Software update: Moving the R package sommer to multivariate mixed models for genome-assisted prediction

Abstract

9In the last decade the use of mixed models has become a pivotal part in the 10 implementation of genome-assisted prediction in plant and animal breeding programs. 11Exploiting the use genetic correlation among traits through multivariate predictions has 12 been proposed in recent years as a way to boost prediction accuracy and understand 13 pleiotropy and other genetic and ecological phenomena better. Multiple mixed model 14 solvers able to use relationship matrices or deal with marker-based incidence matrices 15 have been released in the last years but multivariate versions are scarse. Such solvers 16 have become quite popular in plant and animal breeding thanks to user-friendly platforms 17 such as R. Among such software one of the most recent and popular is the sommer 18 package. In this short communication we discuss the update of the package that is able to 19 run multivariate mixed models with multiple random effects and different covariance 20 structures at the level of random effects and trait-to-trait covariance along with other 21 functionalities for genetic analysis and field trial analysis to enhance the genome-assisted 22 prediction capabilities of researchers. 23 24 Introduction 25 26 Currently, linear mixed models play an important role in science to better understand 27 different biological and non-biological phenomena (Bolker 2009; Gianola and Rosa 28 2015). In brief, linear mixed models are extensions of linear models (Hastie et al., 2009). 29 Among many statistical tools used to test hypothesis and estimate parameters, linear 30 mixed models are particularly robust and flexible, which has given a pivoting role in 31 Biological sciences such as plant and animal breeding and ecology (Bolker et al. 2009; 32 Hadfield 2010; Gianola and Rosa 2015). For example, in the Genetics field, most of the 33 published genome wide association and quantitative genetics studies are based on mixed 34 models and REML estimation (Bush et al. 2012; Hirschhorn et al. 2005; Wang et al. 35 2005; Kang et al. 2008, 2010). 36 37 In general terms, current mixed model solvers use Frequentist or Bayesian approaches to 38 estimate parameters and solve the linear equations, and both approach rely on statistical 39 assumptions (i.e. distributions) on the response or covariates to be modeled (Gianola and 40 Rosa 2015; Hastie et al., 2009). Among Frequentist software and different optimization 41 techniques, mixed models are usually solved by either of the two most popular 42 REML/ML methods; mixed model equation-based (MME) algorithms, based on 43 Henderson and Searle ideas (Gilmour et al 1995; Henderson 1975; Searle 1993) and 44 direct-inversion-based (DI) algorithms which is the natural solution for the linear 45 equations in the mixed model context (Lee et al 2016; Maier 2015). Both methods require 46 iterative procedures to estimate the variance-covariance parameters and coefficients 47 number of observations and dense covariance structures are used (p > n). On the other 53 hand, Bayesian approaches implement Markov chain Mo...

Multivariate GBLUP Improves Accuracy of Genomic Selection for Yield and Fruit Weight in Biparental Populations of Vaccinium macrocarpon Ait

“…The terms, and denote the genetic (or any of the kth random terms) and residual variance of trait “i,” respectively and σ u k ij and σ e ij the genetic (or any of the kth random terms) and residual covariance between traits “i” and “j” ( i = 1,…,t, and j = 1,…,t). For more details about the multivariate algorithm used in sommer please look at Covarrubias-Pazaran ( 2018 ). The genetic correlation among years was calculated using the by-environment genotype predictions as the multivariate response.…”

“…The model used has the typical mixed model form; y = X β + Z u + ε, where y is the response variable, X and Z are incidence matrices for fixed and random effects, respectively, β is the vector of fixed effects (intercept only), u is the vector of random effects associated to the genotypic effects with the corresponding relationship matrices. For the multivariate GBLUP model only the additive relationship matrix was used, and the model and distributions follow Covarrubias-Pazaran ( 2018 ). In total, 100 iterations of 5-fold CV were used to test the PA under the different models.…”

Best Prediction of the Additive Genomic Variance in Random-Effects Models

“…We conducted all calculations with the free software R (R Development Core Team 2017). For each dataset, we used the gBLUP-method in the equivalent version (computational advantages) implemented in the R -package “sommer” (Covarrubias-Pazaran 2017) to fit a REM. We used REML to obtain estimates (trueσ^g2 and trueσ^ε2) for the variance components.…”

Genomic selection for heterobothriosis resistance concurrent with body size in the tiger pufferfish, Takifugu rubripes

“…Heritability and genetic correlation were calculated by a multivariate linear mixed model as follows: where is a vector of phenotypes for trait i ( i = 1 for transformed HC and 2 for SL); and are incidence matrices for fixed effects and random effects , respectively. The model assumes the random effects ( ) follow a multivariate normal distribution as and the residuals ( ) follow ; where and are the variance–covariance matrices of random effects and residuals for the two traits, respectively; is the additive genetic relationship matrix constructed by A.mat function in R/sommer-4.0.1 59 , 60 with the default settings; is the identity matrix; means the operation of Kronecker product. The model was solved by mmer function in R/sommer-4.0.1 to solve the equations.…”