2.5.1. Ridge Regression Best Linear Unbiased Prediction (RRBLUP)

RRBLUP was included here as the benchmark for comparing its performance with other models due to frequent use in wheat breeding and ease of implementation. The model assumes that all markers contribute to the trait and has a constant effect variance. Marker effects and variance patterns are estimated using the restricted estimated maximum likelihood (REML) function based on phenotypic and marker data [37]. The RRBLUP model was implemented with the R package rrBLUP using the mixed.solve function. The model can be represented as

where µ is the overall mean; y is the vector of adjusted means; u is a vector with normally distributed random marker effects with constant variance as u ~ N(0, Iσ²_u); Z is an N × M matrix of markers; and e is the residual error distributed as e ~ N(0, Iσ²_e). The solution for mixed equation can be written as

where u, Z, and y are explained above; I is an identity matrix; and λ is represented as λ = σ²_e/σ²_u and is the ridge regression parameter [37].