| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
ANIMAL GENETICS |
Department of Animal Science, Michigan State University, East Lansing 48824
Abstract
The primary objective of this study was to demonstrate the utility of a hierarchical Bayes implementation of a multiple-breed animal model (MBAM) to estimate breed composition means and additive genetic variances as well as on variances due to the segregation between breeds. The MBAM and a conventional animal model (AM) were both applied to five simulated data sets derived from each of two different populations. Population I consisted of crosses between two breeds having a twofold difference in genetic variance and a nonzero segregation variance. Population II had the same population structure as Population I, except that the two breeds had the same genetic variance with no segregation variance; that is, Population II was essentially single breed in its genetic architecture. For Population I, posterior means of all variance components obtained by MBAM were unbiased, with 95% posterior probability intervals (PPI) having the expected coverage based on five replicates. The MBAM showed a slightly superior performance over the AM for genetic predictions in Population I, although there was no evidence that the use of the MBAM translated into greater genetic gains relative to the use of the AM. Nevertheless, the MBAM was clearly demonstrated to have superior fit to the data using pseudo-Bayes factors (PBF) as the basis for model choice. As expected, the MBAM and AM performed equally well in Population II. A data set consisting of 22,717 postweaning gain (PWG) records of a Nelore-Hereford population (40,082 animals in the pedigree) also was analyzed. The MBAM inference on Nelore and Hereford additive heritabilities 
substantially differed. Herefords had a posterior mean 
of 0.20 with a 95% PPI of 0.15 to 0.25, whereas the corresponding values for the Nelores were 0.07 and 0.04 to 0.11, respectively. The posterior mean genetic variance due to the segregation between these breeds was 8.4 kg2, with a 95% PPI of 2.3 to 24.8 kg2, and represented 35.4% of the Nelore but only 9.9% of the Hereford posterior mean genetic variance. The posterior mean using the AM was 0.15, presumed common across the two breeds, with a 95% PPI of 0.11 to 0.19. The PBF heavily favored the MBAM over the AM for the PWG data. Accordingly, the MBAM represents a viable alternative to AM for multiple-breed genetic evaluations, providing the necessary flexibility in modeling heteroskedastic genetic variances of breed composition groups.
Key Words: Bayesian Inference • Beef Cattle • Crossbreeding • Genetic Evaluation • Multiple-Breed • Postweaning Gain
Introduction
Crossbreeding increases efficiency of livestock production by exploiting heterosis and complementarity between breeds (Gregory et al., 1999
). Moreover, crossbreeding acts synergistically with selection and management in improving beef production. As selection response is proportional to the accuracy of genetic merit predictions (Falconer and Mackay, 1996), this accuracy, in turn, necessarily depends on properly specified genetic covariances between purebred or crossbred animals (Fernando, 1999).
The genetic merit of an animal comprises the mean of its breed composition group plus its individually specific deviation from the group (Arnold et al., 1992; Elzo, 1994; Klei et al., 1996). Other than residual effects, individual deviations are due to random additive and nonadditive genetic effects, which can be estimated using performance records on the individual and its relatives. In order to most efficiently use these records, it is important to properly model genetic covariances between relatives as clearly specified by Lo et al. (1993) for an additive genetic effects model. A model that includes fixed additive and nonadditive genetic effects with random additive individual deviations is computationally tractable and likely to be satisfactorily parsimonious for the genetic evaluation of multiple-breed populations. Hierarchical Bayes model constructions (Sorensen and Gianola, 2002) effectively combine data and prior information and provide a particularly useful framework for mixed model inference (Hobert, 2000) as necessary for genetic evaluations.
The objectives of this study were 1) to formalize a hierarchical Bayes construction of the multiple-breed animal model derived by Lo et al. (1993) to estimate fixed and random genetic effects when breed and segregation variance components are unknown and 2) to compare the fit of this model with the conventional animal model for simulated data and a data set of postweaning gains on Nelore-Hereford crosses.
Material and Methods
Hierarchical Bayes Model Construction
First Stage.
The first stage of the model specifies the conditional sampling density of the n x 1 data vector y = {yj}. Although this development can be readily extended to multiple records per animal, the case of at most one record per animal is presented for pedagogical reasons. The component of the joint density on y due to a single record yj on individual j is
 | [1] |
Here, ß is a p x 1 vector of nongenetic fixed effects (e.g., gender and age of dam). Also 
is a 3 x 1 vector of genetic fixed effects with additive (A), dominance (D), and A x A components, represented respectively by 
Ab, Dbb', and AAbb', with b and b' denoting the breed(s) corresponding to these effects, for b = 1, ..., B; b' = b + 1, ..., B; b < b'. Furthermore, u is a t x 1 vector of nongenetic random effects (e.g., contemporary groups), and a is a q x 1 vector of animal additive genetic effects. Also, 
, 
, 
, and 
, are known row incidence vectors with the elements of being determined by genetic effects coefficients for individual j. Using Kinghorn (1980) and Wolf et al. (1995), the elements of in column order are as follows: 1) fb, defined as the proportion of alleles from the bth breed and corresponding to Ab; 2) fbb' being the probability that for a randomly chosen locus from an individual j, one allele is derived from Breed b and the other allele is derived from Breed b', associated with Dbb'; and 3) 2fbfb' corresponding to AAbb'. Now S represents the sample set of size n of animals having records; typically, n < q because a includes effects for ancestor animals without records. Finally, 
represents the residual variance, assumed to be homogeneous across breed composition groups.
Second Stage.
As with conventional linear mixed-model specifications, multivariate normal structural prior densities are specified for both sets of random effects, specifically
) and a|
~ N(0, G()), where It is an identity matrix of order t. Extending the model to accommodate additional sets of random effects is naturally straightforward. In Bayesian specifications, one might also specify subjective priors on fixed effects, in which case the second stage would also involve the following specifications: ß|ßo, Vß ~ N(ßo, Vß) and |o, V ~ N(o, V) (Sorensen and Gianola, 2002) with ßo, o, Vß, and V being known hyperparameters as specified by the data analyst. Alternatively, uniform bounded priors may be considered on ß and .
The additive genetic variance-covariance matrix G() is a function of more than one dispersion parameter in for crossbred populations. Using Lo et al. (1993), elements of G() can be computed by the tabular method, with the jth diagonal element being determined as
 | [2] |
for j = 1, 2,..., q. Here 
, 
, and 
represent, respectively, the individual, sire, and dam proportion of alleles from Breed b; 
and 
denote, respectively, the additive genetic effect of the sire and the dam of j; 
is the additive variance of Breed b; and 
is variance due to the segregation between Breed b and b' or the additional variance observed in the F2 generation over the F1. That is,
 |
defines all the genetic components of variance. Following Quaas (1988), Lo et al. (1993) showed that the inverse of G() can be computed using
 |
where I is an identity matrix, P is a matrix relating progeny to parents and 
() is a diagonal matrix with the jth diagonal element defined as
 | [3] |
which is a linear function of elements of .
Third Stage.
Scaled inverted 
2 prior densities are specified on the variance components as follows:
Here, 
, and 
are specified hyperparameters. Alternatively, bounded uniform priors may be placed on some or all of these variance components (Sorensen and Gianola, 2002).
Joint and Fully Conditional Posterior Densities.
The joint posterior density of all unknown parameters is obtained by the product of the sampling density in Eq. [1]
and all prior densities as specified in the second and third stages. From this joint posterior density, fully conditional densities (FCD) of all unknown parameters/quantities or blocks thereof necessary to conduct Markov chain Monte Carlo (MCMC) inference are derived in the appendix of this paper.
Simulation Study
A two-breed crossing system simulation study was conducted. Five replicate data sets were generated from each of two populations. In Population I, two breeds had different additive genetic variances with a nonzero segregation variance, whereas Population II had the same two breed crossing design except that the additive genetic variances were equal with zero segregation variance, thereby being genetically equivalent to a single breed population. In Population I, the additive genetic variance of Breed 1 
and Breed 2 
was set to 100.0 and 50.0, respectively; furthermore, the segregation variance between Breeds 1 and 2 
was set to 20.0 and the residual variance 
set to 100.0. Alternatively, these parameters for Population II were 
, 
, and 
.
The same breeding design was used for each replicate data set in both populations. First, data were generated on individuals from each of the two base purebreds, P1 and P2, consisting of 240 animals each. Six sires per each base purebred group were randomly selected such that each base sire was mated to 20 base dams of the other breed to produce 120 offspring of the F1 generation. These matings then generated 240 F1 animals (120 P1 sired and 120 P2 sired) from which 12 sires and 40 dams were then randomly selected. From these animals, six F1 males derived from P1 sires were mated to F1 females (20 total) derived from P2 sires to produce 120 F2 individuals and also mated to P1 dams to produce 120 backcrosses to P1 (BC1). The other six F1 males (P2 sired) were mated to F1 females (20 in total) from P1 sires to produce 120 F2 individuals and to P2 dams to produce 120 backcrosses to P2 (BC2). Additionally, 120 BC1 and 120 BC2 individuals were obtained by mating, respectively, the six P1 sires to F1 dams from P2 sires and the six P2 sires to F1 dams from P1 sires. At this point, the population comprised 240 individuals of each of the P1, P2, F1, F2, BC1, and BC2 breed compositions. In a final stage, six sires and 20 dams from each of the backcross groups and 12 sires and 40 dams from the F2 group were randomly selected and then all selected males and females were randomly mated, producing additional purebred, F1, F2, and backcross individuals and an advanced generation of intercross animals with several different breed compositions. The total number of animals was set to 4,000, but the number of animals in each breed composition group was random and unbalanced within each data set. There were approximately 290 animals for each purebred, 350 F1, 430 F2, 450 animals for each backcross, and 1,650 advanced intercross animals, having the following breed compositions based on expected proportion of P1 derived alleles (with number of individuals within parenthesis): 
(90), 
(240), 
(330) 
(420) 
(330), 
(240), 
(90). Note that -bred, -bred, and -bred individuals of this advanced generation do not necessarily have the same genetic variances as BC2, F1/F2, and BC1 individuals, respectively. For instance, a noninbred -bred individual derived from a BC1 x BC2 mating has a genetic variance Var(aj) = 90, whereas Var(aj) = 75 for an F1 individual and Var(aj) = 95 for an F2 individual using Eq. [2] and the parameters specified for Population I. Inbreeding was avoided in all matings and the gender of each offspring was randomly attributed with equal probability.
A single record on each animal was generated based on a mixed-effects model with simple fixed-effects specifications including an arbitrarily chosen overall mean (µ = 100) and an additive but no nonadditive fixed genetic effects (A1 = 25, A2 = 0, D12 = 0, and AA12 = 0). Random effects included an additive genetic effect and a normally, identically, and independently distributed residual effect. The additive genetic effect of an animal j was generated using 
, where zj ~ N(0, 1) with 
j as defined in Eq. [3], for j = 1, 2,..., 4,000. The parental contributions from and are null when parents were not identified, as in the case of base population animals, such that j is then the corresponding breed-specific additive genetic variance.
Inference was based on MCMC and two models: the multiple-breed animal model (MBAM) described in this study and a conventional animal model (AM) that assumed equal breed genetic variances with no between-breed segregation variance. Uniform bounded priors were utilized for the variance components. The length of chain was G = 60,000 cycles after a burn-in period comprising 10,000 cycles. Samples were saved every 10 cycles. For each variance component, the initial monotone sequence approach (Geyer, 1992) was used to calculate effective sample size (ESS), which estimates the number of independent samples with information content equivalent to that contained within the 6,000 dependent samples (Sorensen et al., 1995).
Pearson and Kendall rank correlations between posterior mean (âj) and true (aj) additive genetic effects were used to compare MBAM and AM in terms of ordering animals for selection. In addition, the average aj for animals ranked in the top 10% based on their âj obtained by MBAM and the average aj for animals ranked in the top 10% based on their âj obtained by AM, were computed to assess potential differences in selection response obtained by the two competing models. These quantities were also computed for the top 5% of animals.
Model Choice Criterion.
The pseudo-Bayes factor (PBF) (Gelfand, 1996) was used as the criterion for model choice. The PBF involves the evaluation of the first stage density in [1] for each MCMC sample. For comparing MBAM with AM, the corresponding PBF was determined to be
 |
where p(yj|y(–j))MBAM and p(yj|y(–j))AM are model specific conditional predictive ordinates for observation yj conditional on all other observations y(–j). A MCMC approximation for the conditional predictive ordinate for yj, using model M, is obtained by a harmonic mean:
 |
where ß(l), (l), u(l), a(l), and 
are the post-burn-in MCMC samples for ß, , a, u, and , respectively, l = 1, 2,..., G.
Application to Field Data
Postweaning gain (PWG) records of Nelore-Hereford cattle under genetic evaluation in Brazil were analyzed to demonstrate a comparison of the MBAM with the conventional AM on field data. From a total of 61,656 PWG records collected between 1974 and 2000 by a large-scale breeding program called Conexão Delta G, a subset of 22,717 records remained after deleting records on animals with uncertain paternity (47.5%), contemporary groups with less than 10 animals (14.9%), and sires with less than five offspring (0.7%). There were 40,082 animals represented in the pedigree file. Animals were raised under extensive pasture conditions in 15 herds and three regions, two of which were tropical and one subtropical. Region 1 comprised two farms located between 14°S and 16°S latitude with 5,410 records (23.8%), Region 2 had three farms located between 21°S and 23°S with 3,110 records (13.7%), and Region 3 had 10 farms located between 30°S and 32°S with 14,197 records (62.5%). The mean PWG was 98.2 kg ± 41.2 across all breed compositions, and breed group-specific means are presented in Table 1. Ages of calves at weaning ranged from 114 to 300 d with a mean of 208 d, whereas the postweaning test periods ranged from 106 to 483 d with a mean of 280 d.
|
Nelore; however, purebred Herefords and F1 had nearly 90% of the records (Table 1) with no purebred Nelore animals having any records. Dams were mostly represented by purebreds of either breed. Most animals with records in the tropical part of the country (Regions 1 and 2) were F1 (Table 1) as these herds belonged to Hereford breeders from southern Brazil and Nelore breeders from central Brazil cooperating on the production of crossbred Braford animals.
Nongenetic fixed effects in both models included the main effects of region, gender, length of the postweaning test period, and linear and quadratic age of dam effects, the latter being included to model possible compensatory growth due to age of dam. The elements of were specified based on the same epistatic loss model (Kinghorn, 1980) presented earlier for a two-breed scenario, including an additive effect represented by the Nelore breed proportion (A1), a Nelore-Hereford dominance effect (D12), and an A x A Nelore-Hereford interaction effect (AA12). The fixed-effects portion of the model was further augmented to allow for interactions between gender, length of test, and age of dam polynomial effects with breed proportion. Region x breed proportion interaction was initially considered but not eventually modeled owing to multicollinearity problems because nearly all animals with records in Regions 1 and 2 were F1, as previously noted. Contemporary groups (herd, year, season, and management subclasses) were modeled as uncorrelated random effects. Due to the lack of objective prior information on this population, bounded uniform priors were adopted on ß and , and conjugate yet relatively noninformative specifications were adopted on variance components, specifically, vA1 = vA2 = vS12 = vu = ve = 5; 
; 
, 
, and 
.
The length of the MCMC chain for PWG was 200,000 cycles after 15,000 burn-in cycles, with samples being saved every 10 cycles for both MBAM and AM. Posterior means, modes, key percentiles, and standard deviations of the parameters as well as the ESS were obtained from these samples. Kendall rank correlations were computed between MBAM and AM for posterior means of additive genetic effects based on all animals by breed composition group.
Results and Discussion
Simulation Study
Multiple-Breed Additive Genetic Variances.
Posterior means, modes, standard deviations, and 95% posterior probability intervals (PPI) for variance components averaged across the five replicates of Population I as obtained by MCMC using MBAM or AM are presented in Table 2. The average posterior mean and mode of all variance components obtained by MBAM indicated these point estimates to be essentially unbiased. The bounds of the individual 95% PPI included the true parameter value in 19 out of 20 cases (based on four variance components estimated within each of five replicates), thereby falling within probabilistic expectation. The single additive genetic variance estimated using AM for Population I was slightly greater than the average of the two true values for the breed-specific genetic variances, the deviation being likely due to the nonzero between-breed segregation variance. Considering the complexity of the multiple-breed population structure and the sample sizes in this study, the results in Table 2 indicate that a MBAM analysis based on MCMC reliably infers upon additive genetic variance components in populations consisting of crossbred animals.
|
, the true and the estimated additive genetic variance
for different breed groups are presented. The genetic variance of each breed group using the MBAM is a function of breed-specific variances and the segregation variance; for example, assuming no inbreeding, the genetic variance of the F2 group 
is obtained by 
and, in general, for breed group g by 
(Lo et al., 1993), whereas a common genetic variance is attributed to all breed groups in AM.
|
Predicted Additive Genetic Effects.
Kendall rank and Pearson correlations between posterior mean and true additive genetic effects are shown in Table 4. In Population I, Kendall rank correlations were higher between MBAM posterior mean and true additive genetic effects than between AM posterior mean and true additive genetic effects for all selection intensities (top 5%, top 10%, or all animals) considered. The differences, however, were small as there was no evidence of a difference (P > 0.05) between the average aj of animals ranked in the top 10% (and 5%) by their âj using MBAM and the average aj for animals ranked in the top 10% (and 5%) by their âj using AM (data not shown). It should be noted, however, that genetic drift and Mendelian sampling likely limit the power of this assessment in a manner similar to an assessment of the effectiveness of marker-assisted selection (Dekkers and Hospital, 2002). There were virtually no differences in rank correlations between models for the data sets generated in Population II. Moreover, Kendall rank correlations between âj obtained by MBAM and âj obtained by AM were high in both Populations I and II. Pearson correlation results follow a pattern similar to that discussed for Kendall rank correlations, having just somewhat larger magnitudes (Table 4). As expected, the conventional AM and MBAM performed most similarly when breeds had the same genetic variance with no variance due to segregation between breeds as in Population II.
|
Genetic Fixed Effects.
Posterior means and standard deviations of genetic fixed effects on PWG obtained by MBAM and AM using Kinghorn’s epistatic loss parameterization (Kinghorn, 1980) are shown in Table 5. Inference obtained using either model was similar with mean PWG decreasing as the Nelore proportion increased. As expected, dominance favorably affected PWG whereas additive x additive interaction (i.e., epistatic loss) adversely affected PWG. An attempt was made to fit the full two-locus model for fixed genetic effects (Hill, 1982; Wolf et al., 1995); however, this analysis was not successful due to extremely high correlations between coefficients of genetic effects ranging from 0.92 between additive x additive (2f1f2) and dominance x dominance
to a maximum of 0.99 between 
and additive x dominance (f1 f12) coefficients. A similar problem was observed by Birchmeier et al. (2002), who eventually decided on a model with only additive and dominance effects but no epistatic effects. Nonadditive genetic fixed effects are generally difficult to estimate using field data because of confounding and multicollinearity (Klei et al., 1996; Birchmeier et al., 2002), particularly when various epistatic effects are modeled. Therefore, genetic fixed effects in proposed multiple-breed genetic evaluation systems have generally included only additive and dominance effects (Cunningham, 1987; Klei et al., 1996; Miller and Wilton, 1999; Sullivan et al., 1999). The specification of prior information on genetic fixed effects, as available from the literature, might be useful for analyses of poorly structured data sets derived from crossbred populations (Quaas and Pollak, 1999) as it mitigates the effects of multicollinearity among genetic effects coefficients. Reliable estimates of dominance effects are available from the literature (e.g., Gregory et al., 1999), whereas reported estimates of epistatic effects are much rarer (Koch et al., 1985; Arthur et al., 1999). Our recombination loss specification for (Dickerson, 1973; Kinghorn, 1980; Kinghorn, 1987) provides a useful compromise between the full two-locus model (Hill, 1982; Wolf et al., 1995) and currently used additive/dominance models.
|
Nelore) in Region 3 (between 30°S and 32°S) are presented in Figure 1. These means were very similar under MBAM and AM, varying from a minimum difference of 0.10 kg for Herefords to a maximum difference of 1.00 kg observed on F1. Incidentally, the same mean differences would be observed in the other two regions since interaction between region and breed composition effects was not modeled.
|
presents variance components for PWG estimated by MBAM and AM. The genetic variances obtained for the Nelore and Hereford breeds by MBAM substantially differed in magnitude. Herefords had a posterior mean genetic variance that was almost fourfold greater than that obtained for the Nelore breed with no apparent overlap between the 95% PPI obtained for these parameters (Table 6). Using the conventional AM, a posterior mean intermediate between the Nelore and Hereford posterior mean genetic variances in the MBAM was obtained, as shown in the posterior densities of these variance components (Figure 2). The posterior mean of the variance due to the segregation between the Hereford and Nelore breeds was about 35.4% of the Nelore posterior mean genetic variance, but represented only 9.9% of the Hereford posterior mean genetic variance (Table 6). These percentages were larger than those found for birth and weaning weights of crosses between Angus and Brahman in Florida, ranging from 1.4 to 3.1% (Elzo and Wakeman, 1998). The magnitude of the segregation variance relative to the Hereford genetic variance (9.9%) was, however, somewhat smaller compared with results obtained for birth weight of Hereford-Nelore crosses in Argentina (16.5%) (Birchmeier et al., 2002). The Nelore variance was 73.5% of the magnitude of the Hereford variance in birth weight in their study, whereas Nelores had a genetic variance for PWG that was only 27.9% than for Herefords (Table 6) in our study. The existence of a scale effect on phenotypic variances, which is observed when groups with larger phenotypic mean have also larger phenotypic variance (Falconer and Mackay, 1996), could have exacerbated the difference between Hereford and Nelore genetic variances. That is, a homoskedastic residual variance assumption would bias both estimates of breed genetic variances due to the highly negative sampling correlations that exist between their MCMC samples and samples of (data not shown). However, there did not appear to be any evidence of such a scale effect from Table 1. All groups, except F1, have somewhat similar phenotypic standard deviations with no apparent variance vs. mean relationship.
|
|
. Heritabilities were determined by not including the contemporary group variance as part of the phenotypic variance to allow comparisons with previously reported within-herd heritability estimates in which contemporary groups are fitted as fixed effects (Meyer, 1992; Koots et al., 1994; Eler et al., 1995). The posterior means of the additive heritability obtained by MBAM ranged from a minimum of 0.07 for purebred Nelores to a maximum of 0.20 for purebred Herefords, with other breed composition groups having intermediate values. As expected, the posterior mean based on the AM had an intermediate value of 0.15. These estimates are considerably smaller than the average value of 0.31 determined for PWG based on 177 studies (Koots et al., 1994). The Hereford estimate is, however, within the expected range for the extensive production environments in Brazil and is larger than the heritability estimate of 0.16 for yearling weight of Hereford cattle raised on pasture production systems in Australia (Meyer, 1992). The posterior mean obtained for for Nelores was very low and less than the corresponding estimate of 0.16 observed for yearling weight in another Brazilian Nelore purebred data set (Eler et al., 1995). Since purebred Nelores were only represented by parents without records in the studied population, it may not be entirely appropriate to compare these two different sets of results owing to, for example, different population structures. As pointed out by one of the reviewers of this paper, other reasons for the low heritability estimates are plausible. For example, selection bias is particularly likely since animals with uncertain paternity were not considered leaving only progeny of AI sires or individually controlled matings. Low heritabilities, however, do not invalidate the model comparisons considered in this study because, if anything, they should diminish differences in model performance between MBAM and AM.
|
; Birchmeier et al., 2002), several recently proposed models (Klei et al., 1996; Miller and Wilton, 1999; Quaas and Pollak, 1999; Sullivan et al., 1999) assume that all breeds have the same additive genetic variance with no genetic variance due to segregation between breeds in advanced crosses. One potential advantage of the Bayesian MBAM over the specifications of Elzo (1994) and Birchmeier et al. (2002) is the ability to incorporate prior information on breed-specific and segregation genetic variance components. Even though existing prior information for segregation variances is limited (Elzo and Wakeman, 1998; Birchmeier et al., 2002), there is extensive information available on breed-specific variances (e.g., Meyer, 1992; Koots et al., 1994) that could be incorporated through the prior densities specified in the third stage of MBAM.
Animal Additive Genetic Effects.
Ranking animals for genetic merit and eventual selection is a chief objective in breeding programs. Kendall rank correlations between posterior mean additive genetic effects obtained by MBAM and by AM across all breed groups and for the most frequent breed groups in the data set are presented in Table 8. The rank correlation across all animals was high, indicating reasonable agreement between MBAM and AM ranks. Nonetheless, because the correlation is not close to 1, there will be some differences between MBAM and AM models when selecting top animals as breeding stock. Ranks were less affected within breed groups than across all animals. This was anticipated due to the different genetic variances (and consequently dispersion of additive genetic effects) that were estimated for each breed composition group under MBAM as opposed to a common genetic variance under the conventional AM. The scatter plots in Figure 3 of the posterior mean additive genetic effects for MBAM vs. AM for Hereford, Nelore, and F1 provide additional evidence of the difference between models in terms of accommodating the breed group-specific variability of genetic effects; it is apparent from Figure 3 that these breed groups would have different slopes for a linear regression of AM posterior mean genetic effects on MBAM posterior mean genetic effects, and, therefore, the ranks will be more similar within than across the breed groups. Across-breed group rank differences are expected to be larger for animals with extreme genetic effects, which are, in general, of greater interest for selection.
|
|
; Elzo, 1994; Klei et al., 1996; Sullivan et al., 1999). The coefficients for the fixed genetic effects (additive, dominance, etc.) will depend on the mate’s genotype, and, therefore, comparison between candidates for selection should be made for specific breed compositions of the mates. The additive genetic effect corresponds to the general combining ability of the individual and does not depend on the genotype of the mates. The determination of specific combining abilities requires the estimation of nonadditive genetic variances. Even though theory for a full additive and dominance two-breed genetic model is available (Lo et al., 1995), this model requires a much larger number of variance components to be estimated, up to 25 when inbreeding is present, and thus may be cumbersome for practical applications.
Further Developments and Implementation.
Generalization of the MBAM model for the analysis of multiple-traits or additive-maternal genetic effects is possible using multiple-breed variance-covariance genetic matrices as proposed by Cantet and Fernando (1995) following Lo et al. (1993). Another potentially important generalization of the MBAM would be to accommodate situations of residual heteroskedasticity using a structural model as in Foulley and Quaas (1995).
Due to the computational issues involving MCMC inference, MBAM could be implemented for genetic evaluation of large beef cattle population using an empirical Bayes approach. In this case, variance components could be estimated for a subset of the data on the population of interest using MCMC. The mixed-model equations as from Eq. [A1], in the appendix herein, could then be used to provide empirical best linear unbiased estimates for elements of ß and and BLUP of u and a for all individuals within the population.
Implications
Multiple-breed genetic evaluations have been previously developed using conventional animal model assumptions, most notably common genetic variances across breed groups and no genetic variance due to segregation between breeds. The multiple-breed animal model is a viable alternative because it specifies the additive genetic variance of each breed composition group to be a function of breed-specific and segregation variances. Accordingly, this model has enhanced flexibility for characterizing the dispersion of genetic merit within breed groups and is demonstrated to be a much better fit compared with the conventional animal model for data on postweaning records derived from multiple-breed groups. A simulation study also demonstrated a somewhat superior performance of the proposed model in terms of ranking individuals in multiple-breed groups for selection; however, the study was not designed to demonstrate a significant effect on genetic gain.
Appendix
Fully Conditional Densities
In what follows, the fully conditional densities (FCD) are presented using the notation "ELSE" to denote the data vector y and all other parameters treated as known in the FCD in question.
Fixed and Random Location Parameters.
Let
 |
where 
and 
. Uniform priors on ß and are equivalent to specifying 
and = 
, respectively. Following Wang et al. (1994), it can be shown the location parameters have the following joint FCD,
 | [A1] |
where 
.
Residual Variance.
The FCD of error variance can be shown to be scaled inverted 2:
 | [A2] |
A uniform prior on is equivalent to specifying ve = –2 and 
in [A2].
Contemporary Group Variance.
The FCD of contemporary group variance can also be shown to be scaled inverted 2:
 | [A3] |
A uniform prior on 
is equivalent to specifying vu = –2 and 
in [A3].
Genetic Variances.
The FCD of genetic variances presented below are not of standard forms:
 | [A4] |
 | [A5] |
Uniform priors can be invoked in [A4] and [A5] in the same manner described previously for and .
The Metropolis-Hastings (MH) algorithm can be used to sample from [A4] and [A5]. Our MH implementation was based on a random walk specification (Chib and Greenberg, 1995) with a scaled inverted 2 proposal density. A proposal value 
is generated from a scaled inverted 2 distribution with scaling factor equal to the product of the parameter value in the previous cycle 
with the degrees of freedom 
. This distribution has the following density:
 |
To improve mixing, was tuned during the burning period such that the acceptance of proposal values was near 40% (Chib and Greenberg, 1995). The acceptance rate is given by
 |
where
is the vector of all parameters but 
, c = Ab, Sbb'. Finally, to facilitate the computations, the facts that 
and that a'(G())–1a can be computed as 
have been used, where qb denotes the number of base animals (animals with both parents unknown).
Footnotes
1 Funded by CAPES—Brasília/Brazil and the Michigan Agric. Exp. Stn. (Project MICL 01822). We are grateful to I. Misztal for making available Sparsem90 and Fspak90, and to the Brazilian genetic evaluation alliance, "Conexão Delta G," for providing the Nelore-Hereford data. 
2 CAPES fellow. Currently at EMBRAPA Pecuária Sul (Brazilian Agric. Res. Corp. Southern Cattle & Sheep Center), Bagé, RS 96401-970, BRAZIL.
3 Correspondence: 1205 Anthony Hall, East Lansing, MI 48824-1225 (phone: 517-355-8445; fax: 517-353-1699; e-mail: tempelma{at}msu.edu).
Received for publication July 22, 2003. Accepted for publication February 9, 2004.
Literature Cited
Arnold, J. W., J. K. Bertrand, and L. L. Benyshek. 1992. Animal-model for genetic evaluation of multibreed data. J. Anim. Sci. 70:3322–3332.[Abstract]
Arthur, P. F., H. Hearnshaw, and P. D. Stephenson. 1999. Direct and maternal additive and heterosis effects from crossing Bos Indicus and Bos Taurus cattle: Cow and calf performance in two environments. Livest. Prod. Sci. 57:231–241.
Birchmeier, A. N., R. J. C. Cantet, R. L. Fernando, C. A. Morris, F. Holgado, A. Jara, and M. S. Cristal. 2002. Estimation of segregation variance for birth weight in beef cattle. Livest. Prod. Sci. 76:27–35.
Cantet, R. J. C., and R. L. Fernando. 1995. Prediction of breeding values with additive animal-models for crosses from 2 populations. Genet. Sel. Evol. 27:323–334.
Chib, S., and E. Greenberg. 1995. Understanding the Metropolis-Hastings algorithm. Am. Stat. 49:327–335.
Cunningham, E. P. 1987. Crossbreeding: The Greek temple model. J. Anim. Breed. Genet. 104:2–11.
Dekkers, J. C. M., and F. Hospital. 2002. The use of molecular genetics in the improvement of agricultural populations. Nature Rev. Genet. 3:22–32.[Medline]
Dickerson, G. E. 1973. Inbreeding and heterosis in animals. Pages 54–77 in Proc. Anim. Breeding Genet. Symp. in Honor of Dr. J. L. Lush, Champaign, IL.
Eler, J. P., L. D. Van Vleck, J. B. S. Ferraz, and R. B. Lobo. 1995. Estimation of variances due to direct and maternal effects for growth traits of Nelore cattle. J. Anim. Sci. 73:3253–3258.[Abstract]
Elzo, M. A. 1994. Restricted maximum-likelihood procedures for the estimation of additive and nonadditive genetic variances and covariances in multibreed populations. J. Anim. Sci. 72:3055–3065.[Abstract]
Elzo, M. A., and D. L. Wakeman. 1998. Covariance components and prediction for additive and nonadditive preweaning growth genetic effects in an Angus-Brahman multibreed herd. J. Anim. Sci. 76:1290–1302.
Falconer, D. S., and T. F. C. Mackay. 1996. Introduction to Quantitative Genetics. 4th ed. Longman Group Ltd., Harlow, England.
Fernando, R. L. 1999. Theory for analysis of multi-breed data. Pages 1–16 in 7th Genet. Pred. Workshop, Kansas City, MO.
Foulley, J. L., and R. L. Quaas. 1995. Heterogeneous variances in Gaussian linear mixed models. Genet. Sel. Evol. 27:211–228.
Gelfand, A. E. 1996. Model determination using sampling-based methods. Pages 145–161 in Markov Chain Monte Carlo in Practice. W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, ed. Chapman & Hall, London, England.
Geyer, C. J. 1992. Practical Markov chain Monte Carlo. Stat. Sci. 7:473–511.
Gregory, K. E., L. V. Cundiff, and R. M. Koch. 1999. Composite breeds to use heterosis and breed differences to improve efficiency of beef population. Tech. Bulletin No. 1875, MARC-USDA-ARS, Clay Center, NE.
Hill, W. G. 1982. Dominance and epistasis as components of heterosis. J. Anim. Breed. Genet. 99:161–168.
Hobert, J. P. 2000. Hierarchical models: A current computational perspective. J. Amer. Statist. Assoc. 95:1312–1316.
Kinghorn, B. 1980. The expression of recombination loss in quantitative traits. J. Anim. Breed. Genet. 97:138–143.
Kinghorn, B. P. 1987. The nature of 2-locus epistatic interactions in animals: Evidence from Sewall Wright guinea-pig data. Theor. Appl. Genet. 73:595–604.
Klei, L., R. L. Quaas, E. J. Pollak, and B. E. Cunningham. 1996. Multiple-breed evaluation. Pages 93–105 in Beef Imp. Fed. 28th Annu. Res. Symp. & Annu. Mtg., Birmingham, AL.
Koch, R. M., G. E. Dickerson, L. V. Cundiff, and K. E. Gregory. 1985. Heterosis retained in advanced generations of crosses among Angus and Hereford cattle. J. Anim. Sci. 60:1117–1132.
Koots, K. R., J. P. Gibson, C. Smith, and J. W. Wilton. 1994. Analyses of published genetic parameter estimates for beef cattle production traits. 1. Heritability. Anim. Breed. Abstr. 62:309–338.
Lo, L. L., R. L. Fernando, R. J. C. Cantet, and M. Grossman. 1995. Theory for modeling means and covariances in a 2-breed population with dominance inheritance. Theor. Appl. Genet. 90:49–62.
Lo, L. L., R. L. Fernando, and M. Grossman. 1993. Covariance between relatives in multibreed populations—additive-model. Theor. Appl. Genet. 87:423–430.
Meyer, K. 1992. Variance-components due to direct and maternal effects for growth traits of Australian beef-cattle. Livest. Prod. Sci. 31:179–204.
Miller, S. P., and J. W. Wilton. 1999. Genetic relationships among direct and maternal components of milk yield and maternal weaning gain in a multibreed beef herd. J. Anim. Sci. 77:1155–1161.
Quaas, R. L. 1988. Additive genetic model with groups and relationships. J. Dairy Sci. 71:1338–1345.
Quaas, R. L., and E. J. Pollak. 1999. Application of a multi-breed genetic evaluation. Pages 30–34 in 7th Genet. Pred. Workshop, Kansas City, MO.
Sorensen, D. A., S. Andersen, D. Gianola, and I. Korsgaard. 1995. Bayesian-inference in threshold models using Gibbs sampling. Genet. Sel. Evol. 27:229–249.
Sorensen, D. A., and D. Gianola. 2002. Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics. Springer-Verlag, New York.
Sullivan, P. G., J. W. Wilton, S. P. Miller, and L. R. Banks. 1999. Genetic trends and breed overlap derived from evaluations of beef cattle for multiple-breed genetic growth traits. J. Anim. Sci. 77:2019–2027.
Wang, C. S., J. J. Rutledge, and D. Gianola. 1994. Bayesian-analysis of mixed linear-models via Gibbs sampling with an application to litter size in Iberian pigs. Genet. Sel. Evol. 26:91–115.
Wolf, J., O. Distl, J. Hyanek, T. Grosshans, and G. Seeland. 1995. Crossbreeding in farm-animals. 5. Analysis of crossbreeding plans with secondary crossbred generations. J. Anim. Breed. Genet. 112:81–94.
This article has been cited by other articles:
|
|
 |
|
  J. P. Sanchez, I. Misztal, I. Aguilar, and J. K. Bertrand Genetic evaluation of growth in a multibreed beef cattle population using random regression-linear spline models J Anim Sci, February 1, 2008; 86(2): 267 - 277. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
F. F. Cardoso, G. J. M. Rosa, and R. J. Tempelman Accounting for outliers and heteroskedasticity in multibreed genetic evaluations of postweaning gain of Nelore-Hereford cattle J Anim Sci, April 1, 2007; 85(4): 909 - 918. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
F. F. Cardoso, G. J. M. Rosa, and R. J. Tempelman Multiple-breed genetic inference using heavy-tailed structural models for heterogeneous residual variances J Anim Sci, August 1, 2005; 83(8): 1766 - 1779. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP |
), Nelores (
), and F1 (
) obtained by a multiple-breed animal model (MBAM) vs. those obtained by a conventional animal model (AM).