Technical note: Calculation of standard errors of estimates of genetic parameters with the multiple-trait derivative-free restricted maximal likelihood programs

The multiple-trait derivative-free REML set of programs was written to handle partially missing data for multiple-trait analyses as well as single-trait models. Standard errors of genetic parameters were reported for univariate models and for multiple-trait analyses only when all traits were measured on animals with records. In addition to estimating (co)variance components for multiple-trait models with partially missing data, this paper shows how the multiple-trait derivative-free REML set of programs can also estimate SE by augmenting the data file when not all animals have all traits measured. Although the standard practice has been to eliminate records with partially missing data, that practice uses only a subset of the available data. In some situations, the elimination of partial records can result in elimination of all the records, such as one trait measured in one environment and a second trait measured in a different environment. An alternative approach requiring minor modifications of the original data and model was developed that provides estimates of the SE using an augmented data set that gives the same residual log likelihood as the original data for multiple-trait analyses when not all traits are measured. Because the same residual vector is used for the original data and the augmented data, the resulting REML estimators along with their sampling properties are identical for the original and augmented data, so that SE for estimates of genetic parameters can be calculated.     

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.     

Confirmation of quantitative trait loci using a low-density single nucleotide polymorphism map for twinning and ovulation rate on bovine chromosome 5

Traditional genetic selection in cattle for traits with low heritability, such as reproduction, has had very little success. With the addition of DNA technologies to the genetic selection toolbox for livestock, the opportunity may exist to improve reproductive efficiency more rapidly in cattle. The US Meat Animal Research Center Production Efficiency Population has 9,186 twinning and 29,571 ovulation rate records for multiple generations of animals, but a significant number of these animals do not have tissue samples available for DNA genotyping. The objectives of this study were to confirm QTL for twinning and ovulation rate previously found on BTA5 and to evaluate the ability of GenoProb to predict genotypic information in a pedigree containing 16,035 animals when using genotypes for 24 SNP from 3 data sets containing 48, 724, or 2,900 animals. Marker data for 21 microsatellites on BTA5 with 297 to 3,395 animals per marker were used in conjunction with each data set of genotyped animals. Genotypic probabilities for females were used to calculate independent variables for regressions of additive, dominance, and imprinting effects. Genotypic regressions were fitted as fixed effects in a 2-trait mixed model analysis by using multiple-trait derivative-free REML. Each SNP was analyzed individually, followed by backward selection fitting all individually significant SNP simultaneously and then removing the least significant SNP until only significant SNP were left. Five significant SNP associations were detected for twinning rate and 3 were detected for ovulation rate. Two of these SNP, 1 for each trait, were significant for imprinting. Additional modeling of paternal and maternal allelic effects confirmed the initial results of imprinting done by contrasting heterozygotes. These results are supported by comparative mapping of mouse and human imprinted genes to this region of bovine chromosome 5.     

Estimation of genetic parameters and effects of cytoplasmic line on scrotal circumference and semen quality traits in Angus bulls

The purpose of this study was to estimate the heritability of scrotal circumference (SC) and semen traits, genetic correlations between SC and semen quality traits, and the effect of cytoplasmic line on SC and semen traits. Breeding soundness exam (BSE) data were collected on registered Angus bulls at 4 ranches over 7 yr. The American Angus Association provided historical pedigree information to estimate the effect of cytoplasmic line on SC and semen quality traits. After editing, the evaluated data set contained 1,281 bulls with breeding soundness exam data that traced back to 100 founder dams. Data were analyzed using a 2-trait animal model to obtain heritability, genetic correlation between SC and semen quality traits, as well as the effect of cytoplasmic line as a random effect for SC, percent motility (MOT), percent primary abnormalities (PRIM), percent secondary abnormalities (SEC), and percent total abnormalities (TOT) using multiple-trait derivative-free REML. Fixed effects included source ranch and collection year, and test age was used as a covariate. Estimates of heritability for SC, MOT, PRIM, SEC, and TOT were 0.46, 0.05, 0.27, 0.23, and 0.25, respectively. Genetic correlations between SC and MOT, PRIM, SEC, and TOT were 0.36, -0.19, -0.11, and -0.23, respectively. The proportions of phenotypic variance accounted for by cytoplasmic line for SC, MOT, PRIM, SEC, and TOT were <0.001, 0.013, 0.023, 0.002, and <0.001, respectively. Genetic correlations between SC and semen quality traits were low to moderate and favorable. Cytoplasmic line may have a marginal effect on MOT and PRIM, but is likely not a significant source of variation for SC, SEC, or TOT.     

Multiple-trait restricted maximum likelihood for simulated measures of ovulation rate with underlying multivariate normal distributions.

A data set that was used to estimate covariance components with REML for an animal model with eight measures of ovulation rate treated as separate traits was used as a template to simulate data sets of eight multivariate normal traits that were then truncated to binomial traits. The model for simulation included eight measures on 610 animals with 1,071 animals in the numerator relationship matrix. Heritabilities were equal for the eight measures, and both genetic and phenotypic correlations among the measures were equal. Ten replications for each combination of heritability (.15, .25, and .35) and genetic correlation (.50, .66, and .90) were simulated on the normal scale. For each replicate, estimates of the eight heritabilities and 28 genetic correlations were obtained by multiple-trait REML. The usual transformation of heritability estimated on the binomial scale overestimated heritability on the normal scale. Genetic correlations on the binomial scale seriously underestimated the correlations on the normal scale. Standard errors of the estimates obtained by replication were somewhat larger than the approximate SE from REMLPK (the multi-trait REML program of K. Meyer). A final set of 10 simulated replications with heritability of .25 and genetic correlation of 1.00 resulted in average estimates of .18 for heritability and of .66 for genetic correlation that agree closely with those from the analysis of measures of ovulation at eight estrous cycles used as a template; averages for heritability of .16 and for genetic correlation of .66 were obtained.     

Spirometric performance in Belgian Blue calves: II. Analysis of environmental factors and estimation of genetic parameters

Genetic parameters and environmental effects for spirometric variables (SV) in calves were estimated using 734 Belgian Blue calves (15 to 297 d of age), sired by 20 AI bulls. For each calf, the following SV were measured: 1) the average ventilation (l/min) recorded during the 15 s of maximal ventilatory changes induced by lobeline administration (0.25 mg/kg, i.v.) (15-s MV(L)); 2) the vital capacity, and the maximal peak expiratory and inspiratory flows recorded after lobeline administration; and 3) the ventilatory reserve (15-s MV(L) - ventilation at rest). Analysis of environmental factors showed age of calf, herd, sex, and vaccination status had significant effects on SV. A sire model and a multiple-trait derivative-free REML procedure were used to estimate genetic parameters for SV, body weight, and muscling score. Heritabilities for SV ranged from 0.28 +/- 0.11 to 0.44 +/- 0.16. Genetic correlations among SV varied from 0.76 to 0.98 and environmental correlations from 0.69 to 0.80. Genetic correlations of SV with body weight (0.25 to 0.56) and with muscling score (0.21 to 0.76) were positive, as were environmental correlations of SV with body weight (0.44 to 0.70) and muscling score (0.09 to 0.25). These results suggest that selection may improve SV without impairing other traits of economic importance.     

Genetic parameters and relationships between hip height and weight in Brahman cattle

Genetic parameters for weaning hip height (WHH), weaning weight (WWT), postweaning hip height growth (PHG), and hip height at 18 mo of age (HH18) and their relationships were estimated for Brahman cattle born from 1984 to 1994 at the Subtropical Agricultural Research Station, Brooksville, FL. Records per trait were 889 WHH, 892 WWT, and 684 HH18. (Co)variances were estimated using REML with a derivative-free algorithm and fitting three two-trait animal models (i.e., WHH-WWT, WHH-PHG, and WWT-HH18). Heritability estimates of WHH direct effects were 0.73 and 0.65 for models WHH-WWT and WHH-PHG and were 0.29 and 0.33 for WWT direct for models WHH-WWT and WWT-HH18, respectively. Estimates of heritability for PHG and HH18 direct were 0.13 and 0.87, respectively. Heritability estimates for maternal effects were 0.10 and 0.09 for WHH for models WHH-WWT and WHH-PHG and 0.18 and 0.18 for WWT for models WHH-WWT and WWT-HH18, respectively. Heritability estimates for PHG and HH18 maternal were 0.00 and 0.03. Estimates of the genetic correlation between direct effects for the different traits were moderate and positive; they were also positive between WHH and WWT maternal and WWT and HH18 maternal but negative (-0.19) between WHH and PHG maternal, which may indicate the existence of compensatory growth. Negative genetic correlations existed between direct and maternal effects for WHH, WWT, PHG, and HH18. The correlation between direct and WWT maternal effects was low and negative, moderate and negative between WHH direct and PHG maternal, and high and negative (-0.80) between WWT direct and HH18 maternal. There is a strong genetic relationship between hip height and weight at weaning that also affects hip height at 18 mo of age. Both product-moment and rank correlations between estimated breeding values (EBV) for direct values indicate that almost all of the same animals would be selected for PHG EBV if the selection criterion used was WHH EBV, and that it is possible to accomplish a preliminary selection for HH18 EBV using WHH EBV. Correlations between breeding values for WHH, WWT, and HH18 indicate that it will be possible to identify animals that will reduce, maintain, or increase hip height while weaning weight is increased. Thus, if the breeding objective is to manipulate growth to 18 mo of age, implementation of multiple-trait breeding programs considering hip height and weight at weaning will help to predict hip height at 18 mo of age.     

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.     

应用多性状动物模型及DFREML法对罗曼褐壳蛋鸡纯系的遗传分析

采用多性状动物模型BLUP和非求导约束最大似然法（MTDFREML）估计了引进罗曼蛋鸡纯系配套组合中4个品系9个性状的方差组分和遗传参数,并估计了固定效应的BLUE值和动物个体的加性遗传效应值（育种值）,分析了测定性状的世代间遗传趋势和表型趋势。结果表明,4个系各性状的遗传力基本一致,体重、蛋重以及不同周期的产蛋量之间存在显著的正相关。蛋重与产蛋量存在较强的遗传负相关。六个世代选育后,各品系蛋重、产蛋量的平均育种值均有提高,表明对产蛋量和蛋重的选择是有效的。     

Association of a genetic marker with blood serum insulin-like growth factor-I concentration and growth traits in Angus cattle

The objective of this research was to evaluate a biallelic genetic marker identified in the first promoter region of the bovine IGF-I gene. The point mutation was identified as a T-to-C transition by sequencing the polymorphic fragments. A PCR-RFLP procedure was developed for determining the marker genotypes. Marker genotypes were determined for 760 Angus calves from divergent lines that were created by selection for high or low serum IGF-I concentration (allele A: 63.9%, B: 36.1%). Data were analyzed using the multiple-trait derivative-free restricted maximum likelihood computer programs with animal models. The full animal model included fixed effects of marker genotype, birth year, season of birth, sex, age of dam, and selection line; random effects of animal, maternal genetic, and maternal permanent environmental effects; and a covariate for age of calf. Traits analyzed included blood serum IGF-I concentrations on d 28, 42, and 56 of the postweaning test, mean IGF-I concentration, birth weight, weaning weight, on-test weight, off-test weight, off-test hip height, postweaning gain, and weight gain during the 20-d period immediately after weaning. Results from the analysis across selection lines showed a significant association of the BB genotype with higher weight gain during the first 20 d after weaning and a slight dominance effect of the marker on postweaning gain. Analysis within the low IGF-I line also showed a significant association of the BB genotype with higher weight gain during the first 20 d after weaning and with on-test weight, although analysis within the high IGF-I line did not show any significant association. The associated effects of the marker need to be verified in other cattle populations.     

Divergent selection for immune responsiveness in chickens: estimation of realized heritability with an animal model.

With the aim of improving general disease resistance, chickens were divergently selected for their antibody titers 5 d after immunization with sheep red blood cells for nine generations. Selected and control lines differed significantly for primary and secondary responses after three generations. Heritability of the antibody titer was estimated by REML fitting an animal model using a derivative-free algorithm. The heritability estimate using data on all lines simultaneously was .31. Realized heritability of the antibody titer in the selected lines was estimated by using either the phenotypic cumulative response as the deviation from the control line or the mean breeding values obtained with an animal model. Values from the two methods were consistent, giving a realized heritability of .21 and .25 in the high and low lines, respectively. The genetic trend was not linear and the response to selection tended to accelerate over generations.     

Estimation of variance components including competitive effects of Large White growing gilts

Records of on-test ADG of Large White gilts were analyzed to estimate variance components of direct and associative genetic effects. Models included the effects of contemporary group (farm-barn-batch), birth litter, pen group, and direct and associative additive genetic effects. The area of each pen was 14 m2. The additive genetic variance was a function of the number of competitors in a group, the additive relationships between the animal performing the record and its pen mates, and the additive relationships between pen mates. To partially account for differences in the number of pen mates, a covariable (qi = 1, 1/n, or 1/n(1/2)) was added to the associative genetic effect. There were 4,946 records from 2,409 litters and 362 pen groups. Pen group size ranged from 12 to 16 gilts. Analyses by REML converged very slowly. A grid search showed that the likelihood function was almost flat when the additive genetic associative effect was fitted. Estimates of direct and associative heritability were 0.15 and 0.03, respectively. Within the BLUPF90 family of programs, the mixed-model equations can be set up directly. For variance component estimation, simple programs (REMLF90 and GIBBSF90) worked without modifications, but more optimized programs did not. Estimates obtained using the three values of qi were similar. With the data structure available for this study and under an environment with relative low competition among animals, accurate estimation of associative genetic effects was not possible. Estimation of competitive effects with large pen size is difficult. The magnitude of competition effects may be larger in commercial populations, where housing is denser and food is limited.     

Evaluation of strategies for selection for lean growth rate in pigs

Lean growth rate (LGR) in pigs is a nonlinear biological function of growth rate and lean quantity. According to animal breeding theory, genetic progress for LGR is maximized with selection on a linear index of its component traits, but selection on direct EBV for LGR is also common. In this study, the performance of five criteria for selection on estimated LGR in pigs was evaluated through simulation over five generations: linear indexes of multiple-trait EBV of component traits with or without updating index weights in each generation; a nonlinear index of multiple-trait EBV of component traits; and direct selection on EBV for LGR from a single-trait model or a multiple-trait model that included LGR and component traits. The nonlinear index yielded the highest response in LGR in Generation 5, but the linear index with updating performed almost as well. Not updating weights for the linear index reduced response in LGR by 1.1% in Generation 5 (P < 0.05). Direct selection on single-trait EBV for LGR yielded the lowest responses in Generation 5. Direct selection on EBV for LGR from a multiple-trait animal model yielded a 3.1% greater response in LGR in Generation 5 than direct selection on EBV for LGR based on a single-trait animal model (P < 0.05), but yielded a 1.9% lower response than the nonlinear index. Although differences in response in LGR were limited, alternative selection criteria resulted in substantially different responses in component traits. Linear index selection for LGR placed more emphasis on lean quantity, whereas direct selection for LGR emphasized growth rate. Based on the relative changes in the responses in LGR, selection for estimated LGR based on a nonlinear index or a linear index with updating is recommended for use in the swine industry.     

Genetic parameter estimates for serum insulin-like growth factor-I concentration and carcass traits in Angus beef cattle

Divergent selection for serum insulin-like growth factor-I (IGF-I) concentration began at the Eastern Ohio Resource Development Center (EORDC) in 1989 using 100 spring-calving (50 high line and 50 low line) and 100 fall-calving (50 high line and 50 low line) purebred Angus cows. Following weaning, bull and heifer calves were fed in drylot for a 140-d postweaning period. At the conclusion of the postweaning test, bulls not selected for breeding were slaughtered and carcass data were collected at a commercial abbatoir. At the time of this analysis, IGF-I measurements were available for 1,283 bull and heifer calves, and carcass data were available for 452 bulls. A set of multiple-trait, derivative-free, restricted maximum likelihood (MTDFREML) computer programs were used for data analysis. Estimates of direct heritability for IGF-I concentration at d 28, 42, and 56 of the postweaning period, and for mean IGF-I concentration were .32, .59, .31, and .42, respectively. Direct heritabilities for carcass traits ranged from .27 to 1.0, .26 to 1.0, and .23 to 1.0 when the age-, fat-, and weight-constant end points, respectively, were used, with marbling score having the smallest heritability and longissimus muscle area having the highest heritability in each case. Maternal heritability and the proportion of phenotypic variance due to permanent environmental effect of dam generally were < or = .21 for IGF-I concentrations and for carcass traits other than longissimus muscle area. Additive genetic correlations of IGF-I concentrations with backfat thickness, longissimus muscle area, hot carcass weight, marbling score, quality grade, and yield grade averaged -.26, .19, -.04, -.53, -.45, and -.27, respectively, when carcass data were adjusted to an age-constant end point. Bulls with lower IGF-I concentrations had higher marbling scores and quality grades, but also had higher backfat thickness and yield grades regardless of the slaughter end point. Serum IGF-I concentration may be a useful selection criterion when efforts are directed toward improvement of marbling scores and quality grades of beef cattle.     

Use of the preconditioned conjugate gradient algorithm as a generic solver for mixed-model equations in animal breeding applications

Utility of the preconditioned conjugate gradient algorithm with a diagonal preconditioner for solving mixed-model equations in animal breeding applications was evaluated with 16 test problems. The problems included single- and multiple-trait analyses, with data on beef, dairy, and swine ranging from small examples to national data sets. Multiple-trait models considered low and high genetic correlations. Convergence was based on relative differences between left- and right-hand sides. The ordering of equations was fixed effects followed by random effects, with no special ordering within random effects. The preconditioned conjugate gradient program implemented with double precision converged for all models. However, when implemented in single precision, the preconditioned conjugate gradient algorithm did not converge for seven large models. The preconditioned conjugate gradient and successive overrelaxation algorithms were subsequently compared for 13 of the test problems. The preconditioned conjugate gradient algorithm was easy to implement with the iteration on data for general models. However, successive overrelaxation requires specific programming for each set of models. On average, the preconditioned conjugate gradient algorithm converged in three times fewer rounds of iteration than successive overrelaxation. With straightforward implementations, programs using the preconditioned conjugate gradient algorithm may be two or more times faster than those using successive overrelaxation. However, programs using the preconditioned conjugate gradient algorithm would use more memory than would comparable implementations using successive overrelaxation. Extensive optimization of either algorithm can influence rankings. The preconditioned conjugate gradient implemented with iteration on data, a diagonal preconditioner, and in double precision may be the algorithm of choice for solving mixed-model equations when sufficient memory is available and ease of implementation is essential.     

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.     

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.     

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.     

Evaluation and interpretation of results for three cross-over designs.

In cross-over designs, individual sequences of treatments are applied to the animals. Within such designs it is possible that every treatment could modify the effect of the subsequent treatment applied to the same animal. We compared three cross-over designs each with three treatments, three periods, and two blocks. This comparison was done with respect to the variance of the estimations of the effects and its biases caused by the interactions between the treatment and the carry over effect of the foregoing treatment. Moreover, different methods of estimating variance components and calculating the degrees of freedom were compared by means of simulation. If the animal variance component is small, then the bias of the REML estimator of the variance components is greater than one of the widespread ANOVA-estimator called 'TYPE3'. But nevertheless, the mean squared error of this estimation is smaller in the case of REML in comparison to ANOVA. Therefore, the REML method should be preferred. For calculating the degrees of freedom, the Kenward-Roger method should be used. After applying this method, the true significance level is almost equal to its required value, but if the Satterthwaite method is used, the true significance level will be too high. If the interaction (treatment x carry over) is ignored in the model although it exists, the standard error of the treatment effect estimation is too great, and, therefore, the true significance level is too small. The methods which have been evaluated are available in the SAS-procedure MIXED (SAS Institute, 1999a). To assist the investigation of cross-over designs by using this software, we developed programs for data management and data analysis. These programs are available from the first author.     

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.