Reduced animal model with differential genetic grouping for direct and maternal effects.

Mixed-model equations for the reduced animal model with maternal effects and different genetic grouping of unknown parents for additive direct and maternal effects are derived. The matrices that relate the expected value and the variance of the breeding values of non-parents to the parents, as well as the different contributions of parental and non-parental breeding values, to the resulting mixed-model equations are presented. Mis-specification of additive maternal variance and the additive covariance between direct and maternal effects, arising from missing information on the dams of known individuals with records, is discussed. To avoid an incorrect specification of the variance-covariance matrix of the records without having to invert a nondiagonal variance of the residual terms, the breeding values of the unknown dams of individuals with records are included in the equations. Breeding values of non-parents are back-solved after the solutions for genetic groups and breeding values of parents are computed as simply as in cases in which maternal effects are absent. A numerical example is included to illustrate the derivations.     

Simple preconditioners for the conjugate gradient method: experience with test day models

Preconditioned conjugate gradient method can be used to solve large mixed model equations quickly. Convergence of the method depends on the quality of the preconditioner. Here, the effect of simple preconditioners on the number of iterations until convergence was studied by solving breeding values for several test day models. The test day records were from a field data set, and several simulated data sets with low and high correlations among regression coefficients. The preconditioner matrices had diagonal or block diagonal parts. Transformation of the mixed model equations by diagonalization of the genetic covariance matrix was studied as well. Preconditioner having the whole block of the fixed effects was found to be advantageous. A block diagonal preconditioner for the animal effects reduced the number of iterations the higher the correlations among animal effects, but increased memory usage of the preconditioner. Diagonalization of the animal genetic covariance matrix often reduced the number of iterations considerably without increased memory usage.     

Covariances among sire by breed group of dam interaction effects in multibreed sire evaluation procedures

In multibreed populations, bulls need to be evaluated for additive and nonadditive genetic effects. When the nonadditive genetic effects associated with a bull are defined as sire x breed-group-of-dam interactions, they can be expressed as linear combinations of interactions between alleles of one or more breeds at one or more loci. If these specific allelic interactions are assumed to be independent, then variances and covariances between sire x breed-group-of-dam interaction subclasses can be shown to be linear combinations of variances and covariances of specific intra- and interlocus intra- and interbreed allelic interactions. Furthermore, covariances between sire x breed-group-of-dam interactions due to specific interactions at one, two, or more loci are zero. If dams are assumed to be unrelated to bulls and among themselves, except through their sires and maternal grandsires, efficient procedures to compute the inverse of the covariance matrices of nonadditive genetic effects can be devised, both in subclass and in regression models. Recursive procedures developed make possible the evaluation of large numbers of bulls for nonadditive genetic effects using mixed-model methodology. For completeness, recursive procedures to compute nonadditive covariance matrices in subclass and in regression models also were developed. The prediction of nonadditive genetic values for bulls, in addition to their additive genetic values, will help plan matings, make selection decisions more accurate and, possibly, make economic projections better.     

Multiple trait prediction for a type of model with heterogeneous genetic and residual covariance structures

A restricted set of models is defined that allows for heterogeneous genetic and residual covariance structures. Multiple trait models and models with multiple random factors are included. The restriction on the model is that the correlations among genetic effects in different classes are the same. Equivalently, the genetic covariance matrices are assumed to differ between classes due to scaling. This assumption greatly reduces the number of parameters that must be specified and does not adversely affect the computational burden of a mixed model analysis. An application of the model for genetic evaluation of beef cattle is described and illustrated numerically.     

Random regression models to estimate genetic parameters for test‐day milk yield in Brazilian Murrah buffaloes

The objective of this work was to estimate covariance functions for additive genetic and permanent environmental effects and, subsequently, to obtain genetic parameters for buffalo’s test‐day milk production using random regression models on Legendre polynomials (LPs). A total of 17 935 test‐day milk yield (TDMY) from 1433 first lactations of Murrah buffaloes, calving from 1985 to 2005 and belonging to 12 herds located in São Paulo state, Brazil, were analysed. Contemporary groups (CGs) were defined by herd, year and month of milk test. Residual variances were modelled through variance functions, from second to fourth order and also by a step function with 1, 4, 6, 22 and 42 classes. The model of analyses included the fixed effect of CGs, number of milking, age of cow at calving as a covariable (linear and quadratic) and the mean trend of the population. As random effects were included the additive genetic and permanent environmental effects. The additive genetic and permanent environmental random effects were modelled by LP of days in milk from quadratic to seventh degree polynomial functions. The model with additive genetic and animal permanent environmental effects adjusted by quintic and sixth order LP, respectively, and residual variance modelled through a step function with six classes was the most adequate model to describe the covariance structure of the data. Heritability estimates decreased from 0.44 (first week) to 0.18 (fourth week). Unexpected negative genetic correlation estimates were obtained between TDMY records at first weeks with records from middle to the end of lactation, being the values varied from ?0.07 (second with eighth week) to ?0.34 (1st with 42nd week). TDMY heritability estimates were moderate in the course of the lactation, suggesting that this trait could be applied as selection criteria in milking buffaloes.     

On estimating non‐additive genetic parameters in chickens 1

1. Several economically important traits in two Leghorn populations (over 9000 birds) were examined for additive and non‐additive components of genetic variance and sex‐linked effects. Data were analysed by two different statistical models based on least‐squares procedures.

2. Six different covariances were first calculated between relatives; i.e., full‐sibs, 3/4‐sisters, half‐sisters, dam‐daughters, grandam‐granddaughters and aunt‐nieces.

3. From the covariances, weighted least‐squares equations were used to obtain estimates of variance components for additive genetic, dominance, maternal and sex‐linkage effects.

4. The estimates of non‐additive components were highly variable but generally small compared with the additive genetic estimates.

5. In general this study suggests that for most traits, with the possible exception of rate of egg production, there is relatively little non‐additive genetic variation.

6. The consequences of possible negative correlations between additive effects and maternal effects are considered as they might apply to egg production in poultry.     

Impact of variation in length of individual testing periods on estimation of (co)variance components of a random regression model for feed intake of growing pigs

A simulation study was conducted to assess the influence of differences in the length of individual testing periods on estimates of (co)variance components of a random regression model for daily feed intake of growing pigs performance tested between 30 and 100 kg live weight. A quadratic polynomial in days on test with fixed regressions for sex, random regressions for additive genetic and permanent environmental effects and a constant residual variance was used for a bivariate simulation of feed intake and daily gain. (Co)variance components were estimated for feed intake only by means of a Bayesian analysis using Gibbs sampling and restricted maximum likelihood (REML). A single trait random regression model analogous to the one used for data simulation was used to analyse two versions of the data: full data sets with 18 weekly means of feed intake per animal and reduced data sets with the individual length of testing periods determined when tested animals reached 100 kg live weight. Only one significant difference between estimates from full and reduced data (REML estimate of genetic covariance between linear and quadratic regression parameters) and two significant differences from expected values (Gibbs estimates of permanent environmental variance of quadratic regression parameters) occurred. These differences are believed to be negligible, as the number lies within the expected range of type I error when testing at the 5% level. The course of test day variances calculated from estimates of additive genetic and permanent environmental covariance matrices also supports the conclusion that no bias in estimates of (co)variance components occurs due to the individual length of testing periods of performance‐tested growing pigs. A lower number of records per tested animal only results in more variation among estimates of (co)variance components from reduced compared with full data sets. Compared with the full data, the effective sample size of Gibbs samples from the reduced data decreased to 18% for residual variance and increased up to five times for other (co)variances. The data structure seems to influence the mixing of Gibbs chains.     

Estimates of genetic and phenotypic covariance functions for postweaning growth and mature weight of beef cows

Annual weights of cows from 19 to 119 months of age in two herds were analysed fitting a random regression model, regressing on orthogonal polynomials of age in months. Estimates of covariances between random regression coefficients were obtained by restricted maximum likelihood, and the resulting estimates of covariance functions were used to construct covariance matrices for all ages in the data. Analyses were carried out fitting regression coefficients corresponding to overall animal effects only and fitting regressions for animals' additive genetic and permanent, environmental effects. Different definitions of fixed effects subclasses were examined. Models were compared using likelihood ratio tests and estimated standard deviations for the ages in the data. Cubic regressions were sufficient to model both population trajectories and individual growth curves. Random regression coefficients were highly correlated, so that estimation forcing their covariance matrices to have reduced rank (2 or 3) did not reduce likelihoods significantly, allowing parsimonious modelling. Results showed that records were clearly not repeated measurements of a single trait with constant variances. As cows grew up to about 5 years of age, variances. As cows grew up to about 5 years of age, variances increased. Estimates of genetic correlations between 3-year-old and older cows were close to unity in one herd but more erratic in the other. For both herds, genetic correlations between weights on 2-year-old cows and older animals were clearly less than unity.     

On estimating components of genetic variance in diallel matings 1

1. The relative importance of additive and non‐additive genetic effects on body weight, egg weight, maturity and rate of egg production were studied from diallel matings in a Leghorn population.

2. From analyses of variance, heritability estimates of the additive fractions, based on half‐sib variances, and the non‐additive or dominance fractions, based on the sire x dam interaction component were obtained.

3. Non‐additive genetic effects were not statistically significant for any of the traits, though for rate of egg production at 32 and 62 weeks, the non‐additive effects as proportion of total variances were 0.29 (P<0.10) and 0.20 (P<0.16), respectively, compared additive effects of 0.08 (NS) and 0.11 (P<0.05).

4. The ratios of non‐additive to additive variances, 1.89 and 3.62 respectively, give support to inbreeding and hybridisation or reciprocal recurrent selection as methods of genetic improvement of egg production.     

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.     

Genomic additive and dominance variance of milk performance traits

Milk performance traits are likely influenced by both additive and non‐additive (e.g. dominance) genetic effects. Genetic variation can be partitioned using genomic information. The objective of this study was to estimate genetic variance components of production and milk component traits (e.g. acetone, fatty acids), which are particularly important for milk processing or which can provide information on the health status of cows. A genomic relationship approach was applied to phenotypic and genetic information of 1295 Holstein cows for estimating additive genetic and dominance variance components. Most of the 17 investigated traits were mainly affected by additive genetic effects, but protein content and casein content also showed a significant contribution of dominance. The ratio of dominance to additive variance was estimated as 0.64 for protein content and 0.56 for casein content. This ratio was highest for SCS (1.36) although dominance was not significant. Dominance effects were negligible in other moderately heritable milk traits.     

Alternative animal models with maternal effects and foster dams

Effects of foster dams can be included in genetic evaluations using animal models with maternal effects in several ways. The alternatives discussed involve minor changes in computing strategies from strategies used with reduced animal models that predict breeding values for direct and maternal effects. The easiest alternative is to assign foster dams to groups by breed and time period and add equations for fixed effects of breed-period. Random and, assumed, independent effects of foster dams can be nested in breed-period groups. If foster dams do not repeat, then those effects can be absorbed into equations for other fixed effects, additive direct breeding value and breed-period effects by slightly modifying least squares contributions to coefficients of those equations. A third alternative for foster dams of the same breed is to add breeding values for foster dams for direct and maternal effects to solution vectors for breeding values. Equations are similar to those without foster dams, except that least squares contributions to coefficient matrix and right-hand sides are to equations for maternal breeding values and nongenetic maternal effects of foster dams rather than biological dams. Relationships and covariance between direct and maternal effects contribute mixed-model coefficients to direct and maternal breeding value equations of biological dams. This alternative basically requires only larger solution vectors for direct and maternal breeding values to accommodate foster dams that might not be included. The fourth alternative includes a vector of maternal breeding values for foster dams of each breed of foster dams and would require using rules of Westell to calculate coefficients due to relationships and fixed maternal genetic groups within each breed of foster dam. These alternatives do not require much additional computational effort compared with full or reduced animal model equations when the transformation to predict breeding values is used with Westell's rules to calculate coefficients due to relationships and genetic group effects due to prior genetic selection.     

Bayesian inference strategies for the prediction of genetic merit using threshold models with an application to calving ease scores in Italian Piemontese cattle

First parity calving difficulty scores from Italian Piemontese cattle were analysed using a threshold mixed effects model. The model included the fixed effects of age of dam and sex of calf and their interaction and the random effects of sire, maternal grandsire, and herd‐year‐season. Covariances between sire and maternal grandsire effects were modelled using a numerator relationship matrix based on male ancestors. Field data consisted of 23 953 records collected between 1989 and 1998 from 4741 herd‐year‐seasons. Variance and covariance components were estimated using two alternative approximate marginal maximum likelihood (MML) methods, one based on expectation‐maximization (EM) and the other based on Laplacian integration. Inferences were compared to those based on three separate runs or sequences of Markov Chain Monte Carlo (MCMC) sampling in order to assess the validity of approximate MML estimates derived from data with similar size and design structure. Point estimates of direct heritability were 0.24, 0.25 and 0.26 for EM, Laplacian and MCMC (posterior mean), respectively, whereas corresponding maternal heritability estimates were 0.10, 0.11 and 0.12, respectively. The covariance between additive direct and maternal effects was found to be not different from zero based on MCMC‐derived confidence sets. The conventional joint modal estimates of sire effects and associated standard errors based on MML estimates of variance and covariance components differed little from the respective posterior means and standard deviations derived from MCMC. Therefore, there may be little need to pursue computation‐intensive MCMC methods for inference on genetic parameters and genetic merits using conventional threshold sire and maternal grandsire models for large datasets on calving ease.     

Geno‐Diver: A combined coalescence and forward‐in‐time simulator for populations undergoing selection for complex traits

Geno‐Diver is a combined coalescence and forward‐in‐time simulator designed to simulate complex traits with a quantitative and/or fitness component and implement multiple selection and mating strategies utilizing pedigree or genomic information. The simulation is carried out in two steps. The first step generates whole‐genome sequence data for founder individuals. A variety of trait architectures can be generated for quantitative and fitness traits along with their covariance. The second step generates new individuals forward‐in‐time based on a variety of selection and mating scenarios. Genetic values are predicted for individuals utilizing pedigree or genomic information. Relationship matrices and their associated inverses are generated using computationally efficient routines. We benchmarked Geno‐Diver with a previous simulation program and described how to simulate a traditional quantitative trait along with a quantitative and fitness trait. A user manual with examples, source code in C++11 and executable versions of Geno‐Diver for Linux are freely available at https://github.com/jeremyhoward/Geno-Diver .     

Role of inbreeding depression,non‐inbred dominance deviations and random year‐season effect in genetic trends for prolificacy in closed rabbit lines

In closed rabbit lines selected for prolificacy at the Polytechnic University of Valencia, genetic responses are predicted using BLUP. With a standard additive BLUP model and year‐season (YS) effects fitted as fixed, genetic trends were overestimated compared to responses estimated using control populations obtained from frozen embryos. In these lines, there is a confounding between genetic trend, YS effects and inbreeding, and the role of dominance is uncertain. This is a common situation in data from reproductively closed selection lines. This paper fits different genetic evaluation models to data of these lines, aiming to identify the source of these biases: dominance, inbreeding depression and/or an ill‐conditioned model due to the strong collinearity between YS, inbreeding and genetic trend. The study involved three maternal lines (A, V and H) and analysed two traits, total born (TB) and the number of kits at weaning (NW). Models fitting YS effect as fixed or random were implemented, in addition to additive genetic, permanent environment effects and non‐inbred dominance deviations effects. When YS was fitted as a fixed effect, the genetic trends were overestimated compared to control populations, inbreeding had an apparent positive effect on litter size and the environmental trends were negative. When YS was fitted as random, the genetic trends were compatible with control populations results, inbreeding had a negative effect (lower prolificacy) and environmental trends were flat. The model fitting random YS, inbreeding and non‐inbred dominance deviations yielded the following ratios of additive and dominance variances to total variance for NW: 0.06 and 0.01 for line A, 0.06 and 0.00 for line V and 0.01 and 0.08 for line H. Except for line H, dominance deviations seem to be of low relevance. When it is confounded with inbreeding as in these lines, fitting YS effect as random allows correct estimation of genetic trends.     

Estimation of genetic parameters for lactational milk yields using two-dimensional random regressions on parities and days in milk in Chinese Simmental cattle

A two‐dimensional random regression model with regressions on days in milk (DIM) and parity number was applied to lactational milk yields in Chinese Simmental cattle. Random regressions were fitted for additive genetic and permanent environmental effects using a two‐dimensional polynomial on DIM and parity number. A total of 4340 lactational milk yields from Chinese Simmental cattle which calved between 1980 and early 2000 were used in this study. Variance components were estimated using Bayesian methodology via Gibbs sampling. Variances of random regression coefficients associated with all terms of the polynomials were significant. A covariance function showed that heritabilities of lactational milk yields between 200 and 400 DIM over parities varied between 0.25 and 0.45. Heritabilities of 305‐day milk yields from 1st to 6–8th parities were 0.28, 0.30, 0.32 0.32, 0.32, and 0.31, respectively. Ratios of permanent environment variances to total variances at each DIM were greater than corresponding heritabilities. Generally, genetic correlations were higher between lactational milk yields with similar DIM and parity number.     

Random regression analyses using B-splines functions to model growth from birth to adult age in Canchim cattle

The objective of this work was to estimate covariance functions using random regression models on B-splines functions of animal age, for weights from birth to adult age in Canchim cattle. Data comprised 49,011 records on 2435 females. The model of analysis included fixed effects of contemporary groups, age of dam as quadratic covariable and the population mean trend taken into account by a cubic regression on orthogonal polynomials of animal age. Residual variances were modelled through a step function with four classes. The direct and maternal additive genetic effects, and animal and maternal permanent environmental effects were included as random effects in the model. A total of seventeen analyses, considering linear, quadratic and cubic B-splines functions and up to seven knots, were carried out. B-spline functions of the same order were considered for all random effects. Random regression models on B-splines functions were compared to a random regression model on Legendre polynomials and with a multitrait model. Results from different models of analyses were compared using the REML form of the Akaike Information criterion and Schwarz' Bayesian Information criterion. In addition, the variance components and genetic parameters estimated for each random regression model were also used as criteria to choose the most adequate model to describe the covariance structure of the data. A model fitting quadratic B-splines, with four knots or three segments for direct additive genetic effect and animal permanent environmental effect and two knots for maternal additive genetic effect and maternal permanent environmental effect, was the most adequate to describe the covariance structure of the data. Random regression models using B-spline functions as base functions fitted the data better than Legendre polynomials, especially at mature ages, but higher number of parameters need to be estimated with B-splines functions.     

Estimation of genetic parameters for heat stress,including dominance gene effects,on milk yield in Thai Holstein dairy cattle

Heat stress in tropical regions is a major cause that strongly negatively affects to milk production in dairy cattle. Genetic selection for dairy heat tolerance is powerful technique to improve genetic performance. Therefore, the current study aimed to estimate genetic parameters and investigate the threshold point of heat stress for milk yield. Data included 52 701 test‐day milk yield records for the first parity from 6247 Thai Holstein dairy cattle, covering the period 1990 to 2007. The random regression test day model with EM‐REML was used to estimate variance components, genetic parameters and milk production loss. A decline in milk production was found when temperature and humidity index (THI) exceeded a threshold of 74, also it was associated with the high percentage of Holstein genetics. All variance component estimates increased with THI. The estimate of heritability of test‐day milk yield was 0.231. Dominance variance as a proportion to additive variance (0.035) indicated that non‐additive effects might not be of concern for milk genetics studies in Thai Holstein cattle. Correlations between genetic and permanent environmental effects, for regular conditions and due to heat stress, were ? 0.223 and ? 0.521, respectively. The heritability and genetic correlations from this study show that simultaneous selection for milk production and heat tolerance is possible.     

Additional considerations to the use of single‐step genomic predictions in a dominance setting

Recent publications indicate that single‐step models are suitable to estimate breeding values, dominance deviations and total genetic values with acceptable quality. Additive single‐step methods implicitly extend known number of allele information from genotyped to non‐genotyped animals. This theory is well derived in an additive setting. It was recently shown, at least empirically, that this basic strategy can be extended to dominance with reasonable prediction quality. Our study addressed two additional issues. It illustrated the theoretical basis for extension and validated genomic predictions to dominance based on single‐step genomic best linear unbiased prediction theory. This development was then extended to include inbreeding into dominance relationships, which is a currently not yet solved issue. Different parametrizations of dominance relationship matrices were proposed. Five dominance single‐step inverse matrices were tested and described as C¹ , C² , C³ , C⁴ and C⁵ . Genotypes were simulated for a real pig population (n = 11,943 animals). In order to avoid any confounding issues with additive effects, pseudo‐records including only dominance deviations and residuals were simulated. SNP effects of heterozygous genotypes were summed up to generate true dominance deviations. We added random noise to those values and used them as phenotypes. Accuracy was defined as correlation between true and predicted dominance deviations. We conducted five replicates and estimated accuracies in three sets: between all ( S₁ ), non‐genotyped ( S₂ ) and inbred non‐genotyped ( S₃ ) animals. Potential bias was assessed by regressing true dominance deviations on predicted values. Matrices accounting for inbreeding ( C³ , C⁴ and C⁵ ) best fit. Accuracies were on average 0.77, 0.40 and 0.46 in S₁ , S₂ and S₃ , respectively. In addition, C³ , C⁴ and C⁵ scenarios have shown better accuracies than C¹ and C² , and dominance deviations were less biased. Better matrix compatibility (accuracy and bias) was observed by re‐scaling diagonal elements to 1 minus the inbreeding coefficient ( C⁵ ).