Covariance between relatives in a multibreed population was derived for an additive model with multiple unlinked loci. An efficient algorithm to compute the inverse of the additive genetic covariance matrix is given. For an additive model, the variance for a crossbred individual is a function of the additive variances for the pure breeds, the covariance between parents, and segregation variances. Provided that the variance of a crossbred individual is computed as presented here, the covariance between crossbred relatives can be computed using formulae for purebred populations. For additive traits the inverse of the genotypic covariance matrix given here can be used both to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood in multibreed populations. For nonadditive traits, the procedure currently used to analyze multibreed data can be improved using the theory presented here to compute additive covariances together with a suitable approximation for nonadditive covariances.

Covariance between relatives in multibreed populations: additive model.

Implementation and accuracy of genomic selection

Variance components were estimated using alternative structures for the additive genetic covariance matrix (G0), for height (m) of trees measured at 10 unequally spaced ages in an open-pollinated progeny test. These structures reflected unstructured, autoregressive, banded correlation and random regressions models. The residual matrix ( R0) was unstructured, and the block and plot strata matrices were autoregressive. The best model for G0 considering the likeli- hood value and number of parameters was the autoregressive correlation form with age-specific variances and time on a natural logarithm basis. The genetic correlation between successive measures ranged from 0.93 at age 1 to 0.99 at age 14 years. Heritability increased with age from 0.09 (age 1) to 0.24 (age 7) and then declined to 0.13 at age 15. Heritabilities from the unstructured model were similar, while heritabilities assuming banded correlations were lower after age 7. The covariance structure implicit in the random regressions model was considered unsatisfactory. Using structures in G0 facilitated model fitting and convergence of the likelihood maximisation algorithm. Fitting a structured matrix that reflects the relationships present in repeated measures may overcome problems of nonpositive definiteness of unstructured matrices from longitudinal data, especially when genetic variation is small. Resume : Les auteurs ont estime les composantes de la variance a l'aide de structures d'ajustement a la matrice de co - variances genetiques additives (G0) pour la hauteur (m) d'arbres mesures a 10 intervalles non reguliers au sein d'un test de descendances issues de pollinisation libre. Ces structures refletaient des modeles non structures, auto-regressifs, par correlations stratifiees et par regressions aleatoires. La matrice residuelle ( R0) n'etait pas structuree, et les matrices a l'echelle des blocs et des parcelles etaient auto-regressives. En considerant la valeur de maximum de vraisemblance et le nombre de parametres, le meilleur modele d'ajustement de G0 etait celui des correlations auto-regressives avec les variances specifiques a chaque âge et le temps suivant l'echelle logarithmique naturelle. Les correlations genetiques entre les mesures successives variaient de 0,9 3a1a n jusqu'a 0,99 a 14 ans. L'heritabilite augmentait avec l'âge, allant de 0,09 (a 1 an) a 0,24 (a 7 ans) pour decroitre a 0,13 a 15 ans. Les heritabilites decoulant du modele non structure etaient similaires, alors que les heritabilites decoulant du modele par correlations stratifiees etaient plus faibles apres 7 ans. La structure de covariances implicite au modele de regressions aleatoires n'a pas ete jugee satisfaisante. La structuration de G0 a facilite l'ajustement des modeles ainsi que la convergence de l'algorithme de maximisation de la vraisemblance. Les auteurs en concluent que l'ajustement d'une matrice structuree refletant les liens de dependance parmi les mesures repetees peut permettre de resoudre les problemes de manque de d'exactitude presents dans les ma- trices non structurees decoulant de donnees longitudinales, specialement lorsque la variabilite genetique est faible.

/pdf/variance-modelling-of-longitudinal-height-data-from-a-pinus-3qw0fkzpjd.pdf

Variance modelling of longitudinal height data from a Pinus radiata progeny test

Variance components for egg production traits (No of eggs produced between 19 and 26, 26 and 38 and 26 and 54 wk of age), egg characteristics (average egg weight at 2 different ages and egg density) and body weight of hens at 40 wk of age were estimated in two strains of a breeding company by univariate and multivariate Restricted Maximum Likelihood (REML) applied to a Reduced Animal Model (RAM). To allow tridiagonalization of the coefficient matrix when RAM is considered, the approach of Thompson and Meyer (1990) using an imaginary random effect with negative variance

/pdf/estimation-of-genetic-parameters-of-egg-production-traits-of-yng7pxfyva.pdf

Estimation of genetic parameters of egg production traits of laying hens by restricted maximum likelihood applied to a multiple-trait reduced animal model

Several methods have recently been developed to identify regions of the genome that have been exposed to strong selection. However, recent theoretical and empirical work suggests that polygenic models are required to identify the genomic regions that are more moderately responding to ongoing selection on complex traits. We examine the effects of multi-trait selection on the genome of a population of US registered Angus beef cattle born over a 50-year period representing approximately 10 generations of selection. We present results from the application of a quantitative genetic model, called Birth Date Selection Mapping, to identify signatures of recent ongoing selection. We show that US Angus cattle have been systematically selected to alter their mean additive genetic merit for most of the 16 production traits routinely recorded by breeders. Using Birth Date Selection Mapping, we estimate the time-dependency of allele frequency for 44,817 SNP loci using genomic best linear unbiased prediction, generalized least squares, and BayesCπ analyses. Finally, we reconstruct the primary phenotypes that have historically been exposed to selection from a genome-wide analysis of the 16 production traits and gene ontology enrichment analysis. We demonstrate that Birth Date Selection Mapping utilizing mixed models corrects for time-dependent pedigree sampling effects that lead to spurious SNP associations and reveals genomic signatures of ongoing selection on complex traits. Because multiple traits have historically been selected in concert and most quantitative trait loci have small effects, selection has incrementally altered allele frequencies throughout the genome. Two quantitative trait loci of large effect were not the most strongly selected of the loci due to their antagonistic pleiotropic effects on strongly selected phenotypes. Birth Date Selection Mapping may readily be extended to temporally-stratified human or model organism populations.

https://link.springer.com/content/pdf/10.1186%2F1471-2164-13-606.pdf

Richard L. Quaas

Papers

BLUP school handbook. Use of mixed models for prediction and for estimation of (co)variance components.