Absorption of equations for non-parents for an animal model with maternal effects and genetic groups

Rules for forming the mixed-model equations for the reduced animal model with all relationships and including maternal effects have been set out by Quaas and Pollak. They also have shown how to simplify the mixed-model equations when genetic group effects are included in the model with what has become known as the Q-P transformation. Westell has given rules for calculating the coefficients for the Q-P transformed equations that are associated with the inverse of the numerator relationship matrix and genetic group effects. Those rules can be extended to include maternal effects and genetic groups for maternal as well as direct effects. As with the rules of Quaas and Pollak for the equations for the reduced animal model, a similar set of rules can be obtained for the genetic groups model after the Q-P transformation. The rules are derived easily by examining the algebraic results of absorbing the direct and maternal breeding value equations for non-parents into the parent breeding value, group and fixed effects equations. These rules involve Westell's rules and the inverse elements of the genetic (co)variance matrix for direct and maternal additive genetic effects. The rules make calculation of breeding values for parents for models including direct and maternal genetic group effects nearly as easy as for models without genetic group effects. Back solution for direct and maternal breeding values of non-parents similarly is as simple as when genetic group effects are not in the model.     

Illustration of the maternal animal model used for genetic evaluation of beef cattle

National cattle evaluation programs for weaning weight in most beef breed associations involve implementation of the maternal animal model to predict direct and maternal EPD. With this model, direct breeding values are predicted for all animals with records or pedigree ties to animals with records, or both. Even though maternal genetic value is expressed only in animals that become dams, these effects are transmitted by all parents and inherited from parents by all animals, leading to maternal breeding values being predicted for all animals as well. A small example data set was simulated involving 12 parents, 8 nonparents, and 13 animals with weaning weight records. The pedigree was developed to include paternal and maternal half-sib families, full-sibs, and some inbreeding, similar to field populations of beef cattle. Assembly of the mixed model equations and solutions for the maternal animal model are illustrated explicitly to assist animal breeding students in their understanding of the properties of the maternal animal model and to explicitly implement the model. Model parameters and moments, fixed contemporary group solutions, adjustment of breeding values for merit of mates, interpretation of maternal permanent environmental effect solutions, and alternatives for the assembly of the equations are shown. This example should lead to increased student and producer understanding of genetic improvement programs for weaning weight in beef cattle.     

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.     

Variance components and breeding values for growth traits from different statistical models.

Estimates of genetic parameters resulting from various analytical models for birth weight (BWT, n = 4,155), 205-d weight (WWT, n = 3,884), and 365-d weight (YWT, n = 3,476) were compared. Data consisted of records for Line 1 Hereford cattle selected for postweaning growth from 1934 to 1989 at ARS-USDA, Miles City, MT. Twelve models were compared. Model 1 included fixed effects of year, sex, age of dam; covariates for birth day and inbreeding coefficients of animal and of dam; and random animal genetic and residual effects. Model 2 was the same as Model 1 but ignored inbreeding coefficients. Model 3 was the same as Model 1 and included random maternal genetic effects with covariance between direct and maternal genetic effects, and maternal permanent environmental effects. Model 4 was the same as Model 3 but ignored inbreeding. Model 5 was the same as Model 1 but with a random sire effect instead of animal genetic effect. Model 6 was the same as Model 5 but ignored inbreeding. Model 7 was a sire model that considered relationships among males. Model 8 was a sire model, assuming sires to be unrelated, but with dam effects as uncorrelated random effects to account for maternal effects. Model 9 was a sire and dam model but with relationships to account for direct and maternal genetic effects; dams also were included as uncorrelated random effects to account for maternal permanent environmental effects. Model 10 was a sire model with maternal grandsire and dam effects all as uncorrelated random effects. Model 11 was a sire and maternal grandsire model, with dams as uncorrelated random effects but with sires and maternal grandsires assumed to be related using male relationships. Model 12 was the same as Model 11 but with all pedigree relationships from the full animal model for sires and maternal grandsires. Rankings on predictions of breeding values were the same regardless of whether inbreeding coefficients for animal and dam were included in the models. Heritability estimates were similar regardless of whether inbreeding effects were in the model. Models 3 and 9 best fit the data for estimation of variances and covariances for direct, maternal genetic, and permanent environmental effects. Other models resulted in changes in ranking for predicted breeding values and for estimates of direct and maternal heritability. Heritability estimates of direct effects were smallest with sire and sire-maternal grandsire models.     

Estimation of genetic effects in the presence of multicollinearity in multibreed beef cattle evaluation

Breed additive, dominance, and epistatic loss effects are of concern in the genetic evaluation of a multibreed population. Multiple regression equations used for fitting these effects may show a high degree of multicollinearity among predictor variables. Typically, when strong linear relationships exist, the regression coefficients have large SE and are sensitive to changes in the data file and to the addition or deletion of variables in the model. Generalized ridge regression methods were applied to obtain stable estimates of direct and maternal breed additive, dominance, and epistatic loss effects in the presence of multicollinearity among predictor variables. Preweaning weight gains of beef calves in Ontario, Canada, from 1986 to 1999 were analyzed. The genetic model included fixed direct and maternal breed additive, dominance, and epistatic loss effects, fixed environmental effects of age of the calf, contemporary group, and age of the dam x sex of the calf, random additive direct and maternal genetic effects, and random maternal permanent environment effect. The degree and the nature of the multicollinearity were identified and ridge regression methods were used as an alternative to ordinary least squares (LS). Ridge parameters were obtained using two different objective methods: 1) generalized ridge estimator of Hoerl and Kennard (R1); and 2) bootstrap in combination with cross-validation (R2). Both ridge regression methods outperformed the LS estimator with respect to mean squared error of predictions (MSEP) and variance inflation factors (VIF) computed over 100 bootstrap samples. The MSEP of R1 and R2 were similar, and they were 3% less than the MSEP of LS. The average VIF of LS, R1, and R2 were equal to 26.81, 6.10, and 4.18, respectively. Ridge regression methods were particularly effective in decreasing the multicollinearity involving predictor variables of breed additive effects. Because of a high degree of confounding between estimates of maternal dominance and direct epistatic loss effects, it was not possible to compare the relative importance of these effects with a high level of confidence. The inclusion of epistatic loss effects in the additive-dominance model did not cause noticeable reranking of sires, dams, and calves based on across-breed EBV. More precise estimates of breed effects as a result of this study may result in more stable across-breed estimated breeding values over the years.     

Genetic and environmental trends for litter size in swine

Best linear unbiased predictors (BLUP) of breeding values for additive direct and additive maternal genetic effects were estimated from 3,944 purebred Yorkshire and Landrace first-parity litters recorded on the Quebec Record of Performance Sow Productivity Program and born between 1977 and 1987. Breeding values for gilts, dams, and sires were estimated using an individual animal model for measures of litter size of total number born (NOBN), number born alive (NOBA), and number weaned (NOWN). Environmental trends were estimated from average herd-year solutions, and genetic trends were estimated by regression of estimated breeding value on year of birth. Environmental trends were positive for all traits in both breeds but were significant only for NOWN in Landrace (.051 +/- .021 pigs/yr). Genetic trends were very small but were mainly negative for direct breeding value and combined direct and maternal breeding value. Significant estimates of genetic trends (P less than .05) were observed only within the Yorkshire breed, and these ranged from -.012 +/- .004 to .004 +/- .002 pigs/yr.     

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.     

Additive, dominance, and epistatic loss effects on preweaning weight gain of crossbred beef cattle from different Bos taurus breeds

(Co)variance components, direct and maternal breed additive, dominance, and epistatic loss effects on preweaning weight gain of beef cattle were estimated. Data were from 478,466 animals in Ontario, Canada, from 1986 to 1999, including records of both purebred and crossbred animals from Angus, Blonde d'Aquitaine, Charolais, Gelbvieh, Hereford, Limousin, Maine-Anjou, Salers, Shorthorn, and Simmental breeds. The genetic model included fixed direct and maternal breed additive, dominance, and epistatic loss effects, fixed environmental effects of age of the calf, contemporary group, and age of the dam x sex of the calf, random additive direct and maternal genetic effects, and random maternal permanent environment effects. Estimates of direct and maternal additive genetic, maternal permanent environmental and residual variances, expressed as proportions of the phenotypic variance, were 0.32, 0.20, 0.12, and 0.52, respectively. Correlation between direct and maternal additive genetic effects was -0.63. Breed ranking was similar to previous studies, but estimates showed large SE. The favorable effects of direct and maternal dominance (P < 0.05) on preweaning gain were equivalent to 1.3 and 2.3% of the phenotypic mean of purebred calves, respectively. The same features for direct and maternal epistatic loss effects were -2.2% (P < 0.05) and -0.1% (P > 0.05). The large SE of breed effects were likely due to multicollinearity among predictor variables and deficiencies in the dataset to separate direct and maternal effects and may result in a less reliable ranking of the animals for across breed comparisons. Further research to identify the causes of the instability of estimates of breed additive, dominance, and epistatic loss genetic effects, and application of alternative statistical methods is recommended.     

Multi-breed genetic evaluation in a Gelbvieh population

A multi-breed model was presented for the genetic evaluation of growth traits in beef cattle. In addition to the fixed effects, random direct and maternal genetic effects, and random maternal permanent environmental effects are considered; the model also fits direct and maternal heterosis and direct and maternal breed-of-founder (BOF) x generation group effects using a Bayesian approach that weights prior literature estimates relative to information supplied by the dataset to which the model will be applied. The multi-breed evaluation procedures also allow the inclusion of external evaluations for animals of other breeds. The multi-breed model was applied to a dataset provided by the American Gelbvieh Association. Different analyses were conducted by varying the weights given to the prior literature relative to the information provided by the dataset. Large differences were observed for the heterosis estimates, the BOF x generation group effect estimates, and the predicted breeding values across breeds due to the weights posed on prior literature estimates versus estimates derived directly from data. However, the rankings within breed were observed to be relatively robust to the different weights on prior information.     

Estimation of genetic parameters for hip dysplasia in Czech Labrador Retrievers

The objective of this study was to estimate the genetic parameters, genetic trends and breeding values using linear model (LM) and threshold model (TM) for the development of hip dysplasia (HD) in Labrador Retrievers in the Czech Republic (n = 3151). The right and left hip joints were evaluated separately using the Fédération Cynologique Internationale scoring system. Four linear and four TMs were tested for the correct estimation of genetic parameters. All the tested models utilized fixed effects of sex, assessor, year of birth, regression of age at evaluation, random direct genetic effects and the effect of the animals' permanent environments. The models differed in the inclusion of the following effects: fixed effects of regression of inbreeding coefficient, random maternal effect and random effect of the maternal permanent environment. Compared to the TM, the LM provided lower coefficients of direct (0.25-0.29 versus 0.26-0.35) and maternal heritability (0.01-0.02 versus 0.03-0.05), repeatability (0.76-0.77 versus 0.78-0.83) and of the correlation between direct and maternal effects (-0.55 to -0.21 versus -0.80 to -0.27). In the tested models, no statistical significance was found for fixed regression of inbreeding coefficients or for the random effect of the permanent maternal environment. In spite of the similarity of the LM and TM results, the TM is recommended as the more suitable model for estimating genetic parameters and subsequent breeding values for HD in Labrador Retrievers in the Czech Republic.     

Genetic parameters for direct and maternal calving ease in Walloon dairy cattle based on linear and threshold models

Calving ease scores from Holstein dairy cattle in the Walloon Region of Belgium were analysed using univariate linear and threshold animal models. Variance components and derived genetic parameters were estimated from a data set including 33 155 calving records. Included in the models were season, herd and sex of calf × age of dam classes × group of calvings interaction as fixed effects, herd × year of calving, maternal permanent environment and animal direct and maternal additive genetic as random effects. Models were fitted with the genetic correlation between direct and maternal additive genetic effects either estimated or constrained to zero. Direct heritability for calving ease was approximately 8% with linear models and approximately 12% with threshold models. Maternal heritabilities were approximately 2 and 4%, respectively. Genetic correlation between direct and maternal additive effects was found to be not significantly different from zero. Models were compared in terms of goodness of fit and predictive ability. Criteria of comparison such as mean squared error, correlation between observed and predicted calving ease scores as well as between estimated breeding values were estimated from 85 118 calving records. The results provided few differences between linear and threshold models even though correlations between estimated breeding values from subsets of data for sires with progeny from linear model were 17 and 23% greater for direct and maternal genetic effects, respectively, than from threshold model. For the purpose of genetic evaluation for calving ease in Walloon Holstein dairy cattle, the linear animal model without covariance between direct and maternal additive effects was found to be the best choice.     

The impact of data structure on genetic (co)variance components of early growth in sheep,estimated using an animal model with maternal effects

Several studies have noted high negative correlations between maternal genetic and direct additive effects and their influence on additive and maternal heritability of early growth traits in sheep. Multigeneration data from the Suffolk Sire Reference Scheme (SSRS) were used to investigate the effect of data structure on estimates of direct and maternal (co)variances for lamb 8-wk weight. In all analyses the additive, maternal genetic, maternal environmental, and residual effects were fitted along with the covariance between direct and maternal additive effects. The contributions of particular genetic relationships to the estimates were studied by analyzing subsets of the SSRS data. A further eight subsets were formed having 10% or 50% of the dams with their own records and having one or two, three or four, five or six, and more than six offspring per dam. Analysis of data having only 10% of the dams with their own record and one or two offspring records yielded a high negative correlation (-0.99) between direct and maternal genetic effects. However, the seven other data sets with more records per dam or a higher proportion of dams with their own records produced values of -0.35 to -0.51. Data structure and the number of dams and granddams with records are important determinants of estimated direct and maternal effects in early growth traits.     

Weaning weight and post-weaning gain genetic parameters and genetic trends in a Blanco Orejinegro–Romosinuano–Angus–Zebu multibreed cattle population in Colombia

Genetic parameters and genetic trends for weaning weight adjusted to 240 d of age (WW240), and weight gain from weaning to 24 mo of age (GW730) were estimated in a Colombian beef cattle population composed of Blanco Orejinegro, Romosinuano, Angus, and Zebu straightbred and crossbred animals. Calves were born and weaned in a single farm, and moved to 14 farms postweaning. Data were analyzed using a multiple trait mixed model procedures. Estimates of variance components and genetic parameters were obtained by Restricted Maximum Likelihood. The 2-trait model included the fixed effects of contemporary group (herd–year–season–sex), age of dam (WW240 only), breed direct genetic effects (as a function of breed fractions of calves), breed maternal genetic effects (as a function of breed fractions of dams; WW240 only), individual heterosis (as a function of calf heterozygosity), and maternal heterosis (as a function of dam heterozygosity; WW240 only). Random effects for WW240 were calf direct genetic, dam maternal genetic, permanent environmental maternal, and residual. Random effects for GW730 were calf direct genetic and residual. All relationships among animals were accounted for. Program AIREML was used to perform computations. Estimates of heritabilities for additive direct genetic effects were 0.20 ± 0.003 for WW240, and 0.32 ± 0.004 for GW730. Maternal heritability was 0.14 ± 0.002 for WW240. Estimates of heritability suggest that selection for preweaning and postweaning growth in this population is feasible. Low direct and maternal preweaning heritabilities suggest that nutrition and management should be improved to allow fuller expressions of calf direct growth and cow maternal ability. The genetic correlation between direct additive and maternal additive effects for WW240 was − 0.42 ± 0.009, indicating an antagonistic relationship between these effects. The correlation between additive direct genetic effects for WW240 and GW730 was almost zero (− 0.04 ± 0.009), suggesting that genes affecting growth preweaning may differ from those influencing growth postweaning. Trends were negative for direct WW240 and GW730 weighted yearly means of calves, sires, and dams from 1995 to 2006. Maternal WW240 showed near zero trends during these years. Trends for calf direct WW240 and GW730 followed sire trends closely, suggesting that more emphasis was placed on choosing sires than on dam replacements.     

Evaluation models and genetic parameters for calving difficulty in beef cattle

Calving difficulty was analyzed under threshold and linear models considering either a fixed or random herd-year effect. The aim of the study was to compare models for predicting breeding values according to the size of herd-year groups. When simulating data sets with small herds, in order to obtain an unbiased evaluation under a nonrandom and negative association of sire and herd effects, the best model for a practical evaluation was the fixed linear model. Field data included 246,576 records of the largest Charolais herds in France. Models were compared using the correlations of estimated breeding values between the different models. Although the best model from a theoretical point of view was a threshold model with a fixed herd-year effect, a linear model with a fixed herd-year effect was the best choice from a practical point of view for predicting direct effects for calving difficulty in beef cattle and was a sufficient choice for predicting the associated maternal effects for data set with large herds. Correlations between direct estimated breeding values under the reference model and the fixed linear model and the random threshold model were 0.94 and 0.91, respectively. Correlations between the corresponding maternal estimated breeding values were 0.94 and 0.98. Heritabilities of direct effects were 0.27 and 0.14 under fixed threshold and fixed linear models, respectively. The corresponding heritabilities of maternal effects were 0.18 and 0.13, and the genetic correlation between direct and maternal effects were -0.36 and -0.34, respectively.     

Synthesis of direct and maternal genetic components of economically important traits from beef breed-cross evaluations

Published information on relative performance of beef breed crosses was used to derive combined estimates of purebred breed values for predominant temperate beef breeds. The sources of information were largely from the United States, Canada, and New Zealand, although some European estimates were also included. Emphasis was on maternal traits of potential economic importance to the suckler beef production system, but some postweaning traits were also considered. The estimates were taken from comparison studies undertaken in the 1970s, 1980s and 1990s, each with representative samples of beef breeds used in temperate agriculture. Weighting factors for breed-cross estimates were derived using the number of sires and offspring that contributed to that estimate. These weights were then used in a weighted multiple regression analysis to obtain single purebred breed effects. Both direct additive and maternal additive genetic effects were estimated for preweaning traits. Important genetic differences between the breeds were shown for many of the traits. Significant regression coefficients were estimated for the effect of mature weight on calving ease, both maternal and direct additive genetic, survival to weaning direct, and birth weight direct. The breeds with greater mature weight were found to have greater maternal genetic effects for calving ease but negative direct genetic effects on calving ease. A negative effect of mature weight on the direct genetic effect of survival to weaning was observed. A cluster analysis was done using 17 breeds for which information existed on nine maternal traits. Regression was used to predict breed-cross-specific heterosis using genetic distance. Only five traits, birth weight, survival to weaning, cow fertility, and preweaning and postweaning growth rate had enough breed-cross-specific heterosis estimates to develop a prediction model. The breed biological values estimated provide a basis to predict the biological value of crossbred suckler cows and their offspring.     

Comparison of threshold vs linear and animal vs sire models for predicting direct and maternal genetic effects on calving difficulty in beef cattle

This study compared the accuracy of several models for obtaining genetic evaluations of calving difficulty. The models were univariate threshold animal (TAM), threshold sire-maternal grandsire (TSM), linear animal (LAM), and linear sire-maternal grandsire (LSM) models and bivariate threshold-linear animal (TLAM), threshold-linear sire-maternal grandsire (TLSM), linear-linear animal (LLAM), and linear-linear sire-maternal grandsire (LLSM) models for calving difficulty and birth weight. Data were obtained from the American Gelbvieh Association and included 84,420 first-parity records of both calving difficulty and birth weight. Calving difficulty scores were distributed as 73.4% in the first category (no assistance), 18.7% in the second, 6.3% in the third, and 1.6% in the fourth. Included in the animal models were fixed sex of calf by age of dam subclasses, random herd-year-season effects, and random animal direct and maternal breeding values. Sire-maternal grandsire models were similar to the animal models, with animal and maternal effects replaced by sire and maternal grandsire effects. Models were compared using a data splitting technique based on the correlation of estimated breeding values from two samples, with one-half of the calving difficulty records discarded randomly in the first sample and the remaining calving difficulty records discarded in the second sample. Reported correlations are averages of 10 replicates. The results obtained using animal models confirmed the slight advantage of TAM over LAM (0.69 vs 0.63) and TLAM over LLAM (0.90 vs 0.86). Bivariate analyses greatly improved the accuracy of genetic prediction of direct effects on calving difficulty relative to univariate analyses. Similar ranking of the models was found for maternal effects, but smaller correlations were obtained for bivariate models. For sire-maternal grandsire models, no differences between sire or maternal grandsire correlations were observed for TLSM compared to LLSM, and small differences were observed between TSM and LSM. The threshold model offered advantages over the linear model in animal models but not in sire-maternal grandsire models. For genetic evaluation of calving difficulty in beef cattle, the threshold-linear animal model seems to be the best choice for predicting both direct and maternal effects.     

Accounting for unknown foster dams in the genetic evaluation of embryo transfer progeny

Animals born by embryo transfer (ET) are usually not included in the genetic evaluation of beef cattle for preweaning growth if the recipient dam is unknown. This is primarily to avoid potential bias in the estimation of the unknown age of dam. We present a method that allows including records of calves with unknown age of dam. Assumptions are as follows: (i) foster cows belong to the same breed being evaluated, (ii) there is no correlation between the breeding value (BV) of the calf and the maternal BV of the recipient cow, and (iii) cows of all ages are used as recipients. We examine the issue of bias for the fixed level of unknown age of dam (AOD) and propose an estimator of the effect based on classical measurement error theory (MEM) and a Bayesian approach. Using stochastic simulation under random mating or selection, the MEM estimating equations were compared with BLUP in two situations as follows: (i) full information (FI); (ii) missing AOD information on some dams. Predictions of breeding value (PBV) from the FI situation had the smallest empirical average bias followed by PBV obtained without taking measurement error into account. In turn, MEM displayed the highest bias, although the differences were small. On the other hand, MEM showed the smallest MSEP, for either random mating or selection, followed by FI, whereas ignoring measurement error produced the largest MSEP. As a consequence from the smallest MSEP with a relatively small bias, empirical accuracies of PBV were larger for MEM than those for full information, which in turn showed larger accuracies than the situation ignoring measurement error. It is concluded that MEM equations are a useful alternative for analysing weaning weight data when recipient cows are unknown, as it mitigates the effects of bias in AOD by decreasing MSEP.     

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.     

Maternal animal model with correlation between maternal environmental effects of related dams

A procedure to take into account the nongenetic relationship between maternal effects in adjacent generations is presented. It considers a correlation between maternal environments provided by a dam and its daughters (lambda). The dispersion structure of the maternal animal model was modified to include a correlation matrix (E) that relates the maternal permanent environmental effects. The structures of the E matrix and its inverse (E(-1)) are described. Both matrices are completely defined by the correlation coefficient lambda. An algorithm to compute these matrices from pedigree information was also developed. Furthermore, a Bayesian analysis of this model including the lambda parameter was developed using Gibbs sampling, with Metropolis steps for the nonstandard conditional distributions. With simulated data, the proposed model reduced the bias in all estimates of dispersion parameters when an antagonism between the maternal effects received by a daughter and its future maternal environment existed. This model also provides an estimate of the environmental relationship between the maternal effects of dams and daughters by the lambda parameter. The same Bayesian analysis was also carried out with weaning weight data of the Bruna dels Pirineus breed. The posterior means (standard deviation) of (co)variance ratios were .214 (.081) for direct heritability (h2d), .107 (.033) for maternal heritability (h2m), .047 (.020) for the proportion of variance due to maternal environmental effects (c2m), and -.034 (.043) for the genetic correlation between direct and maternal effects (r(dm)). The posterior mean of lambda parameter was -.190, and 76% of its marginal posterior distribution took negative values. As occurred with simulated data, considering the maternal environmental correlation in the analysis implied higher h2m estimates, lower c2m and h2d estimates, and less negative values for the marginal posterior distribution of r(dm). These results were considered as evidence of the environmental antagonism between maternal effects provided by a dam and its daughters to weaning weight of their progeny in the Bruna dels Pirineus breed.     

Effects of embryo transfer in beef cattle on genetic evaluation methodology

Because female cattle may be superovulated and their embryos dispersed to many surrogate dams, both the recording system and the models for genetic evaluation need to be modified. The recording system must keep track of the breed of the surrogate dams as well as their age and registration numbers. The model for evaluation of beef cattle for traits in which maternal effects are important, for example, must distinguish between natural and surrogate dams. The potential for interactions of breed and age of surrogate dam with sex of calf and the possibility that these may differ from age of natural dam x sex of calf interactions needs further study. A discussion of these issues is given. An example is given to illustrate how surrogate dams might be modeled. Assuming that only the breed of surrogate dam is known and that each surrogate dam gives birth to only one progeny, the mixed model equations for genetic evaluations can be simplified. These are illustrated.