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.     

Parameter estimation in selected populations with missing data

This study proposes a procedure to estimate genetic parameters in populations where a selection process results in the loss of an unknown number of observations. The method was developed under the Bayesian inference scope following the missing data theory approach. Its implementation requires slight modifications to the Gibbs sampler algorithm. In order to show the efficiency of this option, a simulation study was conducted.     

Estimation of additive genetic variance when base populations are selected

A population of size 40 was simulated 1,000 times for 10 generations. Five out of twenty males were selected each generation, and each male was mated to four females to have two progeny. The additive genetic variance (sigma 2a) before selection was 10, and the initial heritability was .5. Due to covariances among animals, inbreeding and gametic disequilibrium, the genetic variance was reduced to 6.72 after 10 generations of selection. Reduction of variance was lower in another population simulated with size 400 and 10% of the males selected. Restricted Maximum Likelihood was used to estimate sigma 2a using an animal model. The estimate of sigma 2a was empirically unbiased when all data and all relationships were used. Omitting data from selected ancestors caused biased estimates of sigma 2a due to not accounting for all gametic disequilibrium. Including additional relationships between assumed base animals adjusted for inbreeding and for covariances. Bias from gametic disequilibrium decreased slightly with the use of more relationship information, and it was smaller in the small population and(or) when selection had been practiced for just a few generations.     

Reliable computing in estimation of variance components

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

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.     

Variance of actual inbreeding derived from the number and variance of chromosome segment

The inbreeding coefficient (F) is used as a central parameter inferring a proportion of alleles identical by descent within an individual and by genetic variability within a population. The actual inbreeding coefficient varies around a central value, the inbreeding coefficient. C ockerham and W eir (1983) derived the method for computing the variance of inbreeding while reviewing several other methods. The variance of inbreeding in their report was considered to be of two components: one within population and the other between population of varied pedigrees. If pedigree is fixed, F is easily computed for an individual by the standard method (F alconer 1989). For domestic animals, pedigree information is usually available because it is requisite for a programme of genetic improvement. In this study, the variance of inbreeding coefficient was derived for an individual with a pedigree having a single path to a foundation animal.     

Desert island papers—A life in variance parameter and quantitative genetic parameter estimation reviewed using 16 papers

I review my scientific career in terms of eight areas and 16 papers. The first two areas are associated with childhood. The other six are associated with residual maximum likelihood (REML), canonical transformation, inbreeding in selected populations, average information residual maximum likelihood (AIREML), the computer program ASReml and sampling‐based estimation.     

Maximization of total genetic variance in breed conservation programmes

The preservation of the maximum genetic diversity in a population is one of the main objectives within a breed conservation programme. We applied the maximum variance total (MVT) method to a unique population in order to maximize the total genetic variance. The function maximization was performed by the annealing algorithm. We have selected the parents and the mating scheme at the same time simply maximizing the total genetic variance (a mate selection problem). The scenario was compared with a scenario of full-sib lines, a MVT scenario with a rate of inbreeding restriction, and with a minimum coancestry selection scenario. The MVT method produces sublines in a population attaining a similar scheme as the full-sib sublining that agrees with other authors that the maximum genetic diversity in a population (the lowest overall coancestry) is attained in the long term by subdividing it in as many isolated groups as possible. The application of a restriction on the rate of inbreeding jointly with the MVT method avoids the consequences of inbreeding depression and maintains the effective size at an acceptable minimum. The scenario of minimum coancestry selection gave higher effective size values, but a lower total genetic variance. A maximization of the total genetic variance ensures more genetic variation for extreme traits, which could be useful in case the population needs to adapt to a new environment/production system.     

On estimating components of genetic variance in diallel matings 1

1. The relative importance of additive and non‐additive genetic effects on body weight, egg weight, maturity and rate of egg production were studied from diallel matings in a Leghorn population.

2. From analyses of variance, heritability estimates of the additive fractions, based on half‐sib variances, and the non‐additive or dominance fractions, based on the sire x dam interaction component were obtained.

3. Non‐additive genetic effects were not statistically significant for any of the traits, though for rate of egg production at 32 and 62 weeks, the non‐additive effects as proportion of total variances were 0.29 (P<0.10) and 0.20 (P<0.16), respectively, compared additive effects of 0.08 (NS) and 0.11 (P<0.05).

4. The ratios of non‐additive to additive variances, 1.89 and 3.62 respectively, give support to inbreeding and hybridisation or reciprocal recurrent selection as methods of genetic improvement of egg production.     

Multivariate analysis of litter size for multiple parities with production traits in pigs: I. Bayesian variance component estimation

A total of 66,620 records from the first six parities for number of piglets born alive (NBA) from 20,120 Landrace sows and 24,426 records for weight (WT) and backfat thickness (BT) at 175 d of age were analyzed to estimate genetic parameters. The pedigree consisted of 47,186 individuals, including 392 sires and 5,394 dams. Estimates were based on marginal posterior distribution of the genetic parameters obtained using Bayesian inference implemented via the Gibbs sampling procedure with a Data Augmentation step. The posterior means and posterior standard deviation (PSD) for heritability of NBA ranged from 0.064 (PSD 0.005) in the first parity to 0.146 (PSD 0.019) in the sixth parity, always increasing with the order of the parity. The posterior means for genetic correlations of litter size between adjacent parities were, in most cases, greater than 0.80. However, genetic correlation were much lower between nonadjacent parities. For example, the genetic correlation was 0.534 (PSD 0.061) between the fourth and the sixth parity for NBA. The posterior means of heritability for WT and BT were 0.229 (PSD 0.018) and 0.350 (PSD 0.019), respectively. Posterior mean for genetic correlation between WT and BT was 0.339 (PSD 0.044). The posterior means for genetic correlation between production (WT and BT) and reproduction traits (NBA in different parities) were close to zero in most cases. Results from this study suggest that different parities should be considered as different traits. Moreover, selection for growth and backfat should result in no or very little correlated response in litter size.     

Effect of including relationships in the estimation of genetic parameters of beef calves.

Variances and covariances for birth weight, gain from birth to weaning (ADG), and 205-d weight were obtained from a sire-dam model and a sire-maternal grandsire model for a herd of Angus and a herd of Hereford cattle. Estimates of direct additive genetic variance (sigma 2A), maternal additive genetic variance (sigma 2M), covariance between direct and maternal additive genetic effects (sigma AM), permanent environmental variance (sigma 2PE), and residual variance (sigma 2e) were obtained both with and without the inverse of the numerator relationship matrix (A-1) included. Estimates of heritability for direct genetic effects (h2A), maternal genetic effects (h2M), and the correlation between direct and maternal effects (rAM) for birth weight were .37, .18, and -.01 in Angus and .53, .23, and -.19 in Herefords, respectively, for the analyses without A-1. For the analyses with A-1, estimates of h2A, h2M, and rAM were .42, .22, and -.12 for Angus and .58, .22, and -.13 for Herefords, respectively. Estimates of h2A, h2M, and rAM for ADG were .43, .15, and -.44 in Angus and .52, .38, and -.03 in Herefords, respectively, without A-1. With A-1, estimates of h2A, h2M, and rAM were .57, .15, and -.32 for Angus and .58, .39, and -.05 for Herefords, respectively. Estimates of h2A, h2M, and rAM for 205-d weight were .49, .15, and -.46 for Angus and .58, .43, and -.06 for Herefords, respectively, without A-1. With A-1, estimates of h2A, h2M, and rAM were .63, .16, and -.36 for Angus and .66, .43, and -.08 for Herefords, respectively. Estimates of h2A were higher with A-1 than without A-1, but estimates of h2M were similar. Using variances and covariances obtained from analyses including A-1 generally gave higher estimates of direct breeding values than using variances and covariances obtained from analyses not including A-1. Both Pearson product-moment and Spearman rank correlations were high (.99) between estimates of breeding values from the two analyses, although some changes in rank did occur.     

Effects of nonrandom parental selection on estimation of variance components

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

Unexpected estimates of variance components with a true model containing genetic competition effects

Simulation of a model containing genetic competition effects was initiated to determine how well REML could untangle variances due to direct and competition genetic effects and pen effects. A two-generation data set was generated with six unrelated males that were each mated to five unrelated females to produce 300 progeny, from which 30 females (one per mating in previous generation) were mated to six unrelated males to produce 300 more progeny. Progeny were randomly assigned, six per pen, to 50 pens per generation. Parameters were V(g), V(c), C(gc), V(p), and V(e), representing direct and competition genetic variance with covariance, and pen and residual variance. Eight statistical models were used to analyze each of 400 replicates of 16 sets of parameters. Both V(g) and V(e) were fixed at 16.0. Values of C(gc) were -2.0, -1.0, 0.1, 1.0, and 2.0. Values of V(c) were 1.0 and 4.0, and values of V(p) were 0.1, 1.0, and 10.0. With the full model, average estimates resembled true parameters, except that V(p) was consistently overestimated when small (0.1 and 1.0), which in turn slightly changed other estimates. The most unexpected result was overestimation of V(p) when V(c) and Cgc were ignored. Overestimation depended on V(c) and the number of competitors in common between records in a pen. Upward bias was somewhat greater when Cg(c) was positive than when it was negative. For example, with C(gc) = 2, V(c) = 4, and V(p) = 0.1, the mean estimate of V(p) was 20.4 when C(gc) and V(c) were dropped from the model and 15.3 when C(gc) = -2.0. When V(p) was ignored, estimates of both C(gc) and V(c) increased in proportion with V(p). Also V(g) increased more with greater V(p). Another unexpected result occurred when pen was considered fixed. Empirical sampling standard errors of estimates of C(gc) and V(c) were decreased generally by factors of 2 to 30. Based on these results, we conclude a high estimate of pen variance may indicate genetic competition effects are important, and ignoring either the pen or competition effects will bias estimates of other components.     

Estimates of genetic variance and variance of predicted genetic merits using pedigree or genomic relationship matrices in six Brown Swiss cattle populations for different traits

The amount of variance captured in genetic estimations may depend on whether a pedigree‐based or genomic relationship matrix is used. The purpose of this study was to investigate the genetic variance as well as the variance of predicted genetic merits (PGM) using pedigree‐based or genomic relationship matrices in Brown Swiss cattle. We examined a range of traits in six populations amounting to 173 population‐trait combinations. A main aim was to determine how using different relationship matrices affect variance estimation. We calculated ratios between different types of estimates and analysed the impact of trait heritability and population size. The genetic variances estimated by REML using a genomic relationship matrix were always smaller than the variances that were similarly estimated using a pedigree‐based relationship matrix. The variances from the genomic relationship matrix became closer to estimates from a pedigree relationship matrix as heritability and population size increased. In contrast, variances of predicted genetic merits obtained using a genomic relationship matrix were mostly larger than variances of genetic merit predicted using pedigree‐based relationship matrix. The ratio of the genomic to pedigree‐based PGM variances decreased as heritability and population size rose. The increased variance among predicted genetic merits is important for animal breeding because this is one of the factors influencing genetic progress.     

Simulation of genetic control of reproduction in beef cows. IV. Within-herd breeding value estimation with pasture mating

Procedures for breeding value estimation for reproductive traits under pasture mating conditions were developed and tested using a computer simulation model of genetic control of bovine reproduction. The model generated annual calving rates (BCR) (0 or 1) and calving dates (CD) for each cow as a function of underlying genetic variation in two independent traits: single-service conception rate, which was indicative of the ability to conceive when estrus occurs, and postpartum interval (PPI) from calving to first estrus. Observed values for BCR and CD were shown to be complex, nonlinear functions of breeding values for ability to conceive (CRG) and for postpartum interval (PPIG) and of the previous CD. Effects of CRG on BCR and CD were small at high values of CRG, but these effects increased as CRG declined. Effects of PPIG on BCR and CD were small for cows that previously calved within the first 21 d of the calving season, but these effects increased for cows that calved after d 21. Previous CD had substantial nongenetic carryover effects on both BCR and CD. Unbiased estimates of CRG and PPIG could not be derived in the absence of breeding information. However, CD were reasonably highly correlated with breeding values for ability to conceive, provided information on open cows was included in the evaluation. Calving dates were only weakly associated with breeding values for PPI, in part because of the relatively short mean PPI (70 d) that was simulated.     

Optimized management of genetic variability in selected pig populations

Controlling the increase of coancestry and inbreeding coefficients in selected populations is made possible through calculation of the optimal contributions allowed to breeding animals, given the current situation with regard to genetic diversity, and further, through optimal design of matings. The potential of such an approach for pig breeding was tested by retrospective optimization on the French Landrace population in reference to the matings actually carried out during a 21-week test period. The major constraint was that the average overall estimated breeding value (EBV) should be the same as the observed one, for not decreasing short-term genetic gain. Optimizing breeding allocations to boars would have led one to decrease coancestry and inbreeding coefficients by approximately 20%. This decrease would have even increased to approximately 30%, would have replacements and disposals been optimized after accounting for genetic variability, keeping the same constraint of genetic level identical to the observed one. These results showed the potential value, in the future, of completing each periodical calculation of EBVs by optimizations considering genetic variability and of releasing corresponding information to breeders, in order to enhance maintenance of genetic variability.     

The impact of data structure on genetic (co)variance components of early growth in sheep,estimated using an animal model with maternal effects

Several studies have noted high negative correlations between maternal genetic and direct additive effects and their influence on additive and maternal heritability of early growth traits in sheep. Multigeneration data from the Suffolk Sire Reference Scheme (SSRS) were used to investigate the effect of data structure on estimates of direct and maternal (co)variances for lamb 8-wk weight. In all analyses the additive, maternal genetic, maternal environmental, and residual effects were fitted along with the covariance between direct and maternal additive effects. The contributions of particular genetic relationships to the estimates were studied by analyzing subsets of the SSRS data. A further eight subsets were formed having 10% or 50% of the dams with their own records and having one or two, three or four, five or six, and more than six offspring per dam. Analysis of data having only 10% of the dams with their own record and one or two offspring records yielded a high negative correlation (-0.99) between direct and maternal genetic effects. However, the seven other data sets with more records per dam or a higher proportion of dams with their own records produced values of -0.35 to -0.51. Data structure and the number of dams and granddams with records are important determinants of estimated direct and maternal effects in early growth traits.     

Variance component estimation on the frequency of pathologic changes in the navicular bones of Hanoverian Warmblood horses

The results of a standardized radiological examination of 3748 young Hanoverian Warmblood horses selected for sale at auction as riding horses were used to quantify the influence of systematic effects on and to estimate genetic parameters for the prevalence of pathologic changes in the navicular bones. Radiographic findings in the navicular bones of the front limbs were analyzed as all‐or‐none traits. The pathologic changes were mostly classified as slight [PCN(I); 14.9%], less often as moderate [PCN(II); 5.3%] or severe [PCN(III); 1.8%]. Date and year of auction had a significant influence on the prevalence of documented radiographic findings. The prevalence of PCN(I) was further significantly dependent on the examiner, the type and the quality of auction. PCN(II) was significantly more prevalent in male than in female horses. The age, the anticipated suitability and the region of origin of the horses did not have any significant influence on the prevalence of pathologic changes in navicular bones. A higher percentage of genes of the Hanoverian and the Holstein Warmblood horse increased the probability of PCN(I) classification. A significant influence of the sire was found for PCN(I) and PCN(II), and of the male founder for PCN(II) and PCN(III). The female founder was significant only for PCN(II). In general, radiographic findings of any severity in front left and right navicular bones were significantly correlated with each other. Restricted maximum likelihood (REML) was used for the estimation of genetic parameters. The analyses were performed multivariately in linear animal and sire models including height at withers as a separate trait. Heritability estimates for the prevalence of PCN(I), PCN(II) and PCN(III) of horses of both sexes ranged between h² = 0.09 and 0.21. When distinguishing between findings in males and females, somewhat implausible estimates were obtained for PCN(II) in females, which might have been caused by their low prevalence. The additive genetic correlations between the investigated traits indicated that radiographic findings consistent with navicular syndrome have a uniform genetic pattern in males and in females, and irrespective of their severity. However, their genetic correlation to height at withers was found to be inconsistent and, therefore, not to be utilizable for selection.