Ridge Regression and Other Kernels for Genomic Selection with R Package rrBLUP

Many important traits in plant breeding are polygenic and therefore recalcitrant to traditional marker-assisted selection. Genomic selection addresses this complexity by including all markers in the prediction model. A key method for the genomic prediction of breeding values is ridge regression (RR), which is equivalent to best linear unbiased prediction (BLUP) when the genetic covariance between lines is proportional to their similarity in genotype space. This additive model can be broadened to include epistatic effects by using other kernels, such as the Gaussian, which represent inner products in a complex feature space. To facilitate the use of RR and nonadditive kernels in plant breeding, a new software package for R called rrBLUP has been developed. At its core is a fast maximum-likelihood algorithm for mixed models with a single variance component besides the residual error, which allows for effi cient prediction with unreplicated training data. Use of the rrBLUP software is demonstrated through several examples, including the identifi cation of optimal crosses based on superior progeny value. In cross-validation tests, the prediction accuracy with nonadditive kernels was signifi cantly higher than RR for wheat (Triticum aestivum L.) grain yield but equivalent for several maize (Zea mays L.) traits. THE ABILITY TO PREDICT COMPLEX TRAITS from marker data is becoming increasingly important in plant breeding (Bernardo, 2008). Th e earliest attempts, now over 20 years old, involved fi rst identifying signifi cant markers and then combining them in a multiple regression model (Lande and Th ompson, 1990). Th e focus over the last decade has been on genomic selection methods, in which all markers are included in the prediction model (Bernardo and Yu, 2007; Heff ner et al., 2009; Jannink et al., 2010). One of the fi rst methods proposed for genomic selection was ridge regression (RR), which is equivalent to best linear unbiased prediction (BLUP) in the context of mixed models (Whittaker et al., 2000; Meuwissen et al., 2001). Th e basic RR-BLUP model is