Slope of predictions.

<p>Dashed lines and small squares represent fixed weights with <i>N</i> = 7,000. Solid lines and small circles represent fixed weights with <i>N</i> = 14,487. The large square represents the sampled weight for <i>N</i> = 7,000. The large circle represents the sampled weight for <i>N</i> = 14,487. The optimum value is one.</p

Correlation of predictions.

<p>Dashed lines and small squares represent fixed weights with <i>N</i> = 7,000. Solid lines and small circles represent fixed weights with <i>N</i> = 14,487. The large square represents the sampled weight for <i>N</i> = 7,000. The large circle represents the sampled weight for <i>N</i> = 14,487.</p

Reparameterization of the Bayesian RKHS (reproducing kernel Hilbert spaces) and the G-BLUP. Adapted from De los Campos et al. [27].

<p>Reparameterization of the Bayesian RKHS (reproducing kernel Hilbert spaces) and the G-BLUP. Adapted from De los Campos et al. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0093424#pone.0093424-DelosCampos2" target="_blank">[27]</a>.</p

Combining Genomic and Genealogical Information in a Reproducing Kernel Hilbert Spaces Regression Model for Genome-Enabled Predictions in Dairy Cattle

<div><p>Genome-enhanced genotypic evaluations are becoming popular in several livestock species. For this purpose, the combination of the pedigree-based relationship matrix with a genomic similarities matrix between individuals is a common approach. However, the weight placed on each matrix has been so far established with <i>ad hoc</i> procedures, without formal estimation thereof. In addition, when using marker- and pedigree-based relationship matrices together, the resulting combined relationship matrix needs to be adjusted to the same scale in reference to the base population. This study proposes a semi-parametric Bayesian method for combining marker- and pedigree-based information on genome-enabled predictions. A kernel matrix from a reproducing kernel Hilbert spaces regression model was used to combine genomic and genealogical information in a semi-parametric scenario, avoiding inversion and adjustment complications. In addition, the weights on marker- <i>versus</i> pedigree-based information were inferred from a Bayesian model with Markov chain Monte Carlo. The proposed method was assessed involving a large number of SNPs and a large reference population. Five phenotypes, including production and type traits of dairy cattle were evaluated. The reliability of the genome-based predictions was assessed using the correlation, regression coefficient and mean squared error between the predicted and observed values. The results indicated that when a larger weight was given to the pedigree-based relationship matrix the correlation coefficient was lower than in situations where more weight was given to genomic information. Importantly, the posterior means of the inferred weight were near the maximum of 1. The behavior of the regression coefficient and the mean squared error was similar to the performance of the correlation, that is, more weight to the genomic information provided a regression coefficient closer to one and a smaller mean squared error. Our results also indicated a greater accuracy of genomic predictions when using a large reference population.</p></div

Mean squared error of predictions.

<p>Dashed lines and small squares represent fixed weights with <i>N</i> = 7,000. Solid lines and small circles represent fixed weights with <i>N</i> = 14,487. The large square represents the sampled weight for <i>N</i> = 7,000. The large circle represents the sampled weight for <i>N</i> = 14,487.</p

Approximated function of for different with <i>N</i> = 7,000 and <i>N</i> = 14,487.

<p>Approximated function of for different with <i>N</i> = 7,000 and <i>N</i> = 14,487.</p

Posterior mean of the inferred lambda for each trait evaluated and reference population size.

<p>Posterior mean of the inferred lambda for each trait evaluated and reference population size.</p

Rates of change in inbreeding, molecular homozygosity, coancestry and molecular similarity per year (Δ<i>F</i>_(y), Δ<i>H</i>_(y), Δ<i>f</i>_(y) and Δ<i>S</i>_(y)), respectively, and per generation (Δ<i>F</i>, Δ<i>H</i>, Δ<i>f</i> and Δ<i>S</i>) using different sources of information, and estimates of effective population sizes obtained from Δ<i>F</i> (<i>N_e</i>_<i>F</i>), Δ<i>H</i> (<i>N_e</i>_<i>H</i>), Δ<i>f</i> (<i>N_e</i>_<i>f</i>) and from Δ<i>S</i> (<i>N_e</i>_<i>S</i>).

<p>Rates of change in inbreeding, molecular homozygosity, coancestry and molecular similarity per year (Δ<i>F</i>_(y), Δ<i>H</i>_(y), Δ<i>f</i>_(y) and Δ<i>S</i>_(y)), respectively, and per generation (Δ<i>F</i>, Δ<i>H</i>, Δ<i>f</i> and Δ<i>S</i>) using different sources of information, and estimates of effective population sizes obtained from Δ<i>F</i> (<i>N_e</i>_<i>F</i>), Δ<i>H</i> (<i>N_e</i>_<i>H</i>), Δ<i>f</i> (<i>N_e</i>_<i>f</i>) and from Δ<i>S</i> (<i>N_e</i>_<i>S</i>).</p

Intercept (<i>a</i>), regression coefficient (<i>b</i>) and correlation (<i>R</i>) between different estimates.

<p><i>F</i>_<i>PED</i>: pedigree-based inbreeding; <i>F</i>_<i>ROH</i>: inbreeding based on ROHs; <i>H</i>_<i>SNP</i>: SNP-by-SNP based homozygosity; <i>f</i>_<i>PED</i>: pedigree-based coancestry; <i>f</i>_<i>SEG</i>: coancestry based on IBD segments; <i>S</i>_<i>SNP</i>: SNP-by-SNP based similarity</p><p>Intercept (<i>a</i>), regression coefficient (<i>b</i>) and correlation (<i>R</i>) between different estimates.</p

Distribution of genotyped animals by year of birth.

<p>Distribution of genotyped animals by year of birth.</p