Technical note: Use of marker-based relationships with multiple-trait derivative-free restricted maximal likelihood

The widespread use of the set of multiple-trait derivative-free REML programs for prediction of breeding values and estimation of variance components has led to significant improvement in traits of economic importance. The initial version of this software package, however, was generally limited to pedigree-based relationships. With continued advances in genomic research and the increased availability of genotyping, relationships based on molecular markers are obtainable and desirable. The addition of a new program to the set of multiple-trait derivative-free REML programs is described that allows users the flexibility to calculate relationships using standard pedigree files or an arbitrary relationship matrix based on genetic marker information. The strategy behind this modification and its design is described. An application is illustrated in a QTL association study for canine hip dysplasia.     

Two methods for parameter estimation using multiple-trait models and beef cattle field data

Two methods are presented for estimating variances and covariances from beef cattle field data using multiple-trait sire models. Both methods require that the first trait have no missing records and that the contemporary groups for the second trait be subsets of the contemporary groups for the first trait; however, the second trait may have missing records. One method uses pseudo expectations involving quadratics composed of the solutions and the right-hand sides of the mixed model equations. The other method is an extension of Henderson's Simple Method to the multiple trait case. Neither of these methods requires any inversions of large matrices in the computation of the parameters; therefore, both methods can handle very large sets of data. Four simulated data sets were generated to evaluate the methods. In general, both methods estimated genetic correlations and heritabilities that were close to the Restricted Maximum Likelihood estimates and the true data set values, even when selection within contemporary groups was practiced. The estimates of residual correlations by both methods, however, were biased by selection. These two methods can be useful in estimating variances and covariances from multiple-trait models in large populations that have undergone a minimal amount of selection within contemporary groups.     

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.     

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.     

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.     

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.     

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.     

Modeling final scores in US Holsteins as a function of year of classification using a random regression model

Conformation final scores in Holsteins were used to assess genetic changes over the years due to various factors such as selection and changes in trait definition. The model included management group, age group, and stage of lactation as fixed effects; additive genetic effects with random regressions on year of classification using Legendre polynomials with order from linear to cubic; and residual effects assuming heterogeneous variances. Two sets of simulated data were used to test the feasibility of variance component estimation in situations where the definition of the trait of interest changes continuously over time. Estimated variances from the simulated data sets were unbiased. Empirical tests involved 30,041 records of cows with single records scored in 1981-1999. Heritability estimates and additive genetic variances from field data decreased while residual variances increased over time. Differences among estimates of variance components from linear, quadratic and cubic random-regression models were small. Genetic correlations among final scores at years of classification estimated with the multiple-trait model that treated different groups of years as separate traits and with linear, quadratic and cubic random-regression models decreased from 1.0 to a minimum of 0.91, as the distance between the years increased. Although there were no significant differences among estimates of variance components from random-regression models, genetic correlations between different years estimated with higher order random-regression models were closer to those with the multiple trait model that treated different group of years as separate traits. Genetic changes in a trait over time can be studied with a random-regression model.     

Parameter estimation for carcass traits including growth information of Simmental beef cattle using restricted maximum likelihood with a multiple-trait model.

(Co)variance component estimates were computed for retail cuts per day of age (kilograms per day), cutability (percentage of carcass weight), and marbling score (1 through 11) using a multiple-trait sire model. Restricted maximum likelihood estimates of (co)variance components were obtained via an expectation-maximization algorithm. Carcass data consisted of 8,265 progeny records collected by U.S. Simmental producers. Growth trait information (birth weight, weaning weight, and[or] postweaning gain) for those progeny with carcass data and an additional 5,405 contemporaries formed the complete data set for analysis. A total of 420 sires were represented. Three models differing in number of traits were investigated: 1) carcass traits with growth traits, 2) carcass traits only, and 3) single trait. The final models did not include postweaning gain because of convergence problems. Parameter estimates for all three models were essentially the same. Heritability estimates were .30, .18, and .23 for retail cuts per day, cutability, and marbling score, respectively. Correlations between growth and carcass traits were low except for those with retail cuts per day, which were moderate and positive. The additional information gained by adding growth traits to the carcass-traits-only evaluation lowered prediction error variances most for retail cuts per day. Little change in prediction error variances was found for cutability and marbling score. Inclusion of growth traits in future sire evaluations for carcass traits will benefit the evaluation of retail cuts per day but have considerably less effect on cutability and marbling score.     

Parameter estimates for number of lambs born at different ages and for 18-month body weight of Rambouillet sheep

Genetic parameters were estimated using REML with animal models for number of lambs born and 18-mo body weight in Rambouillet sheep. Number of lambs born was modeled either as repeated measurements on the same trait or as different traits at different ages. The original data for number of lambs born were separated according to age of the ewe into two classes: 2 and 3 yr, and older than 3 yr. Numbers of ewes with lambing records for the age classes were 653 and 466 with 1,106 and 1,118 records, respectively. For the data set that included all ages, the number of ewes was 684 with 2,224 records, and for 18-mo body weight the number of ewes measured was 557. For number of lambs born, the animal model included random genetic, permanent environmental, and residual environmental effects and fixed effects for age of ewe, year of lambing, and month of year of lambing. Lambing day within season was used as a covariate. For 18-mo body weight, year of birth of ewe was used as a fixed effect. Actual age in days when the ewe was weighed was used as a covariate. Estimates of heritability for number of lambs born by age group were .04, for 2- and 3-yr old ewes, and .06, for ewes greater than 3 yr old, from the two-trait (two age of ewe classes) analyses and .06 when all ages were included. Estimates of heritability for number of lambs born from the single-trait analyses were somewhat less than estimates from two-trait analyses. Estimate of genetic correlation between number of lambs born for the 2 and 3 yr and the >3 yr classes was near unity (1.00), which suggests that a repeated measures model for number of lambs born is adequate for making selection decisions. Estimate of genetic correlation between number of lambs born and 18-mo body weight was .35 with a heritability estimate of .48 for 18-mo body weight. The estimate of genetic correlation suggests that selection for increased number of lambs born would result in increased 18-mo body weight.     

"Quasi-REML" correlation estimates between production and health traits in the presence of selection and confounding: a simulation study

Performance of the "quasi-REML" method for estimating correlations between a continuous trait and a categorical trait, and between two categorical traits, was studied with Monte Carlo simulations. Three continuous, correlated traits were simulated for identical populations and three scenarios with either no selection, selection for one moderately heritable trait (Trait 1, h2 = .25), and selection for the same trait plus confounding between sires and management groups. The "true" environmental correlations between Traits 2 (h2 = .10) and 3 (h2 = .05) were always of the same absolute size (.20), but further data scenarios were generated by setting the sign of environmental correlation to either positive or negative. Observations for Traits 2 and 3 were then reassigned to binomial categories to simulate health or reproductive traits with incidences of 15 and 5%, respectively. Genetic correlations (r(g12), r(g13), and r(g23) and environmental correlations (r(e12), r(e13), and r(e23)) were estimated for the underlying continuous scale (REML) and the visible categorical scales ("quasi-REML") with linear multiple-trait sire and animal models. Contrary to theory, practically all "quasi-REML" genetic correlations were underestimated to some extent with the sire and animal models. Selection inflated this negative bias for sire model estimates, and the sign of r(e23) noticeably affected r(g23) estimates for the animal model, with greater bias and SD for estimates when the "true" r(e23) was positive. Transformed "quasi-REML" environmental correlations between a continuous and a categorical trait were estimated with good efficiency and little bias, and corresponding correlations between two categorical traits were systematically overestimated. Confounding between sires and contemporary groups negatively affected all correlation estimates on the underlying and the visible scales, especially for sire model "quasi-REML" estimates of genetic correlation. Selection, data structure, and the (co)variance structure influences how well correlations involving categorical traits are estimated with "quasi-REML" methods.     

Parameter estimates for greasy fleece weight of Rambouillet sheep at different ages

Variance components for greasy fleece weight in Rambouillet sheep were estimated. Greasy fleece weight was modeled either as repeated measurements on the same trait or as different traits at different ages. The original data were separated according to the age of the ewe at shearing into three classes; 1 yr, 2 and 3 yr, and older than 3 yr. An animal model was used to obtain estimates of genetic parameters with a REML algorithm. Total numbers of animals in pedigrees for the different age classes were 696, 729, and 573, respectively, and 822 for the repeated measures model across ages. The animal model included direct genetic, permanent environmental, and residual environmental random effects and fixed effects for age of ewe, shearing date as contemporary group, and number of lambs born. Days between shearings was used as a covariate. Single-trait analyses were initially done to obtain starting values for multiple-trait analyses. A repeated measures model across ages was also used. Estimates of heritability by age group were .42, .50, and .58 from three-trait (age class) analyses and for the repeated measures model the estimate was .57. Estimates of genetic correlations between fleece yields for 1 yr and 2 and 3 yr, 1 yr and >3 yr, and 2 and 3 yr and >3 yr classes were .88, .89, and .97, respectively. These estimates of genetic correlations suggest that a repeated measures model for greasy fleece weight is adequate for making selection decisions.     

Estimates of genetic parameters and genetic change for reproduction,weight, and wool characteristics of Columbia sheep

Genetic parameters from both single-trait and bivariate analyses for prolificacy, weight and wool traits were estimated using REML with animal models for Columbia sheep from data collected from 1950 to 1998 at the U.S. Sheep Experiment Station (USSES), Dubois, ID. Breeding values from both single-trait and seven-trait analyses calculated using the parameters estimated from the single-trait and bivariate analyses were compared with respect to genetic trends. Number of observations were 31,401 for litter size at birth and litter size at weaning, 24,741 for birth weight, 23,903 for weaning weight, 29,572 for fleece weight and fleece grade, and 2,449 for staple length. Direct heritability estimates from single-trait analyses were 0.09 for litter size at birth, 0.06 for litter size at weaning, 0.27 for birth weight, 0.16 for weaning weight, 0.53 for fleece weight, 0.41 for fleece grade, and 0.55 for staple length. Estimate of direct genetic correlation between littersize at birth and weaning was 0.84 and between birth and weaning weights was 0.56. Estimate of genetic correlation between fleece weight and staple length was positive (0.55) but negative between fleece weight and fleece grade (-0.47) and between staple length and fleece grade (-0.70). Estimates of genetic correlations were positive but small between birth weight and litter size traits and moderate and positive between weaning weight and litter size traits. Fleece weight was lowly and negatively correlated with both litter size traits. Fleece grade was lowly and positively correlated with both litter size traits, while staple length was lowly and negatively correlated with the litter size traits. Estimates of correlations between weight traits and fleece weight were positive and low to moderate. Estimates of correlations between weight traits and fleece grade were negative and small. Estimates of correlations between staple length and birth weight (0.05) and weaning weight were small (-0.04). Estimated breeding values averaged by year of birth from both the single-trait and multiple-trait analyses for the prolificacy and weight traits increased over time, but were unchanged for the wool traits. Estimated changes in breeding values over time did not differ substantially for single-trait and multiple-trait analyses, except for traits highly correlated with another trait that was responding to selection.     

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.     

Phenotypic and genetic parameter estimates for racing traits of Arabian horses in Turkey

The racing records for Arabian horses used in the study were obtained from the Turkish Jockey Club. The traits used in the study were racing time, best racing time, rank, annual earnings, earnings per start, log annual earnings and log earnings per start. Genetic parameters were estimated by the restricted maximum likelihood (REML) procedure using the DFREML program. The effects of age, sex and origin of horse were significant for each trait. The effect of year was significant on time and earning traits, but not rank. The effect of month on time traits was also significant. Heritability estimates of the entire data set were 0.280, 0.281, 0.069, 0.139, 0.174, 0.152 and 0.171 for racing time, best racing time, rank, annual earnings, earnings per start, log annual earnings and log earnings per start respectively. Estimates of repeatability varied from 0.349 to 0.500 for racing time, from 0.430 to 0.524 for best racing time and from 0.129 to 0.171 for rank depending on the data set used in the analyses. Best racing time was the most appropriate trait for selection in this study, as this might lead to genetic improvement than other traits.     

Genetics of litter size in three maternal lines of rabbits: repeatability versus multiple-trait models

Variance components were estimated in 3 lines of rabbits selected for litter size at weaning (A, Prat, and V) to test one of the assumptions of the models used for selection: that litter size data at different parities are repeated measurements of the same trait. Multiple-trait analyses were performed for the total number of kits born (TB), the number of kits born alive (BA), and the number of kits weaned (NW) per litter. Estimates were obtained by REML in multivariate analyses, including all of the information of the selection criteria, under a repeatability model or a multiple-trait model, considering litter size at the first, second, and subsequent parities as different traits. Models included the fixed effects of the physiological status of the female and the year-season of mating day, buck and doe random permanent environmental effects, and doe additive genetic effects. Results indicated that prolificacy was determined mainly by doe components and that the service sire had a very small effect. Heritabilities for the first and second parities were greater than the estimates obtained under the repeatability model (0.04 to 0.14 for the repeatability model). In the A and V lines, similar values of heritability were found at the first and second parities, but in the Prat line heritability at the second parity was always greater than at the first and greater parities (values of 0.21, 0.17, and 0.15 for TB, BA, and NW, respectively, in second parities of the Prat line). Genetic correlations between the same traits at different parities were approximately 0.8 for all traits in line A, but much lower in the other 2 lines. On average, the values were 0.64 for TB, 0.48 for BA, and 0.39 for NW between the first and second parities, and 0.65 for TB, 0.56 for BA, and 0.45 for NW between the first and third and greater parities. Genetic correlations between the second and greater parities showed the greatest values (approximately 0.8) in lines A and Prat for all traits, but they were lower in line V (0.63 for BA and 0.37 for NW). The heterogeneity of heritabilities and genetic correlations between parities lower than 0.9 suggests that litter size at different parities could be considered as different traits when genetic evaluations are performed. However, when the accuracies of predicted breeding values under a multiple-trait and a repeatability model were calculated, assuming the first to be the true model, the values obtained were nearly the same for all traits in all lines.     

Multiple trait genetic evaluation of ewe traits in Icelandic sheep

The prolificacy of the ewes was measured as the number of lambs born per ewe mated (NLB) when the ewes were 1–4 years of age. The ewe productivity related to the same age interval was measured by special ewe production indices (EPI). The genetic parameters for these traits were estimated by a series of bivariate REML analyses using animal models. The material used for the genetic analysis contained records on 193 213 ewes. The heritability estimates for NLB were h² = 0.17, 0.13, 0.11, 0.10 for the four respective age classes. Corresponding estimates for EPI were h² = 0.16, 0.17, 0.17, 0.15. The genetic correlations among NLB at different ages ranged from 0.63 to 0.98 and among EPI from 0.82 to 0.99. The genetic correlations between NLB and EPI were generally low. The material used for estimating the breeding values by the MT‐BLUP Animal Model consisted of 1.5 million individuals in the pedigree file. In total 815 782 ewes had records for the NLB and 763 491 ewes had production index (at least 1 year). The records were registered in the years 1990–2006. All possible missing patterns were present in the data. In the iteration process expected values for missing traits were generated and solutions were obtained on canonical transformed scale. The genetic evaluations were run independently for NLB and EPI for computational convenience given the correlations between these traits were negligible.     

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.     

Simultaneous estimation of variances and covariances using REML and Henderson 3 in a selected population of white leghorns

1. Genetic and residual variances and covariances were estimated on performance data from 5943 laying hens from a 7 generation selection experiment for the traits: egg number up to day 270 (EN270), egg weight (EW), body weight at day 215 (BW), egg mass 100 g of food (EMFC), and residual food consumption (RFC) by a Henderson 3 and REML procedure.

2. Simultaneous REML estimates of all 30 components were obtained by a software package is based on numerical optimisation of the log likelihood using a multivariate animal model. Henderson 3 estimates were computed on the basis of a hierarchical sire‐dam model. Estimates were generated beginning with a data set comprising only the first generation, and then successively adding one generation after the other.

3. REML estimates for heritabilities h ² on the basis of all performance records were 0.40, 0.75, 0.62, 0.21 and 0.22 for traits EN270, EW, BW, EMFC, and RFC, respectively. The corresponding Henderson 3 estimates were: 0.30, 0.57, 0.43, 0.21, and 0.20.

4. The results indicate that some REML h ² estimates are substantially different from those obtained by Henderson 3 once the data set included three generations as opposed to those based on Henderson 3.     

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。