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 infection prevalence from correlated binomial samples

Infection prevalence in a population often is estimated from grouped binary data expressed as proportions. The groups can be families, herds, flocks, farms, etc. The observed number of cases generally is assumed to have a Binomial distribution and the estimate of prevalence is then the sample proportion of cases. However, the individual binary observations might not be independent--leading to overdispersion. The goal of this paper was to demonstrate random-effects models for the estimation of infection prevalence from data which are correlated and in particular, to illustrate a nonparametric random-effects model for this purpose. The nonparametric approach is a relatively recent addition to the random-effects class of models and does not appear to have been discussed previously in the veterinary epidemiology literature. The assumptions for a logistic-regression model with a nonparametric random effect were outlined. In a demonstration of the method on data relating to Salmonella infection in Irish pig herds, the nonparametric method resulted in the classification of herds into a small number of distinct prevalence groups (i.e. low, medium and high prevalence) and also estimated the relative frequency of each prevalence category in the population. We compared the estimates from a logistic model with a nonparametric distribution for the random effects with four alternative models: a logistic-regression model with no random effects, a marginal model using a generalised estimating equation (GEE) and two methods of fitting a Normally distributed random effect (the GLIMMIX macro and the NLMIXED procedure both in SAS). Parameter estimates from random-effects models are not readily interpretable in terms of prevalences. Therefore, we outlined two methods for calculating population-averaged estimates of prevalence from random-effects models: one using numerical integration and the other using Monte Carlo simulation.     

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.     

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.     

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

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

Efficient Monte Carlo algorithm for restricted maximum likelihood estimation of genetic parameters

Monte Carlo (MC) methods have been found useful in estimation of variance parameters for large data and complex models with many variance components (VC), with respect to both computer memory and computing time. A disadvantage has been a fluctuation in round‐to‐round values of estimates that makes the estimation of convergence challenging. Furthermore, with Newton‐type algorithms, the approximate Hessian matrix might have sufficient accuracy, but the inaccuracy in the gradient vector exaggerates the round‐to‐round fluctuation to intolerable. In this study, the reuse of the same random numbers within each MC sample was used to remove the MC fluctuation. Simulated data with six VC parameters were analysed by four different MC REML methods: expectation‐maximization (EM), Newton–Raphson (NR), average information (AI) and Broyden's method (BM). In addition, field data with 96 VC parameters were analysed by MC EM REML. In all the analyses with reused samples, the MC fluctuations disappeared, but the final estimates by the MC REML methods differed from the analytically calculated values more than expected especially when the number of MC samples was small. The difference depended on the random numbers generated, and based on repeated MC AI REML analyses, the VC estimates were on average non‐biased. The advantage of reusing MC samples is more apparent in the NR‐type algorithms. Smooth convergence opens the possibility to use the fast converging Newton‐type algorithms. However, a disadvantage from reusing MC samples is a possible “bias” in the estimates. To attain acceptable accuracy, sufficient number of MC samples need to be generated.     

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

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

Two-step and random regression analyses of weight gain of station-tested beef bulls

Our objectives were to compare a two-step model and a joint procedure via random regression model for evaluating weight gain of beef bulls, weighed every 28 d on 140-d test, and to estimate genetic, environmental, and phenotypic parameters. Two-step analysis consisted of fitting fixed linear regressions to weights of each bull to determine weight gain on test. In the second step, gain on test was analyzed by a mixed model that included fixed effects of breed, test group, and starting age and random effects of weaning herd-year group and animal (additive genetic). The random regression model included the same effects as the two-step mixed-model analysis with an additional random animal permanent environment effect. Fourth-order Legendre polynomials of days on test were fitted for all fixed and random effects in the random regression model, except for breed. Breed effects and residual variances varied for each measurement period. Variance components and EBV for gain were obtained from the covariance function and estimates of random regression coefficients for weight, respectively. Random regression heritability estimates for gain on test increased over time, being maximum at end of test (0.38) and equal to two-step estimate. Permanent environment variance ratio estimates also increased over time and were greater than heritability estimates. Estimate of weaning herd-year variance ratio was approximately constant over time, being equal to 0.07 at end of test and similar to two-step estimate. Genetic correlations between gain through different periods on test given by random regression model were high (from 0.81, between 28 and 140-d gain on test, to 0.99, between 112 and 140-d gain on test). Genetic correlations between gain on discrete 28-d intervals were moderate to high (e.g., 0.49 and 0.99 between the last 28 d on test and the first and fourth 28 d, respectively). Rank correlations between EBV for 140-d gain by the two procedures were 0.98, 0.84, and 0.73 for all bulls and the 5% and 1% of bulls with highest random regression EBV, respectively. Results indicated that the two procedures rank top bulls quite differently for 140-d gain on test. Random regression model accounted for changes over time of genetic and environmental effects on the test weight gain curve of the bulls. Use of 112-d instead of a 140-d test provided similar ranking of bulls on the basis of EBV for gain on test.     

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.     

Genetic evaluation of Australian dairy cattle for somatic cell scores using multi-trait random regression test-day model

Summary A multi-trait (MT) random regression (RR) test day (TD) model has been developed for genetic evaluation of somatic cell scores for Australian dairy cattle, where first, second and third lactations were considered as three different but correlated traits. The model includes herd-test-day, year-season, age at calving, heterosis and lactation curves modelled with Legendre polynomials as fixed effects, and random genetic and permanent environmental effects modelled with Legendre polynomials. Residual variance varied across the lactation trajectory. The genetic parameters were estimated using asreml . The heritability estimates ranged from 0.05 to 0.16. The genetic correlations between lactations and between test days within lactations were consistent with most of the published results. Preconditioned conjugate gradient algorithm with iteration on data was implemented for solving the system of equations. For reliability approximation, the method of Tier and Meyer was used. The genetic evaluation system was validated with Interbull validation method III by comparing proofs from a complete evaluation with those from an evaluation based on a data set excluding the most recent 4 years. The genetic trend estimate was in the allowed range and correlations between the two sets of proofs were very high. Additionally, the RR model was compared to the previous test day model. The correlations of proofs between both models were high (0.97) for bulls with high reliabilities. The correlations of bulls decreased with increasing incompleteness of daughter performance information. The correlations between the breeding values from two consecutive runs were high ranging from 0.97 to 0.99. The MT RR TD model was able to make effective use of available information on young bulls and cows, and could offer an opportunity to breeders to utilize estimated breeding values for first and later lactations.     

Comparison of (co)variance component estimates in control populations of red flour beetle (Tribolium castaneum) using restricted maximum likelihood and Gibbs sampling

Mixed model (co)variance component estimates by REML and Gibbs sampling for two traits were compared for base populations and control lines of Red Flour Beetle (Tribolium castaneum). Two base populations (1296 records in the first replication, 1292 in the second) were sampled from laboratory stock. Control lines were derived from corresponding base populations with random selection and mating for 16 generations. The REML estimate of each (co)variance component for both pupa weight and family size was compared with the mean and 95% central interval of the particular (co)variance estimated by Gibbs sampling with three different weights on the given priors: ‘flat’, smallest, and 3.7% degrees of belief. Results from Gibbs sampling showed that flat priors gave a wider and more skewed marginal posterior distribution than the other two weights on priors for all parameters. In contrast, the 3.7% degree of belief on priors provided reasonably narrow and symmetric marginal posterior distributions. Estimation by REML does not have the flexibility of changing the weight on prior information as does the Bayesian analysis implemented by Gibbs sampling. In general, the 95% central intervals from the three different weights on priors in the base populations were similar to those in control lines. Most REML estimates in base populations differed from REML estimates in control lines. Insufficient information from the data, and confounding of random effects contributed to the variability of REML estimates in base populations. Evidence is presented showing that some (co)variance components were estimated with less precision than others. Results also support the hypothesis that REML estimates were equivalent to the joint mode of posterior distribution obtained from a Bayesian analysis with flat priors, but only when there was sufficient information from data, and no confounding among random effects.     

Correlated time to event data: Modeling repeated clinical mastitis data from dairy cattle in New York State

Mastitis is the most prevalent production disease in dairy herds worldwide and is considered to be the most economically important disease of dairy cattle. Modeling the risk of cows contracting mastitis is therefore of great interest for both targeting prevention programs and evaluating treatment protocols. Clinical mastitis (CM) is a disease of recurrent nature, thus correlation between the subsequent events within one cow may be present. This would violate the assumption behind most statistical time-to-event models. In the case of time to event models, the semi-parametric Cox regression models have become the default tool in modeling the time to an event. Limited methods are currently available to evaluate marginal and random (frailty) effects to account for multiple correlation sources. The objective of this study was to explore the implications of using several Cox or related semi-parametric or parametric models to estimate the hazard for CM in the presence of correlation between events. We evaluated the Andersen-Gill model which uses robust standard errors to account for the correlation, the Conditional Anderson-Gill model that uses stratification to account for event dependence, the Frailty model that introduces a random term to account for unobserved (cow level) heterogeneity, and a related generalized linear mixed model that uses Poisson regression to allow multi-level modeling of time-to-event data. We analyzed data on the occurrence of CM from five dairy farms in New York State. Data were from 8206 cows with 721, 275, 119, and 57 first, second, third, and fourth occurrences of CM, respectively, in the same lactation. The analysis of our sample dataset demonstrated that both cow- and farm-level correlation are present in the case of CM. The Conditional Frailty model was able to model one source of correlation in a random effect and one in a fixed effect. Poisson modeling allowed for simultaneous estimation of within cow correlation and within herd correlation.     

Genetic parameters for various random regression models to describe total sperm cells per ejaculate over the reproductive lifetime of boars

The objective of this study was to model the variances and covariances of total sperm cells per ejaculate (TSC) over the reproductive lifetime of AI boars. Data from boars (n = 834) selected for AI were provided by Smithfield Premium Genetics. The total numbers of records and animals were 19,629 and 1,736, respectively. Parameters were estimated for TSC by age of boar classification with a random regression model using the Simplex method and DxMRR procedures. The model included breed, collector, and year-season as fixed effects. Random effects were additive genetic, permanent environmental effect of boar, and residual. Observations were removed when the number of data at a given age of boar classification was < 10 records. Preliminary evaluations showed the best fit with fifth-order polynomials, indicating that the best model would have fifth-order fixed regression and fifth-order random regressions for animal and permanent environmental effects. Random regression models were fitted to evaluate all combinations of first- through seventh-order polynomial covariance functions. Goodness of fit for the models was tested using Akaike's Information Criterion and the Schwarz Criterion. The maximum log likelihood value was observed for sixth-, fifth-, and seventh-order polynomials for fixed, additive genetic, and permanent environmental effects, respectively. However, the best fit as determined by Akaike's Information Criterion and the Schwarz Criterion was by fitting sixth-, fourth-, and seventh-order polynomials; and fourth-, second-, and seventh-order polynomials for fixed, additive genetic, and permanent environmental effects, respectively. Heritability estimates for TSC ranged from 0.27 to 0.48 across age of boar classifications. In addition, heritability for TSC tended to increase with age of boar classification.     

Cluster and meta‐analyses of genetic parameters for feed intake traits in growing beef cattle

A data set based on 50 studies including feed intake and utilization traits was used to perform a meta‐analysis to obtain pooled estimates using the variance between studies of genetic parameters for average daily gain (ADG); residual feed intake (RFI); metabolic body weight (MBW); feed conversion ratio (FCR); and daily dry matter intake (DMI) in beef cattle. The total data set included 128 heritability and 122 genetic correlation estimates published in the literature from 1961 to 2012. The meta‐analysis was performed using a random effects model where the restricted maximum likelihood estimator was used to evaluate variances among clusters. Also, a meta‐analysis using the method of cluster analysis was used to group the heritability estimates. Two clusters were obtained for each trait by different variables. It was observed, for all traits, that the heterogeneity of variance was significant between clusters and studies for genetic correlation estimates. The pooled estimates, adding the variance between clusters, for direct heritability estimates for ADG, DMI, RFI, MBW and FCR were 0.32 ± 0.04, 0.39 ± 0.03, 0.31 ± 0.02, 0.31 ± 0.03 and 0.26 ± 0.03, respectively. Pooled genetic correlation estimates ranged from ?0.15 to 0.67 among ADG, DMI, RFI, MBW and FCR. These pooled estimates of genetic parameters could be used to solve genetic prediction equations in populations where data is insufficient for variance component estimation. Cluster analysis is recommended as a statistical procedure to combine results from different studies to account for heterogeneity.     

Comparison of multiplicative heterogeneous variance adjustment models for genetic evaluations

Two heterogeneous variance adjustment methods and two variance models were compared in a simulation study. The method used for heterogeneous variance adjustment in the Nordic test‐day model, which is a multiplicative method based on Meuwissen (J. Dairy Sci., 79, 1996, 310), was compared with a restricted multiplicative method where the fixed effects were not scaled. Both methods were tested with two different variance models, one with a herd‐year and the other with a herd‐year‐month random effect. The simulation study was built on two field data sets from Swedish Red dairy cattle herds. For both data sets, 200 herds with test‐day observations over a 12‐year period were sampled. For one data set, herds were sampled randomly, while for the other, each herd was required to have at least 10 first‐calving cows per year. The simulations supported the applicability of both methods and models, but the multiplicative mixed model was more sensitive in the case of small strata sizes. Estimation of variance components for the variance models resulted in different parameter estimates, depending on the applied heterogeneous variance adjustment method and variance model combination. Our analyses showed that the assumption of a first‐order autoregressive correlation structure between random‐effect levels is reasonable when within‐herd heterogeneity is modelled by year classes, but less appropriate for within‐herd heterogeneity by month classes. Of the studied alternatives, the multiplicative method and a variance model with a random herd‐year effect were found most suitable for the Nordic test‐day model for dairy cattle evaluation.     

The empirical bias of estimates by restricted maximum likelihood, Bayesian method, and method R under selection for additive, maternal, and dominance models.

Bayesian analysis via Gibbs sampling, restricted maximum likelihood (REML), and Method R were used to estimate variance components for several models of simulated data. Four simulated data sets that included direct genetic effects and different combinations of maternal, permanent environmental, and dominance effects were used. Parents were selected randomly, on phenotype across or within contemporary groups, or on BLUP of genetic value. Estimates by Bayesian analysis and REML were always empirically unbiased in large data sets. Estimates by Method R were biased only with phenotypic selection across contemporary groups; estimates of the additive variance were biased upward, and all the other estimates were biased downward. No empirical bias was observed for Method R under selection within contemporary groups or in data without contemporary group effects. The bias of Method R estimates in small data sets was evaluated using a simple direct additive model. Method R gave biased estimates in small data sets in all types of selection except BLUP. In populations where the selection is based on BLUP of genetic value or where phenotypic selection is practiced mostly within contemporary groups, estimates by Method R are likely to be unbiased. In this case, Method R is an alternative to single-trait REML and Bayesian analysis for analyses of large data sets when the other methods are too expensive to apply.     

Parameter estimates for greasy fleece weight of Rambouillet sheep at different ages

Variance components for greasy fleece weight in Rambouillet sheep were estimated. Greasy fleece weight was modeled either as repeated measurements on the same trait or as different traits at different ages. The original data were separated according to the age of the ewe at shearing into three classes; 1 yr, 2 and 3 yr, and older than 3 yr. An animal model was used to obtain estimates of genetic parameters with a REML algorithm. Total numbers of animals in pedigrees for the different age classes were 696, 729, and 573, respectively, and 822 for the repeated measures model across ages. The animal model included direct genetic, permanent environmental, and residual environmental random effects and fixed effects for age of ewe, shearing date as contemporary group, and number of lambs born. Days between shearings was used as a covariate. Single-trait analyses were initially done to obtain starting values for multiple-trait analyses. A repeated measures model across ages was also used. Estimates of heritability by age group were .42, .50, and .58 from three-trait (age class) analyses and for the repeated measures model the estimate was .57. Estimates of genetic correlations between fleece yields for 1 yr and 2 and 3 yr, 1 yr and >3 yr, and 2 and 3 yr and >3 yr classes were .88, .89, and .97, respectively. These estimates of genetic correlations suggest that a repeated measures model for greasy fleece weight is adequate for making selection decisions.     

Adjusting for heterogeneity of experimental data in genetic evaluation of dry matter intake in dairy cattle

The objectives of the present study were (i) to find the best fitted model for repeatedly measured daily dry matter intake (DMI) data obtained from different herds and experiments across lactations and (ii) to get better estimates of the genetic parameters and better genetic evaluations. After editing, there were 572,512 daily DMI records of 3,495 animals (Holstein cows) from 11 different herds across 13 lactations and the animals were under 110 different nutritional experiments. The fitted model for this data set was a univariate repeated‐measure animal model (called model 1) in which additive genetic and permanent environmental (within and across lactations) effects were fitted as random. Model 1 was fitted as two distinct models (called models 2 and 3) based on alternative fixed effect corrections. For unscaled data, each model (models 2 and 3) was fitted as a homoscedastic (HOM) model first and then as a heteroscedastic (HET) model. Then, data were scaled by multiplying with particular herd‐scaling factors, which were calculated by accounting for heterogeneity of phenotypic within‐herd variances. Models were selected based on cross‐validation and prediction accuracy results. Scaling factors were re‐estimated to determine the effectiveness of accounting for herd heterogeneity. Variance components and respective heritability and repeatability were estimated based on a pedigree‐based relationship matrix. Results indicated that the model fitted for scaled data showed better fit than the models (HOM or HET) fitted for unscaled data. The heritability estimates of the models 2 and 3 fitted for scaled data were 0.30 and 0.08, respectively. The repeatability estimates of the model fitted for scaled data ranged from 0.51 to 0.63. The re‐estimated scaling factor after accounting for heterogeneity of residual variances was close to 1.0, indicating the stabilization of residual variances and herd accounted for most of the heterogeneity. The rank correlation of EBVs between scaled and unscaled data ranged from 0.96 to 0.97.     

Estimates of genetic covariance functions for growth of Angus cattle

Estimates of covariance functions and genetic parameters were obtained for growth of Angus cattle from birth to 820 days of age. Data comprised 84 533 records on 20 731 animals in 43 herds, with a high proportion of animals with 4 or more weights recorded. Changes in weights were modelled through random regression on orthogonal polynomials of age at recording. A total of 11 combinations of quadratic, cubic, quartic and quintic polynomials to model direct and maternal genetic effects and permanent environmental effects were considered. Results showed good agreement for all models at ages with many records, but differed at the highest ages and at very early ages with few weights available. Cubic polynomials appeared to be most problematic. The order of polynomial fit for permanent environmental effects of the animal dominated estimates of phenotypic variances and mean squares for residual errors. A model fitting a quartic polynomial for these effects and quadratic polynomials for the other random effects, appeared to be the best compromise between detailedness of the model which could be supported by the data, plausibility of results, and fit, measured as mean square error.     

Multiplicative random regression model for heterogeneous variance adjustment in genetic evaluation for milk yield in Simmental.

A multiplicative random regression (M-RRM) test-day (TD) model was used to analyse daily milk yields from all available parities of German and Austrian Simmental dairy cattle. The method to account for heterogeneous variance (HV) was based on the multiplicative mixed model approach of Meuwissen. The variance model for the heterogeneity parameters included a fixed region x year x month x parity effect and a random herd x test-month effect with a within-herd first-order autocorrelation between test-months. Acceleration of variance model solutions after each multiplicative model cycle enabled fast convergence of adjustment factors and reduced total computing time significantly. Maximum Likelihood estimation of within-strata residual variances was enhanced by inclusion of approximated information on loss in degrees of freedom due to estimation of location parameters. This improved heterogeneity estimates for very small herds. The multiplicative model was compared with a model that assumed homogeneous variance. Re-estimated genetic variances, based on Mendelian sampling deviations, were homogeneous for the M-RRM TD model but heterogeneous for the homogeneous random regression TD model. Accounting for HV had large effect on cow ranking but moderate effect on bull ranking.