On identifiability of (co)variance components in animal models with competition effects

There is an increased interest in estimating the (co)variance components of additive animal models with direct and competition effects (AMC). However, some attempts to estimate the dispersion parameters in different animal species faced problems of convergence or inaccurate estimates when pen effects entered the model. We argue that the problem relates to lack of identifiability of the (co)variance components in some AMC. The check for identifiability of the dispersion parameters in mixed models with linear (co)variance structure requires that all the eigenvalues of the restricted maximum likelyhood information matrix ( I ( θ )) be positive. We show, by way of simple numerical examples, that the singularity of I ( θ ) is due to confounding between fixed pen effects and the additive competition effects (SBVs). It is also observed that setting pen effects as random does not always remedy the collinearity with SBVs. An alternative AMC is presented in which the incidence matrix of the SBVs can be written as a function of the ‘intensity of competition’ (IC) among animals in the same pen. Examples are presented in which the ICs are related to time. The distribution of families of full and half sibs across pens also plays a role in the identifiability and asymptotic variances of the (co)variance components.     

Efficiency of alternative MCMC strategies illustrated using the reaction norm model.

The Markov chain Monte Carlo (MCMC) strategy provides remarkable flexibility for fitting complex hierarchical models. However, when parameters are highly correlated in their posterior distributions and their number is large, a particular MCMC algorithm may perform poorly and the resulting inferences may be affected. The objective of this study was to compare the efficiency (in terms of the asymptotic variance of features of posterior distributions of chosen parameters, and in terms of computing cost) of six MCMC strategies to sample parameters using simulated data generated with a reaction norm model with unknown covariates as an example. The six strategies are single-site Gibbs updates (SG), single-site Gibbs sampler for updating transformed (a priori independent) additive genetic values (TSG), pairwise Gibbs updates (PG), blocked (all location parameters are updated jointly) Gibbs updates (BG), Langevin-Hastings (LH) proposals, and finally Langevin-Hastings proposals for updating transformed additive genetic values (TLH). The ranking of the methods in terms of asymptotic variance is affected by the degree of the correlation structure of the data and by the true values of the parameters, and no method comes out as an overall winner across all parameters. TSG and BG show very good performance in terms of asymptotic variance especially when the posterior correlation between genetic effects is high. In terms of computing cost, TSG performs best except for dispersion parameters in the low correlation scenario where SG was the best strategy. The two LH proposals could not compete with any of the Gibbs sampling algorithms. In this study it was not possible to find an MCMC strategy that performs optimally across the range of target distributions and across all possible values of parameters. However, when the posterior correlation between parameters is high, TSG, BG and even PG show better mixing than SG.     

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.     

Estimation of genetic parameters for average daily gain using models with competition effects

Components of variance for ADG with models including competition effects were estimated from data provided by the Pig Improvement Company on 11,235 pigs from 4 selected lines of swine. Fifteen pigs with average age of 71 d were randomly assigned to a pen by line and sex and taken off test after approximately 89 d (off-test BW ranged from 61 to 158 kg). Models included fixed effects of line, sex, and contemporary group and initial test age as a covariate, with random direct genetic, competition (genetic and environmental), pen, litter, and residual effects. With the full model, variances attributable to direct, direct-competition, genetic competition, and litter (co)variance components could be partitioned; genetic competition variance was small but statistically significantly different from zero. Variances attributable to environmental competition, pen, and residual effects could not be partitioned, but combinations of these environmental variances were estimable. Variances could be partitioned with either pen effects or environmental competition effects in the model. Environmental competition effects seemed to be the source of variance associated with pens. With pen as a fixed effect and without environmental competition effects in the model, genetic components of variance could not be partitioned, but combinations of genetic (co)variances were estimable. With both pen and environmental competition effects ignored, estimates of direct-competition and genetic competition (co)variance components were greatly inflated. With competition (genetic and environmental) effects ignored, the estimate of pen variance increased by 39%, with little change in estimates of direct genetic or residual variance. When both pen and competition (genetic and environmental) effects were dropped from the model, variance attributable to direct genetic effects was inflated. Estimates of variance attributable to competition effects were small in this study. Including environmental competition effects as permanent environmental effects in the model did not change estimates of genetic (co)variances. We concluded that including either pen effects or environmental competition effects as random effects in the model avoids bias in estimates of genetic variances but that including pen effects is much easier.     

Monthly model for genetic evaluation of laying hens 1. Fixed regression.

1. This paper addresses the possibility of using a monthly model for the genetic evaluation of laying hens, based on the definition of a test day model with fixed regression as used in dairy cattle, in which monthly records were treated as repeated measurements of the same trait. 2. Production records of 6450 hens, daughters of 180 sires and 1335 dams were analysed using an animal model with restricted maximum likelihood (REML). The traits considered were individual monthly egg production and cumulative egg production in 11 months. Four different models were fitted to various combinations of monthly and cumulative records. The covariates were derived from the regression of Ali and Schaeffer (1987). 3. Spearman rank correlations were computed to compare breeding values from different models. Two types of correlations were computed: between individual breeding values and between sire breeding values based on subsets of full-sib records. 4. The results indicated that a monthly model with nested covariates produced higher heritability and permanent environmental variance than the models with non-nested or without covariates. The estimates of heritability obtained from monthly model were lower than the estimates from the cumulative model. The monthly model resulted in higher correlations of sire breeding values between two subsets of full-sib records than those from cumulative models. 5. In conclusion, the monthly model with nested covariates appears to be better than the model with non-nested covariates or without covariate. Although the heritability estimates obtained from the monthly model were lower, the monthly model with nested covariates could be better than the cumulative model for genetic evaluation of laying hens in the 1st cycle of laying period when using either full or part records. The use of information from odd months of production could be of interest for the evaluation of full records.     

Genetic parameters estimated with multitrait and linear spline-random regression models using Gelbvieh early growth data

Estimates of direct and maternal genetic parameters in beef cattle were obtained with a random regression model with a linear spline function (SFM) and were compared with those obtained by a multitrait model (MTM). Weight data of 18,900 Gelbvieh calves were used, of which 100, 75, and 17% had birth (BWT), weaning (WWT), and yearling (YWT) weights, respectively. The MTM analysis was conducted with a three-trait maternal animal model. The MTM included an overall linear partial fixed regression on age at recording for WWT and YWT, and direct-maternal genetic and maternal permanent environmental effects. The SFM included the same effects as MTM, plus a direct permanent environmental effect and heterogeneous residual variance. Three knots, or breakpoints, were set to 1, 205, and 365 d. (Co)variance components in both models were estimated with a Bayesian implementation via Gibbs sampling using flat priors. Because BWT had no variability of age at recording, there was good agreement between corresponding components of variance estimated from both models. For WWT and YWT, with the exception of the sum of direct permanent environmental and residual variances, there was a general tendency for SFM estimates of variances to be lower than MTM estimates. Direct and maternal heritability estimates with SFM tended to be lower than those estimated with MTM. For example, the direct heritability for YWT was 0.59 with MTM, and 0.48 with SFM. Estimated genetic correlations for direct and maternal effects with SFM were less negative than those with MTM. For example, the direct-maternal correlation for WWT was -0.43 with MTM and -0.33 with SFM. Estimates with SFM may be superior to MTM due to better modeling of age in both fixed and random effects.     

Dairy sire evaluation fitting some of the herd-year-season effects as random

Three models of sire evaluation using different environmental groupings were compared. Effects fitted were herd, period (either 6 or 12 months) within herd, season (either 1 or 2 or 4 months) within period within herd, sire and linear and quadratic regressions on age at calving. Models differed in fitting (1) the effect of herd-period-season fixed, or (2) herd-period fixed and herd-period-season random, or (3) herd fixed, herd-period and herd-period-season random. The overall effects of period and season of calving were regarded as fixed, and were removed by precorrection. Records of first lactation fat yield on 49 242 progeny of 69 widely used proven Friesian-Holstein sires in 1628 herds in England and Wales were used.Compared to Model 1, Model 2 required about four-fifths and Model 3 required two-thirds of the effective number of daughters to give the equivalent variance of the estimates of sire effects. Using random effects models the relative advantage, in terms of a smaller variance of sire effects, increased as the size of herd-period-season subclass decreased.In herd-period-season fixed effects models subclasses with a single of few records, or subclasses with all or almost all records of the same sire, contribute nothing or little to the progeny group comparisons. The random effects models could avoid these losses, and were considered to be useful especially where herds are small, provided sires can be assumed as randomly distributed over environmental subclasses.     

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.     

Comparison of models for estimation of genetic parameters for mature weight of Hereford cattle

Genetic parameters of mature weight are needed for effective selection and genetic evaluation. Data for estimating these parameters were collected from 1963 to 1985 and consisted of 32,018 mature weight records of 4,175 Hereford cows that were in one control and three selection lines that had been selected for weaning weight, for yearling weight, or for an index combining yearling weight and muscle score for 22 yr. Several models and subsets of the data were considered. The mature weight records consisted of a maximum of three seasonal weights taken each year, at brand clipping (February and March), before breeding (May and June), and at palpation (August and September). Heritability estimates were high (0.49 to 0.86) for all models considered, which suggests that selection to change mature weight could be effective. The model that best fit the data included maternal genetic and maternal permanent environmental effects in addition to direct genetic and direct permanent environmental effects. Estimates of direct heritability with this model ranged from 0.53 to 0.79, estimates of maternal heritability ranged from 0.09 to 0.21, and estimates of the genetic correlation between direct and maternal effects ranged from -0.16 to -0.67 for subsets of the data based on time of year that mature weight was measured. For the same subsets, estimates of the proportions of variance due to direct permanent environment and maternal permanent environment ranged from 0.00 to 0.09 and 0.00 to 0.06, respectively. Using a similar model that combined all records and included an added fixed effect of season of measurement of mature weight, direct heritability, maternal heritability, genetic correlation between direct and maternal effects, proportion of variance due to direct permanent environmental effects, and proportion of variance due to maternal permanent environmental effects were estimated to be 0.69, 0.13, -0.65, 0.00, and 0.04, respectively. Mature weight is a highly heritable trait that could be included in selection programs and maternal effects should not be ignored when analyzing mature weight data.     

Estimates of covariance functions for growth of Nelore cattle applying a parametric correlation structure to model within-animal correlations

A total of 20,065 weights recorded on 3016 Nelore animals were used to estimate covariance functions for growth from birth to 630 days of age, assuming a parametric correlation structure to model within-animal correlations. The model of analysis included fixed effects of contemporary groups and age of dam as quadratic covariable. Mean trends were taken into account by a cubic regression on orthogonal polynomials of animal age. Genetic effects of the animal and its dam and maternal permanent environmental effects were modelled by random regressions on Legendre polynomials of age at recording. Changes in direct permanent environmental effect variances were modelled by a polynomial variance function, together with a parametric correlation function to account for correlations between ages. Stationary and nonstationary models were used to model within-animal correlations between different ages. Residual variances were considered homogeneous or heterogeneous, with changes modelled by a step or polynomial function of age at recording. Based on Bayesian information criterion, a model with a cubic variance function combined with a nonstationary correlation function for permanent environmental effects, with 49 parameters to be estimated, fitted best. Modelling within-animal correlations through a parametric correlation structure can describe the variation pattern adequately. Moreover, the number of parameters to be estimated can be decreased substantially compared to a model fitting random regression on Legendre polynomial of age.     

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.     

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.     

Assessing the convergence of Markov Chain Monte Carlo methods: an example from evaluation of diagnostic tests in absence of a gold standard

The accessibility of Markov Chain Monte Carlo (MCMC) methods for statistical inference have improved with the advent of general purpose software. This enables researchers with limited statistical skills to perform Bayesian analysis. Using MCMC sampling to do statistical inference requires convergence of the MCMC chain to its stationary distribution. There is no certain way to prove convergence; it is only possible to ascertain when convergence definitely has not been achieved. These methods are rather subjective and not implemented as automatic safeguards in general MCMC software. This paper considers a pragmatic approach towards assessing the convergence of MCMC methods illustrated by a Bayesian analysis of the Hui–Walter model for evaluating diagnostic tests in the absence of a gold standard. The Hui–Walter model has two optimal solutions, a property which causes problems with convergence when the solutions are sufficiently close in the parameter space. Using simulated data we demonstrate tools to assess the convergence and mixing of MCMC chains using examples with and without convergence. Suggestions to remedy the situation when the MCMC sampler fails to converge are given. The epidemiological implications of the two solutions of the Hui–Walter model are discussed.     

Effect of competition on gain in feedlot bulls from Hereford selection lines

This study examined competition effects on ADG in the feedlot of 1,882 Hereford bulls representing 8 birth years from a selection experiment. Each year, 8 feedlot pens were used to feed bulls in groups, with 2 pens nested within each of the 4 selection lines. Gains were recorded for up to 8 periods of 28 d. Models for analyses included pen effects (fixed or random), fixed effects such as year and line, and random direct genetic, competition genetic (and in some analyses competition environmental), and environmental effects. Each pen mate as a competitor affected the records of all others in the pen. All lines traced to common foundation animals, so the numerator relationships among and within pens were the bases for separating direct and competition genetic effects and pen effects. For this population and pen conditions (average of 30 bulls per pen), the major results were 1) competition genetic effects seemed present for the first 28-d period but not for the following 7 periods; 2) models with pens considered as fixed effects could not separate variances and covariance due to direct and competition genetic effects; 3) models without competition effects had large estimates of the variance component due to pen effects for gain through 8 periods; and 4) models with genetic and environmental competition effects accounted for nearly all of the variance traditionally attributed to pen effects (even though estimates of the competition variance component were small, the estimates of pen variance were near zero).     

Estimation of litter size variability phenotypes in Large White sows

The objective of this study was to obtain new phenotypes of phenotypic variability for the total number born (TNB) in pigs using the residual variance of TNB. The analysis was based on 246,799 Large White litter observations provided by Topigs Norsvin. Three animal models were used to obtain estimates of residual variance for TNB: the basic model (BM) containing fixed effects of farm–year and season and random effects of animal and permanent environmental sow, the basic model with an additional fixed effect of parity (BMP) and a random regression model (RRM). The within-individual variance of the residuals was calculated and log-transformed to obtain three new variability traits: LnVarBM, LnVarBMP and LnVarRRM. Then, (co)variance components, heritability, the genetic coefficient of variation at the standard deviation level (GCV_SDe) and genetic correlations between the three LnVar's and between the LnVar's and mean total number born (mTNB) were estimated with uni-, bi- and trivariate models. Results indicated that genetically LnVar's are the same trait and are positively correlated with the mTNB (~0.60). Thus, both traits should be included in breeding programmes to avoid an increase in TNB variability while selecting for increased TNB. Heritability of the LnVar's was estimated at 0.021. The GCV_SDe for LnVar's showed that a change of 8% in residual standard deviation of TNB could be obtained per generation. Those results indicate that phenotypic variability of litter size is under genetic control, thus it may be improved by selection.     

Direct genetic, maternal genetic, and heterozygosity effects on weaning weight in a Colombian multibreed beef cattle population

The (co)variance components of BW at weaning (WW) were estimated for a Colombian multibreed beef cattle population. A single-trait animal model was used. The model included the fixed effect of contemporary group (sex, season, and year), and covariates including age of calf at weaning, age of cow, individual and maternal heterozygosity proportions, and breed percentage. Direct genetic, maternal genetic, permanent environmental, and residual effects were included as random effects. Direct, maternal, and total heritabilities were 0.23 +/- 0.047, 0.15 +/- 0.041, and 0.19, respectively. The genetic correlation between direct and maternal effects was -0.42 +/- 0.131, indicating that there may be antagonism among genes for growth and genes for maternal ability, which in turn suggests that improving WW by direct and maternal EPD may be difficult. A greater value for the direct heterosis effect compared with the maternal heterosis effect was found. Furthermore, the greater the proportion of Angus, Romosinuano, and Blanco Orejinegro breeds, the less the WW.     

Comparison of models to estimate genetic effects of weaning weight of Angus cattle

Weaning weights from nine sets of Angus field data from three regions of the United States were analyzed. Six animal models were used to compare two approaches to account for an environmental dam-offspring covariance and to investigate the effects of sire x herd-year interaction on the genetic parameters. Model 1 included random direct and maternal genetic, maternal permanent environmental, and residual effects. Age at weaning was a covariate. Other fixed effects were age of dam and a herd-year-management-sex combination. Possible influence of a dam's phenotype on her daughter's maternal ability was modeled by including a regression on maternal phenotype (fm) (Model 3) or by fitting grandmaternal genetic and grandmaternal permanent environmental effects (Model 5). Models 2, 4, and 6 were based on Models 1, 3, and 5, respectively, and additionally included sire x herd-year (SH) interaction effects. With Model 3, estimates of fm ranged from -.003 to .014, and (co)variance estimates were similar to those from Model 1. With Model 5, grandmaternal heritability estimates ranged from .02 to .07. Estimates of maternal heritability and direct-maternal correlation (r(am)) increased compared with Model 1. With models including SH, estimates of the fraction of phenotypic variance due to SH interaction effects were from .02 to .10. Estimates of direct and maternal heritability were smaller and estimates of r(am) were greater than with models without SH interaction effects. Likelihood values showed that SH interaction effects were more important than fm and grandmaternal effects. The comparisons of models suggest that r(am) may be biased downward if SH interaction and(or) grandmaternal effects are not included in models for weaning weight.     

Success at first insemination in Australian Angus cattle: analysis of uncertain binary responses

Field data from Australian Angus herds were used to investigate 2 methods of analyzing uncertain binary responses for success or failure at first insemination. A linear mixed model that included herd, year, and month of mating as fixed effects; unrelated service sire, additive animal, and residual as random effects; and linear and quadratic effects of age at mating as covariates was used to analyze binary data. An average gestation length (GL) derived from artificial insemination data was used to assign an insemination date to females mated to natural service sires. Females that deviated from this average GL led to uncertain binary responses. Two analyses were carried out: 1) a threshold model fitted to uncertain binary data, ignoring uncertainty (M1); and 2) a threshold model fitted to uncertain binary data, accounting for uncertainty via fuzzy logic classification (M2). There was practically no difference between point estimates obtained from M1 and M2 for service sire and herd variance; however, when uncertain binary data were analyzed ignoring uncertainty (M1), additive variance and heritability estimates were greater than with M2. Pearson correlations indicated that no major reranking would be expected for service sire effects and animal breeding values using M1 and M2. Given the results of the current study, a threshold model contemplating uncertainty is suggested for noisy binary data to avoid bias when estimating genetic parameters.     

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.     

Estimates of direct and maternal (co)variance components as well as genetic parameters of growth traits in Nellore sheep

In the present study, (co)variance components and genetic parameters in Nellore sheep were obtained by restricted maximum likelihood (REML) method using six different animal models with various combinations of direct and maternal genetic effects for birth weight (BW), weaning weight (WW), 6-month weight (6MW), 9-month weight (9MW) and 12-month weight (YW). Evaluated records of 2075 lambs descended from 69 sires and 478 dams over a period of 8 years (2007–2014) were collected from the Livestock Research Station, Palamaner, India. Lambing year, sex of lamb, season of lambing and parity of dam were the fixed effects in the model, and ewe weight was used as a covariate. Best model for each trait was determined by log-likelihood ratio test. Direct heritability for BW, WW, 6MW, 9MW and YW were 0.08, 0.03, 0.12, 0.16 and 0.10, respectively, and their corresponding maternal heritabilities were 0.07, 0.10, 0.09, 0.08 and 0.11. The proportions of maternal permanent environment variance to phenotypic variance (Pe²) were 0.07, 0.10, 0.07, 0.06 and 0.10 for BW, WW, 6MW, 9MW and YW, respectively. The estimates of direct genetic correlations among the growth traits were positive and ranged from 0.44(BW-WW) to 0.96(YW-9MW), and the estimates of phenotypic and environmental correlations were found to be lower than those of genetic correlations. Exclusion of maternal effects in the model resulted in biased estimates of genetic parameters in Nellore sheep. Hence, to implement optimum breeding strategies for improvement of traits in Nellore sheep, maternal effects should be considered.