G. de los Campos

Semi-parametric genomic-enabled prediction of genetic values using reproducing kernel Hilbert spaces methods

Prediction of genetic values is a central problem in quantitative genetics. Over many decades, such predictions have been successfully accomplished using information on phenotypic records and family structure usually represented with a pedigree. Dense molecular markers are now available in the genome of humans, plants and animals, and this information can be used to enhance the prediction of genetic values. However, the incorporation of dense molecular marker data into models poses many statistical and computational challenges, such as how models can cope with the genetic complexity of multi-factorial traits and with the curse of dimensionality that arises when the number of markers exceeds the number of data points. Reproducing kernel Hilbert spaces regressions can be used to address some of these challenges. The methodology allows regressions on almost any type of prediction sets (covariates, graphs, strings, images, etc.) and has important computational advantages relative to many parametric approaches. Moreover, some parametric models appear as special cases. This article provides an overview of the methodology, a discussion of the problem of kernel choice with a focus on genetic applications, algorithms for kernel selection and an assessment of the proposed methods using a collection of 599 wheat lines evaluated for grain yield in four mega environments.

Reproducing kernel Hilbert spaces regression: a general framework for genetic evaluation.

Reproducing kernel Hilbert spaces (RKHS) methods are widely used for statistical learning in many areas of endeavor. Recently, these methods have been suggested as a way of incorporating dense markers into genetic models. This note argues that RKHS regression provides a general framework for genetic evaluation that can be used either for pedigree- or marker-based regressions and under any genetic model, infinitesimal or not, and additive or not. Most of the standard models for genetic evaluation, such as infinitesimal animal or sire models, and marker-assisted selection models appear as special cases of RKHS methods.

Predictive ability of direct genomic values for lifetime net merit of Holstein sires using selected subsets of single nucleotide polymorphism markers

The objective of the present study was to assess the predictive ability of subsets of single nucleotide polymorphism (SNP) markers for development of low-cost, low-density genotyping assays in dairy cattle. Dense SNP genotypes of 4,703 Holstein bulls were provided by the USDA Agricultural Research Service. A subset of 3,305 bulls born from 1952 to 1998 was used to fit various models (training set), and a subset of 1,398 bulls born from 1999 to 2002 was used to evaluate their predictive ability (testing set). After editing, data included genotypes for 32,518 SNP and August 2003 and April 2008 predicted transmitting abilities (PTA) for lifetime net merit (LNM

Accuracy of direct genomic values derived from imputed single nucleotide polymorphism genotypes in Jersey cattle

), the latter resulting from progeny testing. The Bayesian least absolute shrinkage and selection operator method was used to regress August 2003 PTA on marker covariates in the training set to arrive at estimates of marker effects and direct genomic PTA. The coefficient of determination (R(2)) from regressing the April 2008 progeny test PTA of bulls in the testing set on their August 2003 direct genomic PTA was 0.375. Subsets of 300, 500, 750, 1,000, 1,250, 1,500, and 2,000 SNP were created by choosing equally spaced and highly ranked SNP, with the latter based on the absolute value of their estimated effects obtained from the training set. The SNP effects were re-estimated from the training set for each subset of SNP, and the 2008 progeny test PTA of bulls in the testing set were regressed on corresponding direct genomic PTA. The R(2) values for subsets of 300, 500, 750, 1,000, 1,250, 1,500, and 2,000 SNP with largest effects (evenly spaced SNP) were 0.184 (0.064), 0.236 (0.111), 0.269 (0.190), 0.289 (0.179), 0.307 (0.228), 0.313 (0.268), and 0.322 (0.291), respectively. These results indicate that a low-density assay comprising selected SNP could be a cost-effective alternative for selection decisions and that significant gains in predictive ability may be achieved by increasing the number of SNP allocated to such an assay from 300 or fewer to 1,000 or more.

Genome-enabled prediction of genetic values using radial basis function neural networks

The objective of the present study was to evaluate the predictive ability of direct genomic values for economically important dairy traits when genotypes at some single nucleotide polymorphism (SNP) loci were imputed rather than measured directly. Genotypic data consisted of 42,552 SNP genotypes for each of 1,762 Jersey sires. Phenotypic data consisted of predicted transmitting abilities (PTA) for milk yield, protein percentage, and daughter pregnancy rate from May 2006 for 1,446 sires in the training set and from April 2009 for 316 sires in the testing set. The SNP effects were estimated using the Bayesian least absolute selection and shrinkage operator (LASSO) method with data of sires in the training set, and direct genomic values (DGV) for sires in the testing set were computed by multiplying these estimates by corresponding genotype dosages for sires in the testing set. The mean correlation across traits between DGV (before progeny testing) and PTA (after progeny testing) for sires in the testing set was 70.6% when all 42,552 SNP genotypes were used. When genotypes for 93.1, 96.6, 98.3, or 99.1% of loci were masked and subsequently imputed in the testing set, mean correlations across traits between DGV and PTA were 68.5, 64.8, 54.8, or 43.5%, respectively. When genotypes were also masked and imputed for a random 50% of sires in the training set, mean correlations across traits between DGV and PTA were 65.7, 63.2, 53.9, or 49.5%, respectively. Results of this study indicate that if a suitable reference population with high-density genotypes is available, a low-density chip comprising 3,000 equally spaced SNP may provide approximately 95% of the predictive ability observed with the BovineSNP50 Beadchip (Illumina Inc., San Diego, CA) in Jersey cattle. However, if fewer than 1,500 SNP are genotyped, the accuracy of DGV may be limited by errors in the imputed genotypes of selection candidates.

Obesity and mortality: are the risks declining? Evidence from multiple prospective studies in the United States

The availability of high density panels of molecular markers has prompted the adoption of genomic selection (GS) methods in animal and plant breeding. In GS, parametric, semi-parametric and non-parametric regressions models are used for predicting quantitative traits. This article shows how to use neural networks with radial basis functions (RBFs) for prediction with dense molecular markers. We illustrate the use of the linear Bayesian LASSO regression model and of two non-linear regression models, reproducing kernel Hilbert spaces (RKHS) regression and radial basis function neural networks (RBFNN) on simulated data and real maize lines genotyped with 55,000 markers and evaluated for several trait–environment combinations. The empirical results of this study indicated that the three models showed similar overall prediction accuracy, with a slight and consistent superiority of RKHS and RBFNN over the additive Bayesian LASSO model. Results from the simulated data indicate that RKHS and RBFNN models captured epistatic effects; however, adding non-signal (redundant) predictors (interaction between markers) can adversely affect the predictive accuracy of the non-linear regression models.

The relevance of purebred information for predicting genetic merit of survival at birth of crossbred piglets.

We evaluated whether the obesity‐associated years of life lost (YLL) have decreased over calendar time. We implemented a meta‐analysis including only studies with two or more serial body mass index (BMI) assessments at different calendar years. For each BMI category (normal weight: BMI 18.5 to <25 [reference]; overweight: BMI 25 to <30; grade 1 obesity: BMI 30 to <35; and grade 2–3 obesity: BMI ≥ 35), we estimated the YLL change between 1970 and 1990. Because of low sample sizes for African–American, results are reported on Caucasian. Among men aged ≤60 years YLL for grade 1 obesity increased by 0.72 years (P < 0.001) and by 1.02 years (P = 0.01) for grade 2–3 obesity. For men aged >60, YLL for grade 1 obesity decreased by 1.02 years (P < 0.001) and increased by 0.63 years for grade 2–3 obesity (P = 0.63). Among women aged ≤60, YLL for grade 1 obesity decreased by 4.21 years (P < 0.001) and by 4.97 years (P < 0.001) for grade 2–3 obesity. In women aged >60, YLL for grade 1 obesity decreased by 3.98 years (P < 0.001) and by 2.64 years (P = 0.001) for grade 2–3 obesity. Grade 1 obesitys association with decreased longevity has reduced for older Caucasian men. For Caucasian women, there is evidence of a decline in the obesity YLL association across all ages.

Assessment of Poisson, Probit and linear models for genetic analysis of presence and number of black spots in Corriedale sheep

The objective of this study was to infer (co)variance components for piglet survival at birth in purebred and crossbred pigs. Data were from 13,643 (1,213 litters) crossbred and 30,919 (3,162 litters) purebred pigs, produced by mating the same 168 purebred boars to 460 Large White-derived crossbred females and 1,413 purebred sows, respectively. The outcome variable was piglet survival at birth as a binary trait. A Bayesian bivariate threshold model was implemented via Gibbs sampling. Flat priors were assigned to the effects of sex, parity of the dam, litter size, and year-month of birth. Gaussian priors were assigned to litter, dam, and sire effects. Marginal posterior means (SD) of the sire and dam variances for liability of piglet survival in purebred were 0.018 (0.008) and 0.077 (0.020), respectively. For crossbred, sire and dam variance estimates were 0.030 (0.018) and 0.120 (0.034), respectively. The posterior means (SD) of the heritability of liability of survival in purebred and crossbred and of the genetic correlation between these traits were 0.049 (0.023), 0.091 (0.054), and 0.248 (0.336), respectively. The greatest 95% confidence region (-0.406, 0.821) for the genetic correlation between purebred and crossbred liabilities of piglet survival included zero. Results suggest that the expected genetic progress for piglet survival in crossbreds when selection is based on purebred information may be nil.

On the genomic analysis of data from structured populations

Black skin spots are associated with pigmented fibres in wool, an important quality fault. Our objective was to assess alternative models for genetic analysis of presence (BINBS) and number (NUMBS) of black spots in Corriedale sheep. During 2002-08, 5624 records from 2839 animals in two flocks, aged 1 through 6 years, were taken at shearing. Four models were considered: linear and probit for BINBS and linear and Poisson for NUMBS. All models included flock-year and age as fixed effects and animal and permanent environmental as random effects. Models were fitted to the whole data set and were also compared based on their predictive ability in cross-validation. Estimates of heritability ranged from 0.154 to 0.230 for BINBS and 0.269 to 0.474 for NUMBS. For BINBS, the probit model fitted slightly better to the data than the linear model. Predictions of random effects from these models were highly correlated, and both models exhibited similar predictive ability. For NUMBS, the Poisson model, with a residual term to account for overdispersion, performed better than the linear model in goodness of fit and predictive ability. Predictions of random effects from the Poisson model were more strongly correlated with those from BINBS models than those from the linear model. Overall, the use of probit or linear models for BINBS and of a Poisson model with a residual for NUMBS seems a reasonable choice for genetic selection purposes in Corriedale sheep.

Genomic variance estimates: With or without disequilibrium covariances?

The availability of dense SNP panels allows assessing similarity between distantly related individuals, including those with different genetic background. This has renewed the interest in the analysis of data from heterogeneous populations. Examples include the genomic analysis of data from multiple breeds (Hayes et al. 2009, Gen. Sel. Evol. 41, 51), or from structured populations (e.g. de los Campos et al. 2009, Genetics 182, 375–385; Daetwyler et al. 2012, J. Anim. Sci. 90, 3375–3384). Whole genome regression (WGR) methods (Meuwissen, Hayes, and Goddard, 2001, Genetics 157, 1819– 1829), where allele substitution effects are assumed to be homogeneous across subjects, have been used for the analysis of data from structured populations (e.g. Hayes et al. 2009, Gen. Sel. Evol. 41, 51; de los Campos et al. 2009, Genetics 182, 375–385). This approach allows borrowing information across groups and, under some circumstances, can increase prediction accuracy. For example, in a combined analysis of Holstein and Jersey data, Hayes et al. (2009, Gen. Sel. Evol. 41, 51) showed that prediction accuracy of estimated breeding values could be increased, relative to a within-breed analysis, in Jerseys but not in Holsteins. However, assuming that marker effects are constant across groups ignores the fact that dominance, epistasis or differences in the marker–QTL LD (linkage disequilibrium) patterns can lead to group-specific marker effects. Principal Components (PCs) methods are commonly used in genome-wide association studies to account for population structure (e.g. Price et al. 2006, Nat. Gen. 38, 34–41; Marchini et al. 2004, Nat. Gen. 36, 512–517). Drawing on these ideas, some authors suggested expanding WGRs such as the G-BLUP (genomic best linear unbiased predictor) by adding marker-derived PCs as fixed effect covariates. This approach has been used to account for stratification in the estimation of variance components (e.g. Yang et al., 2010, Nat. Genet. 42, 565–569) and in the prediction of breeding values (e.g. Daetwyler et al., 2012, J. Anim. Sci. 90, 3375–3384). However, Janss et al. (2012, Genetics 192, 693–704) demonstrated that adding eigenvectors as fixed effects in G-BLUP can create important inferential problems. Indeed, Gaussian processes, including the G-BLUP, are equivalent to a random regression on all marker-derived PCs (e.g. de los Campos et al., 2010, Genetics Research 92, 295–308). Therefore, the PCs that are added as fixed effects in the G-BLUP, typically those with the largest eigenvalue, enter twice in the model, and this can have adverse effects on inferences on variance components. The problem is aggravated by the fact that in G-BLUP, despite the random nature, the effects of eigenvectors with large eigenvalues are effectively estimated as fixed effects. In their article, Janss et al. showed how the standard G-BLUP, parameterized using PCs, can be used to draw inferences and predictions based on all or some PCs in a coherent statistical framework. This approach should be preferred over the one using PCs as fixed effects in a G-BLUP model. However, regardless of how PCs are dealt with, when only a subset of PCs is used for inferences, the connection with the original model is lost and parameters have no genetic interpretation.