Ole F. Christensen

Genomic prediction when some animals are not genotyped

BackgroundThe use of genomic selection in breeding programs may increase the rate of genetic improvement, reduce the generation time, and provide higher accuracy of estimated breeding values (EBVs). A number of different methods have been developed for genomic prediction of breeding values, but many of them assume that all animals have been genotyped. In practice, not all animals are genotyped, and the methods have to be adapted to this situation.ResultsIn this paper we provide an extension of a linear mixed model method for genomic prediction to the situation with non-genotyped animals. The model specifies that a breeding value is the sum of a genomic and a polygenic genetic random effect, where genomic genetic random effects are correlated with a genomic relationship matrix constructed from markers and the polygenic genetic random effects are correlated with the usual relationship matrix. The extension of the model to non-genotyped animals is made by using the pedigree to derive an extension of the genomic relationship matrix to non-genotyped animals. As a result, in the extended model the estimated breeding values are obtained by blending the information used to compute traditional EBVs and the information used to compute purely genomic EBVs. Parameters in the model are estimated using average information REML and estimated breeding values are best linear unbiased predictions (BLUPs). The method is illustrated using a simulated data set.ConclusionsThe extension of the method to non-genotyped animals presented in this paper makes it possible to integrate all the genomic, pedigree and phenotype information into a one-step procedure for genomic prediction. Such a one-step procedure results in more accurate estimated breeding values and has the potential to become the standard tool for genomic prediction of breeding values in future practical evaluations in pig and cattle breeding.

Genomic relationships and speciation times of human, chimpanzee, and gorilla inferred from a coalescent hidden Markov model.

The genealogical relationship of human, chimpanzee, and gorilla varies along the genome. We develop a hidden Markov model (HMM) that incorporates this variation and relate the model parameters to population genetics quantities such as speciation times and ancestral population sizes. Our HMM is an analytically tractable approximation to the coalescent process with recombination, and in simulations we see no apparent bias in the HMM estimates. We apply the HMM to four autosomal contiguous human–chimp–gorilla–orangutan alignments comprising a total of 1.9 million base pairs. We find a very recent speciation time of human–chimp (4.1 ± 0.4 million years), and fairly large ancestral effective population sizes (65,000 ± 30,000 for the human–chimp ancestor and 45,000 ± 10,000 for the human–chimp–gorilla ancestor). Furthermore, around 50% of the human genome coalesces with chimpanzee after speciation with gorilla. We also consider 250,000 base pairs of X-chromosome alignments and find an effective population size much smaller than 75% of the autosomal effective population sizes. Finally, we find that the rate of transitions between different genealogies correlates well with the region-wide present-day human recombination rate, but does not correlate with the fine-scale recombination rates and recombination hot spots, suggesting that the latter are evolutionarily transient.

Estimating Additive and Non-Additive Genetic Variances and Predicting Genetic Merits Using Genome-Wide Dense Single Nucleotide Polymorphism Markers

Non-additive genetic variation is usually ignored when genome-wide markers are used to study the genetic architecture and genomic prediction of complex traits in human, wild life, model organisms or farm animals. However, non-additive genetic effects may have an important contribution to total genetic variation of complex traits. This study presented a genomic BLUP model including additive and non-additive genetic effects, in which additive and non-additive genetic relation matrices were constructed from information of genome-wide dense single nucleotide polymorphism (SNP) markers. In addition, this study for the first time proposed a method to construct dominance relationship matrix using SNP markers and demonstrated it in detail. The proposed model was implemented to investigate the amounts of additive genetic, dominance and epistatic variations, and assessed the accuracy and unbiasedness of genomic predictions for daily gain in pigs. In the analysis of daily gain, four linear models were used: 1) a simple additive genetic model (MA), 2) a model including both additive and additive by additive epistatic genetic effects (MAE), 3) a model including both additive and dominance genetic effects (MAD), and 4) a full model including all three genetic components (MAED). Estimates of narrow-sense heritability were 0.397, 0.373, 0.379 and 0.357 for models MA, MAE, MAD and MAED, respectively. Estimated dominance variance and additive by additive epistatic variance accounted for 5.6% and 9.5% of the total phenotypic variance, respectively. Based on model MAED, the estimate of broad-sense heritability was 0.506. Reliabilities of genomic predicted breeding values for the animals without performance records were 28.5%, 28.8%, 29.2% and 29.5% for models MA, MAE, MAD and MAED, respectively. In addition, models including non-additive genetic effects improved unbiasedness of genomic predictions.

An Introduction to Model-Based Geostatistics

The term geostatistics identifies the part of spatial statistics which is concerned with continuous spatial variation, in the following sense. The scientific focus is to study a spatial phenomenon, s(x)say, which exists throughout a continuous spatial region A ⊂ ℝ2 and can be treated as if it were a realisation of a stochastic process S(·) = {S(x): x ∈ A}. In general, S(·) is not directly observable. Instead, the available data consist of measurements Y 1,..., Y n taken at locations x 1,..., x n sampled within A, and Y i is a noisy version of S(x i ). We shall assume either that the sampling design for x 1,..., x n is deterministic or that it is stochastic but independent of the process S(·), and all analyses are carried out conditionally on x 1,...,x n .

Single-step methods for genomic evaluation in pigs

Genetic evaluation based on information from phenotypes, pedigree and markers can be implemented using a recently developed single-step method. In this paper we compare accuracies of predicted breeding values for daily gain and feed conversion ratio (FCR) in Danish Duroc pigs obtained from different versions of single-step methods, the traditional pedigree-based method and the genomic BLUP (GBLUP) method. In particular, we present a single-step method with an adjustment of the genomic relationship matrix so that it is compatible to the pedigree-based relationship matrix. Comparisons are made for both genotyped and non-genotyped animals and univariate and bivariate models. The results show that the three methods with marker information (two single-step methods and GBLUP) produce more accurate predictions of genotyped animals than the pedigree-based method. In addition, single-step methods provide more accurate predictions for non-genotyped animals. The results also show that the single-step method with adjusted genomic relationship matrix produce more accurate predictions than the original single-step method. Finally, the results for the bivariate analyses show a somewhat improved accuracy and reduced inflation of predictions for FCR for the two single-step methods compared with the univariate analyses. The conclusions are: first, the methods with marker information improve prediction compared with the pedigree-based method; second, a single-step method, contrary to GBLUP, provides improved predictions for all animals compared to the pedigree-based method; and third, a single-step method should be used with an adjustment of the genomic relationship matrix.

Genomic prediction for Nordic Red Cattle using one-step and selection index blending

This study investigated the accuracy of direct genomic breeding values (DGV) using a genomic BLUP model, genomic enhanced breeding values (GEBV) using a one-step blending approach, and GEBV using a selection index blending approach for 15 traits of Nordic Red Cattle. The data comprised 6,631 bulls of which 4,408 bulls were genotyped using Illumina Bovine SNP50 BeadChip (Illumina, San Diego, CA). To validate reliability of genomic predictions, about 20% of the youngest genotyped bulls were taken as test data set. Deregressed proofs (DRP) were used as response variables for genomic predictions. Reliabilities of genomic predictions in the validation analyses were measured as squared correlations between DRP and genomic predictions corrected for reliability of DRP, based on the bulls in the test data sets. A set of weighting (scaling) factors was used to construct the combined relationship matrix among genotyped and nongenotyped bulls for one-step blending, and to scale DGV and its expected reliability in the selection index blending. Weighting (scaling) factors had a small influence on reliabilities of GEBV, but a large influence on the variation of GEBV. Based on the validation analyses, averaged over the 15 traits, the reliability of DGV for bulls without daughter records was 11.0 percentage points higher than the reliability of conventional pedigree index. Further gain of 0.9 percentage points was achieved by combining information from conventional pedigree index using the selection index blending, and gain of 1.3 percentage points was achieved by combining information of genotyped and nongenotyped bulls simultaneously applying the one-step blending. These results indicate that genomic selection can greatly improve the accuracy of preselection for young bulls in Nordic Red population, and the one-step blending approach is a good alternative to predict GEBV in practical genetic evaluation program.

Comparison on genomic predictions using three GBLUP methods and two single-step blending methods in the Nordic Holstein population

BackgroundA single-step blending approach allows genomic prediction using information of genotyped and non-genotyped animals simultaneously. However, the combined relationship matrix in a single-step method may need to be adjusted because marker-based and pedigree-based relationship matrices may not be on the same scale. The same may apply when a GBLUP model includes both genomic breeding values and residual polygenic effects. The objective of this study was to compare single-step blending methods and GBLUP methods with and without adjustment of the genomic relationship matrix for genomic prediction of 16 traits in the Nordic Holstein population.MethodsThe data consisted of de-regressed proofs (DRP) for 5 214 genotyped and 9 374 non-genotyped bulls. The bulls were divided into a training and a validation population by birth date, October 1, 2001. Five approaches for genomic prediction were used: 1) a simple GBLUP method, 2) a GBLUP method with a polygenic effect, 3) an adjusted GBLUP method with a polygenic effect, 4) a single-step blending method, and 5) an adjusted single-step blending method. In the adjusted GBLUP and single-step methods, the genomic relationship matrix was adjusted for the difference of scale between the genomic and the pedigree relationship matrices. A set of weights on the pedigree relationship matrix (ranging from 0.05 to 0.40) was used to build the combined relationship matrix in the single-step blending method and the GBLUP method with a polygenetic effect.ResultsAveraged over the 16 traits, reliabilities of genomic breeding values predicted using the GBLUP method with a polygenic effect (relative weight of 0.20) were 0.3% higher than reliabilities from the simple GBLUP method (without a polygenic effect). The adjusted single-step blending and original single-step blending methods (relative weight of 0.20) had average reliabilities that were 2.1% and 1.8% higher than the simple GBLUP method, respectively. In addition, the GBLUP method with a polygenic effect led to less bias of genomic predictions than the simple GBLUP method, and both single-step blending methods yielded less bias of predictions than all GBLUP methods.ConclusionsThe single-step blending method is an appealing approach for practical genomic prediction in dairy cattle. Genomic prediction from the single-step blending method can be improved by adjusting the scale of the genomic relationship matrix.

Deregressed EBV as the response variable yield more reliable genomic predictions than traditional EBV in pure-bred pigs

BackgroundGenomic selection can be implemented by a multi-step procedure, which requires a response variable and a statistical method. For pure-bred pigs, it was hypothesised that deregressed estimated breeding values (EBV) with the parent average removed as the response variable generate higher reliabilities of genomic breeding values than EBV, and that the normal, thick-tailed and mixture-distribution models yield similar reliabilities.MethodsReliabilities of genomic breeding values were estimated with EBV and deregressed EBV as response variables and under the three statistical methods, genomic BLUP, Bayesian Lasso and MIXTURE. The methods were examined by splitting data into a reference data set of 1375 genotyped animals that were performance tested before October 2008, and 536 genotyped validation animals that were performance tested after October 2008. The traits examined were daily gain and feed conversion ratio.ResultsUsing deregressed EBV as the response variable yielded 18 to 39% higher reliabilities of the genomic breeding values than using EBV as the response variable. For daily gain, the increase in reliability due to deregression was significant and approximately 35%, whereas for feed conversion ratio it ranged between 18 and 39% and was significant only when MIXTURE was used. Genomic BLUP, Bayesian Lasso and MIXTURE had similar reliabilities.ConclusionsDeregressed EBV is the preferred response variable, whereas the choice of statistical method is less critical for pure-bred pigs. The increase of 18 to 39% in reliability is worthwhile, since the reliabilities of the genomic breeding values directly affect the returns from genomic selection.

Allele coding in genomic evaluation

BackgroundGenomic data are used in animal breeding to assist genetic evaluation. Several models to estimate genomic breeding values have been studied. In general, two approaches have been used. One approach estimates the marker effects first and then, genomic breeding values are obtained by summing marker effects. In the second approach, genomic breeding values are estimated directly using an equivalent model with a genomic relationship matrix. Allele coding is the method chosen to assign values to the regression coefficients in the statistical model. A common allele coding is zero for the homozygous genotype of the first allele, one for the heterozygote, and two for the homozygous genotype for the other allele. Another common allele coding changes these regression coefficients by subtracting a value from each marker such that the mean of regression coefficients is zero within each marker. We call this centered allele coding. This study considered effects of different allele coding methods on inference. Both marker-based and equivalent models were considered, and restricted maximum likelihood and Bayesian methods were used in inference.ResultsTheoretical derivations showed that parameter estimates and estimated marker effects in marker-based models are the same irrespective of the allele coding, provided that the model has a fixed general mean. For the equivalent models, the same results hold, even though different allele coding methods lead to different genomic relationship matrices. Calculated genomic breeding values are independent of allele coding when the estimate of the general mean is included into the values. Reliabilities of estimated genomic breeding values calculated using elements of the inverse of the coefficient matrix depend on the allele coding because different allele coding methods imply different models. Finally, allele coding affects the mixing of Markov chain Monte Carlo algorithms, with the centered coding being the best.ConclusionsDifferent allele coding methods lead to the same inference in the marker-based and equivalent models when a fixed general mean is included in the model. However, reliabilities of genomic breeding values are affected by the allele coding method used. The centered coding has some numerical advantages when Markov chain Monte Carlo methods are used.

Monte Carlo Maximum Likelihood in Model-Based Geostatistics

When using a model-based approach to geostatistical problems, often, due to the complexity of the models, inference relies on Markov chain Monte Carlo methods. This article focuses on the generalized linear spatial models, and demonstrates that parameter estimation and model selection using Markov chain Monte Carlo maximum likelihood is a feasible and very useful technique. A dataset of radionuclide concentrations on Rongelap Island is used to illustrate the techniques. For this dataset we demonstrate that the log-link function is not a good choice, and that there exists additional nonspatial variation which cannot be attributed to the Poisson error distribution. We also show that the interpretation of this additional variation as either micro-scale variation or measurement error has a significant impact on predictions. The techniques presented in this article would also be useful for other types of geostatistical models.