Estimation of breeding values from large‐sized routine carcass data in Japanese Black cattle using Bayesian analysis

Volumes of official data sets have been increasing rapidly in the genetic evaluation using the Japanese Black routine carcass field data. Therefore, an alternative approach with smaller memory requirement to the current one using the restricted maximum likelihood (REML) and the empirical best linear unbiased prediction (EBLUP) is desired. This study applied a Bayesian analysis using Gibbs sampling (GS) to a large data set of the routine carcass field data and practically verified its validity in the estimation of breeding values. A Bayesian analysis like REML‐EBLUP was implemented, and the posterior means were calculated using every 10th sample from 90 000 of samples after 10 000 samples discarded. Moment and rank correlations between breeding values estimated by GS and REML‐EBLUP were very close to one, and the linear regression coefficients and the intercepts of the GS on the REML‐EBLUP estimates were substantially one and zero, respectively, showing a very good agreement between breeding value estimation by the current GS and the REML‐EBLUP. The current GS required only one‐sixth of the memory space with REML‐EBLUP. It is confirmed that the current GS approach with relatively small memory requirement is valid as a genetic evaluation procedure using large routine carcass data.     

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.     

Variational bayesian method of estimating variance components

We developed a Bayesian analysis approach by using a variational inference method, a so‐called variational Bayesian method, to determine the posterior distributions of variance components. This variational Bayesian method and an alternative Bayesian method using Gibbs sampling were compared in estimating genetic and residual variance components from both simulated data and publically available real pig data. In the simulated data set, we observed strong bias toward overestimation of genetic variance for the variational Bayesian method in the case of low heritability and low population size, and less bias was detected with larger population sizes in both methods examined. The differences in the estimates of variance components between the variational Bayesian and the Gibbs sampling were not found in the real pig data. However, the posterior distributions of the variance components obtained with the variational Bayesian method had shorter tails than those obtained with the Gibbs sampling. Consequently, the posterior standard deviations of the genetic and residual variances of the variational Bayesian method were lower than those of the method using Gibbs sampling. The computing time required was much shorter with the variational Bayesian method than with the method using Gibbs sampling.     

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.     

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.     

Bayesian inference for genetic parameter estimation on growth traits for Nelore cattle in Brazil,using the Gibbs sampler

This data set consisted of over 29 245 field records from 24 herds of registered Nelore cattle born between 1980 and 1993, with calves sires by 657 sires and 12 151 dams. The records were collected in south‐eastern and midwestern Brazil and animals were raised on pasture in a tropical climate. Three growth traits were included in these analyses: 205‐ (W205), 365‐ (W365) and 550‐day (W550) weight. The linear model included fixed effects for contemporary groups (herd‐year‐season‐sex) and age of dam at calving. The model also included random effects for direct genetic, maternal genetic and maternal permanent environmental (MPE) contributions to observations. The analyses were conducted using single‐trait and multiple‐trait animal models. Variance and covariance components were estimated by restricted maximum likelihood (REML) using a derivative‐free algorithm (DFREML) for multiple traits (MTDFREML). Bayesian inference was obtained by a multiple trait Gibbs sampling algorithm (GS) for (co)variance component inference in animal models (MTGSAM). Three different sets of prior distributions for the (co)variance components were used: flat, symmetric, and sharp. The shape parameters (ν) were 0, 5 and 9, respectively. The results suggested that the shape of the prior distributions did not affect the estimates of (co)variance components. From the REML analyses, for all traits, direct heritabilities obtained from single trait analyses were smaller than those obtained from bivariate analyses and by the GS method. Estimates of genetic correlations between direct and maternal effects obtained using REML were positive but very low, indicating that genetic selection programs should consider both components jointly. GS produced similar but slightly higher estimates of genetic parameters than REML, however, the greater robustness of GS makes it the method of choice for many applications.     

Reliable computing in estimation of variance components

The purpose of this study is to present guidelines in selection of statistical and computing algorithms for variance components estimation when computing involves software packages. For this purpose two major methods are to be considered: residual maximal likelihood (REML) and Bayesian via Gibbs sampling. Expectation‐Maximization (EM) REML is regarded as a very stable algorithm that is able to converge when covariance matrices are close to singular, however it is slow. However, convergence problems can occur with random regression models, especially if the starting values are much lower than those at convergence. Average Information (AI) REML is much faster for common problems but it relies on heuristics for convergence, and it may be very slow or even diverge for complex models. REML algorithms for general models become unstable with larger number of traits. REML by canonical transformation is stable in such cases but can support only a limited class of models. In general, REML algorithms are difficult to program. Bayesian methods via Gibbs sampling are much easier to program than REML, especially for complex models, and they can support much larger datasets; however, the termination criterion can be hard to determine, and the quality of estimates depends on a number of details. Computing speed varies with computing optimizations, with which some large data sets and complex models can be supported in a reasonable time; however, optimizations increase complexity of programming and restrict the types of models applicable. Several examples from past research are discussed to illustrate the fact that different problems required different methods.     

Genetic parameters and evaluation of the Tunisian dairy cattle population for milk yield by Bayesian and BLUP analyses

A Markov Chain Monte Carlo Bayesian method and BLUP analyses were used on Tunisian dairy cattle data. Data included 92,106 lactation records collected on 37,536 animals over 19 freshening years, from 1983 to 2001. Each record was partitioned into the fixed effects of herd-year, month of calving, and age-parity, a permanent environmental effect, an additive genetic effect, and a residual effect.

Posterior conditional distributions were determined for variance components and model effects. Solutions (BLUE) and posterior means for levels of herd-year, month of calving, and age-parity showed similar patterns. Posterior means of heritability and repeatability were 0.17 ± 18 × 10^− 5 and 0.39 ± 8 × 10^− 5, respectively. Posterior means of bull's breeding values were compared to bull's BLUP solutions. BLUP solutions were obtained using 0.17 and 0.39, estimated from the data, and 0.25 and 0.40 estimates for heritability and repeatability, respectively. Rank correlations between bull's posterior means and BLUP breeding values were 0.998 and 0.994 using genetic parameters estimated from the data and from the literature, respectively. This correlation coefficient was 0.995 between bull's BLUP solutions using either of the two sets of genetic parameters.     

Implementation issues for Markov Chain Monte Carlo methods in random regression test-day models

Markov Chain Monte Carlo methods made possible estimation of parameters for complex random regression test‐day models. Models evolved from single‐trait with one set of random regressions to multiple‐trait applications with several random effects described by regressions. Gibbs sampling has been used for models with linear (with respect to coefficients) regressions and normality assumptions for random effects. Difficulties associated with implementations of Markov Chain Monte Carlo schemes include lack of good practical methods to assess convergence, slow mixing caused by high posterior correlations of parameters and long running time to generate enough posterior samples. Those problems are illustrated through comparison of Gibbs sampling schemes for single‐trait random regression test‐day models with different model parameterizations, different functions used for regressions and posterior chains of different sizes. Orthogonal polynomials showed better convergence and mixing properties in comparison with ‘lactation curve’ functions of the same number of parameters. Increasing the order of polynomials resulted in smaller number of independent samples for covariance components. Gibbs sampling under hierarchical model parameterization had a lower level of autocorrelation and required less time for computation. Posterior means and standard deviations of genetic parameters were very similar for chains of different size (from 20 000 to 1 000 000) after convergence. Single‐trait random regression models with large data sets can be analysed by Markov Chain Monte Carlo methods in relatively short time. Multiple‐trait (lactation) models are computationally more demanding and better algorithms are required.     

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

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

Effects of nonrandom parental selection on estimation of variance components

Bayesian estimation via Gibbs sampling, REML, and Method R were compared for their empirical sampling properties in estimating genetic parameters from data subject to parental selection using an infinitesimal animal model. Models with and without contemporary groups, random or nonrandom parental selection, two levels of heritability, and none or 15% randomly missing pedigree information were considered. Nonrandom parental selection caused similar effects on estimates of variance components from all three methods. When pedigree information was complete, REML and Bayesian estimation were not biased by nonrandom parental selection for models with or without contemporary groups. Method R estimates, however, were strongly biased by nonrandom parental selection when contemporary groups were in the model. The bias was empirically shown to be a consequence of not fully accounting for gametic phase disequilibrium in the subsamples. The joint effects of nonrandom parental selection and missing pedigree information caused estimates from all methods to be highly biased. Missing pedigree information did not cause biased estimates in random mating populations. Method R estimates usually had greater mean square errors than did REML and Bayesian estimates.     

Estimation of genetic parameters and prediction of breeding values for multivariate threshold and continuous data in a simulated horse population using Gibbs sampling and residual maximum likelihood

Simulated horse data were used to compare multivariate estimation of genetic parameters and prediction of breeding values (BV) for categorical, continuous and molecular genetic data using linear animal models via residual maximum likelihood (REML) and best linear unbiased prediction (BLUP) and mixed linear-threshold animal models via Gibbs sampling (GS). Simulation included additive genetic values, residuals and fixed effects for one continuous trait, liabilities of four binary traits, and quantitative trait locus (QTL) effects and genetic markers with different recombination rates and polymorphism information content for one of the liabilities. Analysed data sets differed in the number of animals with trait records and availability of genetic marker information. Consideration of genetic marker information in the model resulted in marked overestimation of the heritability of the QTL trait. If information on 10,000 or 5,000 animals was used, bias of heritabilities and additive genetic correlations was mostly smaller, correlation between true and predicted BV was always higher and identification of genetically superior and inferior animals was - with regard to the moderately heritable traits, in many cases - more reliable with GS than with REML/BLUP. If information on only 1,000 animals was used, neither GS nor REML/BLUP produced genetic parameter estimates with relative bias 50% for all traits. Selection decisions for binary traits should rather be based on GS than on REML/BLUP breeding values.     

Genetic and environmental influences on size of first litter in sheep, estimated by the REML method

Fixed effects of age at first litter and of season of lambing as well as variance components for additive genetic, flock × year and sire of litter effects on size of first litter were estimated for an animal model by a derivative-free REML procedure. Data of the four Swiss sheep breeds White Alpine (WAS), Brown-headed Meat (BFS), Black-Brown Mountain (SBS) and Valais Black Nose (SN) were available. Number of first litters used were 21 384, 21 607, 15 013, 12 394 and 18 110 the WAS (two data sets, WAS1, WAS2), BFS, SBS and SN, respectively. Litter size of ewes lambing the first time at 2 years of age was 0.27, 0.31, 0.41, 0.46 and 0.26 lambs larger than of ewes lambing the first time at 1 year of age. The largest increase occurred for the two breeds (BFS, SBS) with the lowest average age at first litter. The largest difference between any two lambing seasons within breed were 0.16, 0.16, 0.29, 0.22 and 0.07 lambs. Estimates of additive genetic variance of size of first litter were between 0.0269 (SN) and 0.0765 (SBS). Heritability estimates for this trait were 0.171, 0.156, 0.114, 0.225 and 0.122 for WAS1, WAS2, BFS, SBS and SN, respectively. A large flock × year component (relative to phenotypic variance) of 0.148 was found for SN, compared with estimates between 0.042 and 0.067 for the other breeds. A sire of litter component (relative to phenotypic variance) of 0.066 was found for SN, compared with estimates between 0.016 and 0.039 for the other breeds. It can be concluded that all nongenetic effects investigated should be taken into account for the estimation of additive genetic variance and breeding values for size of first litter, and that considerable variation in size of genetic and nongenetic effects exists in the sheep breeds under consideration.     

Bayesian conjugate analysis using a generalized inverted Wishart distribution accounts for differential uncertainty among the genetic parameters--an application to the maternal animal model

Consider the estimation of genetic (co)variance components from a maternal animal model (MAM) using a conjugated Bayesian approach. Usually, more uncertainty is expected a priori on the value of the maternal additive variance than on the value of the direct additive variance. However, it is not possible to model such differential uncertainty when assuming an inverted Wishart (IW) distribution for the genetic covariance matrix. Instead, consider the use of a generalized inverted Wishart (GIW) distribution. The GIW is essentially an extension of the IW distribution with a larger set of distinct parameters. In this study, the GIW distribution in its full generality is introduced and theoretical results regarding its use as the prior distribution for the genetic covariance matrix of the MAM are derived. In particular, we prove that the conditional conjugacy property holds so that parameter estimation can be accomplished via the Gibbs sampler. A sampling algorithm is also sketched. Furthermore, we describe how to specify the hyperparameters to account for differential prior opinion on the (co)variance components. A recursive strategy to elicit these parameters is then presented and tested using field records and simulated data. The procedure returned accurate estimates and reduced standard errors when compared with non-informative prior settings while improving the convergence rates. In general, faster convergence was always observed when a stronger weight was placed on the prior distributions. However, analyses based on the IW distribution have also produced biased estimates when the prior means were set to over-dispersed values.     

Use of robust multivariate linear mixed models for estimation of genetic parameters for carcass traits in beef cattle

Assumptions of normality of residuals for carcass evaluation may make inferences vulnerable to the presence of outliers, but heavy‐tail densities are viable alternatives to normal distributions and provide robustness against unusual or outlying observations when used to model the densities of residual effects. We compare estimates of genetic parameters by fitting multivariate Normal (MN) or heavy‐tail distributions (multivariate Student's t and multivariate Slash, MSt and MS) for residuals in data of hot carcass weight (HCW), longissimus muscle area (REA) and 12th to 13th rib fat (FAT) traits in beef cattle using 2475 records from 2007 to 2008 from a large commercial operation in Nebraska. Model comparisons using deviance information criteria (DIC) favoured MSt over MS and MN models, respectively. The posterior means (and 95% posterior probability intervals, PPI) of v for the MSt and MS models were 5.89 ± 0.90 (4.35, 7.86) and 2.04 ± 0.18 (1.70, 2.41), respectively. Smaller values of posterior densities of v for MSt and MS models confirm that the assumption of normally distributed residuals is not adequate for the analysis of the data set. Posterior mean (PM) and posterior median (PD) estimates of direct genetic variances were variable with MSt having the highest mean value followed by MS and MN, respectively. Posterior inferences on genetic variance were, however, comparable among the models for FAT. Posterior inference on additive heritabilities for HCW, REA and FAT using MN, MSt and MS models indicated similar and moderate heritability comparable with the literature. Posterior means of genetic correlations for carcass traits were variable but positive except for between REA and FAT, which showed an antagonistic relationship. We have demonstrated that genetic evaluation and selection strategies will be sensitive to the assumed model for residuals.     

Genetic correlations between osseous fragments in fetlock and hock joints, deforming arthropathy in hock joints and pathologic changes in the navicular bones of Warmblood riding horses

The results of a standardized radiological examination of 5231 Hanoverian Warmblood horses were used to investigate heritability of and genetic correlations between prevalent radiographic findings in the equine limbs. Radiographic findings were categorized by joint location and type of visible alterations and analyzed as all-or-none traits. Heritabilities and correlations were estimated multivariately for most prevalent radiographic findings in equine limbs using Residual Maximum Likelihood (REML) and Gibbs Sampling (GS). Linear animal models and linear sire models were used for REML; sire threshold models were used for GS analyses. Heritabilities and residual correlations from linear model analyses were transformed from observed scale to underlying liability scale. Osseous fragments were seen in fetlock joints (OFF) of 23.5% and in hock joints (OFH) of 9.2% of investigated horses. Deforming arthropathy in hock joints (DAH) was diagnosed in 12.0% and pathologic changes in navicular bones (PCN) in 25.8% of investigated horses. Heritabilities differed little between analyses with animal and sire models and with REML and GS. Ranges of heritability estimates were h² = 0.16–0.44 with REML and h² = 0.07–0.43 with GS. Genetic correlation estimates were larger in GS than in REML analyses. Additive genetic correlation between OFF and DAH was positive (r_g = 0.25 to 0.77). Negative additive genetic correlations were determined between OFF and OFH (r_g = − 0.17 to − 0.82), between OFH and DAH (r_g = − 0.14 to − 0.81), and between OFH and PCN (r_g = − 0.19 to − 0.26). No relevant additive genetic correlations were estimated between PCN and OFF, and between PCN and DAH. The results of the present study indicate that the prevalences of common radiographic findings in the limbs of young riding horses are relevantly influenced by genetics and probably caused by different genes. Genetic correlations between radiological health traits therefore deserve closer attention in horse breeding. The quantitatively most important radiographic findings should be concurrently considered as individual traits in order to provide for general improvement of radiological health of the limbs of young Warmblood riding horses.     

Analysis of beef cattle longitudinal data applying a nonlinear model

The objective of this work was to evaluate the Nelore beef cattle, growth curve parameters using the Von Bertalanffy function in a nested Bayesian procedure that allowed estimation of the joint posterior distribution of growth curve parameters, their (co)variance components, and the environmental and additive genetic components affecting them. A hierarchical model was applied; each individual had a growth trajectory described by the nonlinear function, and each parameter of this function was considered to be affected by genetic and environmental effects that were described by an animal model. Random samples of the posterior distributions were drawn using Gibbs sampling and Metropolis-Hastings algorithms. The data set consisted of a total of 145,961 BW recorded from 15,386 animals. Even though the curve parameters were estimated for animals with few records, given that the information from related animals and the structure of systematic effects were considered in the curve fitting, all mature BW predicted were suitable. A large additive genetic variance for mature BW was observed. The parameter a of growth curves, which represents asymptotic adult BW, could be used as a selection criterion to control increases in adult BW when selecting for growth rate. The effect of maternal environment on growth was carried through to maturity and should be considered when evaluating adult BW. Other growth curve parameters showed small additive genetic and maternal effects. Mature BW and parameter k, related to the slope of the curve, presented a large, positive genetic correlation. The results indicated that selection for growth rate would increase adult BW without substantially changing the shape of the growth curve. Selection to change the slope of the growth curve without modifying adult BW would be inefficient because their genetic correlation is large. However, adult BW could be considered in a selection index with its corresponding economic weight to improve the overall efficiency of beef cattle production.     

Non-iterative variance component estimation in QTL analysis

In variance component quantitative trait loci (QTL) analysis, a mixed model is used to detect the most likely chromosome position of a QTL. The putative QTL is included as a random effect and a method is needed to estimate the QTL variance. The standard estimation method used is an iterative method based on the restricted maximum likelihood (REML). In this paper, we present a novel non-iterative variance component estimation method. This method is based on Henderson's method 3, but relaxes the condition of unbiasedness. Two similar estimators were compared, which were developed from two different partitions of the sum of squares in Henderson's method 3. The approach was compared with REML on data from a European wild boar × domestic pig intercross. A meat quality trait was studied on chromosome 6 where a functional gene was known to be located. Both partitions resulted in estimated QTL variances close to the REML estimates. From the non-iterative estimates, we could also compute good approximations of the likelihood ratio curve on the studied chromosome.     

Estimation of variance components for carcass traits in Japanese Black cattle using 50K SNP genotype data

Genomic selection using high‐density single nucleotide polymorphism (SNP) genotype data may accelerate genetic improvements in livestock animals. In this study, we attempted to estimate the variance components of six carcass traits in fattened Japanese Black steers using SNP genotype data. Six hundred and seventy‐three steers were genotyped using an Illumina Bovine SNP50 BeadChip and phenotyped for cold carcass weight, ribeye area, rib thickness, subcutaneous fat thickness, estimated yield percent and marbling score. Additive polygenic variance and the variance attributable to a set of SNPs that had statistically significant effects on the trait were estimated via Gibbs sampling with two models: (i) a model with the chosen SNPs and the additive polygenic effects; and (ii) a model with the polygenic effects alone. The proportion of the estimated variance attributable to the SNPs became higher as the number of SNP effects that fit increased. High correlations between breeding values estimated with the model containing the polygenic effect alone and those estimated by chosen SNPs were obtained. No fraction of the total genetic variance was explained by SNPs associated with the trait at P ≥ 0.1. Our results suggest that for the carcass traits of Japanese Black cattle, a maximum of half of the total additive genetic variance may be explained by SNPs between 100 several tens to several 100s.     

Employing a Monte Carlo algorithm in expectation maximization restricted maximum likelihood estimation of the linear mixed model

Multiple‐trait and random regression models have multiplied the number of equations needed for the estimation of variance components. To avoid inversion or decomposition of a large coefficient matrix, we propose estimation of variance components by Monte Carlo expectation maximization restricted maximum likelihood (MC EM REML) for multiple‐trait linear mixed models. Implementation is based on full‐model sampling for calculating the prediction error variances required for EM REML. Performance of the analytical and the MC EM REML algorithm was compared using a simulated and a field data set. For field data, results from both algorithms corresponded well even with one MC sample within an MC EM REML round. The magnitude of the standard errors of estimated prediction error variances depended on the formula used to calculate them and on the MC sample size within an MC EM REML round. Sampling variation in MC EM REML did not impair the convergence behaviour of the solutions compared with analytical EM REML analysis. A convergence criterion that takes into account the sampling variation was developed to monitor convergence for the MC EM REML algorithm. For the field data set, MC EM REML proved far superior to analytical EM REML both in computing time and in memory need.