A Bayesian Decision Theory Approach for Genomic Selection

Abstract

Plant and animal breeders are interested in selecting the best individuals from a candidate set for the next breeding cycle. In this paper, we propose a formal method under the Bayesian decision theory framework to tackle the selection problem based on genomic selection (GS) in single- and multi-trait settings. We proposed and tested three univariate loss functions (Kullback-Leibler, KL; Continuous Ranked Probability Score, CRPS; Linear-Linear loss, LinLin) and their corresponding multivariate generalizations (Kullback-Leibler, KL; Energy Score, EnergyS; and the Multivariate Asymmetric Loss Function, MALF). We derived and expressed all the loss functions in terms of heritability and tested them on a real wheat dataset for one cycle of selection and in a simulated selection program. The performance of each univariate loss function was compared with the standard method of selection (Std) that does not use loss functions. We compared the performance in terms of the selection response and the decrease in the population’s genetic variance during recurrent breeding cycles. Results suggest that it is possible to obtain better performance in a long-term breeding program using the single-trait scheme by selecting 30% of the best individuals in each cycle but not by selecting 10% of the best individuals. For the multi-trait approach, results show that the population mean for all traits under consideration had positive gains, even though two of the traits were negatively correlated. The corresponding population variances were not statistically different from the different loss function during the 10th selection cycle. Using the loss function should be a useful criterion when selecting the candidates for selection for the next breeding cycle.

Simultaneous selection for grain yield and protein content in genomics-assisted wheat breeding

“…maintaining grain yield or protein content but favoured one trait at cost of the other. Accordingly, it can be recommended to focus on genomic selection indices that correspond to deviations from regression line when conducting a simultaneous selection for grain yield and protein content and for finding the desirable outliers from the common trend, although it should be noticed that other methods such as using the multi-optimization framework by setting optimal compromise solutions or from the Bayesian decision theory have also shown great promise (Akdemir et al 2018 ; de Villar-Hernández et al 2018 ).…”

Application of multi-trait Bayesian decision theory for parental genomic selection

“…Genomic approaches (1) estimate the GEBV through a statistical model and information about the unobserved (genotyped) individuals (candidate population) using the phenotypic and genotypic data of their parents, (2) rank the lines based on GEBV, and (3) select the top-ranking lines. Recently, Villar-Hernández et al (2018) proposed a method based on Bayesian decision theory (BDT) for selecting the best candidates (in a single trait or in multi-trait) that maximize R ; results were obtained by simulating a breeding program. For a single trait, and assuming the candidates have the same amount of information and are identically distributed, R could be expressed in terms of the selection differential , the difference between the mean of the selected individuals, , and the mean of the original population, ) multiplied by the heritability .…”

“…Thus, R= S , and when R S (maximum expected response to selection, minimum expected loss in the decision of which candidates to select based on our breeding goals), whereas when , R ≪ S (minimum expected response to selection, maximum expected loss) . The BDT methodology proposed by Villar-Hernández et al (2018) considers the variance-covariance matrix between traits and the trait mean while minimizing the posterior expected distance between the distribution of the offspring of the selected individuals (distribution of the candidates) and the distribution of the selected individuals (distribution of the selected parents), and therefore maximizing the expected response to selection ( R ) given the phenotypic, genotypic and genomic information at hand. Minimizing the distance between the distribution of the parental candidates and the progeny distribution increases the accuracy of selection (assuming equal selection intensity).…”

“…The selection of multi-traits can be facilitated by ranking lines based on a single number; for example, genomic selection indices (SI) score lines based on a weighted average of GEBV, and then select those lines with high scores ( Cerón-Rojas and Crossa 2020 ). The approach of Villar-Hernández et al (2018) ranks the lines based on the posterior expected loss (PEL, a single number) given our breeding goals, in terms of the mean and the genetic variance-covariance matrix. Thus, those candidates whose distributions are closer to the theoretical parental distribution will have the lowest expected loss value (high R ), and the decision should be to advance those lines given that they reach the desired mean and keep the genetic variance as much as possible (high ).…”

“…As described in Villar-Hernández et al (2018) , the LF is the vehicle to go from the action space (candidate lines) to the resulting space (selected lines) given a Bayesian action (an action that guarantees a minimum PEL given our preferences). Of the three multivariate LFs used by Villar-Hernández et al (2018) , the Kullback–Leibler (KL) LF is easier to understand.…”