On the accuracy in high‐dimensional linear models and its application to genomic selection

Abstract

Genomic selection is today a hot topic in genetics. It consists in predicting breeding values of selection candidates, using the large number of genetic markers now available owing to the recent progress in molecular biology. One of the most popular methods chosen by geneticists is ridge regression. We focus on some predictive aspects of ridge regression and present theoretical results regarding the accuracy criteria, that is, the correlation between predicted value and true value. We show the influence of singular values, the regularization parameter, and the projection of the signal on the space spanned by the rows of the design matrix. Asymptotic results in a high‐dimensional framework are given; in particular, we prove that the convergence to optimal accuracy highly depends on a weighted projection of the signal on each subspace. We discuss on how to improve the prediction. Last, illustrations on simulated and real data are proposed.

Training set optimization of genomic prediction by means of EthAcc

“…This accuracy expectation was the goal of the different formulas derived from the works of [15, 16]. Via estimation of the QTL locations and effects, the authors of [27] compared the expectation of the theoretical accuracy to these analytical formulas and found that the former performed better than the other formulas. The most important issue with the theoretical accuracy formula is the estimation of the causal-QTL locations and effects.…”

“…The most important issue with the theoretical accuracy formula is the estimation of the causal-QTL locations and effects. The authors of [27] compared different penalized regression methods and revealed that adaptive LASSO [32] performed the best. We also compared different methods by plugging them into the theoretical accuracy formula, but in contrast to [27], we did not use the penalized regression estimators of the QTL effects because they are known to be biased.…”

“…The authors of [27] compared different penalized regression methods and revealed that adaptive LASSO [32] performed the best. We also compared different methods by plugging them into the theoretical accuracy formula, but in contrast to [27], we did not use the penalized regression estimators of the QTL effects because they are known to be biased. We used a two-step procedure to avoid this bias, whereas MLMM and penalized regression methods were employed to locate the QTLs, and their effects were estimated by classical ordinary least square methods.…”

“…The authors of [27] used penalized regression methods to detect the QTLs and to estimate their effects though it is well known that penalized regression yields biased estimators of QTL effects. In this paper, we used a two-step procedure, the first step was to locate the QTLs via multi-QTL methods developed for a genome-wide association study (GWAS), then the QTL effects were estimated via a classical linear model by the ordinary least square method as well as the error variance in the causal model .…”

“…The EN α parameter was set to 0.5 and 0.1. To complete the penalized regression methods, we added the adaptive LASSO [32] that gives the most accurate results when QTL effects are estimated by penalized regression estimators [27].…”