Multivariate methods

Multi-task group-EN (MTV_EN, Hastie and Qian 2016) is a multivariate extension of EN, it solves the following equation: BMTV_EN^=argminλ||Y−Xβ||F2+(1−α)λ||β||F2+αλ||β||2, F being the Frobenius norm. It assumes that each predictor variable has either a zero or nonzero effect across all traits, allowing nonzero effects to have different values among traits. λ and α parameters were tuned using cv.glmnet (family = “mgaussian”). MTV_RR is the multivariate extension of RR, also tuned with cv.glmnet (family = “mgaussian,” α  =  0). MTV_LASSO is the multivariate extension of LASSO, also tuned with cv.glmnet (family = “mgaussian,” α  =  1). The implementation of these three methods is identical.

The multivariate structured penalized regression (called SPRING in Chiquet et al. 2017) applies a L1−penalty (λ₁ parameter) for controlling sparsity (like LASSO) and a smooth L2−penalty (λ₂ parameter) for controlling the amount of structure among predictor variables (L) to add in the model, i.e., the correlation between markers according to their position on the genetic map. Both parameters λ₁ and λ₂ were tuned by cross-validation using cv.spring function (from R/spring package, version 0.1-0). The regression equation can be written as: Y=XB+ϵ with ϵ∼N(0,R), R is the covariance matrix of residuals (Gaussian noise). The allelic effects are: B=−ΩXyΩyy−1 and they comprise both direct effects Ω_Xy and indirect ones Ω_yy.

SPRING solves the following equation: (ΩXy^,Ωyy^)=argmin−1nlog ℓ(ΩXy,Ωyy)+λ22tr(ΩyXLΩXyΩyy−1)+λ1||ΩXy||1. Unlike multi-task group-EN, SPRING selects specific predictors for each trait, i.e., a selected predictor can have a nonzero effect for a subset of the traits. Moreover, SPRING allows the distinction between direct and indirect effects by using conditional Gaussian graphical modeling. These effects are due to covariance of the noise such as environmental effects affecting several traits simultaneously. This distinction results in two kinds of estimated allelic effects: the direct ones, re-estimated with OLS, which are best suited for QTL detection (we called the corresponding prediction method spring.dir.ols) and the regression ones, which involve both direct and indirect effects and are best suited for prediction (spring.reg method).