Conditioning on the causal network prevents indirect response to selection

Multiple trait animal models (MTM) allow to estimate the breeding values (BV) of several traits simultaneously while accounting for genetic and environmental correlations among them. However, relationships among traits may not be reciprocal but rather causal in nature. In these cases, and given a causal network, structural equations models (SEM) arise as a more appropriate methodology. Although MTM and SEM have been shown to be parametrically equivalent, the estimated breeding value (EBV) obtained from either one or the other should be interpreted differently. In this study, we investigated the impact of using these estimates on the response to selection for a causal network comprising five different traits through a stochastic simulation experiment. Three different selection targets were assayed, involving traits located upstream, midstream and downstream this causal network. We first considered the case in which traits were causally related but not genetically correlated. The current results support our hypothesis that MTM will absorb causal relationships as genetic correlations and, consequently, change the response to selection achieved as compared with SEM. We found no differences on the response to selection when the target trait was located at the top of the causal network, but noticeable differences were detected on upstream traits when selection pressure was placed on midstream or downstream traits. We also assayed a scenario in which causal effects and genetic correlations act simultaneously and found that selection based on BVs estimated using SEM diminished the indirect response in traits upstream the causal network.

Causal inference for the covariance between breeding values under identity disequilibrium

Background

The covariance matrix of breeding values is at the heart of prediction methods. Prediction of breeding values can be formulated using either an “observed” or a theoretical covariance matrix, and a major argument for choosing one or the other is the reduction of the computational burden for inverting such a matrix. In this regard, covariance matrices that are derived from Markov causal models possess properties that deliver sparse inverses.

Results

By using causal Markov models, we express the breeding value of an individual as a linear regression on ancestral breeding values, plus a residual term, which we call residual breeding value (RBV). The latter is a noise term that accounts for the uncertainty in prediction due to lack of fit of the linear regression. A notable property of these models is the parental Markov condition, through which the multivariate distribution of breeding values is uniquely determined by the distribution of the mutually independent RBV. Animal breeders have long been relying on a causal Markov model, while using the additive relationship matrix as the covariance matrix structure of breeding values, which is calculated assuming gametic equilibrium. However, additional covariances among breeding values arise due to identity disequilibrium, which is defined as the difference between the covariance matrix under the multi-loci probability of identity-by-descent (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma}$$\end{document}Σ) and its expectation under gametic phase equilibrium, i.e., A. The disequilibrium term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma}$$\end{document}Σ−A is considered in the model for predicting breeding values called the “ancestral regression” (AR), a causal Markov model. Here, we introduce the “ancestral regression to parents” (PAR) causal Markov model, which reduces the computational burden of the AR approach. By taking advantage of the conditional independence property of the PAR Markov model, we derive covariances between the breeding values of grandparents and grand-offspring and between parents and offspring. In addition, we obtain analytical expressions for the covariance between collateral relatives under the PAR model, as well as for the inbreeding coefficient.

Conclusions

We introduced the causal PAR Markov model that captures identity disequilibrium in the covariances among breeding values and produces a sparse inverse covariance matrix to build and solve a set of mixed model equations.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12711-022-00750-6.

Genome-wide scan for signatures of selection in the Brangus cattle genome

Brangus is a composite cattle breed developed with the objective of combining the advantages of Angus and Zebuine breeds (Brahman, mainly) in tropical climates. The aim of this work was to estimate breed composition both genome-wide and locally, at the chromosome level, and to uncover genomic regions evidencing positive selection in the Argentinean Brangus population/nucleus. To do so, we analysed marker data from 478 animals, including Brangus, Angus and Brahman. Average breed composition was 35.0% ± 9.6% of Brahman, lower than expected according to the theoretical fractions deduced by the usual cross-breeding practice in this breed. Local ancestry analysis evidenced that breed composition varies between chromosomes, ranging from 19.6% for BTA26 to 56.1% for BTA5. Using approaches based on allelic frequencies and linkage disequilibrium, genomic regions with putative selection signatures were identified in several chromosomes (BTA1, BTA5, BTA6 and BTA14). These regions harbour genes involved in horn development, growth, lipid metabolism, reproduction and immune response. We argue that the overlapping of a chromosome segment originated in one of the parental breeds and over-represented in the sample with the location of a signature of selection constitutes evidence of a selection process that has occurred in the breed since its take off in the 1950s. In this regard, our results could contribute to the understanding of the genetic mechanisms involved in cross-bred cattle adaptation and productivity in tropical environments.

Comparing Genomic Prediction Models by Means of Cross Validation

In the two decades of continuous development of genomic selection, a great variety of models have been proposed to make predictions from the information available in dense marker panels. Besides deciding which particular model to use, practitioners also need to make many minor choices for those parameters in the model which are not typically estimated by the data (so called "hyper-parameters"). When the focus is placed on predictions, most of these decisions are made in a direction sought to optimize predictive accuracy. Here we discuss and illustrate using publicly available crop datasets the use of cross validation to make many such decisions. In particular, we emphasize the importance of paired comparisons to achieve high power in the comparison between candidate models, as well as the need to define notions of relevance in the difference between their performances. Regarding the latter, we borrow the idea of equivalence margins from clinical research and introduce new statistical tests. We conclude that most hyper-parameters can be learnt from the data by either minimizing REML or by using weakly-informative priors, with good predictive results. In particular, the default options in a popular software are generally competitive with the optimal values. With regard to the performance assessments themselves, we conclude that the paired k-fold cross validation is a generally applicable and statistically powerful methodology to assess differences in model accuracies. Coupled with the definition of equivalence margins based on expected genetic gain, it becomes a useful tool for breeders.

Single-step genomic prediction of Eucalyptus dunnii using different identity-by-descent and identity-by-state relationship matrices

Genomic selection based on the single-step genomic best linear unbiased prediction (ssGBLUP) approach is becoming an important tool in forest tree breeding. The quality of the variance components and the predictive ability of the estimated breeding values (GEBV) depends on how well marker-based genomic relationships describe the actual genetic relationships at unobserved causal loci. We investigated the performance of GEBV obtained when fitting models with genomic covariance matrices based on two identity-by-descent (IBD) and two identity-by-state (IBS) relationship measures. Multiple-trait multiple-site ssGBLUP models were fitted to diameter and stem straightness in five open-pollinated progeny trials of Eucalyptus dunnii, genotyped using the EUChip60K. We also fitted the conventional ABLUP model with a pedigree-based covariance matrix. Estimated relationships from the IBD estimators displayed consistently lower standard deviations than those from the IBS approaches. Although ssGBLUP based in IBS estimators resulted in higher trait-site heritabilities, the gain in accuracy of the relationships using IBD estimators has resulted in higher predictive ability and lower bias of GEBV, especially for low-heritability trait-site. ssGBLUP based on IBS and IBD approaches performed considerably better than the traditional ABLUP. In summary, our results advocate the use of the ssGBLUP approach jointly with the IBD relationship matrix in open-pollinated forest tree evaluation.

Additive genetic variance and covariance between relatives in synthetic wheat crosses with variable parental ploidy levels

Cultivated bread wheat (Triticum aestivum L.) is an allohexaploid species resulting from the natural hybridization and chromosome doubling of allotetraploid durum wheat (T. turgidum) and a diploid goatgrass Aegilops tauschii Coss (Ae. tauschii). Synthetic hexaploid wheat (SHW) was developed through the interspecific hybridization of Ae. tauschii and T. turgidum, and then crossed to T. aestivum to produce synthetic hexaploid wheat derivatives (SHWDs). Owing to this founding variability, one may infer that the genetic variances of native wild populations vs improved wheat may vary due to their differential origin and evolutionary history. In this study, we partitioned the additive variance of SHW and SHWD with respect to their breed origin by fitting a hierarchical Bayesian model with heterogeneous covariance structure for breeding values to estimate variance components for each breed category, and segregation variance. Two data sets were used to test the proposed hierarchical Bayesian model, one from a multi-year multi-location field trial of SHWD and the other comprising the two species of SHW. For the SHWD, the Bayesian estimates of additive variances of grain yield from each breed category were similar for T. turgidum and Ae. tauschii, but smaller for T. aestivum. Segregation variances between Ae. tauschii-T. aestivum and T. turgidum-T. aestivum populations explained a sizable proportion of the phenotypic variance. Bayesian additive variance components and the Best Linear Unbiased Predictors (BLUPs) estimated by two well-known software programs were similar for multi-breed origin and for the sum of the breeding values by origin for both data sets. Our results support the suitability of models with heterogeneous additive genetic variances to predict breeding values in wheat crosses with variable ploidy levels.

Mapping Recombination Rate on the Autosomal Chromosomes Based on the Persistency of Linkage Disequilibrium Phase Among Autochthonous Beef Cattle Populations in Spain

In organisms with sexual reproduction, genetic diversity, and genome evolution are governed by meiotic recombination caused by crossing-over, which is known to vary within the genome. In this study, we propose a simple method to estimate the recombination rate that makes use of the persistency of linkage disequilibrium (LD) phase among closely related populations. The biological material comprised 171 triplets (sire/dam/offspring) from seven populations of autochthonous beef cattle in Spain (Asturiana de los Valles, Avileña-Negra Ibérica, Bruna dels Pirineus, Morucha, Pirenaica, Retinta, and Rubia Gallega), which were genotyped for 777,962 SNPs with the BovineHD BeadChip. After standard quality filtering, we reconstructed the haplotype phases in the parental individuals and calculated the LD by the correlation -r- between each pair of markers that had a genetic distance < 1 Mb. Subsequently, these correlations were used to calculate the persistency of LD phase between each pair of populations along the autosomal genome. Therefore, the distribution of the recombination rate along the genome can be inferred since the effect of the number of generations of divergence should be equivalent throughout the genome. In our study, the recombination rate was highest in the largest chromosomes and at the distal portion of the chromosomes. In addition, the persistency of LD phase was highly heterogeneous throughout the genome, with a ratio of 25.4 times between the estimates of the recombination rates from the genomic regions that had the highest (BTA18-7.1 Mb) and the lowest (BTA12-42.4 Mb) estimates. Finally, an overrepresentation enrichment analysis (ORA) showed differences in the enriched gene ontology (GO) terms between the genes located in the genomic regions with estimates of the recombination rate over (or below) the 95^th (or 5^th) percentile throughout the autosomal genome.

82 Compensation of the Growth and Development of Individually Transferred Bovine Bisected Embryos

Embryo bisection has been used in cattle for decades to multiply embryos. Recent articles assert that each bisected hemi-embryo continues its growth and differentiation without regenerating the original number of cells, producing smaller embryos. This size difference has been verified until Day 17, when the embryo has already elongated. The objective of this research was to evaluate fetal morphometric differences between embryos and hemi-embryos during the intrauterine period and at birth. For this purpose, 6 Angus cows were superovulated using the FSH/LH protocol (Pluset®, Hertape Calier, Minas Gerais, Brazil) and all were artificially inseminated with the sire ‘Las Lilas 1931 Líder Max’. At Day 7 post-AI, 45 embryos were recovered using a non-surgical technique. In the laboratory, 30 of them were bisected under a stereoscopic magnifier using a portable micromanipulator with a microscale attached. Each hemi-embryo (N = 60) and the remaining integer embryos (N = 15, used as controls) were immediately transferred individually to a recipient. The growth of the fetuses was evaluated ultrasonographically using a Esaote Aquila Pro vet ultrasound (Esaote SpA, Genoa, Italy). Transrectal measurements were performed at 60 and 90 days, recording femur length (only at 60 days), biparietal distance, and abdominal and umbilical cord diameter. At birth, animals were weighed, measured for length and height, and the circumferences of the skull, thorax, abdomen, and metatarsus were recorded. Data were analysed by the t-test implemented in InfoStat Version 1.1 software (https://www.infostat.com.ar/). Table 1 shows the average and standard deviation of all measurements performed. No significant differences (P < 0.05) were observed between hemi-embryos and control at any stage, except for biparietal diameter on Day 90 of gestation, which was not reflected in skull circumference at birth. In conclusion, individually transferred hemi-embryos generated 60-day fetuses with almost no differences in growth and development compared with the control embryos. The compensation phenomenon should have occurred before Day 60 of gestation. Table 1.Mean and standard deviation of all measurements performed on hemi-embryos and control embryos at 60 and 90 days and at birth

Performance of genomic selection under a single-step approach in autochthonous Spanish beef cattle populations

This study evaluated different strategies for implementing a single-step genomic selection programme in two autochthonous Spanish beef cattle populations (Pirenaica-Pi and Rubia Gallega-RG). The strategies were compared in terms of accuracy attained under different scenarios by simulating genomic data over the known genealogy. Several genotyping approaches were tested, as well as, other factors like marker density, effective population size, mutation rate and heritability of the trait. The results obtained showed gains in accuracy with respect to pedigree BLUP evaluation in all cases. The greatest benefit was obtained when the candidates to selection had their genotypes included in the evaluation. Moreover, genotyping the individuals with the most accurate predictions maximized the gains but other suboptimal strategies also yielded satisfactory results. Furthermore, the gains in accuracy increased with the marker density reaching a plateau at around 50,000 markers. Likewise, the effective population size and the mutation rate have also shown an effect, both increasing the accuracy with decreasing values of these population parameters. Finally, the results obtained for the RG population showed greater gains compared to the Pi population, probably attributed to the wider implantation of artificial insemination.

Detección de regiones genómicas con elevado desequilibrio de ligamiento en poblaciones de vacuno de carne españolas con análisis de BovineHD BeadChip

El objetivo de este trabajo fue evaluar el patrón de desequilibrio de ligamiento a lo largo del genoma en siete poblaciones españolas autóctonas de vacuno de carne (Asturiana de los Valles, Avileña Negra-Ibérica, Bruna dels Pirineus, Morucha, Pirenaica, Retinta y Rubia Gallega). Para ello, se utilizó el BovineHD BeadChip con el que se genotiparon 171 tríos formados por individuo/padre/madre. Después del filtrado, se dispuso de 573.134 SNP. A partir de esta información se definió un parámetro que mide el desequilibrio medio del genoma por regiones de 1Mb en cada una de las poblaciones. Los resultados mostraron que el desequilibrio de ligamiento es muy heterogéneo a lo largo del genoma y que, además, esta heterogeneidad es consistente entre poblaciones. Las causas de esta heterogeneidad pueden ser, o bien estructurales y atribuibles a una menor tasa de mutación y/o recombinación, o bien consecuencia de procesos de selección estabilizadora.

186 SUPPLEMENTATION WITH LOW DOSES OF DIMETHYL SULFOXIDE DURING IN VITRO MATURATION RESULTS IN IMPROVED IN VITRO EMBRYO PRODUCTION IN CATTLE

Oocyte in vitro maturation (IVM) is crucial for subsequent in vitro embryo production. It involves acquisition of competence for fertilization and embryo development. Therefore, its optimization could have a direct impact on in vitro embryo development. Dimethyl sulfoxide (DMSO) is commonly used as solvent or vehicle, but also increases the membrane permeability and behaves as a scavenger of cytotoxic free radicals. The aim of this study was to evaluate the effect of DMSO supplementation during bovine oocyte maturation on subsequent in vitro embryo development and to determine the optimal usage dose with no toxic effect. To this aim, cumulus-oocyte complexes were collected from slaughterhouse ovaries and IVM in TCM 199 containing 10% fetal bovine serum, 10 µg mL−1 of FSH, 0.3 mM sodium pyruvate, 100 mM cysteamine, and 2% antibiotic-antimycotic. The oocytes were incubated for 24 h at 6.5% CO2 in humidified air at 38.5°C. For Experiment 1, IVM medium was supplemented with DMSO at concentrations of 0, 0.1, 0.5, 1, or 10% (vol/vol) DMSO (n = 241, 195, 42, 192, 172 oocytes) and IVM rate was determined by presence of the first polar body. For Experiment 2, 0, 0.1, 0.25, 0.5, 0.75, 1, or 10% (vol/vol) DMSO (n = 446, 322, 65, 194, 77, 250, 39 oocytes) was supplemented to IVM medium and cleavage and blastocyst rates were determined to establish the optimal usage dose. In vitro fertilization was performed according to Brackett and Oliphant (1975), with 16 × 106 spermatozoa/mL for 5 h. Afterwards, presumptive zygotes were cultured in SOF for 7 days at 38.5°C and 5% O2. Cleavage and blastocyst rates were determined on Days 2 and 7, respectively. Results were statistically analysed using Fisher’s exact test by GraphPad Prism software (GraphPad Software Inc., La Jolla, CA, USA). Also, the percentage of blastocyst was adjusted to DMSO concentration using the R software quadratic regression model. The optimum usage dose was determined by calculating the maximum of the estimated predictive equation. In vitro maturation in 10% DMSO resulted in significantly lower first polar body extrusion rates (0% = 74%a, 0.1% = 73%a, 0.5% = 83%a, 1% = 66%a, and 10% = 8%b; different letters indicate statistical differences) and lower cleavage rates (0% = 75%a, 0.1% = 77%a, 0.25% = 80%a, 0.5% = 79%a, 0.75% = 78%a, 1% = 77%a, and 10% = 3%b) than the other treatments. Furthermore, blastocyst production was higher for the 0.25 and 0.5% (vol/vol) supplemented DMSO groups (0% = 26%b, 0.1% = 37%ab, 0.25% = 40%a, 0.5% = 41%a, 0.75% = 34%ab, 1% = 23%b, and 10% = 0%c). The predictive equation results indicate that the maximum percentage of blastocysts is obtained with a concentration of 0.458% (vol/vol) of DMSO. In conclusion, DMSO supplementation during IVM of bovine oocytes had a positive effect on in vitro development. Further studies will be carried out to elucidate its mechanism of action.

On the performance of tests for the detection of signatures of selection: a case study with the Spanish autochthonous beef cattle populations

Background

Procedures for the detection of signatures of selection can be classified according to the source of information they use to reject the null hypothesis of absence of selection. Three main groups of tests can be identified that are based on: (1) the analysis of the site frequency spectrum, (2) the study of the extension of the linkage disequilibrium across the length of the haplotypes that surround the polymorphism, and (3) the differentiation among populations. The aim of this study was to compare the performance of a subset of these procedures by using a dataset on seven Spanish autochthonous beef cattle populations.

Results

Analysis of the correlations between the logarithms of the statistics that were obtained by 11 tests for detecting signatures of selection at each single nucleotide polymorphism confirmed that they can be clustered into the three main groups mentioned above. A factor analysis summarized the results of the 11 tests into three canonical axes that were each associated with one of the three groups. Moreover, the signatures of selection identified with the first and second groups of tests were shared across populations, whereas those with the third group were more breed-specific. Nevertheless, an enrichment analysis identified the metabolic pathways that were associated with each group; they coincided with canonical axes and were related to immune response, muscle development, protein biosynthesis, skin and pigmentation, glucose metabolism, fat metabolism, embryogenesis and morphology, heart and uterine metabolism, regulation of the hypothalamic–pituitary–thyroid axis, hormonal, cellular cycle, cell signaling and extracellular receptors.

Conclusions

We show that the results of the procedures used to identify signals of selection differed substantially between the three groups of tests. However, they can be classified using a factor analysis. Moreover, each canonical factor that coincided with a group of tests identified different signals of selection, which could be attributed to processes of selection that occurred at different evolutionary times. Nevertheless, the metabolic pathways that were associated with each group of tests were similar, which suggests that the selection events that occurred during the evolutionary history of the populations probably affected the same group of traits.

Electronic supplementary material

The online version of this article (doi:10.1186/s12711-016-0258-1) contains supplementary material, which is available to authorized users.

Estimates of the actual relationship between half-sibs in a pig population

Genomic relationships based on markers capture the actual instead of the expected (based on pedigree) proportion of genome shared identical by descent (IBD). Several methods exist to estimate genomic relationships. In this research, we compare four such methods that were tested looking at the empirical distribution of the estimated relationships across 6704 pairs of half-sibs from a cross-bred pig population. The first method based on multiple marker linkage analysis displayed a mean and standard deviation (SD) in close agreement with the expected ones and was robust to changes in the minor allele frequencies (MAF). A single marker method that accounts for linkage disequilibrium (LD) and inbreeding came second, showing more sensitivity to changes in the MAF. Another single marker method that considers neither inbreeding nor LD showed the smallest empirical SD and was the most sensible to changes in MAF. A higher mean and SD were displayed by VanRaden's method, which was not sensitive to changes in MAF. Therefore, the method based on multiple marker linkage analysis and the single marker method that considers LD and inbreeding performed closer to theoretical values and were consistent with the estimates reported in literature for human half-sibs.

Genetic diversity and divergence among Spanish beef cattle breeds assessed by a bovine high-density SNP chip

The availability of SNP chips for massive genotyping has proven to be useful to genetically characterize populations of domestic cattle and to assess their degree of divergence. In this study, the Illumina BovineHD BeadChip genotyping array was used to describe the genetic variability and divergence among 7 important autochthonous Spanish beef cattle breeds. The within-breed genetic diversity, measured as the marker expected heterozygosity, was around 0.30, similar to other European cattle breeds. The analysis of molecular variance revealed that 94.22% of the total variance was explained by differences within individuals whereas only 4.46% was the result of differences among populations. The degree of genetic differentiation was small to moderate as the pairwise fixation index of genetic differentiation among breeds (F) estimates ranged from 0.026 to 0.068 and the Nei's D genetic distances ranged from 0.009 to 0.016. A neighbor joining (N-J) phylogenetic tree showed 2 main groups of breeds: Pirenaica, Bruna dels Pirineus, and Rubia Gallega on the one hand and Avileña-Negra Ibérica, Morucha, and Retinta on the other. In turn, Asturiana de los Valles occupied an independent and intermediate position. A principal component analysis (PCA) applied to a distance matrix based on marker identity by state, in which the first 2 axes explained up to 17.3% of the variance, showed a grouping of animals that was similar to the one observed in the N-J tree. Finally, a cluster analysis for ancestries allowed assigning all the individuals to the breed they belong to, although it revealed some degree of admixture among breeds. Our results indicate large within-breed diversity and a low degree of divergence among the autochthonous Spanish beef cattle breeds studied. Both N-J and PCA groupings fit quite well to the ancestral trunks from which the Spanish beef cattle breeds were supposed to derive.

Longitudinal data analysis of polymorphisms in the κ-casein and β-lactoglobulin genes shows differential effects along the trajectory of the lactation curve in tropical dairy goats

The κ-casein (CSN-3) and β-lactoglobulin (BLG) genes are extensively polymorphic in ruminants. Several association studies have estimated the effects of polymorphisms in these genes on milk yield, milk composition, and cheese-manufacturing properties. Usually, these results are based on production integrated over the lactation curve or on cross-sectional studies at specific days in milk (DIM). However, as differential expression of milk protein genes occurs over lactation, the effect of the polymorphisms may change over time. In this study, we fitted a mixed-effects regression model to test-day records of milk yield and milk quality traits (fat, protein, and total solids yields) from Colombian tropical dairy goats. We used the well-characterized A/B polymorphisms in the CSN-3 and BLG genes. We argued that this approach provided more efficient estimators than cross-sectional designs, given the same number and pattern of observations, and allowed exclusion of between-subject variation from model error. The BLG genotype AA showed a greater performance than the BB genotype for all traits along the whole lactation curve, whereas the heterozygote showed an intermediate performance. We observed no such constant pattern for the CSN-3 gene between the AA homozygote and the heterozygote (the BB genotype was absent from the sample). The differences among the genotypic effects of the BLG and the CSN-3 polymorphisms were statistically significant during peak and mid lactation (around 40-160 DIM) for the BLG gene and only for mid lactation (80-145 DIM) for the CSN-3 gene. We also estimated the additive and dominant effects of the BLG locus. The locus showed a statistically significant additive behavior along the whole lactation trajectory for all quality traits, whereas for milk yield the effect was not significant at later stages. In turn, we detected a statistically significant dominance effect only for fat yield in the early and peak stages of lactation (at about 1-45 DIM). The longitudinal analysis of test-day records allowed us to estimate the differential effects of polymorphisms along the lactation curve, pointing toward stages that could be affected by the gene.

Estimation of the environmental dam-offspring correlation in beef cattle

A long standing controversy in animal breeding is related to the strong negative estimates of the direct-maternal genetic correlation obtained when fitting data on maternally influenced traits. In this article, we focused on a model that introduces a new correlation parameter among dam-offspring records. The extant theory allows estimation of the parameter when dams have at most a single offspring. Our goal was to develop an inferential procedure in a more general setting. To do so, we applied a Bayesian approach and we showed that the estimation could be accomplished by introducing a Markov chain Monte Carlo (MCMC) step embedded into a regular Gibbs sampler program. The procedure was implemented by means of an MCMC algorithm known as the Griddy-Gibbs sampler, and a Fortran 90 library was created to accomplish the task. The computer program is available from http://www.agro.uba.ar/catedras/mg_animal/software/RDBLK. With this tool at hand, we applied the inferential procedure to weaning weight records on beef cattle calves from an Argentinean Hereford herd, and we estimated the marginal distribution of the environmental dam-offspring correlation parameter. The distribution was unimodal and symmetric with a mean value of -0.14 (±0.03) and a 95% high posterior density interval between -0.20 and -0.07, indicating that the model placed a huge mass on negative values of the parameter. Noticeably, the magnitude of the direct-maternal genetic correlation diminished from -0.61 to -0.37 with respect to the standard maternal animal model. This result reinforces the idea that environmental covariances among dam-offspring records may bias the estimate of the direct-maternal genetic correlation.

Polynomial order selection in random regression models via penalizing adaptively the likelihood

Orthogonal Legendre polynomials (LP) are used to model the shape of additive genetic and permanent environmental effects in random regression models (RRM). Frequently, the Akaike (AIC) and the Bayesian (BIC) information criteria are employed to select LP order. However, it has been theoretically shown that neither AIC nor BIC is simultaneously optimal in terms of consistency and efficiency. Thus, the goal was to introduce a method, 'penalizing adaptively the likelihood' (PAL), as a criterion to select LP order in RRM. Four simulated data sets and real data (60,513 records, 6675 Colombian Holstein cows) were employed. Nested models were fitted to the data, and AIC, BIC and PAL were calculated for all of them. Results showed that PAL and BIC identified with probability of one the true LP order for the additive genetic and permanent environmental effects, but AIC tended to favour over parameterized models. Conversely, when the true model was unknown, PAL selected the best model with higher probability than AIC. In the latter case, BIC never favoured the best model. To summarize, PAL selected a correct model order regardless of whether the 'true' model was within the set of candidates.

Accounting for unknown foster dams in the genetic evaluation of embryo transfer progeny

Animals born by embryo transfer (ET) are usually not included in the genetic evaluation of beef cattle for preweaning growth if the recipient dam is unknown. This is primarily to avoid potential bias in the estimation of the unknown age of dam. We present a method that allows including records of calves with unknown age of dam. Assumptions are as follows: (i) foster cows belong to the same breed being evaluated, (ii) there is no correlation between the breeding value (BV) of the calf and the maternal BV of the recipient cow, and (iii) cows of all ages are used as recipients. We examine the issue of bias for the fixed level of unknown age of dam (AOD) and propose an estimator of the effect based on classical measurement error theory (MEM) and a Bayesian approach. Using stochastic simulation under random mating or selection, the MEM estimating equations were compared with BLUP in two situations as follows: (i) full information (FI); (ii) missing AOD information on some dams. Predictions of breeding value (PBV) from the FI situation had the smallest empirical average bias followed by PBV obtained without taking measurement error into account. In turn, MEM displayed the highest bias, although the differences were small. On the other hand, MEM showed the smallest MSEP, for either random mating or selection, followed by FI, whereas ignoring measurement error produced the largest MSEP. As a consequence from the smallest MSEP with a relatively small bias, empirical accuracies of PBV were larger for MEM than those for full information, which in turn showed larger accuracies than the situation ignoring measurement error. It is concluded that MEM equations are a useful alternative for analysing weaning weight data when recipient cows are unknown, as it mitigates the effects of bias in AOD by decreasing MSEP.