Penalized factorial regression as a flexible and computationally attractive reaction norm model for prediction in the presence of GxE.

Key message: Penalized factorial regression offers a computationally attractive alternative to kernel and deep learning methods for prediction of genotype by environment interactions. For two representative data sets on wheat and maize, prediction accuracies were comparable, while computing requirements and time were clearly lower. A longstanding challenge in plant breeding and genetics is the prediction of yield for new environments in the presence of genotype by environment interaction ( $G\times E$ ). The genotypes in this case are promising candidate varieties at an advanced stage of breeding programs or are part of statutory variety trials or post registration trials. The genotypes have been tested in a limited set of trials and the question is how these genotypes will perform in future growing conditions. A reaction norm approach seems adequate to address this challenge. Reaction norms are functions with genotype-specific parameters that express the phenotype as a function of environmental inputs. $G\times E$ follows from differences in genotype-specific slope or rate parameters. Prediction of yield for new environments requires the identification of suitable reaction norm functions and the estimation of genotype-specific parameters together with knowledge about the environmental conditions. Here, we present penalized factorial regression with simple linear reaction norms for individual genotypes whose slopes are regularized by imposing a penalty upon them. Different types of penalization provide shrinkage, automatic selection of environmental covariates (EC's) and protection against overfitting for prediction of yield with medium to large numbers of EC's. Illustrations of our approach are given for a maize and a wheat data set. For these data, our approach compares well to alternative methods based on Bayesian regression and deep learning with respect to prediction accuracy, while computational demands are clearly lower.