gBLUP 和 rrBLUP

The two models are equivalent!
RR-BLUP and G-BLUP will give us the same results, but they are implemented in a slightly different way.

01 核心公式

\[\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{u} + \boldsymbol{\varepsilon} \]

02 主要区别

rrBLUP：直接估计「每个标记的效应」，然后基于标记效应值计算个体育种值
gBLUP：不估计单个标记，用「基因型关系矩阵 G」估计总体的个体育种值

02.1 随机效应设计矩阵Z矩阵

In RR-BLUP, the random effects design matrix, Z, is the marker data, and the rows of Z must relate to the y vector. We assume that the markers are independent and identically distribute.
矩阵Z是SNP基因型矩阵，维度是n(样本数) ✖ p(标记数)

In G-BLUP we use a relationship matrix and the random effects design matrix Z must relate the y vector to the relationship matrix.
G-BLUP中随机效应设计矩阵 Z 的任务是把表型向量 y 与关系矩阵的行/列对应起来。（0/1，是n × N 的“出现矩阵”）

02.2 随机效应u的分布

\[\mathbf{u} \sim \mathcal{N}(\mathbf{0}, \mathbf{G}\sigma_u^2) \]

RR-BLUP: \(G\)矩阵是单位矩阵
g-BLUP: \(G\)矩阵是亲缘关系\(K\)矩阵

02.3 library(rrBLUP)中核心函数的使用

RR-BLUP:

mod1rr<- mixed.solve(y= y2, Z=Z2, K=NULL, X=X2)

g-BLUP:

mod1<- mixed.solve(y= y2, Z=Z2, K=A, X=X2)

y is the response variable, Z is the random effects design matrix, K is the relationship matrix that specifies the relationship between the genotypes, and X is the fixed effects design matrix. With G-BLUP, the K matrix is the marker-based relationship matrix.