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

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

Selection of locations of knots for linear splines in random regression test‐day models

Using spline functions (segmented polynomials) in regression models requires the knowledge of the location of the knots. Knots are the points at which independent linear segments are connected. Optimal positions of knots for linear splines of different orders were determined in this study for different scenarios, using existing estimates of covariance functions and an optimization algorithm. The traits considered were test‐day milk, fat and protein yields, and somatic cell score (SCS) in the first three lactations of Canadian Holsteins. Two ranges of days in milk (from 5 to 305 and from 5 to 365) were taken into account. In addition, four different populations of Holstein cows, from Australia, Canada, Italy and New Zealand, were examined with respect to first lactation (305 days) milk only. The estimates of genetic and permanent environmental covariance functions were based on single‐ and multiple‐trait test‐day models, with Legendre polynomials of order 4 as random regressions. A differential evolution algorithm was applied to find the best location of knots for splines of orders 4 to 7 and the criterion for optimization was the goodness‐of‐fit of the spline covariance function. Results indicated that the optimal position of knots for linear splines differed between genetic and permanent environmental effects, as well as between traits and lactations. Different populations also exhibited different patterns of optimal knot locations. With linear splines, different positions of knots should therefore be used for different effects and traits in random regression test‐day models when analysing milk production traits.     

Constructing covariance functions for random regression models for growth in Gelbvieh beef cattle

Genetic parameters for a random regression model of growth in Gelbvieh beef cattle were constructed using existing estimates. Information for variances along ages was provided by parameters used for routine Gelbvieh multiple-trait evaluation, and information on correlations among different ages was provided by random regression model estimates from literature studies involving Nellore cattle. Both sources of information were combined into multiple-trait estimates; corrected for continuity, smoothness, and general agreement with literature estimates; and extrapolated to 730 d. Covariance functions using standardized Legendre polynomials were fit for the following effects: additive genetic (direct and maternal), and animal and maternal permanent environment. Residual variances at different ages were fitted using linear splines with three knots. Fit was by least squares. The order of polynomials was varied from third to sixth. Increasing the fit beyond cubic provided small improvements in R2 and increased the number of small eigenvalues of covariance matrices, especially for the additive effect. Parameters for a random regression model in beef cattle can be constructed with negligible artifacts from literature estimates. Formulas can easily be modified for other types of polynomials and splines.     

The non-linear effect (determined by the penalised partial-likelihood approach) of milk-protein concentration on time to first insemination in Belgian dairy cows

The time to first insemination in dairy cows depends partly on the energy balance of the cow. Because milk-protein concentration is related to the energy balance, we investigated whether milk-protein concentration predicted the hazard of being inseminated. The main objective of the paper is to demonstrate that the relationship between milk-protein concentration and the hazard of being inseminated was not linear and that this non-linear relationship was modelled adequately using cubic–splines. The semiparametric Cox model was used to introduce protein concentration into the model as a time-varying covariate and additionally herd was added to the model as a frailty term to adjust for the clustering of the cows within a herd. We extended the penalised partial-likelihood technique to fit the frailty model with cubic–splines for the effect of the protein concentration. The model was fitted for a large database consisting of 5114 multiparous cows from 181 different farms. Low milk-protein concentration (<2.7%) was associated with a negative energy balance and this probably led to the decreased hazard. On the other hand, high milk-protein concentration (>4.0%) was linked with low milk production and it was probably a farmer’s decision not to inseminate such cows, leading to the observed decreased hazard.     

Incorporation of observations with different residual error variances into existing complex test-day models

Automated milking systems (AMS) have become popular on dairy farms. Due to a different test-day recording scheme the variation of test-day observations differ from AMS differ from those of conventional milking system (CMS). An approach is presented for upgrading the genetic evaluation model for production traits milk, protein and fat yield by including residual covariance matrices for AMS and CMS test-day observations. Residual variances were found to be 16–37% smaller for milk and protein yields and 42–47% larger for fat yield when recorded under AMS herds compared to CMS herds. Daily heritability was higher for milk and protein yield and lower for fat yield when traits were recorded under AMS compared to recording under CMS. No difference was found between AMS and CMS in 305-day heritability for milk and protein yield except for second lactation milk yield. 305-day heritability for fat was lower for all lactations under AMS.     

Joint longitudinal modeling of age of dam and age of animal for growth traits in beef cattle

Two methods to jointly model age of dam (AOD) and age of animal in random regression analyses of growth in Gelbvieh cattle were examined. The first method (M1) was analogous to the multiple-trait analysis and consisted of AOD as a nested class variable and a cubic polynomial regression on age nested within birth, weaning, and yearly weights. The second method (M2) used two-dimensional splines, with age knots at 150, 205, 270, 340, and 390 d. The AOD knots were placed at 725, 1,464, and 2,189 d. These selected knots were used to form a two-dimensional grid containing 15 knots, each representing a specific age and AOD combination. A data set containing Gelbvieh growth records was split along contemporary groups into two data sets. Data set 1 contained 316,078 records and was used for prediction by mixed-model equations. Data set 2 contained 164,167 records and was used for cross validation. In the complete data set, only 90 and 30% of animals with birth weight had records on weaning and yearling weights, respectively. Models were evaluated based on R2, average squared error (ASE), percent bias, and plots of solutions. The ASE for weights associated with birth weight, weaning weight, and yearling weight for M1 were 15, 505, and 703 kg2. With M2, large jumps in fixed-effect estimates were observed outside the two-dimensional grid. To eliminate this problem, weighted one-dimensional splines were used for extrapolation beyond the two-dimensional grid. For M2 with weighted spline extrapolation, the ASE were 15, 542, and 777 kg2 for birth weight, weaning weight, and yearling weight, respectively. Creation of optimal two-dimensional splines is difficult when data are clustered. Despite such difficulties, the two-dimensional spline was capable of jointly and continuously modeling AOD and age of animal.     

Properties of random regression models using linear splines

Properties of random regression models using linear splines (RRMS) were evaluated with respect to scale of parameters, numerical properties, changes in variances and strategies to select the number and positions of knots. Parameters in RRMS are similar to those in multiple trait models with traits corresponding to points at knots. RRMS have good numerical properties because of generally superior numerical properties of splines compared with polynomials and sparser system of equations. These models also contain artefacts in terms of depression of variances and predictions in the middle of intervals between the knots, and inflation of predictions close to knots; the artefacts become smaller as correlations corresponding to adjacent knots increase. The artefacts can be greatly reduced by a simple modification to covariables. With the modification, the accuracy of RRMS increases only marginally if the correlations between the adjacent knots are ≥0.6. In practical analyses the knots for each effect in RRMS can be selected so that: (i) they cover the entire trajectory; (ii) changes in variances in intervals between the knots are approximately linear; and (iii) the correlations between the adjacent knots are at least 0.6. RRMS allow for simple and numerically stable implementations of genetic evaluations with artefacts present but transparent and easily controlled.     

Computing simplifications for non-additive genetic models

A limiting factor in the analysis of non‐additive genetic models has been the ability to compute the inverses of non‐additive genetic covariance matrices for large populations. Also, the order of the equations was equal to the number of animals times the number of non‐additive genetic effects that were included in the model. This paper describes a computing algorithm that avoids the inverses of the non‐additive genetic covariance matrices and keeps the size of the equations to be the same as any animal model with only additive genetic effects. Quadratic forms for the non‐additive genetic variances could also be computed without the inverses of the non‐additive genetic covariance matrices.     

Genetic evaluation using random regression models with different covariance functions for test‐day milk yield in an admixture population of Thailand goats

The objectives of this study were to compare covariance functions (CF) and estimate the heritability of milk yield from test‐day records among exotic (Saanen, Anglo‐Nubian, Toggenburg and Alpine) and crossbred goats (Thai native and exotic breed), using a random regression model. A total of 1472 records of test‐day milk yield were used, collected from 112 does between 2003 and 2006. CF of the study were Wilmink function, second‐ and third‐order Legendre polynomials, and linear splines 4 knots located at 5, 25, 90 and 155 days in milk (SP25–90) and 5, 35, 95 and 155 of days in milk (SP35–95). Variance components were estimated by restricted maximum likelihood method (REML). Goodness of fit, Akaike information criterion (AIC), percentage of squared bias (PSB), mean square error (MSE), and empirical correlation (RHO) between the observed and predicted values were used to compare models. The results showed that CF had an impact on (co)variance estimation in random regression models (RRM). The RRM with splines 4 knots located at 5, 25, 90 and 155 of days in milk had the lowest AIC, PSB and MSE, and the highest RHO. The heritability estimated throughout lactation obtained with this model ranged from 0.13 to 0.23.     

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.     

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

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

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.     

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.     

Computing options for multiple‐trait test‐day random regression models while accounting for heat tolerance

Data included 90 242 799 test day records from first, second and third parities of 5 402 484 Holstein cows and 9 326 754 animals in the pedigree. Additionally, daily temperature humidity indexes (THI) from 202 weather stations were available. The fixed effects included herd test day, age at calving, milking frequency and days in milk classes (DIM). Random effects were additive genetic, permanent environment and herd‐year and were fit as random regressions. Covariates included linear splines with four knots at 5, 50, 200 and 305 DIM and a function of THI. Mixed model equations were solved using an iteration on data program with a preconditioned conjugate gradient algorithm. Preconditioners used were diagonal (D), block diagonal due to traits (BT) and block diagonal due to traits and correlated effects (BTCORR). One run included BT with a ‘diagonalized’ model in which the random effects were reparameterized for diagonal (co)variance matrices among traits (BTDIAG). Memory requirements were 8.7 Gb for D, 10.4 Gb for BT and BTDIAG, and 24.3 Gb for BTCORR. Computing times (rounds) were 14 days (952) for D, 10.7 days (706) for BT, 7.7 days (494) for BTDIAG and 4.6 days (289) for BTCORR. The convergence pattern was strongly influenced by the choice of fixed effects. When sufficient memory is available, the option BTCORR is the fastest and simplest to implement; the next efficient method, BTDIAG, requires additional steps for diagonalization and back‐diagonalization.     

A penalized likelihood approach to pooling estimates of covariance components from analyses by parts

Estimates of covariance matrices for numerous traits are commonly obtained by pooling results from a series of analyses of subsets of traits. A penalized maximum‐likelihood approach is proposed to combine estimates from part analyses while constraining the resulting overall matrices to be positive definite. In addition, this provides the scope for ‘improving’ estimates of individual matrices by applying a penalty to the likelihood aimed at borrowing strength from their phenotypic counterpart. A simulation study is presented showing that the new method performs well, yielding unpenalized estimates closer to results from multivariate analyses considering all traits, than various other techniques used. In particular, combining results for all sources of variation simultaneously minimizes deviations in phenotypic estimates if sampling covariances can be approximated. A mild penalty shrinking estimates of individual covariance matrices towards their sum or estimates of canonical eigenvalues towards their mean proved advantageous in most cases. The method proposed is flexible, computationally undemanding and provides combined estimates with good sampling properties and is thus recommended as alternative to current methods for pooling.     

Repeated-measure animal models to estimate genetic components of mature weight,hip height,and body condition score

Information on mature weight, hip height, and body condition score from Angus cows was analyzed to estimate variance components and compare prediction models. Observations from repeated measures were analyzed with animal models with or without condition score as a covariate and with or without an effect for permanent environment. Heritability (repeatability) estimates for mature weight, hip height, and condition score from Method R procedures were 0.40 (0.77), 0.62 (0.81), and 0.11 (0.38), respectively, from animal models containing a permanent environmental effect but without a covariate for condition score. Heritability estimates from animal models without a permanent environmental effect were similar to repeatability estimates from animal models with it, suggesting inflated estimates of genetic variance from models not containing a permanent environmental effect. Regressing mature weight on condition score reduced both additive genetic variance and permanent environmental variance, increasing the heritability estimate of mature weight to 0.54 and altering the biological interpretation of the trait. The covariate for condition score had little effect on hip height. Regressions of mature weight and hip height on condition score were 25.9 kg/unit of body condition score and 0.4 cm/unit, respectively. Least-squares means for mature weight and hip height tended to increase until 7 and 5 yr of age, respectively. Condition score tended to increase until 6 yr of age and decrease after 8 yr of age. Correlations between breeding value solutions for the same trait were high whether or not prediction models included a permanent environmental effect or a covariate for condition score, and whether or not the variance components used were derived from models containing a covariate for condition score. Results suggest the importance of including a permanent environmental effect in genetic prediction models for these traits. Whether mature weight should be adjusted for body condition is arguable, depending on availability of condition score predictions and tools for analyzing mature weight and condition score predictions in an environment-specific context.     

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.     

Use of heterogeneous operation-specific contact parameters changes predictions for foot-and-mouth disease outbreaks in complex simulation models

The role of contact parameters in a complex spatial simulation model of foot-and-mouth disease spread was determined by comparing predictions of number of infected premises, epidemic duration, and relative infection risk for different production sectors between a model that included the Full, heterogeneous (differing by production type) type-specific information about animal, vehicle and personnel movement between premises, and models that used partial and homogeneous (similar across production types) weighted-mean or proxy parameter sets for contacts between premises of all types. The model was run using a dataset of known premises locations in a three-county area in the Central Valley of California and categorized into 13 premises types and six production sectors.Results from models run with homogeneous contact parameters were always different from those obtained from the Full model, demonstrating that model predictions are affected by heterogeneity in contact parameters. Models simplified by using weighted-mean parameters predicted fewer infected premises. Models that were simplified by using medium dairy farm or large swine operation proxy parameters predicted longer epidemics with more infected premises, while those using small beef operation proxy parameters predicted shorter epidemics with fewer infected premises. Simplified-parameter models underestimated the impact on the economically important dairy sector, while overestimating the impact on beef and backyard operations. Results establish a need for heterogeneous, operation-specific contact parameters in complex stochastic simulation models that must be weighed against the cost of obtaining and coding premises type-specific contact information.     

Genetic parameters for various random regression models to describe the weight data of pigs

Various random regression models have been advocated for the fitting of covariance structures. It was suggested that a spline model would fit better to weight data than a random regression model that utilizes orthogonal polynomials. The objective of this study was to investigate which kind of random regression model fits best to weight data of pigs. Two random regression models that described weight of individual pigs, one using orthogonal polynomials, and the other using splines, were compared. A comparison with a multivariate model, Akaike's information criterion, and the Bayesian-Schwarz information criterion were used to select the best model. Genetic, permanent environmental, and total variances increased with age. Heritabilities for the multivariate model ranged from 0.14 to 0.19, and for both random regression models the heritabilities were fluctuating around 0.17. Both genetic and phenotypic correlations decreased when the interval between measurements increased. The spline model needed fewer parameters than the multivariate and polynomial models. Akaike's information criterion was least for the spline model and greatest for the multivariate model. The Bayesian-Schwarz information criterion was least for the polynomial model and greatest for the multivariate model. Residuals of all models were normally distributed. Based on these results, it is concluded that random regression models provide the best fit to pig weight data.     

Multiple-breed genetic inference using heavy-tailed structural models for heterogeneous residual variances

Multiple-breed genetic models recently have been demonstrated to account for the heterogenous genetic variances that exist between different beef cattle breed groups. We extend these models to allow for residual heteroskedasticity (heterogeneous residual variances), specified as a function of fixed effects (e.g., sex, breed proportion, breed group heterozygosity) and random effects such as contemporary groups (CG). We additionally specify the residual distributions to be either Gaussian or based on heavier-tailed alternatives such as the Student's t or Slash densities. For each of these three residual densities using either homoskedastic (homogeneous variance) or heteroskedastic error specifications, we analyzed 22,717 postweaning gain records from a Nelore-Hereford population based on a Markov chain Monte Carlo animal model implementation. The heteroskedastic Student's t error model (with estimated df = 7.33 +/- 0.48) was clearly the best-fitting model based on a pseudo-Bayes factor criterion. Breed group heterozygosity and, to a lesser extent, calf sex seemed to be marginally important sources of residual heteroskedasticity. Specifically, the residual variance in F1 animals was estimated to be 0.70 +/- 0.16 times that for purebreds, whereas the male residual variance was estimated to be 1.13 +/- 0.09 times that for females. The CG effects were important random sources of residual heteroskedasticity (i.e., the coefficient of variation of CG-specific residual variances was estimated to be 0.72 +/- 0.06). Purebred Nelores were estimated to have a larger genetic variance (124.84 +/- 21.75 kg2) compared with Herefords (40.89 +/- 6.70 kg2) under the heteroskedastic Student's t error model; however, the converse was observed from results based on a homoskedastic Student's t error model (46.24 +/- 10.90 kg2 and 60.11 +/- 8.54 kg2, respectively). These results indicate that allowing for robustness to outliers and accounting for heteroskedasticity of residual variances has potentially important implications for variance component and genetic parameter estimates from data on multiple-breed populations.