Background Single-step genomic best linear unbiased prediction (SSGBLUP) is a comprehensive method for genomic prediction. Point estimates of marker effects from SSGBLUP are often used for genome-wide association studies (GWAS) without a formal framework of hypothesis testing. Our objective was to implement p-values for single-marker GWAS studies within the single-step GWAS (SSGWAS) framework by deriving computational algorithms and procedures, and by applying these to a large beef cattle population. Methods P-values were obtained based on the prediction error (co)variances for single nucleotide polymorphisms (SNPs), which were obtained from the prediction error (co)variances of genomic predictions based on the inverse of the coefficient matrix and formulas to estimate SNP effects. Results Computation of p-values took a negligible time for a dataset with almost 2 million animals in the pedigree and 1424 genotyped sires, and no inflation of statistics was observed. The SNPs that passed the Bonferroni threshold of 10 −5.9 were the same as those that explained the highest proportion of additive genetic variance, but even at the same significance levels and effects, some of them explained less genetic variance due to lower allele frequency. Conclusions The use of a p-value for SSGWAS is a very general and efficient strategy to identify quantitative trait loci (QTL). It can be used for complex datasets such as those used in animal breeding, where only a proportion of the pedigreed animals are genotyped.

We prove uniform weighted high frequency estimates for the resolvent of the Laplace-Beltrami operator on connected infinite volume Riemannian manifolds under some natural assumptions on the metric on the ends of the manifold. This extends previous results by Burq [3] and Vodev [8].

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The objectives of this work were to assess alternative linear reaction norm (RN) models for genetic evaluation of Angus cattle in Brazil. That is, we investigated the interaction between genotypes and continuous descriptors of the environmental variation to examine evidence of genotype by environment interaction (G×E) in post-weaning BW gain (PWG) and to compare the environmental sensitivity of national and imported Angus sires. Data were collected by the Brazilian Angus Improvement Program from 1974 to 2005 and consisted of 63,098 records and a pedigree file with 95,896 animals. Six models were implemented using Bayesian inference and compared using the Deviance Information Criterion (DIC). The simplest model was M(1), a traditional animal model, which showed the largest DIC and hence the poorest fit when compared with the 4 alternative RN specifications accounting for G×E. In M(2), a 2-step procedure was implemented using the contemporary group posterior means of M(1) as the environmental gradient, ranging from -92.6 to +265.5 kg. Moreover, the benefits of jointly estimating all parameters in a 1-step approach were demonstrated by M(3). Additionally, we extended M(3) to allow for residual heteroskedasticity using an exponential function (M(4)) and the best fitting (smallest DIC) environmental classification model (M(5)) specification. Finally, M(6) added just heteroskedastic residual variance to M(1). Heritabilities were less at harsh environments and increased with the improvement of production conditions for all RN models. Rank correlations among genetic merit predictions obtained by M(1) and by the best fitting RN models M(3) (homoskedastic) and M(5) (heteroskedastic) at different environmental levels ranged from 0.79 and 0.81, suggesting biological importance of G×E in Brazilian Angus PWG. These results suggest that selection progress could be optimized by adopting environment-specific genetic merit predictions. The PWG environmental sensitivity of imported North American origin bulls (0.046 ± 0.009) was significantly larger (P < 0.05) than that of local sires (0.012 ± 0.013). Moreover, PWG of progeny of imported sires exceeded that of native sires in medium and superior production levels. On the other hand, Angus cattle locally selected in Brazil tended to be more robust to environmental changes and hence be more suitable when production environments for potential progeny is uncertain.

Multiple-breed genetic models recently have been demonstrated to account for the heterogenous genetic variances that exist between different beef cattle breed groups. We extend these models to allow for residual heteroskedasticity (heterogeneous residual variances), specified as a function of fixed effects (e.g., sex, breed proportion, breed group heterozygosity) and random effects such as contemporary groups (CG). We additionally specify the residual distributions to be either Gaussian or based on heavier-tailed alternatives such as the Student's t or Slash densities. For each of these three residual densities using either homoskedastic (homogeneous variance) or heteroskedastic error specifications, we analyzed 22,717 postweaning gain records from a Nelore-Hereford population based on a Markov chain Monte Carlo animal model implementation. The heteroskedastic Student's t error model (with estimated df = 7.33 +/- 0.48) was clearly the best-fitting model based on a pseudo-Bayes factor criterion. Breed group heterozygosity and, to a lesser extent, calf sex seemed to be marginally important sources of residual heteroskedasticity. Specifically, the residual variance in F1 animals was estimated to be 0.70 +/- 0.16 times that for purebreds, whereas the male residual variance was estimated to be 1.13 +/- 0.09 times that for females. The CG effects were important random sources of residual heteroskedasticity (i.e., the coefficient of variation of CG-specific residual variances was estimated to be 0.72 +/- 0.06). Purebred Nelores were estimated to have a larger genetic variance (124.84 +/- 21.75 kg2) compared with Herefords (40.89 +/- 6.70 kg2) under the heteroskedastic Student's t error model; however, the converse was observed from results based on a homoskedastic Student's t error model (46.24 +/- 10.90 kg2 and 60.11 +/- 8.54 kg2, respectively). These results indicate that allowing for robustness to outliers and accounting for heteroskedasticity of residual variances has potentially important implications for variance component and genetic parameter estimates from data on multiple-breed populations.