Karin Meyer

Variance components due to direct and maternal effects for growth traits of Australian beef cattle

Abstract Variance components for birth, weaning, yearling and final weight in Australian Hereford, Angus and Zebu Cross cattle were estimated by Restricted Maximum Likelihood. Six different animal models were fitted for each trait and breed, ranging from a simple model with animals as the only random effect to the most comprehensive model allowing for both genetic and environmental maternal effects and a genetic covariance between direct and maternal effects. The most detailed model generally provided the best fit to the data, though differences between models with at least one maternal effect (genetic or environmental) were often not significant. Ignoring maternal effects, direct heritability (h2) estimates were inflated substantially, in particular for growth till weaning. Significant maternal effects were found in all analyses except for final weight in Angus. There were marked differences between breeds in the relative magnitude ofh2 and the maternal heritability, and the direct-maternal genetic correlation (rAM). For Angus,rAM was low, positive and not significantly different from zero for all traits. For Hereford and Zebu Cross cattle,rAM was negative, moderate to large for weaning weight (−0.59 and −0.78) and somewhat smaller for yearling weight (−0.48 and −0.39). For Herefords, maternal environmental effects were consistently more important than maternal genetic effects.

Restricted maximum likelihood to estimate variance components for animal models with several random effects using a derivative-free algorithm

Summary - A method is described for the simultaneous estimation of variance components due to several genetic and environmental effects from unbalanced data by restricted maximum likelihood (REML). Estimates are obtained by evaluating the likelihood explicitly and using standard, derivative-free optimization procedures to locate its maximum. The model of analysis considered is the so-called Animal Model which includes the additive genetic merit of animals as a random effect, and incorporates all information on relationships between animals. Furthermore, random effects in addition to animals’ additive genetic effects, such as maternal genetic, dominance or permanent environmental effects are taken into account. Emphasis is placed entirely upon univariate analyses. Simulation is employed to investigate the efficacy of three different maximization techniques and the scope for approximation of sampling errors. Computations are illustrated with a numerical example. variance components - restricted maximum likelihood - animal model - additional random effects - derivative - free approach

WOMBAT—A tool for mixed model analyses in quantitative genetics by restricted maximum likelihood (REML)

WOMBAT is a software package for quantitative genetic analyses of continuous traits, fitting a linear, mixed model; estimates of covariance components and the resulting genetic parameters are obtained by restricted maximum likelihood. A wide range of models, comprising numerous traits, multiple fixed and random effects, selected genetic covariance structures, random regression models and reduced rank estimation are accommodated. WOMBAT employs up-to-date numerical and computational methods. Together with the use of efficient compilers, this generates fast executable programs, suitable for large scale analyses. Use of WOMBAT is illustrated for a bivariate analysis. The package consists of the executable program, available for LINUX and WINDOWS environments, manual and a set of worked example, and can be downloaded free of charge from http://agbu.une.edu.au/:_kmeyer/wombat.html

Estimation of genetic and phenotypic covariance functions for longitudinal or ‘repeated’ records by restricted maximum likelihood

Abstract Covariance functions are the equivalent of covariance matrices for traits with many, potentially infinitely many, records in which the covariances are defined as a function of age or time. They can be fitted for any source of variation, e.g. genetic, permanent environment or phenotypic. A suitable family of functions for covariance functions are orthogonal polynomials. These give the covariance between measurements at any two ages as a higher order polynomial of the ages at recording. Polynomials can be fitted to full or reduced order. The former is equivalent to a multivariate analysis estimating covariance components. A reduced order fit involves less parameters and smoothes out differences in estimates of covariances. It gives predicted covariance matrices of rank equal to the order of fit. The coefficients of covariance functions can be estimated by restricted maximum likelihood through a reparameterisation of existing algorithms to estimate covariance components. For a simple animal model with equal design matrices for all traits, computational requirements to estimate covariance functions are proportional to the order of fit for the genetic covariance function. Applications to simulated data and a set of beef cattle data are shown.

Estimating covariance functions for longitudinal data using a random regression model

A method is described to estimate genetic and environmental covariance functions for traits measured repeatedly per individual along some continuous scale, such as time, directly from the data by restricted maximum likelihood. It relies on the equivalence of a covariance function and a random regression model. By regressing on random, orthogonal polynomials of the continuous scale variable, the coefficients of covariance functions can be estimated as the covariances among the regression coefficients. A parameterisation is described which allows the rank of estimated covariance matrices and functions to be restricted, thus facilitating a highly parsimonious description of the covariance structure. The procedure and the type of results which can be obtained are illustrated with an application to mature weight records of beef cows. @ Inra/Elsevier, Paris covariance functions / genetic parameters / longitudinal data / restricted maximum likelihood / random regression model

Estimating variances and covariances for multivariate animal models by restricted maximum likelihood

Summary — Restricted maximum likelihood estimates of variance and covariance components can be obtained by direct maximization of the associated likelihood using standard, derivative-free optimization procedures. In general, this requires a multi-dimensional search and numerous evaluations of the (log) likelihood function. Use of this approach for analyses under an animal model has been described for the univariate case. This model includes animals’ additive genetic merit as random effect and accounts for all relationships between animals. In addition, other random factors such as common environmental or maternal genetic effects can be fitted. This paper describes the extension to multivariate analyses, allowing for missing records. A numerical example is given and simplifications for specific models are discussed. variance component / restricted maximum likelihood / animal model / additional random effect / derivative-free approach / multivariate analysis

Maximum Likelihood Estimation of Variance Components for a Multivariate Mixed Model with Equal Design Matrices

An algorithm is described for estimating variance and covariance components by restricted maximum likelihood for a multivariate mixed two-way classification with equal design matrices. The procedure involves a transformation to canonical scale, effectively reducing a q-variate analysis to q corresponding univariate analyses. A small numerical example is given as well as a large-scale practical application.

Components of variance underlying fitness in a natural population of the great tit Parus major.

Traits that are closely associated with fitness tend to have lower heritabilities (h2) than those that are not. This has been interpreted as evidence that natural selection tends to deplete genetic variation more rapidly for traits more closely associated with fitness (a corollary of Fisher’s fundamental theorem), but Price and Schluter (1991) suggested the pattern might be due to higher residual variance in traits more closely related to fitness. The relationship between 10 different traits for females, seven traits for males, and overall fitness (lifetime recruitment) was quantified for great tits (Parus major) studied in their natural environment of Wytham Wood, England, using data collected over 39 years. Heritabilities and the coefficients of additive genetic and residual variance (CVA and CVR, respectively) were estimated using an “animal model.” For both males and females, a trait’s correlation (r) with fitness was negatively related to its h2 but positively related to its CVR. The CVA was not related to the trait’s correlation with fitness in either sex. This is the third study using directly measured fitness in a wild population to show the important role of residual variation in determining the pattern of lower heritabilities for traits more closely related to fitness.

Estimates of genetic parameters for weaning weight of beef cattle accounting for direct-maternal environmental covariances

Abstract Restricted Maximum Likelihood algorithm estimates of (co)variance components due to maternal effects as well as a regression on maternal phenotype were obtained for seven weaning weight data sets of Australian and New Zealand beef cattle. Fitting such regression, analyses accounted for environmental covariances between dams and their offspring. Results show a substantial, negative regression on maternal phenotype (up to −0.2) for Hereford field data, accompanied by small, negative estimates of a direct-maternal genetic covariance. In contrast, for Angus and Limousins, the direct-maternal genetic covariance was clearly more important than its environmental counterpart, i.e., for these breeds an estimate of the direct-maternal genetic correlation of about −0.5 could not be attributed to a negative environmental relationship which previously had not been modeled correctly. Fitting a sire × herd-year interaction as an additional random effect increased the likelihood dramatically for all data sets. While estimates of the regression on maternal phenotype were little affected, fitting the interaction reduced estimates of the direct-maternal genetic covariance, substantially so for Angus and Limousin, reducing (absolute value) estimates of the corresponding correlations to −0.3 to −0.2.

Genetic variation and covariation for growth, parasite resistance and heat tolerance in tropical cattle

Abstract Genetic parameters for growth, parasite resistance and heat tolerance in zebu-cross cattle in a tropical environment were estimated by restricted maximum likelihood procedures using various forms of an animal model. Heritability estimates for direct additive genetic effects for liveweights at birth, weaning, 12 and 18 months were 0.61, 0.20, 0.25 and 0.26, respectively and corresponding maternal additive genetic effects were 0.11, 0.32, 0.20 and 0.09. Estimates of genetic correlations between direct and maternal additive genetic effects were zero. Weight gains during the dry season were genetically uncorrelated or slightly negatively correlated to birth weight, preweaning weight gain or gains during the wet season. Heritability estimates for rectal temperature, numbers of ticks, buffalo flies and worms were 0.19, 0.34, 0.06 and 0.28, respectively, with corresponding repeatability estimates of 0.23, 0.45, 0.06 and 0.29. Phenotypic correlations between adaptive traits (heat tolerance and parasite resistance) and growth traits were low. Genetic correlations between growth and rectal temperature were negative. Genetic correlations of growth traits with worm and fly burdens were positive and with tick burdens were zero. It was concluded that in reasonably adapted genotypes, selection for growth in the Australian tropics would result in improved heat tolerance and reduced resistance to parasites. The effect of genotype on parameter estimates is discussed.