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.     

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.     

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.     

Comparison of computing properties of derivative and derivative-free algorithms in variance-component estimation by REML

SUMMARY: Computing properties of better derivative and derivative-free algorithms were compared both theoretically and practically. Assuming that the log-likelihood function is approximately quadratic, in a t-trait analysis the number of steps to achieve convergence increases as t(2) in 'better' derivative-free algorithms and is independent of that number in 'better' derivative algorithms. The cost of one step increases as t(3) . Consequently, both classes of algorithms have a similar computational cost for single-trait models. In multiple traits, the computing costs increase as t(3) and t(5) , respectively. The derivative-free algorithms have worse numerical properties. Four programs were used to obtain one-, two-, and three-trait REML estimates from field data. Compared to single-trait analyses, the cost of one run for derivative-free algorithms increased by 27-40 times for two traits and 152-686 times for three traits. A similar increase in rounds of iteration for a derivative algorithm reached 5 and 21, and 1.8 and 2.2 in canonical transformation. Convergence and estimates of derivative algorithms were more predictable and, unlike derivative-free algorithms, were much less dependent on the choice of priors. Well-implemented derivative REML algorithms are less expensive and more reliable in multiple traits than derivative-free ones. ZUSAMMENFASSUNG: Vergleich von Rechen (Computing) merkmalen von abgeleiteten und ableitungsfreien Algorithmen zur Varianzkomponentensch?tzung mittels REML Rechenmerkmale von verbesserten ableitungsfreien und Algorithmen, die Ableitung benutzen, werden theoretisch und praktisch verglichen. Unter der Annahme einer ungef?hr quadratischen log-likelihood Funktion, nimmt in der Analyse von t Merkmalen die Zahl der Rechenschritte bis zu Konvergenz mit t(2) in 'besseren' ableitungsfreien Algorithmen zu und ist davon unabh?ngig von dieser Zahl in der 'besseren' Ableitung. Die Kosten je Schritt steigen mit t(3) . Daher haben beide Berechnungsarten für Einzelmerkmale ?hnliche Rechenkosten. Bei mehreren Merkmalen steigen die Kosten mit t(3) bzw. t(5) und ableitungsfreie Algorithmen haben schlechtere numerische Eigenschagten. Vier Programme haben für ein-, zwei- und drei-Merkmale REML Sch?tzungen von Felddaten erzeugt. Im Vergleich zu Ein-Merkmal Analysen stiegen Kosten für einen Lauf bei ableitungsfreien Algorithmen um das 27-40 fache bei zwei- und um das 152-686 fache bei drei-Merkmalen. Die Steigerungen je Lauf bei auf Ableitung beruhenden Algorithmen waren 5-21 fach und 1.8 und 2.2 fach bei kanonischer Transformation. Konvergenz und Sch?tzwerte von Algorithmen mit Ableitung waren besser vorhersagbar und weniger von der Wahl der priors beeinflu?t. Gut ausgestattete REML Methoden, die Ableitungen benutzen, sind ?konomischer und verl??licher bei Mehrmerkmalsproblemen als ableitungsfreie.     

“Bending” and beyond: Better estimates of quantitative genetic parameters?

Multivariate estimation of genetic parameters involving more than a handful of traits can be afflicted by problems arising through substantial sampling variation. We present a review of underlying causes and proposals to improve estimates, focusing on linear mixed model‐based estimation via restricted maximum likelihood (REML). Both full multivariate analyses and pooling of results from overlapping subsets of traits are considered. It is suggested to impose a penalty on the likelihood designed to reduce sampling variances at the expense of a little additional bias. Simulation results are discussed which demonstrate that this can yield REML estimates that are on average closer to the population values than their unpenalized counterparts. Suitable penalties can be obtained based on assumed prior distributions of selected parameters. Necessary choices of penalty functions and of the stringency of penalization are examined. We argue that scale‐free penalty functions lend themselves to a simple scheme imposing a mild, default penalty which can yield “better” estimates without being likely to incur detrimental effects.     

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.     

Is a Delayed Treatment with GnRH,hCG or Progesterone Beneficial for Reducing Embryonic Mortality in Buffaloes?

The aims of this study were to verify the efficacy of delayed hormonal treatments performed on day 25 post‐insemination on pregnancy rate at 45 and 70 days in buffalo. The trial was performed on 385 buffaloes synchronized by the Ovsynch/TAI protocol and submitted to artificial insemination (AI). Twenty‐five days after AI, pregnant animals were assigned to four treatments: (1) GnRH agonist (n = 52), 12 μg of buserelin acetate; (2) hCG (n = 51), 1500 IU of human chorionic gonadotrophin; (3) Progesterone (n = 47), 341 mg of P4 intramuscular (im) every 4 days for three times; (4) Control (n = 54), treatment with physiological saline (0.9% NaCl). Milk samples were collected on days 10, 20 and 25 after AI in all buffaloes to determine progesterone concentration in whey by radioimmunoassay method. Statistical analysis was performed by anova . Pregnancy rate on day 25 after AI was 52.9%, but declined to 41.8% by day 45, indicating an embryonic mortality (EM) of 21%. If only control group is considered, the incidence of EM was 38.9%. Pregnant buffaloes had higher (p < 0.01) progesterone concentrations on day 20 and 25 after AI than both non‐pregnant buffaloes and buffaloes that showed EM. The treatments on day 25 increased (p < 0.01) pregnancy rate, although in buffaloes with a low whey progesterone concentration on day 20 and 25 after AI (n = 22); all treatments were ineffective to reduce EM.     

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.     

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 for heat stress,including dominance gene effects,on milk yield in Thai Holstein dairy cattle

Heat stress in tropical regions is a major cause that strongly negatively affects to milk production in dairy cattle. Genetic selection for dairy heat tolerance is powerful technique to improve genetic performance. Therefore, the current study aimed to estimate genetic parameters and investigate the threshold point of heat stress for milk yield. Data included 52 701 test‐day milk yield records for the first parity from 6247 Thai Holstein dairy cattle, covering the period 1990 to 2007. The random regression test day model with EM‐REML was used to estimate variance components, genetic parameters and milk production loss. A decline in milk production was found when temperature and humidity index (THI) exceeded a threshold of 74, also it was associated with the high percentage of Holstein genetics. All variance component estimates increased with THI. The estimate of heritability of test‐day milk yield was 0.231. Dominance variance as a proportion to additive variance (0.035) indicated that non‐additive effects might not be of concern for milk genetics studies in Thai Holstein cattle. Correlations between genetic and permanent environmental effects, for regular conditions and due to heat stress, were ? 0.223 and ? 0.521, respectively. The heritability and genetic correlations from this study show that simultaneous selection for milk production and heat tolerance is possible.     

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.     

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.     

Bayesian inference strategies for the prediction of genetic merit using threshold models with an application to calving ease scores in Italian Piemontese cattle

First parity calving difficulty scores from Italian Piemontese cattle were analysed using a threshold mixed effects model. The model included the fixed effects of age of dam and sex of calf and their interaction and the random effects of sire, maternal grandsire, and herd‐year‐season. Covariances between sire and maternal grandsire effects were modelled using a numerator relationship matrix based on male ancestors. Field data consisted of 23 953 records collected between 1989 and 1998 from 4741 herd‐year‐seasons. Variance and covariance components were estimated using two alternative approximate marginal maximum likelihood (MML) methods, one based on expectation‐maximization (EM) and the other based on Laplacian integration. Inferences were compared to those based on three separate runs or sequences of Markov Chain Monte Carlo (MCMC) sampling in order to assess the validity of approximate MML estimates derived from data with similar size and design structure. Point estimates of direct heritability were 0.24, 0.25 and 0.26 for EM, Laplacian and MCMC (posterior mean), respectively, whereas corresponding maternal heritability estimates were 0.10, 0.11 and 0.12, respectively. The covariance between additive direct and maternal effects was found to be not different from zero based on MCMC‐derived confidence sets. The conventional joint modal estimates of sire effects and associated standard errors based on MML estimates of variance and covariance components differed little from the respective posterior means and standard deviations derived from MCMC. Therefore, there may be little need to pursue computation‐intensive MCMC methods for inference on genetic parameters and genetic merits using conventional threshold sire and maternal grandsire models for large datasets on calving ease.     

Apoptosis and mitosis in tumours of the skin and subcutaneous tissues of the dog

The presence of apoptotic cell death was evaluated in routinely processed tissue samples of 39 neoplasms of the skin and subcutaneous tissues of the dog using the method of terminal deoxynucleotidyl transferase (T d T) mediated deoxyuridine-5'-triphosphate (d UTP)-biotin nick end labelling (TUNEL). The degree of apoptosis was related to the frequency of mitosis, an index of cell proliferation. The correlation between the apoptotic index (AI), the percentage of positive cells after randomly enumerating 1000 cells and the mitotic count (MC), the number of mitotic figures in 10 fields at a magnification of 400 times was assessed by the Spearman non-parametric correlation test. TUNEL signals were observed in all types of tumours as brown products detected in non-pyknotic nuclei, in non-identifiable rounded structures (so-called apoptotic bodies) and occasionally in the cytoplasm, either singly or in combination. An inverse relationship between AI and MC was observed in benign tumours, while no correlation was found between AI and MC in either malignant or locally invasive tumours. Among benign tumours, intracutaneous cornifying epithelioma, fibroma, haemangioma and Schwannoma had high AI and low MC, while histiocytomas had low AI and high MC and pilomatrixomas low AI and MC. All malignant tumours had low AI and high MC, except for fibrosarcomas, which had high AI and MC. Finally, higher heterogeneity was observed among locally invasive tumours, as they had high AI and low MC (squamous cell carcinomas), and low AI with either low MC (haemangiopericytomas) or high MC (basal cell tumours). The classification of the tumours according to their AI (>15.8% high and <15.8% low) and MC (>9 high, <9 low) did not reflect the clinical behaviour of some tumour types.     

Analyses of growth curves of nellore cattle by multiple-trait and random regression models

The purpose of this study was to compare estimates of genetic parameters for sequential growth of beef cattle using two models and two data sets. Growth curves of Nellore cattle were analyzed using body weights measured at ages 1 (birth weight) to 733 d. Two data samples were created, one with 71,867 records sampled from all herds (MISS), and the other with 74,601 records sampled from herds with no missing traits (NMISS). Records preadjusted to a fixed age were analyzed by a multiple-trait model (MTM), which included the effects of contemporary group, age of dam class, additive direct, additive maternal, and maternal permanent environment. Analyses were by REML, with five traits at a time. The random regression model (RRM) included the effects of age of animal, contemporary group, age of dam class, additive direct, additive maternal, permanent environment, and maternal permanent environment. All effects were modeled as cubic Legendre polynomials. These analyses were also by REML. Shapes of estimates of variances by MTM were mostly similar for both data sets for all except late ages, where estimates for MISS were less regular, and for birth weight with MISS. Genetic correlations among ages for the direct and maternal effects were less smooth with MISS. Genetic correlations between direct and maternal effects were more negative for NMISS, where few sires were maternal grandsires. Parameter estimates with RRM were similar to MTM cept that estimates of variances showed more artifacts for MISS; the estimates of additive direct-maternal correlations were more negative with both data sets and approached -1.0 for some ages with NMISS. When parameters of a growth model obtained by used for genetic evaluation, these parameters should be examined for consistency with parameters from MTM and prior information, and adjustments may be required to eliminate artifacts.     

方差组分估计方法的比较

本文通过蒙特卡罗模拟产生的八个模拟资料比较了我国常用的方差分析法(不考虑场年季效应)、Henderson方法1、Henderson方法3、最大似然法、改进最大似然法与约束最大似然法。结果表明:方差分析法估计值偏差最大,而约束最大似然法的估计值最准确。     

A comparison via simulation of least squares Lehmann-Scheffé estimators of two variances and heritability with those of restricted maximum likelihood

The objective was to compare the performance of a recently derived, new method of estimating variances and covariances with any mixed linear model and any pattern of missing data with that of restricted maximum likelihood. For each of 96 combinations of six three-herd x four-sire unbalanced designs of 39 offspring each, four heritability values, two ratios of sire variance to interaction variance, and two distributions (multivariate normal and multivariate chi2, 3 df), 15,000 vectors (n = 39) were generated. Least squares Lehmann-Scheffé (LSLS) estimators of sire variance, interaction variance, and heritability were compared to those of REML with the performance measures of percentage of estimates (of the 15,000) that were positive, mean square error, variance, percentage of estimates within +/- 50% of the parameter, bias, maximum value, skewness, and kurtosis. The LSLS method vastly outperformed REML in almost all 96 combinations. Averaged over the 48 combinations with multivariate normal data, the average percentage that REML estimators of heritability performed relative to those of LSLS for the first five of the above listed eight performance measures was -100%. The number of times LSLS was better than REML was 235 out of 240. The analogous values for the 48 combinations with multivariate chi2, 3-df data were -90% and 230 out of 240. The REML maximum values were always larger than the LSLS values. The LSLS skewness and kurtosis values were about the same as those for REML, with the exception of LSLS heritability kurtosis values, which were notably less than those for REML. The explicit expectations of the LSLS estimators showed that the LSLS estimators were surprisingly unbiased given the paucity of data. Explicit coefficients for calculating mean square errors, variances, and biases squared of the LSLS estimators of the three variances were obtained for each design. The LSLS advantage was not quite so large with the multivariate chi2, 3-df data as with the multivariate normal data. Results with a symmetric multinomial distribution were the same as with the multivariate normal. The overall result was that the LSLS estimators produced substantially more non-zero estimates than REML estimators and these more abundant positive estimates were substantially grouped closer to their respective parameters. Results justify efforts to make the LSLS procedure computationally available.     

Comparison of restricted maximum likelihood and method R for estimating heritability and predicting breeding value under selection

Method R and Restricted Maximum Likelihood (REML) were compared for estimating heritability (h2) and subsequent prediction of breeding values (a) with data subject to selection. A single-trait animal model was used to generate the data and to predict breeding values. The data originated from 10 sires and 100 dams and simulation progressed for 10 overlapping generations. In simulating the data, genetic evaluation used the underlying parameter values and sires and dams were chosen by truncation selection for greatest predicted breeding values. Four alternative pedigree structures were evaluated: complete pedigree information, 50% of phenotypes with sire identities missing, 50% of phenotypes with dam identities missing, and 50% of phenotypes with sire and dams identities missing. Under selection and with complete pedigree data, Method R was a slightly less consistent estimator of h2 than REML. Estimates of h2 by both methods were biased downward when there was selection and loss of pedigree information and were unbiased when no selection was practiced. The empirical mean square error (EMSE) of Method R was several times larger than the EMSE of REML. In a subsequent analysis, different combinations of generations selected and generations sampled were simulated in an effort to disentangle the effects of both factors on Method R estimates of h2. It was observed that Method R overestimated h2 when both the sampling that is intrinsic in the method and the selection occurred in generations 6 to 10. In a final experiment, BLUP(a) were predicted with h2 estimated by either Method R or REML. Subsequently, five more generations of selection were practiced, and the mean square error of prediction (MSEP) of BLUP(a) was calculated with estimated h2 by either method, or the true value of the parameter. The MSEP of empirical BLUP(a) using Method R was greater than the MSEP of empirical BLUP(a) using REML. The latter statistic was closer to prediction error variance of BLUP(a) than the MSEP of empirical BLUP(a) using Method R, indicating that empirical BLUP(a) calculated using REML produced accurate predictions of breeding values under selection. In conclusion, the variability of h2 estimates calculated with Method R was greater than the variability of h2 estimates calculated with REML, with or without selection. Also, the MSEP of EBLUP(a) calculated using estimates of h2 by Method R was larger than MSEP of EBLUP(a) calculated with REML estimates of h2.     

Impact of fitting dominance and additive effects on accuracy of genomic prediction of breeding values in layers

Most genomic prediction studies fit only additive effects in models to estimate genomic breeding values (GEBV). However, if dominance genetic effects are an important source of variation for complex traits, accounting for them may improve the accuracy of GEBV. We investigated the effect of fitting dominance and additive effects on the accuracy of GEBV for eight egg production and quality traits in a purebred line of brown layers using pedigree or genomic information (42K single‐nucleotide polymorphism (SNP) panel). Phenotypes were corrected for the effect of hatch date. Additive and dominance genetic variances were estimated using genomic‐based [genomic best linear unbiased prediction (GBLUP)‐REML and BayesC] and pedigree‐based (PBLUP‐REML) methods. Breeding values were predicted using a model that included both additive and dominance effects and a model that included only additive effects. The reference population consisted of approximately 1800 animals hatched between 2004 and 2009, while approximately 300 young animals hatched in 2010 were used for validation. Accuracy of prediction was computed as the correlation between phenotypes and estimated breeding values of the validation animals divided by the square root of the estimate of heritability in the whole population. The proportion of dominance variance to total phenotypic variance ranged from 0.03 to 0.22 with PBLUP‐REML across traits, from 0 to 0.03 with GBLUP‐REML and from 0.01 to 0.05 with BayesC. Accuracies of GEBV ranged from 0.28 to 0.60 across traits. Inclusion of dominance effects did not improve the accuracy of GEBV, and differences in their accuracies between genomic‐based methods were small (0.01–0.05), with GBLUP‐REML yielding higher prediction accuracies than BayesC for egg production, egg colour and yolk weight, while BayesC yielded higher accuracies than GBLUP‐REML for the other traits. In conclusion, fitting dominance effects did not impact accuracy of genomic prediction of breeding values in this population.     

Genetic correlation between serum insulin-like growth factor-1 concentration and performance and meat quality traits in Duroc pigs

This study was intended to examine whether serum IGF-I concentration is appropriate for use as a physiological predictor for genetic improvement of meat production and meat quality traits in pigs. Heritabilities and genetic correlations were estimated for these traits. The Duroc breed used in this study was selected for seven generations for average daily BW gain (DG) from 30 to 105 kg of BW, loin-eye muscle area (EM), backfat thickness (BF), and intramuscular fat (IMF) content. Serum IGF-I concentration of boars and gilts at the fourth generation of selection and that of boars, gilts, and barrows from the fifth to seventh generations of selection were measured at 8 wk (IGFI-8W) for 832 animals and again at the time they reached 105 kg of BW (IGFI-105KG) for 834 animals. A multivariate REML procedure was used to estimate genetic parameters with a model incorporating generation of selection, sex, common environmental effect of litter, and individual additive genetic effects. Heritability estimates for IGFI-8W and IGFI-105KG were 0.23 +/- 0.02 and 0.26 +/- 0.03, respectively. The estimates of common environmental effect for IGFI-8W and IGFI-105KG were 0.20 +/- 0.02 and 0.03 +/- 0.01, respectively. Positive genetic correlations were estimated between IGFI-8W and DG (0.26 +/- 0.08), EM (0.22 +/- 0.10), and IMF (0.32 +/- 0.10). Moreover, the positive genetic correlation between IGFI-105KG and EM was 0.42 +/- 0.08. These results indicate that serum IGF-I concentration at an early stage of growth was effective for prediction of IMF, but it was not a reliable physiological predictor of genetic merit of meat production traits.