Effects of data structure on the estimation of covariance functions to describe genotype by environment interactions in a reaction norm model

Covariance functions have been proposed to predict breeding values and genetic (co)variances as a function of phenotypic within herd-year averages (environmental parameters) to include genotype by environment interaction. The objective of this paper was to investigate the influence of definition of environmental parameters and non-random use of sires on expected breeding values and estimated genetic variances across environments. Breeding values were simulated as a linear function of simulated herd effects. The definition of environmental parameters hardly influenced the results. In situations with random use of sires, estimated genetic correlations between the trait expressed in different environments were 0.93, 0.93 and 0.97 while simulated at 0.89 and estimated genetic variances deviated up to 30% from the simulated values. Non random use of sires, poor genetic connectedness and small herd size had a large impact on the estimated covariance functions, expected breeding values and calculated environmental parameters. Estimated genetic correlations between a trait expressed in different environments were biased upwards and breeding values were more biased when genetic connectedness became poorer and herd composition more diverse. The best possible solution at this stage is to use environmental parameters combining large numbers of animals per herd, while losing some information on genotype by environment interaction in the data.     

Habitat overlap between predatory benthic fish and their invertebrate prey in streams: the relative influence of spatial and temporal factors on predation risk

1. The spatial heterogeneity of ecosystems as well as temporal activity patterns of organisms can have far‐reaching effects on predator–prey relationships. We hypothesised that spatiotemporal constraints in mesohabitat use by benthic fish predators would reduce habitat overlap with benthic invertebrates and lead to mesohabitat‐specific predation risks. 2. We analysed the spatiotemporal activity patterns of two small‐bodied benthivorous fishes, gudgeon (Gobio gobio) and stone loach (Barbatula barbatula), and of benthic invertebrates in a small temperate stream during three 24‐h field experiments. By applying a novel method of field video observation, we monitored the spatiotemporal foraging behaviour of the fish in their natural environment. A parallel analysis of invertebrate mesohabitat use by means of small area Hess sampling allowed a direct estimation of habitat overlap at a pool–riffle scale. 3. Gudgeon showed a dominant spatial activity pattern preferring pools at all times of day, whereas stone loach used both mesohabitats but with a distinct temporal (nocturnal) activity pattern. The patterns of residence were not identical with those of active foraging. Invertebrate community composition differed significantly between mesohabitats but not between times of day. More than half of the total dissimilarity between pools and riffles was accounted for by six invertebrate taxa. Five of these were subject to higher fish predation in pools than in riffles. The total prey consumption of the two fish species together in pools was about three times as high as in riffles. Trophic niche breadth of stone loach and thus its predation range was broader than that of gudgeon. 4. These results indicate that the potential predation risk for stream invertebrates depends on the combination of spatial and temporal patterns of both predator and prey. Given the distinct differences in predation risk found between pools and riffles, we conclude that spatial heterogeneity at the mesohabitat scale can influence mechanisms and consequences of selective predation. We also suggest that the analysis of spatiotemporal predator–prey relationships should not be based on the premise that the main residence habitat and active foraging habitat of a predator are identical.     

Long-term response to genomic selection: effects of estimation method and reference population structure for different genetic architectures

Background

Genomic selection has become an important tool in the genetic improvement of animals and plants. The objective of this study was to investigate the impacts of breeding value estimation method, reference population structure, and trait genetic architecture, on long-term response to genomic selection without updating marker effects.

Methods

Three methods were used to estimate genomic breeding values: a BLUP method with relationships estimated from genome-wide markers (GBLUP), a Bayesian method, and a partial least squares regression method (PLSR). A shallow (individuals from one generation) or deep reference population (individuals from five generations) was used with each method. The effects of the different selection approaches were compared under four different genetic architectures for the trait under selection. Selection was based on one of the three genomic breeding values, on pedigree BLUP breeding values, or performed at random. Selection continued for ten generations.

Results

Differences in long-term selection response were small. For a genetic architecture with a very small number of three to four quantitative trait loci (QTL), the Bayesian method achieved a response that was 0.05 to 0.1 genetic standard deviation higher than other methods in generation 10. For genetic architectures with approximately 30 to 300 QTL, PLSR (shallow reference) or GBLUP (deep reference) had an average advantage of 0.2 genetic standard deviation over the Bayesian method in generation 10. GBLUP resulted in 0.6% and 0.9% less inbreeding than PLSR and BM and on average a one third smaller reduction of genetic variance. Responses in early generations were greater with the shallow reference population while long-term response was not affected by reference population structure.

Conclusions

The ranking of estimation methods was different with than without selection. Under selection, applying GBLUP led to lower inbreeding and a smaller reduction of genetic variance while a similar response to selection was achieved. The reference population structure had a limited effect on long-term accuracy and response. Use of a shallow reference population, most closely related to the selection candidates, gave early benefits while in later generations, when marker effects were not updated, the estimation of marker effects based on a deeper reference population did not pay off.     

Right-hand-side updating for fast computing of genomic breeding values

Background

Since both the number of SNPs (single nucleotide polymorphisms) used in genomic prediction and the number of individuals used in training datasets are rapidly increasing, there is an increasing need to improve the efficiency of genomic prediction models in terms of computing time and memory (RAM) required.

Methods

In this paper, two alternative algorithms for genomic prediction are presented that replace the originally suggested residual updating algorithm, without affecting the estimates. The first alternative algorithm continues to use residual updating, but takes advantage of the characteristic that the predictor variables in the model (i.e. the SNP genotypes) take only three different values, and is therefore termed “improved residual updating”. The second alternative algorithm, here termed “right-hand-side updating” (RHS-updating), extends the idea of improved residual updating across multiple SNPs. The alternative algorithms can be implemented for a range of different genomic predictions models, including random regression BLUP (best linear unbiased prediction) and most Bayesian genomic prediction models. To test the required computing time and RAM, both alternative algorithms were implemented in a Bayesian stochastic search variable selection model.

Results

Compared to the original algorithm, the improved residual updating algorithm reduced CPU time by 35.3 to 43.3%, without changing memory requirements. The RHS-updating algorithm reduced CPU time by 74.5 to 93.0% and memory requirements by 13.1 to 66.4% compared to the original algorithm.

Conclusions

The presented RHS-updating algorithm provides an interesting alternative to reduce both computing time and memory requirements for a range of genomic prediction models.     

Accuracy of imputation to whole-genome sequence data in Holstein Friesian cattle

Background

The use of whole-genome sequence data can lead to higher accuracy in genome-wide association studies and genomic predictions. However, to benefit from whole-genome sequence data, a large dataset of sequenced individuals is needed. Imputation from SNP panels, such as the Illumina BovineSNP50 BeadChip and Illumina BovineHD BeadChip, to whole-genome sequence data is an attractive and less expensive approach to obtain whole-genome sequence genotypes for a large number of individuals than sequencing all individuals. Our objective was to investigate accuracy of imputation from lower density SNP panels to whole-genome sequence data in a typical dataset for cattle.

Methods

Whole-genome sequence data of chromosome 1 (1737 471 SNPs) for 114 Holstein Friesian bulls were used. Beagle software was used for imputation from the BovineSNP50 (3132 SNPs) and BovineHD (40 492 SNPs) beadchips. Accuracy was calculated as the correlation between observed and imputed genotypes and assessed by five-fold cross-validation. Three scenarios S40, S60 and S80 with respectively 40%, 60%, and 80% of the individuals as reference individuals were investigated.

Results

Mean accuracies of imputation per SNP from the BovineHD panel to sequence data and from the BovineSNP50 panel to sequence data for scenarios S40 and S80 ranged from 0.77 to 0.83 and from 0.37 to 0.46, respectively. Stepwise imputation from the BovineSNP50 to BovineHD panel and then to sequence data for scenario S40 improved accuracy per SNP to 0.65 but it varied considerably between SNPs.

Conclusions

Accuracy of imputation to whole-genome sequence data was generally high for imputation from the BovineHD beadchip, but was low from the BovineSNP50 beadchip. Stepwise imputation from the BovineSNP50 to the BovineHD beadchip and then to sequence data substantially improved accuracy of imputation. SNPs with a low minor allele frequency were more difficult to impute correctly and the reliability of imputation varied more. Linkage disequilibrium between an imputed SNP and the SNP on the lower density panel, minor allele frequency of the imputed SNP and size of the reference group affected imputation reliability.     

A comparison of principal component regression and genomic REML for genomic prediction across populations

Background

Genomic prediction faces two main statistical problems: multicollinearity and n ≪ p (many fewer observations than predictor variables). Principal component (PC) analysis is a multivariate statistical method that is often used to address these problems. The objective of this study was to compare the performance of PC regression (PCR) for genomic prediction with that of a commonly used REML model with a genomic relationship matrix (GREML) and to investigate the full potential of PCR for genomic prediction.

Methods

The PCR model used either a common or a semi-supervised approach, where PC were selected based either on their eigenvalues (i.e. proportion of variance explained by SNP (single nucleotide polymorphism) genotypes) or on their association with phenotypic variance in the reference population (i.e. the regression sum of squares contribution). Cross-validation within the reference population was used to select the optimum PCR model that minimizes mean squared error. Pre-corrected average daily milk, fat and protein yields of 1609 first lactation Holstein heifers, from Ireland, UK, the Netherlands and Sweden, which were genotyped with 50 k SNPs, were analysed. Each testing subset included animals from only one country, or from only one selection line for the UK.

Results

In general, accuracies of GREML and PCR were similar but GREML slightly outperformed PCR. Inclusion of genotyping information of validation animals into model training (semi-supervised PCR), did not result in more accurate genomic predictions. The highest achievable PCR accuracies were obtained across a wide range of numbers of PC fitted in the regression (from one to more than 1000), across test populations and traits. Using cross-validation within the reference population to derive the number of PC, yielded substantially lower accuracies than the highest achievable accuracies obtained across all possible numbers of PC.

Conclusions

On average, PCR performed only slightly less well than GREML. When the optimal number of PC was determined based on realized accuracy in the testing population, PCR showed a higher potential in terms of achievable accuracy that was not capitalized when PC selection was based on cross-validation. A standard approach for selecting the optimal set of PC in PCR remains a challenge.

Electronic supplementary material

The online version of this article (doi:10.1186/s12711-014-0060-x) contains supplementary material, which is available to authorized users.     

Genomic prediction of breeding values using previously estimated SNP variances

Background

Genomic prediction requires estimation of variances of effects of single nucleotide polymorphisms (SNPs), which is computationally demanding, and uses these variances for prediction. We have developed models with separate estimation of SNP variances, which can be applied infrequently, and genomic prediction, which can be applied routinely.

Methods

SNP variances were estimated with Bayes Stochastic Search Variable Selection (BSSVS) and BayesC. Genome-enhanced breeding values (GEBV) were estimated with RR-BLUP (ridge regression best linear unbiased prediction), using either variances obtained from BSSVS (BLUP-SSVS) or BayesC (BLUP-C), or assuming equal variances for each SNP. Datasets used to estimate SNP variances comprised (1) all animals, (2) 50% random animals (RAN50), (3) 50% best animals (TOP50), or (4) 50% worst animals (BOT50). Traits analysed were protein yield, udder depth, somatic cell score, interval between first and last insemination, direct longevity, and longevity including information from predictors.

Results

BLUP-SSVS and BLUP-C yielded similar GEBV as the equivalent Bayesian models that simultaneously estimated SNP variances. Reliabilities of these GEBV were consistently higher than from RR-BLUP, although only significantly for direct longevity. Across scenarios that used data subsets to estimate GEBV, observed reliabilities were generally higher for TOP50 than for RAN50, and much higher than for BOT50. Reliabilities of TOP50 were higher because the training data contained more ancestors of selection candidates. Using estimated SNP variances based on random or non-random subsets of the data, while using all data to estimate GEBV, did not affect reliabilities of the BLUP models. A convergence criterion of 10⁻⁸ instead of 10⁻¹⁰ for BLUP models yielded similar GEBV, while the required number of iterations decreased by 71 to 90%. Including a separate polygenic effect consistently improved reliabilities of the GEBV, but also substantially increased the required number of iterations to reach convergence with RR-BLUP. SNP variances converged faster for BayesC than for BSSVS.

Conclusions

Combining Bayesian variable selection models to re-estimate SNP variances and BLUP models that use those SNP variances, yields GEBV that are similar to those from full Bayesian models. Moreover, these combined models yield predictions with higher reliability and less bias than the commonly used RR-BLUP model.

Electronic supplementary material

The online version of this article (doi:10.1186/s12711-014-0052-x) contains supplementary material, which is available to authorized users.