Single-Environment RR-BLUP and G-BLUP Models

The parametric regression model for a single environment j^th (j = 1, …, m) is defined as y_j = 1_njμ_j + X_jβ_j + ε_j, where the vector y_j represents nj independent centered observations of the response variable in the j^th environment; 1_nj is a vector of ones of order nj; μ_j is the overall mean of the j^th environment; X_j is the matrix for the p centered and standardized molecular markers in the j^th environment; vector β_j represents the effect of each of the p markers in the j^th environment, and ε_j is the vector of random errors in the j^th environment with normal distribution and common variance σεj2. The RR-BLUP assumes that the effects of the markers have a multivariate normal distribution βj∼N⁢(0,I⁢σβj2).

Assuming that the effects of the markers β_j and ε_j are independent, and that u_j = X_jβ_j, then the above model for the j^th environment can be written as y_j = 1_njμ_j + u_j + ε_j, where u_j, and ε_j are independent random variables with uj∼N⁢(0,σuj2⁢Kj), and εj∼N⁢(0,σε2⁢I), respectively; σuj2 is the variance of u_j (to be estimated), and K_j is a symmetric matrix representing the covariance of the genetic values. Thus, for a single-environment where the K_j is of the linear form Kj=Gj=Xj⁢Xj′/p the G-BLUP is equivalent to RR-BLUP (Van Raden, 2008).