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.     

Estimation of variance components with dominance and inbreeding in dairy cattle

SUMMARY: The five variance components in the genetic (co)variance among inbred relatives for a quantitative trait with additive and dominance genetic variation were estimated by equating variances among and within different types of families of inbred cows to their expectations. The data used were milk and fat yields of 85,433 U.S. Holstein cows with inbreeding coefficients of 6.25% or higher. When all five parameters were estimated, unrealistic results were obtained. If all quantitative trait loci are biallelic, genetic (co)variance depends on only four parameters. More realistic estimates were obtained under this assumption. There was a substantial negative covariance among breeding values and dominance effects under inbreeding, and the dominance variance in inbred cows was larger than the dominance variance in the noninbred base population. ZUSAMMENFASSUNG: Varianzkomponenteneinsch?tzung bei Dominanz und Inzucht von Milchvieh Die fünf Varianzkomponenten in der genetischen (Ko)Varianz zwischen ingezüchteten Verwandten in einem quantitativen Merkmal wurden gesch?tzt durch Gleichsetzen von Varianzen zwischen und innerhalb verschiedener Familien von ingezüchteter Kühen zu ihren Erwartungswerten. Das Datenmaterial bestand aus den Milch- und Fettmengen von 85,433 U.S. Holstein Kühen mit Inzuchtkoeffizienten von 6.25% oder h?her. Die gleichzeitige Sch?tzung aller fünf Parameter führte zu unrealistischen Ergebnissen. Wenn an allen Genorten des quantitativen Merkmals nur zwei Allele vorkommen, gehen nur vier Parameter in die genetische (Ko)Varianz ein. Die Sch?tzwerte, die unter dieser Annahme berechnet wurden, waren plausibler. Eine betr?chtliche negative Kovarianz zwischen den Zuchtwerten und den Dominanzwerten bei Inzucht wurde gefunden, und die Dominanzvarianz war unter Inzucht gró?er als in der Basispopulation.     

Estimation of variance components for postpartum dysgalactia syndrome in sows

The postpartum dysgalactia syndrome (PDS) represents one of the most important diseases after parturition in sows. The genetic background of the disease has been investigated some time ago and heritability estimates around 0.10 have been obtained. To compute current estimates, a dataset of 1680 sampled sows and their 2001 clinically examined litters was used for variance components estimation with a threshold liability model. Affected sows were defined through clinical examination 12–48 h after parturition. Posterior mean of additive genetic variance was 0.10 and estimated heritability for PDS averaged 0.0879 with a 95% confidence interval of 0.0876 and 0.0881. The results are in agreement with those of other studies and emphasize the importance of considering the genetic predisposition for susceptibility to PDS as well as of additional factors including hygiene and management conditions.     

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

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

Estimation of maternal variance components considering cow-calf contacts under extensive pastoral systems

Traditional methods of variance component estimation for traits under maternal influence consist of partitioning the variance into direct additive genetic, maternal additive genetic, permanent maternal environmental, and error variance components. This partitioning is based on the assumption that each calf is nurtured and fed exclusively by its own dam. However, under extensive pastoral systems, voluntary cross-suckling may occur and could be quantified by using contact loggers recording cow-calf affiliations. A simulation study was conducted to test several variance models for partitioning maternal variation by including information on cow-calf contacts. The results indicated that weighting maternal genetic and permanent maternal environmental effects by the relative time calves spent with particular cows, including their own mothers, is feasible and significantly increased the log-likelihood of the models. However, the interpretation of the variance components in terms of traditional direct and maternal heritability is no longer straightforward. The need for further research and implications for the industry are discussed.     

Effects of excluding a set of random effects on prediction error variance of breeding value

SUMMARY: The effects of excluding a set of random effects (U-effects) uncorrelated to breeding values (BV) on prediction error variance (PEV) is studied analytically. Two situations are considered for model comparison: (a) existence of a 'true' model, (b) uncertainty about which of the competing models is 'true'. Models compared are the 'long' one, which includes BV + U-effects, and the 'short' one which includes BV's as the only random systematic effect. Expressions for PEV(BV) were obtained for the long model (PEVL); the short model (PEVS); and the short model assuming the long model is the correct one (PEVSI). It is shown that in general PEVS ≤ PEVL ≤ PEVSI. Results are exemplified by means of an example including a computer simulation. RESUMEN: En este trabajo se estudia analiticamente el efecto de excluir una variable aleatoria (efecto U) no correlacionada con el valor de cría (BV), sobre la varianza del error de predicción de este último (PEV(BV)). Para ello se utilizan dos enfoques de comparación de modelos: (a) existencia de un modelo 'verdadero', (b) incertidumbre respecto de cuál de ambos modelos alternativos es el correcto. Los modelos que se comparan son: el 'largo', que incluye BV+U, y el 'corto', el cuál solo incluye BV. Se obtienen las expresiones para PEV(BV) en las siguientes situaciones: (1) en el modelo largo (PEVL), (2) en el modelo corto (PEVS), y (3) en el modelo corto pero asumiendo que el largo es el verdadero (PEVSI). Se demuestra que en general PEVS ≤ PEVL ≤ PEVSI. Los resultados obtenidos son ilustrados mediante un ejemplo que incluye una simulación estocástica. ZUSAMMENFASSUNG: Ver?nderung der Fehlervarianz der Zuchtwertvoraussage durch Vernachl?ssigung einer Gruppe zuf?liger Wirkungen. Es wird die Auswirkung der Ausschaltung einer Gruppe zuf?lliger Wirkungen (U-effects), die mit Zuchtwerten (BV) nicht korreliert sind, auf die Varianz des Voraussage-Fehlers (PEV) analytisch untersucht. Zwei Modelle werden betrachtet: (a) Existenz eines "wahren"Modells, (b) Ungwi?heit hinsichtlich des "wahren"Modells. Es werden verglichen: ein langes Modell, das BV und U-Wirkungen umfa?t und ein kurzes, das nur BV als zuf?llige systematische Wirkung aufweist. Für PEV(BV) wurden Gr??en erhalten: (1) für das lange Modell (PEVL), (2) das kurze (PEVS) und (3) für das kurze unter der Annahme, da? das lange Modell das richtige ist (PEVSI). Im allgemeinen gilt PEVS ≤ PEVL ≤ PEVSI. Ergebnisse werden anhand einer Computersimulation erl?utert.     

Estimation of genetic parameters among breeding soundness examination components and growth traits in yearling bulls

Data on breeding soundness examinations (BSE) and performance traits were obtained on 549 yearling beef bulls at the San Juan Basin Research Center, Hesperus, Co from 1976 to 1984. Genetic parameters estimated for components of BSE included percent motility (PMOT), percent primary abnormalities (PPRIM), percent secondary abnormalities (PSEC), percent normal sperm (PNOR), scrotal circumference (SC) and BSE score (BSESC). Performance traits included birth weight, weaning weight, yearling weight and average daily gain. The least squares model included birth year, age of dam and breed as fixed effects, sire/breed as a random variable, and age and percent inbreeding as covariates. Paternal half-sib estimates of heritability were PMOT, .08 +/- .07; PPRIM, .31 +/- .09; PSEC, .02 +/- .05; PNOR, .07 +/- .06; BSESC, .10 +/- .06 and SC, .40 +/- .09. Phenotypic correlations among BSE components and growth traits were generally favorable. Genetic correlations involving percent secondary abnormalities were highly variable with large standard errors. Seminal traits improved as age increased and became poorer as inbreeding increased.     

Estimation of (co)variance components and genetic parameters of greasy fleece weights in Muzaffarnagari sheep

Variance components and genetic parameters for greasy fleece weights of Muzaffarnagari sheep maintained at the Central Institute for Research on Goats, Makhdoom, Mathura, India, over a period of 29 years (1976 to 2004) were estimated by restricted maximum likelihood (REML), fitting six animal models including various combinations of maternal effects. Data on body weights at 6 (W6) and 12 months (W12) of age were also included in the study. Records of 2807 lambs descended from 160 rams and 1202 ewes were used for the study. Direct heritability estimates for fleece weight at 6 (FW6) and 12 months of age (FW12), and total fleece weights up to 1 year of age (TFW) were 0.14, 0.16 and 0.25, respectively. Maternal genetic and permanent environmental effects did not significantly influence any of the traits under study. Genetic correlations among fleece weights and body weights were obtained from multivariate analyses. Direct genetic correlations of FW6 with W6 and W12 were relatively large, ranging from 0.61 to 0.67, but only moderate genetic correlations existed between FW12 and W6 (0.39) and between FW12 and W12 (0.49). The genetic correlation between FW6 and FW12 was very high (0.95), but the corresponding phenotypic correlation was much lower (0.28). Heritability estimates for all traits were at least 0.15, indicating that there is potential for their improvement by selection. The moderate to high positive genetic correlations between fleece weights and body weights at 6 and 12 months of age suggest that some of the genetic factors that influence animal growth also influence wool growth. Thus selection to improve the body weights or fleece weights at 6 months of age will also result in genetic improvement of fleece weights at subsequent stages of growth.     

Estimation of variance components due to imprinting effects with DFREML and VCE: results of a simulation study

Treating gametes as homozygous diploid individuals, TIER and SÖLKNER (Theor. Appl. Genet. 85: 868–872, 1993) proposed a method which manages the use of available computer programs with a common animal model to estimate variance components caused by imprinting effects. Despite some relevant model restrictions, this approach has already been used in some field data analyses by an adapted version of the widely used DFREML computer program, subsequently indicated by DFREML ^a. The main objective of this study was to ascertain the properties of DFREML ^a by computer simulation and to examine other alternative estimation approaches. The most important results may be summarized as follows: (1) Treating gametes as homozygous diploid individuals has the consequence that one‐half of the actually realized gametic effect is totally abstracted in variance component estimation. Thus, an additional adjustment of the phenotypic variance calculated by DFREML ^a is necessary to get correct values of estimated variance component ratios. (2) Adjusted DFREML ^a estimates yielded correct results when animals were unselected and only maternal or paternal imprinting (not both simultaneously) occurred. (3) When the model did not adequately account for the additive genetic component within a maternal lineage, significant upward biases for the cytoplasmic component were observed. (4) The use of a simple dam and sire model with appropriate relationship matrices can be recommended when only the difference of maternal and paternal imprinting effects is of primary interest and the covariance between maternal halfsibs is not substantially increased by common environmental effects. (5) An adequate estimation of variance components for all possible imprinting situations requires the use of an animal model augmented by both maternal and paternal gametic effects. Unfortunately, a computer program on the basis of such a model does not yet exist.     

Top down preselection using marker assisted estimates of breeding values in dairy cattle

Top down preselection of young bulls before entering progeny testing has been proposed as a practicable form of marker‐assisted selection (MAS), especially in dairy cattle populations with large male paternal half‐sib families. Linkage phase between the superior (Q) and the inferior (q) QTL alleles of heterozygous sires (Qq at the QTL) with informative markers is established within each paternal half‐sib family and may be used for selection among grand‐progeny. If, additionally to sires, bulldams are also genotyped and data from consecutive generations are used, then a marker‐assisted best linear unbiased prediction (MA‐BLUP) model can be employed to connect the information of all generations and families of a top down design, and to select across all families. A customized ‘augmented’ sire model (with sires and dams of sires as random effects) is introduced for this purpose. Adapted formulae for the mixed model equations are given and their equivalence to a corresponding animal model and to a certain variant of previously proposed reduced animal models is shown. The application of the augmented sire model in MA‐BLUP estimation from daughter‐yield deviations and effective daughter contributions is presented.     

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.     

Effect of culling on selection response using phenotypic selection or best linear unbiased prediction of breeding values in small, closed herds of swine.

Records from 7,200 separate closed herds with either 12 or 25 sows that were mated to either four or eight boars per year were simulated by computer. Effects of selection method, herd size, and contemporary group variability on average genetic change, genetic variance, and inbreeding over 10 yr of selection were analyzed for traits with heritabilities of .1, .3, and .6. Selection of replacement animals was on individual phenotype or BLUP of breeding value using a reduced animal model. For both of these selection methods, two culling schemes were imposed: 1) based only on involuntary culling because of losses due to conception rate and age and 2) when an available replacement animal was projected to be superior to an existing breeding animal in the herd in addition to the involuntary culling. The contemporary group standard deviation was set at either .1 or .5 of a phenotypic standard deviation. Selection with BLUP gave 72, 36, and 12% more genetic improvement for heritabilities of .1, .3, and .6, respectively, than selection on individual phenotype after 10 yr. However, inbreeding increased 20 to 52% more rapidly and there was a decrease in genetic variance. Culling based on Scheme 2 increased genetic improvement over Scheme 1 by about 75% with coincident increases in inbreeding level and decreases in genetic variance. The largest changes in inbreeding and genetic variance were associated with culling on BLUP. Culling when a superior animal was available with individual phenotype had little effect on inbreeding and genetic variance. Use of four boars rather than eight boars and 25 rather than 12 sows per herd increased genetic response. Use of four boars also increased inbreeding and decreased genetic variance. Genetic variance was higher in herds with 25 sows, but the size of the sow herd had little effect on inbreeding. Contemporary group variation influenced only the genetic response of individual phenotypic selection with culling.     

A method of computing restricted best linear unbiased prediction of breeding values for some animals in a population

Restricted BLUP (R-BLUP) is derived by imposing restrictions directly within a multiple-trait mixed model. As a result, the R-BLUP procedure requires the solution of high-order simultaneous equations. If restrictions are imposed on breeding values for only some animals in a population, calculations become more complex. A new procedure for computing the R-BLUP of breeding values was derived when constraints were imposed on the additive genetic values of only some animals in a population. Rules for including records when proportional constraints are imposed were developed based on the traits that are recorded for an animal. The technique was better than the previous method in both memory requirement and central processing unit time.     

Estimation of animal-level prevalence from pooled samples in animal production

Often, the prevalence of an infection in the animal-production sector is determined at the group level. The prevalence at animal level (p) gives more-precise information on the infection status of the sector. This paper shows that pooled-sample data together with mathematical models allow for estimation of p. For this, model assumptions have to be made on the variation of p between groups separated in space and/or time. Formulas were derived for four models that were based on different assumptions. Model 1 assumed that p has the same value for all groups. Models 2–4 assumed that some of the groups were not infected. In addition, model 2 assumed that p has the same value for all infected groups; model 3 assumed that for an infected group, p is equal to either p₁ or p₂; and model 4 assumed that p was Beta distributed among infected groups. The models were applied to data sets on Salmonella infection in broiler flocks, including serotype data dominated by S. Hadar and S. Paratyphi B, var. Java. Based on likelihood-ratio tests, models 3 and 4 consistently fitted significantly better to the data. The applicability of model 4 is numerically bounded, related to the shape of the Beta distribution of p. Model calculations show that flock-level prevalence of Salmonella is much higher after than before slaughter. This difference (which possibly is related to different types of samples) is much smaller at the animal level. An important result of the estimation of p is that it in turn allows for an estimation of the proportion of false-negative groups — which is important in estimating the effect of veterinary or public-health measures.     

Accuracy of breeding values when using and ignoring the polygenic effect in genomic breeding value estimation with a marker density of one SNP per cM

Genomic selection is based on breeding values that are estimated using genome-wide dense marker maps. The objective of this paper was to investigate the effect of including or ignoring the polygenic effect on the accuracy of total genomic breeding values, when there is coverage of the genome with approximately one SNP per cM. The importance of the polygenic effect might differ for high and low heritability traits, and might depend on the design of the reference dataset. Hence, different scenarios were evaluated using stochastic simulation. Accuracies of the total breeding value of juvenile selection candidates depended on the number of animals included in the reference data. When excluding polygenic effects, those accuracies ranged from 0.38 to 0.55 and from 0.73 to 0.79 for traits with heritabilities of 10 and 50%, respectively. Accuracies were improved by including a polygenic effect in the model for the low heritability trait, when the LD-measure r2 between adjacent markers became smaller than approximately 0.10, while for the high heritability trait there was already a small improvement at r2 between adjacent markers of 0.14. In all situations, the estimated total genetic variance was underestimated, particularly when the polygenic effect was excluded from the model. The haplotype variance was less underestimated when more animals were added in the reference dataset.     

Estimation of (co)variance components for reproductive traits in Angus beef cattle divergently selected for blood serum IGF-I concentration

The objective of this study was to obtain estimates of (co)variance components for reproductive traits and insulin-like growth factor-I (IGF-I) concentration. Data were from a divergent selection experiment for blood serum IGF-I concentration in Angus beef cattle. Numbers of observations for mean IGF-I concentration of three blood samples taken at d 28, 42, and 56 of the 140-d postweaning test, scrotal circumference (SC), percentage of motile sperm cells (PMSC), percentage of morphologically normal sperm cells (PNSC), age of heifers at first calving (AFC), and calving rate (CR) were 1,848, 825, 596, 765, 294, and 2,092, respectively. Total number of animals in the numerator relationship matrix, including base animals, was 2,864, of which 1,861 were inbred. Estimates of direct heritability for IGF-I concentration of three blood samples collected at d 28, 42, and 56 of the postweaning test and for mean IGF-I concentration were 0.43+/-0.08, 0.51+/-0.09, 0.41+/-0.08, and 0.50+/-0.08, respectively. Estimates of direct heritability for SC, PMSC, PNSC, AFC, and CR were 0.51+/-0.13, 0.08+/-0.12, 0.47+/-0.07, 0.26+/-0.28, and 0.11+/-0.05, respectively. With the exception of age at first calving, estimates of maternal heritability and proportion of phenotypic variance that were due to permanent environmental effects of the dams were smaller than 0.21. Observations for calving rate were entered as either 1 (if calved) or 100 (if not calved). Estimates of additive genetic correlations of mean IGF-I concentration with SC, PMSC, PNSC, AFC, and CR were 0.35+/-0.11, 0.43+/-0.32, 0.00+/-0.03, -0.14+/-0.33, and -0.41+/-0.16, respectively. Environmental and phenotypic correlations for all of the traits with IGF-I measurements were smaller than 0.23. These results suggest that selection for increased serum IGF-I concentration should result in increased scrotal circumference, percent motile sperm cells, and calving rate.     

Effects of nonrandom parental selection on estimation of variance components

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

Estimation of 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.     

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.