Efficient Monte Carlo algorithm for restricted maximum likelihood estimation of genetic parameters

Monte Carlo (MC) methods have been found useful in estimation of variance parameters for large data and complex models with many variance components (VC), with respect to both computer memory and computing time. A disadvantage has been a fluctuation in round‐to‐round values of estimates that makes the estimation of convergence challenging. Furthermore, with Newton‐type algorithms, the approximate Hessian matrix might have sufficient accuracy, but the inaccuracy in the gradient vector exaggerates the round‐to‐round fluctuation to intolerable. In this study, the reuse of the same random numbers within each MC sample was used to remove the MC fluctuation. Simulated data with six VC parameters were analysed by four different MC REML methods: expectation‐maximization (EM), Newton–Raphson (NR), average information (AI) and Broyden's method (BM). In addition, field data with 96 VC parameters were analysed by MC EM REML. In all the analyses with reused samples, the MC fluctuations disappeared, but the final estimates by the MC REML methods differed from the analytically calculated values more than expected especially when the number of MC samples was small. The difference depended on the random numbers generated, and based on repeated MC AI REML analyses, the VC estimates were on average non‐biased. The advantage of reusing MC samples is more apparent in the NR‐type algorithms. Smooth convergence opens the possibility to use the fast converging Newton‐type algorithms. However, a disadvantage from reusing MC samples is a possible “bias” in the estimates. To attain acceptable accuracy, sufficient number of MC samples need to be generated.     

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.     

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.     

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.     

Estimation of genetic correlations between racing times recorded at different racing distances by restricted maximum likelihood in Thoroughbred racehorses

SUMMARY: Genetic correlations between racing times on track type (turf and dirt), and at racing distances on turf (1200 m, 1400 m, 1600 m, 1800 m, and/or 2000 m) and dirt (1000 m, 1200 m, 1400 m, 1600 m, 1700 m, and/or 1800 m) tracks, were estimated in Thoroughbred horses. (Co)variance components were estimated using multiple-trait derivative-free restricted maximum likelihood (MTDFREML). The data used were collected by the Japan Racing Association from 1992 to 1993. The generation 2 pedigree information was preferable for (co)variance estimates. The genetic correlations between racing times on turf and dirt tracks ranged from 0.69 to 0.31 (average 0.51). The genetic correlations between racing distances ranged from 0.68 to 1.00 (average 0.85) and from 0.53 to 1.00 (average 0.88) on turf and dirt tracks, respectively. These results suggest that the racing time per 100 m can be used for horse genetic evaluation within one track type. ZUSAMMENFASSUNG: Sch?tzung genetischer Korrelationen zwischen Rennzeiten von Vollblüternüber verschiednen Distanzen mittels restringierter Genetische Korrelationen zwischen Rennzeiten auf Rasen- und Erdbahnen, Renndistanzen auf Rasen- (1200, 1400, 1600, 1800 und 2000 m) und Erdbahnen (1000, 1200, 1400, 1600, 1700 und 2000 m) wurden für Vollblüter gesch?tzt. (Co)Varianzkomponenten wurden mittels Mehr-Merkmal Ableitungsfreier Restringierter Maximum Likelihood (MTDFREML) gesch?tzt. Die Unterlagen wurden von der Japanischen Renn Vereinigung 1992 und 1993 gesammelt. Generation 2 Abstammungsinformation war für die Co-Varianzsch?tzung günstig. Genetische Korrelationen zwischen Rennzeiten auf Rasen und auf Erdbahnen waren zwischen 0.69 und 0.31 (Durchschnitt 0.51), jene zwischen Distanzen zwischen 0.68 und 1.00 (Durchschnitt 0.85) und zwischen 0.53 und 1.00 (Durchschnitt 0.88) auf Rasen und Erdbahnen. Die Ergebnisse zeigen, da? Rennzeit per 100 m zur Bewertung der Pferde geeignet ist.     

Bias in heritability estimates from genomic restricted maximum likelihood methods under different genotyping strategies

We investigated the effects of different strategies for genotyping populations on variance components and heritabilities estimated with an animal model under restricted maximum likelihood (REML), genomic REML (GREML), and single‐step GREML (ssGREML). A population with 10 generations was simulated. Animals from the last one, two or three generations were genotyped with 45,116 SNP evenly distributed on 27 chromosomes. Animals to be genotyped were chosen randomly or based on EBV. Each scenario was replicated five times. A single trait was simulated with three heritability levels (low, moderate, high). Phenotypes were simulated for only females to mimic dairy sheep and also for both sexes to mimic meat sheep. Variance component estimates from genomic data and phenotypes for one or two generations were more biased than from three generations. Estimates in the scenario without selection were the most accurate across heritability levels and methods. When selection was present in the simulations, the best option was to use genotypes of randomly selected animals. For selective genotyping, heritabilities from GREML were more biased compared to those estimated by ssGREML, because ssGREML was less affected by selective or limited genotyping.     

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.     

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.     

Estimates of genetic parameters for kyphosis in two crossbred swine populations

Genetic parameters for degree of kyphosis were estimated from a Duroc-Landrace F(2) population (n = 316) and from a composite population (line C) composed of Duroc, Large White, and 2 sources of Landrace (n = 1,552). Live presentation did not indicate kyphosis in pigs or sows. Degree of kyphosis was measured by scoring the shape of the vertebral column of split carcasses on a scale from 0 (normal) to 3 (severe). Of the animals slaughtered, 75.6 and 68.9% were normal, 11.1 and 23.3% were mild, 11.1 and 6.2% were moderate, and 2.2 and 1.5% were severe in F(2) and line C, respectively. Fixed effects of age, sex, number of ribs, number of lumbar vertebrae, number of nipples, carcass length, and HCW were not significantly associated (P > 0.10) with kyphosis score when using linear models. Estimated heritabilities for kyphosis score were 0.30 and 0.32 in F(2) and line C, respectively, when using an animal model. Estimated genetic correlations between kyphosis score and number of ribs, number of lumbar vertebrae, number of nipples, carcass length, and HCW were 0.05, -0.13, 0.00, 0.05, and 0.03, respectively. Selection to decrease kyphosis should be effective and would not be expected to affect the number of ribs, lumbar vertebrae, nipples, or carcass length. In addition, selection for growth should not affect the incidence of kyphosis.     

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.     

An alternative derivation method of mixed model equations from best linear unbiased prediction (BLUP) and restricted BLUP of breeding values not using maximum likelihood

Two mixed model equations (MME) for best linear unbiased prediction (BLUP) of breeding values and for restricted BLUP of breeding values were derived by maximum likelihood from the joint normal probability distribution of the observations and breeding values. As a result, MME is actually more general than maximum likelihood because we can prove that each set of solutions of MME are identical to BLUP and restricted BLUP of breeding values and then it does not depend on normality. In the present study, the author shows deriving directly each MME from BLUP and restricted BLUP equations for breeding values without assuming the joint normal distribution of the data and random effects. However, if we cannot assume the multivariate normal density distribution of the estimated aggregate breeding value and each breeding value for selected traits, the response to selection by restricted BLUP may deviate from the expected values.     

Selection for placental efficiency in swine: genetic parameters and trends

The objectives of this study were to estimate response to divergent selection for an index of placental efficiency in swine, and to evaluate the effect of placental efficiency on litter size. The selection index (SI) included total born (TB), birth weight (BRWT), and placental weight (PW), and was designed to increase in the high line (H) or decrease in the low line (L) the efficiency of the placental function (PE), defined as the ratio BRWT:PW. (Co)variance components were estimated for direct and maternal additive effects by using an animal model with MTDFREML procedures. Estimated breeding values were calculated by using records on individual BRWT (n = 2,111), PW (n = 2,006), PE (n = 1,677), and SI (n = 1,677). Litter traits were evaluated using records on 193 litters. The model included the fixed effects of contemporary group for all traits, with the addition of sex for individual traits and parity for litter traits. Litter was fitted as an uncorrelated random effect for all traits, and TB was used as a linear and quadratic covariate for BRWT, PW, and PE. Direct heritability estimates from single-trait models were 0.03, 0.25, 0.18, 0.11, and 0.08 for BRWT, PW, PE, SI, and TB, respectively. Estimated breeding values were compared between lines by using a model including generation, line within generation, and replicate within line as the error term. Estimates of genetic divergence were 20.7 +/- 2.7 g, 0.24 +/- 0.03, 0.11 +/- 0.02, and 0.07 +/- 0.02 per generation for PW, PE, SI, and TB, respectively (P < 0.01), but divergence was not significant for BRWT. At Generation 4, direct EBV was higher in L than in H for PW (55.9 +/- 8.7 vs. -24.2 +/- 9.5 g, respectively; P < 0.01) and higher in H than in L for PE (0.58 +/- 0.10 vs. -0.35 +/- 0.09 g, respectively; P < 0.01). However, EBV was not different for BRWT, SI, or TB. These results indicate that PW and PE are susceptible to change by genetic selection; however, the correlated response in TB was an unexpected genetic trend toward a higher TB in L of 0.05 +/- 0.01 piglets per generation (P < 0.01).     

Selection response and genetic parameters for residual feed intake in Yorkshire swine

Residual feed intake (RFI) is a measure of feed efficiency defined as the difference between the observed feed intake and that predicted from the average requirements for growth and maintenance. The objective of this study was to evaluate the response in a selection experiment consisting of a line selected for low RFI and a random control line and to estimate the genetic parameters for RFI and related production and carcass traits. Beginning with random allocation of purebred Yorkshire littermates, in each generation, electronically measured ADFI, ADG, and ultrasound backfat (BF) were evaluated during a approximately 40- to approximately 115-kg of BW test period on approximately 90 boars from first parity and approximately 90 gilts from second parity sows of the low RFI line. After evaluation of first parity boars, approximately 12 boars and approximately 70 gilts from the low RFI line were selected to produce approximately 50 litters for the next generation. Approximately 30 control line litters were produced by random selection and mating. Selection was on EBV for RFI from an animal model analysis of ADFI, with on-test group and sex (fixed), pen within group and litter (random), and covariates for interactions of on- and off-test BW, on-test age, ADG, and BF with generations. The RFI explained 34% of phenotypic variation in ADFI. After 4 generations of selection, estimates of heritability for RFI, ADFI, ADG, feed efficiency (FE, which is the reciprocal of the feed conversion ratio and equals ADG/ ADFI), and ultrasound-predicted BF, LM area (LMA), and intramuscular fat (IMF) were 0.29, 0.51, 0.42, 0.17, 0.68, 0.57, and 0.28, respectively; predicted responses based on average EBV in the low RFI line were -114, -202, and -39 g/d for RFI (= 0.9 phenotypic SD), ADFI (0.9 SD), and ADG (0.4 SD), respectively, and 1.56% for FE (0.5 SD), -0.37 mm for BF (0.1 SD), 0.35 cm(2) for LMA (0.1 SD), and -0.10% for IMF (0.3 SD). Direct phenotypic comparison of the low RFI and control lines based on 92 low RFI and 76 control gilts from the second parity of generation 4 showed that selection had significantly decreased RFI by 96 g/d (P = 0.002) and ADFI by 165 g/d (P < 0.0001). The low RFI line also had 33 g/d lower ADG (P = 0.022), 1.36% greater FE (P = 0.09), and 1.99 mm less BF (P = 0.013). There was not a significant difference in LMA and other carcass traits, including subjective marbling score, despite a large observed difference in ultrasound-predicted IMF (-1.05% with P < 0.0001). In conclusion, RFI is a heritable trait, and selection for low RFI has significantly decreased the feed required for a given rate of growth and backfat.     

Estimates of genetic parameters for carcass measures of body composition and growth in swine

Records for pigs included in an experiment on reciprocal recurrent selection conducted from 1956 through 1971 at the USDA Beltsville Agriculture Research Center were analyzed to obtain estimates of heritabilities and genetic correlations and to derive prediction equations for estimating weight of lean cuts (WTLC) and percentage of lean cuts of shrunk slaughter weight (LCPC). Lean cuts growth rate (LCGR) was then estimated as WTLC/age of pig at slaughter. The base population consisted of two unrelated crossbred strains. A total of 1,294 records of F1 and F2 crossbred pigs were analyzed with one barrow and one gilt from each litter. Estimates of heritabilities and genetic correlations were computed with sire components of variance and covariance from a nested analysis of variance with an assumed model of years, strain-lines within years, sire within strain-lines, dams within sires and residual. Degrees of freedom were 307 for sires in strain-lines, 270 for dams in sires and 646 for residual. Heritability (h2) estimates were .42 +/- .13, .41 +/- .13 and .27 +/- .18 for WTLC, LCPC and LCGR, respectively, and .71 +/- .16, .38 +/- .13, .31 +/- .13 and .25 +/- .15 for carcass length, average backfat thickness, longissimus muscle area and ADG in BW, respectively. These estimates were apparently the first published genetic estimates involving LCGR based on carcass data. It was recommended that prediction equations to estimate WTLC, LCPC and LCGR for use in swine testing programs be derived from current meat-type pigs.     

Purebred-crossbred genetic parameters for reproductive traits in swine

For swine breeding programs, testing and selection programs are usually within purebred (PB) populations located in nucleus units that are generally managed differently and tend to have a higher health level than the commercial herds in which the crossbred (CB) descendants of these nucleus animals are expected to perform. This approach assumes that PB animals selected in the nucleus herd will have CB progeny that have superior performance at the commercial level. There is clear evidence that this may not be the case for all traits of economic importance and, thus, including data collected at the commercial herd level may increase the accuracy of selection for commercial CB performance at the nucleus level. The goal for this study was to estimate genetic parameters for five maternal reproductive traits between two PB maternal nucleus populations (Landrace and Yorkshire) and their CB offspring: Total Number Born (TNB), Number Born Alive (NBA), Number Born Alive > 1 kg (NBA > 1 kg), Total Number Weaned (TNW), and Litter Weight at Weaning (LWW). Estimates were based on single-step GBLUP by analyzing any two combinations of a PB and the CB population, and by analyzing all three populations jointly. The genomic relationship matrix between the three populations was generated by using within-population allele frequencies for relationships within a population, and across-population allele frequencies for relationships of the CB with the PB animals. Utilization of metafounders for the two PB populations had no effect on parameter estimates, so the two PB populations were assumed to be genetically unrelated. Joint analysis of two (one PB plus CB) vs. three (both PB and CB) populations did not impact estimates of heritability, additive genetic variance, and genetic correlations. Heritabilities were generally similar between the PB and CB populations, except for LWW and TNW, for which PB populations had about four times larger estimates than CB. Purebred-crossbred genetic correlations (rpc) were larger for Landrace than for Yorkshire, except for NBA > 1 kg. These estimates of rpc indicate that there is potential to improve selection of PB animals for CB performance by including CB information for all traits in the Yorkshire population, but that noticeable additional gains may only occur for NBA > 1 kg and TNW in the Landrace population.     

Estimation of genetic parameters of feed intake and daily gain in Durocs using data from electronic swine feeders

Genetic parameters for daily feed intake (DFI, g/day) and daily gain (DG, g/day) were estimated using records of 1916 Duroc boars from electronic feeder stations. Management was limited and resulted in varied ranges of age and weight on test. Boars were housed in 102 pens, each equipped with one feeder, and allowed ad libitum feeding. Weekly averages of DFI and DG were used due to large variation in daily records. Six traits were defined as DFI and DG during 85–106 (period 1), 107–128 (period 2) and 129–150 days of age (period 3). A six‐trait model included age as a linear and a quadratic covariate for DFI and a linear covariate for DG with a fixed effect of year–week–pen and random effects of litter, additive genetic animal and permanent environmental animal. Variance components were estimated by a Bayesian approach using Gibbs sampling algorithm. Estimates of heritability for respective periods were 18%, 12% and 10% for DFI and 21%, 11% and 10% for DG. Genetic correlations between DFI and DG in the same period were 0.70, 0.73 and 0.32 for the respective periods. DFI and DG obtained from automatic feeders can be analysed to reveal variation across testing periods by using weekly averages when many monthly averages are incomplete.     

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.     

Estimates of genetic parameters for direct and maternal effects on embryonic survival in swine.

Survival of 16,838 potential embryos was determined by counting corpora lutea and fetuses at 50 d of gestation for 1,081 litters by 225 sires. These data, coded as 1 or 0 depending on whether an ovulation was represented by a fetus, were used to estimate direct and maternal additive genetic variances and their covariance for embryonic survival. Data were from first-parity gilts of a Large White-Landrace composite population subdivided into two lines, one selected for an index of ovulation rate and embryonic survival for seven generations and a contemporary control line. Variance components were obtained by ANOVA and expectations of covariances among relatives and by derivative-free restricted maximum likelihood (DFREML) in an animal model. As a trait of the embryo, heritability of direct effects obtained with ANOVA was 3.8%, heritability of maternal effects was 1.5%, and the genetic correlation between them was -.51. After adjustment of embryonic survival for ovulation rate, lower estimates of each parameter were obtained with ANOVA. Heritability of embryonic survival as a trait of the dam was 9 to 10%. Estimates of heritability of both direct and maternal effects obtained with DFREML were less than 1% and the genetic correlation between them was -.64. When survival of embryos from only those dams with 15 or more ovulations was analyzed, heritability of maternal effects was 4.4%. Estimates of common environmental effects on embryonic survival ranged from 5 to 7%.     

Estimation of direct and maternal genetic variance for litter size in Canadian Yorkshire and Landrace swine using an animal model

Estimates of additive direct heritability (h2a) for traits such as litter size may be biased by maternal effects. The size of these effects was estimated using a derivative-free restricted maximum likelihood procedure under an animal model. First-parity records from Yorkshire (Y) and Landrace (L) gilts were obtained from the Quebec Record of Performance sow productivity program for 21,127 litters born between 1977 and 1987. Direct (sigma 2a) and maternal (sigma 2m) additive genetic variances, their covariance (sigma am) and error variance (sigma 2e) were estimated for total numbers born (NOBN), born alive (NOBA) and weaned (NOWN). Analysis of purebred Y and crossbred litters indicated that estimates of sigma 2a were of similar magnitude for all traits, with h2a ranging from .06 to .13. Except for L litters, estimates of sigma 2m were relatively low for NOBN and NOBA, and increased in size for NOWN, with h2m ranging from 0 to .08. Also, estimates of sigma am were negative, except for NOBN and NOBA with crossbred litters, and became increasingly negative for NOWN. Results from purebred L litters indicated there was a stronger negative correlation between direct and maternal genetic effects for NOBN and NOBA than for NOWN.     

Restricted maximum likelihood estimation of additive genetic variance when selected base animals are considered fixed.

A method to estimate genetic parameters with a model that considers selected base animals as fixed was investigated. The model estimates genetic variance as a conditional variance based on the Mendelian sampling of gametes from the base parents. In a simulation study, 20 sires were selected and each was mated to 20 dams to create 400 animals for the next generation. Selection was for five generations, but only animals of Generations 4 and 5 were assumed to have performance records and known parents. Simulated values for additive genetic and residual variance were 10. Estimated genetic variance was 8.58 when base animals were assumed random and 6.03 when they were assumed fixed. Residual variance was overestimated in the latter case. When males of Generation 4 were not selected to have progeny, estimated genetic variance was 9.91. It was concluded that estimates for genetic parameters in a model with base animals assumed as fixed were not biased by selection of base animals, but a new bias was introduced if descendants of fixed base animals were selected. Estimation of genetic variance from dairy records of daughters of AI test bulls gave differences of up to 8% when the model removed bias from selected base animals.