A Novel Generalized Ridge Regression Method for Quantitative Genetics

Abstract

As the molecular marker density grows, there is a strong need in both genome-wide association studies and genomic selection to fit models with a large number of parameters. Here we present a computationally efficient generalized ridge regression (RR) algorithm for situations in which the number of parameters largely exceeds the number of observations. The computationally demanding parts of the method depend mainly on the number of observations and not the number of parameters. The algorithm was implemented in the R package bigRR based on the previously developed package hglm. Using such an approach, a heteroscedastic effects model (HEM) was also developed, implemented, and tested. The efficiency for different data sizes were evaluated via simulation. The method was tested for a bacteria-hypersensitive trait in a publicly available Arabidopsis data set including 84 inbred lines and 216,130 SNPs. The computation of all the SNP effects required ,10 sec using a single 2.7-GHz core. The advantage in run time makes permutation test feasible for such a whole-genome model, so that a genome-wide significance threshold can be obtained. HEM was found to be more robust than ordinary RR (a.k.a. SNP-best linear unbiased prediction) in terms of QTL mapping, because SNPspecific shrinkage was applied instead of a common shrinkage. The proposed algorithm was also assessed for genomic evaluation and was shown to give better predictions than ordinary RR. HIGH-dimensional problems are increasing in importance in genetics, computational biology, and other fields of research where technological developments have greatly facilitated the collection of data (Hastie et al. 2009). In genome-wide association studies (GWAS) and genomic selection (GS), the number of observations n is generally in the order of hundreds/thousands whereas the number of marker effects to be fitted p is in the order of hundreds of thousands. This is a rather extreme p ) n problem, and the methods developed for analyses of the data need to be computationally feasible. At the same time the models fitted should be flexible enough to capture the important genetic effects that are often quite small .Methodologies regarding high-dimensional genomic data focus on both detection and prediction purposes. There is currently a trend that GWAS and GS could potentially apply the same framework of models. Such models fit the whole genome based on penalized likelihood or Bayesian shrinkage estimation (see the review by de los Campos et al. 2013). Ordinary GWAS usually avoids high-dimensional models and turns the problem into multiple testing instead (e.g., the review by Kingsmore et al. 2008). The tests of all the SNPs (single nucleotide polymorphisms) are dismembered. Such routine sacrifices both detective and predictive power. Using detected QTL (quantitative trait loci that are genome-wide significant), the prediction can be rather poor, which led to the insignificant application of marker-assisted selection (MAS) (Dekkers 2004). GS, however, has been practically use...

Genome-Wide Association Mapping and Genomic Prediction Elucidate the Genetic Architecture of Morphological Traits in Arabidopsis

“…To this end, a heteroscedastic effects model (Shen et al, 2013) was applied to all traits. Prediction accuracy (r 2 ) in 2-fold cross validation increased with increasing numbers of randomly chosen SNPs until a maximum was reached between 500 and 25,000 SNPs for all traits (Supplemental Fig.…”

“…Traits were scaled and the complete marker set was used for the construction of genomic prediction (GP) models using the bigRR library (Shen et al, 2013). GP was performed for different random and sorted sets of markers based on the highest 2log 10 (P) value.…”

“…These models were originally proposed to predict breeding values of unphenotyped individuals, but are increasingly used to identify causal variants (Shen et al, 2013;Moser et al, 2015). Important advantages over single locus methods are that effect estimates are generally more accurate, and that the total genetic variance can be attributed to specific genomic regions (Goddard et al, 2009;Yang et al, 2010;Wood et al, 2014).…”

A comparison of genomic selection models across time in interior spruce (Picea engelmannii × glauca) using unordered SNP imputation methods

“…Mean imputation was carried out using the 'A.mat' function provided in the 'rrBLUP' R package, and refers to imputation using the mean for each marker (Endelman, 2011). (Shen et al, 2013), and BayesCπ (BCπ) (Habier et al, 2011). Further, rrBLUP and BCπ are both homoscedastic effects models using common marker variances for shrinkage, whereas GRR is a heteroscedastic effects model (that is, marker-specific variances are used).…”

“…GRR is a two-step variable selection process, and was carried out using the R package 'bigRR' (Shen et al, 2013). In the first step, initial estimates ofŝ 2 e ,ŝ 2 u , andû were obtained through the same MME as in rrBLUP.…”