Summary: 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.

Ovarian function was compared between nulliparous heifers (n = 29; 10 to 16 mo old) and lactating Holstein cows (n = 31; 55.9 +/- 3.5 d postpartum). Follicular dynamics, corpus luteum growth, and regression, and serum steroid concentrations were evaluated through ultrasonography and daily blood sampling. Most heifers (27 of 29) but only 14 of 31 cows had typical spontaneous estrous cycles after cycles were initiated. Twelve cows had atypical cycles, and 5 became anovulatory during the study. The 12 cows with atypical estrous cycles had low serum estradiol after luteolysis and failed to ovulate the dominant follicle present at luteolysis. Heifers and cows with typical cycles were compared directly. Interovulatory intervals were similar between heifers (22.0 +/- 0.4 d) and cows (22.9 +/- 0.7 d). Those animals had estrous cycles with either 2 (15 heifers; 11 cows), 3 (9 heifers; 2 cows), or 4 follicular waves (3 heifers; 1 cow). Cows ovulated later after luteolysis than heifers (5.2 +/- 0.2 vs. 4.6 +/- 0.1 d, respectively), and had more multiple ovulations (17.9 vs. 1.9%). Maximal serum estradiol concentration preceding ovulation was lower in cows than in heifers (7.9 +/- 0.8 vs. 11.3 +/- 0.6 pg/mL) even though ovulatory follicles were larger in cows (16.8 +/- 0.5 vs. 14.9 +/- 0.2 mm). Similarly, maximal serum progesterone concentration was lower for cows (5.6 +/- 0.5 vs. 7.3 +/- 0.4 ng/mL), whereas maximal volume of luteal tissue was larger for cows than heifers (11,120 +/- 678 vs. 7303 +/- 308 mm3). Thus, higher incidence of reproductive anomalies in lactating cows, such as low conception rate, ovulation failure, delayed ovulation, and multiple ovulations, may be due to lower circulating steroid concentrations in spite of larger ovulatory follicles and luteal structures.

Background: In the study of associations between genomic data and complex phenotypes there may be relationships that are not amenable to parametric statistical modeling. Such associations have been investigated mainly using single-marker and Bayesian linear regression models that differ in their distributions, but that assume additive inheritance while ignoring interactions and non-linearity. When interactions have been included in the model, their effects have entered linearly. There is a growing interest in non-parametric methods for predicting quantitative traits based on reproducing kernel Hilbert spaces regressions on markers and radial basis functions. Artificial neural networks (ANN) provide an alternative, because these act as universal approximators of complex functions and can capture non-linear relationships between predictors and responses, with the interplay among variables learned adaptively. ANNs are interesting candidates for analysis of traits affected by cryptic forms of gene action. Results: We investigated various Bayesian ANN architectures using for predicting phenotypes in two data sets consisting of milk production in Jersey cows and yield of inbred lines of wheat. For the Jerseys, predictor variables were derived from pedigree and molecular marker (35,798 single nucleotide polymorphisms, SNPS) information on 297 individually cows. The wheat data represented 599 lines, each genotyped with 1,279 markers. The ability of predicting fat, milk and protein yield was low when using pedigrees, but it was better when SNPs were employed, irrespective of the ANN trained. Predictive ability was even better in wheat because the trait was a mean, as opposed to an individual phenotype in cows. Non-linear neural networks outperformed a linear model in predictive ability in both data sets, but more clearly in wheat. Conclusion: Results suggest that neural networks may be useful for predicting complex traits using high-dimensional genomic information, a situation where the number of unknowns exceeds sample size. ANNs can capture nonlinearities, adaptively. This may be useful when prediction of phenotypes is crucial.