Estimation of variance and covariance components to determine heritabilities and repeatability of weaning weight in American Simmental cattle

Components of (co)variance for weaning weight were estimated from field data provided by the American Simmental Association. These components were obtained for the observational components of variance corresponding to a sire, maternal grandsire, and dam within maternal grandsire model. From these estimates, direct additive genetic variance (Sigma2A), maternal additive genetic variance (Sigma2M), covariance between direct and maternal additive genetic effects (SigmaAM), variance of permanent environment(Sigma2pe) and temporary environment variance(Sigma2te) were determined. A procedure to approximate restricted maximum likelihood (REML) estimates of the observational components of variance based on the expectation-maximization (EM) algorithm is described. From these results, phenotypic variance ( ) of weaning weight was 667.88 kg2. Values forSigma2A, Sigma2M, Sigma2pe and Sigma2te were 79,30,58,38,49.45, and 469.97 kg2, respectively. Genetic correlation between direct and maternal additive genetic effects was .16.     

Estimating maternal genetic effects in livestock

This study investigates the estimation of direct and maternal genetic (co)variances, accounting for environmental covariances between direct and maternal effects. Estimated genetic correlations between direct and maternal effects presented in the literature have often been strongly negative, and their validity has been questioned. Explanations of extreme estimates have focused on the existence of environmental covariances between dam and offspring. As a solution, models including a regression on dam-phenotype have been proposed, but have yielded biased estimates. The performance of models that implement the variance structure arising from the classical model of Willham, however, has not been evaluated. This study investigated the covariance structure of the parts of the residual term that arise from Willham's model. Results show that a correlation between the residual of the record of an individual and that of its dam is a direct consequence of combining Willham's model with the usual assumption that phenotypic covariances between different traits are the sum of additive genetic and environmental covariances. Stochastic simulations show that fitting this structure yields unbiased estimates of the genetic (co)variances. When correlated residuals were ignored in the cases investigated, the bias in the estimated genetic correlations was approximately equal to the value of the environmental correlation. In contrast to models including a regression on dam-phenotype, there were no difficulties with interpretation of results, and the approach was consistent with standard quantitative genetic theory. The use of Willham's model while accounting for correlated residuals is conceptually appealing and yields unbiased results, with no need for regression on dam phenotype. Inclusion of the ability to fit the residual variance structure required for maternal effects into existing software packages would be helpful to animal breeders.     

A method for the prediction of multitrait breeding values for use in stochastic simulation to compare progeny-testing schemes, with large progeny groups for proven sires

A method of approximating estimated breeding values (EBV) from a multivariate distribution of true breeding values (TBV) and EBV is proposed for use in large-scale stochastic simulation of alternative breeding schemes with a complex breeding goal. The covariance matrix of the multivariate distributions includes the additive genetic (co)variances and approximated prediction error (co)variances at different selection stages in the life of the animal. The prediction error (co)variance matrix is set up for one animal at a time, utilizing information on the selection candidate and its offspring, the parents, as well as paternal and maternal half- sibs. The EBV are a regression on TBV taking individual uncertainty into account, but with additional 'free' variation drawn at random. With the current information included in the calculation of the prediction error variance of a selection candidate, it is concluded that the method can be used to optimize progeny-testing schemes, where the progeny-tested sires are utilized with large progeny groups, e.g. through artificial insemination.     

Influence of population structure on estimates of direct and maternal parameters.

The estimation of (co)variance components for multiple traits with maternal genetic effects was found to be influenced by population structure. Two traits in a closed breeding herd with random mating were simulated over nine generations. Population structures were simulated on the basis of different proportions of dams not having performance records (0, 0.1, 0.5, 0.8 and 0.9): three genetic correlations (-0.5, 0.0 and +0.5) between direct and maternal effects and three genetic correlations (0, 0.3 and 0.8) between two traits. Three ratios of direct to maternal genetic variances, (1:3, 1:1, 3:1), were also considered. Variance components were estimated by restricted maximum likelihood. The proportion of dams without records had an effect on the SE of direct-maternal covariance estimates when the proportion was 0.8 or 0.9 and the true correlation between direct and maternal effects was negative. The ratio of direct to maternal genetic variances influenced the SE of the (co)variance estimates more than the proportion of dams with missing records. The correlation between two traits did not have an effect on the SE of the estimates. The proportion of dams without records and the correlation between direct and maternal effects had the strongest effects on bias of estimates. The largest biases were obtained when the proportion of dams without records was high, the correlation between direct and maternal effects was positive, and the direct variance was greater than the maternal variance, as would be the situation for most growth traits in livestock. Total bias in all parameter estimates for two traits was large in the same situations. Poor population structure can affect both bias and SE of estimates of the direct-maternal genetic correlation, and can explain some of the large negative estimates often obtained.     

Choice of statistical model for estimating genetic parameters using restricted maximum likelihood in swine

Summary Restricted maximum likelihood (REML) was used to determine the choice of statistical model, additive genetic maternal and common litter effects and consequences of ignoring these effects on estimates of variance–covariance components under random and phenotypic selection in swine using computer simulation. Two closed herds of different size and two traits, (i) pre‐weaning average daily gain and (ii) litter size at birth, were considered. Three levels of additive direct and maternal genetic correlations (r_dm) were assumed to each trait. Four mixed models (denoted as GRM1 through GRM4) were used to generate data sets. Model GRM1 included only additive direct genetic effects, GRM2 included only additive direct genetic and common litter effects, GRM3 included only additive direct and maternal genetic effects and GRM4 included all the random effects. Four mixed animal models (defined as EPM1 through EPM4) were defined for estimating genetic parameters similar to GRM. Data from each GRM were fitted with EPM1 through EPM4. The largest biased estimates of additive genetic variance were obtained when EPM1 was fitted to data generated assuming the presence of either additive maternal genetic, common litter effects or a combination thereof. The bias of estimated additive direct genetic variance (VA_d) increased and those of recidual variance (VE) decreased with an increase in level of r_dm when GRM3 was used. EPM1, EPM2 and EPM3 resulted in biased estimation of the direct genetic variances. EPM4 was the most accurate in each GRM. Phenotypic selection substantially increased bias of estimated additive direct genetic effect and its mean square error in trait 1, but decreased those in trait 2 when ignored in the statistical model. For trait 2, estimates under phenotypic selection were more biased than those under random selection. It was concluded that statistical models for estimating variance components should include all random effects considered to avoid bias.     

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.     

Variance and covariance estimates for weaning weight of Senepol cattle.

Variance and covariance components were estimated for weaning weight from Senepol field data for use in the reduced animal model for a maternally influenced trait. The 4,634 weaning records were used to evaluate 113 sires and 1,406 dams on the island of St. Croix. Estimates of direct additive genetic variance (sigma 2A), maternal additive genetic variance (sigma 2M), covariance between direct and maternal additive genetic effects (sigma AM), permanent maternal environmental variance (sigma 2PE), and residual variance (sigma 2 epsilon) were calculated by equating variances estimated from a sire-dam model and a sire-maternal grandsire model, with and without the inverse of the numerator relationship matrix (A-1), to their expectations. Estimates were sigma 2A, 139.05 and 138.14 kg2; sigma 2M, 307.04 and 288.90 kg2; sigma AM, -117.57 and -103.76 kg2; sigma 2PE, -258.35 and -243.40 kg2; and sigma 2 epsilon, 588.18 and 577.72 kg2 with and without A-1, respectively. Heritability estimates for direct additive (h2A) were .211 and .210 with and without A-1, respectively. Heritability estimates for maternal additive (h2M) were .47 and .44 with and without A-1, respectively. Correlations between direct and maternal (IAM) effects were -.57 and -.52 with and without A-1, respectively.     

Estimates of genetic parameters for worm resistance,wool and growth traits in Merino sheep of Uruguay

The genotype of an individual and the environment as the maternal ability of its dam have substantial effects on the phenotype expression of many production traits. The aim of the present study was to estimate the (co)variance components for worm resistance, wool and growth traits in Merino sheep, testing the importance of maternal effects and to determine the most appropriate model for each trait. The traits analyzed were Greasy Fleece Weight (GFW), Clean Fleece Weight (CFW), average Fibre Diameter (FD), Coefficient of Variation of FD (CVFD), Staple Length (SL), Comfort Factor (CF30), Weaning Weight (WWT), Yearling Body Weight (YWT) and Faecal worm Egg Count (FEC). The data were recorded during a 15-year period from 1995 to 2010, from Uruguayan Merino stud flocks. A Bayesian analysis was performed to estimate (co)variance components and genetic parameters. By ignoring or including maternal genetic or environmental effects, five different univariate models were fitted in order to determine the most effective for each trait. For CVFD and YWT, the model fitting the data best included direct additive effects as the only significant random source of variation. For GFW, CFW, FD, SL and CF30 the most appropriate model included direct-maternal covariance; while for FEC included maternal genetics effects with a zero direct-maternal covariance. The most suitable model for WWT included correlated maternal genetic plus maternal permanent environmental effects. The estimates of direct heritability were moderate to high and ranged from 0.15 for log transformed FEC to 0.74 for FD. Most of the direct additive genetic correlation (rg) estimations were in the expected range for Merino breed. However, the estimate of rg between FEC and FD was unfavourable (−0.18±0.03). In conclusion, there is considerable genetic variation in the traits analyzed, indicating the potential to make genetic progress on these traits. This study showed that maternal effects are influencing most of traits analyzed, thus these effects should be considered in Uruguayan Merino breeding programs; since the implementation of an appropriate model of analysis is critical to obtain accurate estimates.     

Bayesian inference for genetic parameter estimation on growth traits for Nelore cattle in Brazil,using the Gibbs sampler

This data set consisted of over 29 245 field records from 24 herds of registered Nelore cattle born between 1980 and 1993, with calves sires by 657 sires and 12 151 dams. The records were collected in south‐eastern and midwestern Brazil and animals were raised on pasture in a tropical climate. Three growth traits were included in these analyses: 205‐ (W205), 365‐ (W365) and 550‐day (W550) weight. The linear model included fixed effects for contemporary groups (herd‐year‐season‐sex) and age of dam at calving. The model also included random effects for direct genetic, maternal genetic and maternal permanent environmental (MPE) contributions to observations. The analyses were conducted using single‐trait and multiple‐trait animal models. Variance and covariance components were estimated by restricted maximum likelihood (REML) using a derivative‐free algorithm (DFREML) for multiple traits (MTDFREML). Bayesian inference was obtained by a multiple trait Gibbs sampling algorithm (GS) for (co)variance component inference in animal models (MTGSAM). Three different sets of prior distributions for the (co)variance components were used: flat, symmetric, and sharp. The shape parameters (ν) were 0, 5 and 9, respectively. The results suggested that the shape of the prior distributions did not affect the estimates of (co)variance components. From the REML analyses, for all traits, direct heritabilities obtained from single trait analyses were smaller than those obtained from bivariate analyses and by the GS method. Estimates of genetic correlations between direct and maternal effects obtained using REML were positive but very low, indicating that genetic selection programs should consider both components jointly. GS produced similar but slightly higher estimates of genetic parameters than REML, however, the greater robustness of GS makes it the method of choice for many applications.     

Estimation of variance components due to imprinting effects with DFREML and VCE: results of a simulation study

Treating gametes as homozygous diploid individuals, TIER and SÖLKNER (Theor. Appl. Genet. 85: 868–872, 1993) proposed a method which manages the use of available computer programs with a common animal model to estimate variance components caused by imprinting effects. Despite some relevant model restrictions, this approach has already been used in some field data analyses by an adapted version of the widely used DFREML computer program, subsequently indicated by DFREML ^a. The main objective of this study was to ascertain the properties of DFREML ^a by computer simulation and to examine other alternative estimation approaches. The most important results may be summarized as follows: (1) Treating gametes as homozygous diploid individuals has the consequence that one‐half of the actually realized gametic effect is totally abstracted in variance component estimation. Thus, an additional adjustment of the phenotypic variance calculated by DFREML ^a is necessary to get correct values of estimated variance component ratios. (2) Adjusted DFREML ^a estimates yielded correct results when animals were unselected and only maternal or paternal imprinting (not both simultaneously) occurred. (3) When the model did not adequately account for the additive genetic component within a maternal lineage, significant upward biases for the cytoplasmic component were observed. (4) The use of a simple dam and sire model with appropriate relationship matrices can be recommended when only the difference of maternal and paternal imprinting effects is of primary interest and the covariance between maternal halfsibs is not substantially increased by common environmental effects. (5) An adequate estimation of variance components for all possible imprinting situations requires the use of an animal model augmented by both maternal and paternal gametic effects. Unfortunately, a computer program on the basis of such a model does not yet exist.     

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.     

Symmetric differences squared and analysis of variance procedures for estimating genetic and environmental variances and covariances for beef cattle weaning weight: II. Estimates from a data set

Analysis of variance (ANOVA) and symmetric differences squared (SDS) methods were used to estimate additive genetic and environmental variances and covariances associated with weaning weight. The two methods were applied to 503 beef records collected over 19 yr from a relatively unselected university Angus herd. The SDS methodology was used with four models. The first model included direct (g) and maternal (gm) additive genetic effects, the genetic covariance between direct and maternal additive genetic effects (sigma ggm), permanent maternal environmental effects (m) and temporary environmental effects (e). The second model also allowed for a nonzero environmental covariance (sigma mem) between dam and offspring weaning weights. Models 3 and 4 were models 1 and 2, respectively, expanded to include a grandmaternal genetic effect (gn) and covariances sigma ggn and sigma gmgn. Two ANOVA solution sets for the parameters of model 4 were based on sire, dam, maternal grandsire, maternal grandam and phenotypic variances and offspring-dam (covOD), offspring-sire (covOS), offspring-grandam (covOGD) and offspring-maternal half-aunt or uncle (covOMH) covariances. Four ANOVA solution sets for the parameters of model 2 were based on sire, dam, within dam and maternal grandsire variances, covOD and either covOS or covOGD. Symmetric differences squared estimates of h2g and h2gm averaged .30 and .16, respectively. All SDS estimates of rho ggm (correlation between direct and maternal genetic effects) were less than -1. Estimates of sigma mem were positive. Both SDS estimates and one of the two ANOVA estimates of the grandmaternal variance were negative. The ANOVA model 4 estimates of h2g were .33. The estimates of h2gm were .44 and .39, while the estimates for rho ggm were -.88 and -.80. Both estimates of sigma mem were positive. The four ANOVA model 2 estimates of h2g and h2gm averaged .33 and .48, respectively. Three of the four estimates of rho ggm were less than -.97; the fourth was .35. Three of the four estimates of sigma mem were positive. Expectations show the extent to which SDS and ANOVA estimators were biased by nonzero grandmaternal components that were not accounted for. The extent to which dominance components bias the ANOVA estimators also is shown. Nonzero grandmaternal effects need to be taken into account in either SDS or ANOVA solution sets, or important biases occur with most of the estimators. More numerous, and generally more severe, biases occur with ANOVA estimators than with SDS estimators in solution sets that do not account for grandmaternal effects.     

Estimation of correlation between maternal permanent environmental effects of related dams in beef cattle

Weaning weights from Gelbvieh (GV; n = 82,138) and Limousin (LM; n = 88,639) calves were used to estimate genetic and environmental variance components with models that included different values for the correlation (lambda) between permanent environmental effects of dams and their daughters. Each analysis included fixed discrete effects of contemporary group, sex of calf, age of dam at calving, and month of calving, a fixed continuous effect of age of calf, random direct and maternal additive genetic effects, permanent environmental effects due to dams, and residual effects. The REML procedure was employed with a "grid search," in which the likelihood was computed for a series of values for lambda. For both breeds, models that included a nonzero value for lambda fitted the data significantly better than the model that did not include lambda. The maximum restricted likelihood was obtained for lambda of approximately -0.2 for both breeds. Estimates of residual and direct genetic variances were similar for all values of lambda, including zero; however, estimates of maternal genetic variance and maternal heritability increased slightly, and maternal permanent environmental variance and the proportion of the maternal variance to the total (phenotypic) variance decreased slightly, when the correlated structure for permanent environmental effects was assumed. As the value of lambda became more negative, absolute values of the direct-maternal genetic covariance and direct-maternal correlation estimates were decreased. Pearson and rank correlations for direct genetic, maternal genetic, and maternal environmental effects estimated with and without lambda were very high (>0.99). These results indicated that the linear relationship between maternal permanent environmental effects of dams and their daughters for weaning weight is negative but low in both breeds. Considering this relationship in the operational model did not significantly affect estimated breeding values, and thus, it may not be important in genetic evaluations.     

Estimation of direct, maternal, and grandmaternal genetic effects for weaning weight in several breeds of beef cattle.

Weaning weights from nine parental breeds and three composites were analyzed to estimate variance due to grandmaternal genetic effects and to compare estimates for variance due to maternal genetic effects from two different models. Number of observations ranged from 794 to 3,465 per population. Number of animals in the pedigree file ranged from 1,244 to 4,326 per population. Two single-trait animal models were used to obtain estimates of covariance components by REML using an average information method. Model 1 included random direct and maternal genetic, permanent maternal environmental, and residual environmental effects as well as fixed sex x year and age of dam effects. Model 2 in addition included random grandmaternal genetic and permanent grandmaternal environmental effects to account for maternal effects of a cow on her daughter's maternal ability. Non-zero estimates of proportion of variance due to grandmaternal effects were obtained for 7 of the 12 populations and ranged from .03 to .06. Direct heritability estimates in these populations were similar with both models. Existence of variance due to grandmaternal effects did not affect the estimates of maternal heritability (m2) or the correlation between direct and maternal genetic effects (r(am)) for Angus and Gelbvieh. For the other five populations, magnitude of estimates increased for both m2 and r(am) when estimates of variance due to grandmaternal effects were not zero. Estimates of the correlation between maternal and grandmaternal genetic effects were large and negative. These results suggest that grand-maternal effects exist in some populations, that when such effects are ignored in analyses maternal heritability may be underestimated, and that the correlation between direct and maternal genetic effects may be biased downward if grandmaternal effects are not included in the model for weaning weight of beef cattle.     

Symmetric differences squared and analysis of variance procedures for estimating genetic and environmental variances and covariances for beef cattle weaning weight: I. Comparison via simulation

Analysis of variance (ANOVA) and symmetric differences squared (SDS) methods for estimating genetic and environmental variances and covariances associated with beef cattle weaning weight were compared via simulation. Simulation was based on the pedigree and record structure of 503 beef weaning weights collected over 19 yr from a university herd. The SDS methodology was used with four models. The simplest model included direct (g) and maternal (gm) additive genetic effects, genetic covariance between direct and maternal additive genetic effects (sigma ggm), permanent maternal environmental effects (m) and temporary environmental effects (e). The second model also allowed for a nonzero environmental covariance (sigma mem) between dam and offspring weaning weights. Models 3 and 4 were models 1 and 2, respectively, expanded to include a grandmaternal genetic effect (gn) and covariances sigma ggn and sigma gmgn. Two ANOVA solution sets for the parameters of model 4 were obtained using sire, dam, maternal grandsire, maternal grandam and phenotypic variances and offspring-dam (covOD), offspring-sire (covOS), offspring-grandam (covOGD), and offspring-maternal half-aunt or uncle (covOMH) covariances. Four ANOVA solution sets for the parameters of model 2 were obtained using sire, dam, within dam and maternal grandsire variances, covOD and either covOS or covOGD. Two sets of 1,000 replicates of the data were simulated. These data were used to compare precision and accuracy of SDS and ANOVA estimators, to estimate correlations among SDS and ANOVA estimators, and to study the importance of taking inbreeding into account with SDS methodology. All ANOVA estimators for rho ggm were biased downward. The SDS procedure had a clear advantage over ANOVA. Averages of SDS estimates were closer to parameter values used to simulate the data and their standard deviations were generally smaller. The standard deviations of both SDS and ANOVA estimates of rho ggm were very large. It is important to allow for a nonzero sigma mem (at least when it is negative) when using SDS methods; otherwise estimators of sigma 2gm and sigma ggm are biased upward and downward, respectively.     

Genetic parameters for nuclear and nonnuclear inheritance in three synthetic lines of beef cattle differing in mature size.

Genetic parameters for nuclear and cytoplasmic genetic effects were estimated from preweaning growth data collected on three synthetic lines of beef cattle differing in mature size. Lines of small-, medium-, and large-framed calves were represented in each of two research herds (Rhodes and McNay). Variance components were estimated separately by herd and size line for birth weight and 205-d weight (WW) by REML with an animal mode using an average of 847 and 427 calf records from Rhodes and McNay, respectively. Model 1 included effects of fixed year, sex of calf, age of dam, and random additive direct (a), additive maternal genetic (m), covariance (a,m), permanent environment affecting the dam, and residual error. Model 2 differed from Model 1 by including random cytoplasmic lineage effects and by ignoring permanent environmental effects. Model 1--direct (maternal) heritability estimates for birth weight at Rhodes were .62(.03) for small, .67(.06) for medium, and .30(.11) for large lines. Genetic correlations between direct and maternal effects for birth weight were .67, -.16, and .48 for the respective size groups. For WW at Rhodes, direct (maternal) heritability estimates were .30(.29), .30(.14), and .10(.16) for small, medium, and large lines, respectively, with genetic correlations of -.34 (small), -.12 (medium), and .17 (large). Heritability estimates at McNay were similar to those at Rhodes, except that maternal genetic heritabilities for WW were smaller (.10, small; .01, medium; .00, large). Model 2--estimates for nuclear genetic effects were consistent with the estimates from Model 1. Cytoplasmic variance accounted for 0 to 5% of the total random variance in birth weight. For WW, cytoplasmic variance was negligible at Rhodes and accounted for 4% of the total random variance in the large line at McNay, averaging less than the permanent environment. Results failed to indicate that cytoplasmic variance was important for preweaning performance.     

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.     

Analysis of beef cattle longitudinal data applying a nonlinear model

The objective of this work was to evaluate the Nelore beef cattle, growth curve parameters using the Von Bertalanffy function in a nested Bayesian procedure that allowed estimation of the joint posterior distribution of growth curve parameters, their (co)variance components, and the environmental and additive genetic components affecting them. A hierarchical model was applied; each individual had a growth trajectory described by the nonlinear function, and each parameter of this function was considered to be affected by genetic and environmental effects that were described by an animal model. Random samples of the posterior distributions were drawn using Gibbs sampling and Metropolis-Hastings algorithms. The data set consisted of a total of 145,961 BW recorded from 15,386 animals. Even though the curve parameters were estimated for animals with few records, given that the information from related animals and the structure of systematic effects were considered in the curve fitting, all mature BW predicted were suitable. A large additive genetic variance for mature BW was observed. The parameter a of growth curves, which represents asymptotic adult BW, could be used as a selection criterion to control increases in adult BW when selecting for growth rate. The effect of maternal environment on growth was carried through to maturity and should be considered when evaluating adult BW. Other growth curve parameters showed small additive genetic and maternal effects. Mature BW and parameter k, related to the slope of the curve, presented a large, positive genetic correlation. The results indicated that selection for growth rate would increase adult BW without substantially changing the shape of the growth curve. Selection to change the slope of the growth curve without modifying adult BW would be inefficient because their genetic correlation is large. However, adult BW could be considered in a selection index with its corresponding economic weight to improve the overall efficiency of beef cattle production.     

Nuclear, cytoplasmic, and environmental effects on growth, fat, and muscle traits in suffolk lambs from a sire referencing scheme

Maternal effects are an important source of variation in early growth and body traits in sheep but are often excluded from genetic analyses. Maternal additive genetic, maternal environmental, and cytoplasmic effects were investigated in a large Suffolk breeding scheme using a range of models involving different combinations of these effects with the direct additive genetic effect. Weights at 8 wk of age and at scanning (mean age 146 d) and ultrasonically measured muscle and fat depth were analyzed using an animal model on 55,683 (8-wk weight) and 28,947 (scanning traits) lamb records. Simple additive models always overestimated the heritability of all traits when compared to more complex models. The successive inclusion of maternal environmental, maternal genetic, and the covariance between direct and maternal additive effects in the model significantly improved the fit for almost all models and all traits, as indicated by a likelihood ratio test. Under the full model, the heritability of both weight traits was low (0.14 and 0.20 for 8-wk and scanning weight, respectively). The maternal additive and maternal environmental effects, as a proportion of the phenotypic variance, were similar (0.10 and 0.08 for 8-wk weight and 0.07 and 0.06 for scanning weight). The two scanning traits had higher heritabilities (0.29 and 0.27 for muscle depth and fat depth, respectively) with low levels of maternal genetic and maternal environmental variance. No evidence was found of a cytoplasmic effect on any of the traits studied under the full model. Breeding schemes for early growth and body traits in sheep should account for maternal effects in their genetic evaluations in order to improve their accuracy. The exact model to use will depend on the trait and individual circumstances of the scheme.     

Effect of including relationships in the estimation of genetic parameters of beef calves.

Variances and covariances for birth weight, gain from birth to weaning (ADG), and 205-d weight were obtained from a sire-dam model and a sire-maternal grandsire model for a herd of Angus and a herd of Hereford cattle. Estimates of direct additive genetic variance (sigma 2A), maternal additive genetic variance (sigma 2M), covariance between direct and maternal additive genetic effects (sigma AM), permanent environmental variance (sigma 2PE), and residual variance (sigma 2e) were obtained both with and without the inverse of the numerator relationship matrix (A-1) included. Estimates of heritability for direct genetic effects (h2A), maternal genetic effects (h2M), and the correlation between direct and maternal effects (rAM) for birth weight were .37, .18, and -.01 in Angus and .53, .23, and -.19 in Herefords, respectively, for the analyses without A-1. For the analyses with A-1, estimates of h2A, h2M, and rAM were .42, .22, and -.12 for Angus and .58, .22, and -.13 for Herefords, respectively. Estimates of h2A, h2M, and rAM for ADG were .43, .15, and -.44 in Angus and .52, .38, and -.03 in Herefords, respectively, without A-1. With A-1, estimates of h2A, h2M, and rAM were .57, .15, and -.32 for Angus and .58, .39, and -.05 for Herefords, respectively. Estimates of h2A, h2M, and rAM for 205-d weight were .49, .15, and -.46 for Angus and .58, .43, and -.06 for Herefords, respectively, without A-1. With A-1, estimates of h2A, h2M, and rAM were .63, .16, and -.36 for Angus and .66, .43, and -.08 for Herefords, respectively. Estimates of h2A were higher with A-1 than without A-1, but estimates of h2M were similar. Using variances and covariances obtained from analyses including A-1 generally gave higher estimates of direct breeding values than using variances and covariances obtained from analyses not including A-1. Both Pearson product-moment and Spearman rank correlations were high (.99) between estimates of breeding values from the two analyses, although some changes in rank did occur.