintes maximisées sous les modèles réduit (hypothèse d'homogénéité) et saturé. Un algorithme d’espérance-maximisation (EM) est proposé pour calculer les estimations du maximum de vraisemblance restreinte (REML) des composantes résiduelles et familiales de variance et de covariance. Les formules EM à appliquer sont itératives et utilisent les statistiques classiques de l’analyse de variance (ANOVA), c’est-à-dire les sommes de carrés et coproduits inter- et intrafamilles. Elles s’appliquent à la fois aux modèles réduit et saturé et garantissent l’appartenance des solutions à l’espace des paramètres. Les méthodes présentées dans cet article sont illustrées par l’analyse de 5 caractères végétatifs et reproductifs mesurés lors d’une expérience portant sur 20 familles de pleins frères testées dans 3 milieux chez la minette (Medicago lupulina L). L’application au maximum de vraisemblance stricto sensu, la généralisation à des dispositifs déséquilibrés ainsi que la comparaison à des approches reposant sur d’autres modèles sont également discutées

Inclusion of genetically identical animals to a numerator relationship matrix and modification of its inverse

In the field of animal breeding, estimation of genetic parameters and prediction of breeding values are routinely conducted by analyzing quantitative traits. Using an animal model and including the direct inverse of a numerator relationship matrix (NRM) into a mixed model has made these analyses possible. However, a method including a genetically identical animal (GIA) in NRM if genetic relationships between pairs of GIAs are not perfect, is still lacking. Here, we describe a method to incorporate GIAs into NRM using a K matrix in which diagonal elements are set to 1.0, off-diagonal elements between pairs of GIAs to (1-x) and the other elements to 0, where x is a constant less than 0.05. The inverse of the K matrix is then calculated directly by a simple formula. Thus, the inverse of the NRM is calculated by the products of the lower triangular matrix that identifies the parents of each individual, its transpose matrix, the inverse of the K matrix and the inverse of diagonal matrix D, in which the diagonal elements comprise a number of known parents and their inbreeding coefficients. The computing method is adaptable to the analysis of a data set including pairs of GIAs with imperfect relationships

Genetic prediction of complex traits: integrating infinitesimal and marked genetic effects

Genetic prediction for complex traits is usually based on models including individual (infinitesimal) or marker effects. Here, we concentrate on models including both the individual and the marker effects. In particular, we develop a ''Mendelian segregation'' model combining infinitesimal effects for base individuals and realized Mendelian sampling in descendants described by the available DNA data. The model is illustrated with an example and the analyses of a public simulated data file. Further, the potential contribution of such models is assessed by simulation. Accuracy, measured as the correlation between true (simulated) and predicted genetic values, was similar for all models compared under different genetic backgrounds. As expected, the segregation model is worthwhile when markers capture a low fraction of total genetic variance. (Résumé d'auteur

Evaluating alternate models to estimate genetic parameters of calving traits in United Kingdom Holstein-Friesian dairy cattle

<p>Abstract</p> <p>Background</p> <p>The focus in dairy cattle breeding is gradually shifting from production to functional traits and genetic parameters of calving traits are estimated more frequently. However, across countries, various statistical models are used to estimate these parameters. This study evaluates different models for calving ease and stillbirth in United Kingdom Holstein-Friesian cattle.</p> <p>Methods</p> <p>Data from first and later parity records were used. Genetic parameters for calving ease, stillbirth and gestation length were estimated using the restricted maximum likelihood method, considering different models i.e. sire (−maternal grandsire), animal, univariate and bivariate models. Gestation length was fitted as a correlated indicator trait and, for all three traits, genetic correlations between first and later parities were estimated. Potential bias in estimates was avoided by acknowledging a possible environmental direct-maternal covariance. The total heritable variance was estimated for each trait to discuss its theoretical importance and practical value. Prediction error variances and accuracies were calculated to compare the models.</p> <p>Results and discussion</p> <p>On average, direct and maternal heritabilities for calving traits were low, except for direct gestation length. Calving ease in first parity had a significant and negative direct-maternal genetic correlation. Gestation length was maternally correlated to stillbirth in first parity and directly correlated to calving ease in later parities. Multi-trait models had a slightly greater predictive ability than univariate models, especially for the lowly heritable traits. The computation time needed for sire (−maternal grandsire) models was much smaller than for animal models with only small differences in accuracy. The sire (−maternal grandsire) model was robust when additional genetic components were estimated, while the equivalent animal model had difficulties reaching convergence.</p> <p>Conclusions</p> <p>For the evaluation of calving traits, multi-trait models show a slight advantage over univariate models. Extended sire models (−maternal grandsire) are more practical and robust than animal models. Estimated genetic parameters for calving traits of UK Holstein cattle are consistent with literature. Calculating an aggregate estimated breeding value including direct and maternal values should encourage breeders to consider both direct and maternal effects in selection decisions.</p