Multi-environment genomic model with a genotype-by-environment term

Formula: Y = 1 mu + X_E beta_E + Z_L b_1 + Z_EL b_2 + eps, b_1 ~ N(0, sigma2_g G), b_2 ~ N(0, Sigma_E (x) G)

Usage

Gxeblup(y, X_E, Z_L, Z_EL, G, sigma2_g, Sigma_E, sigma2_e = 1)

Arguments

y

Response vector of length n.

X_E

Environment design matrix; None or empty uses an intercept only.

Z_L

Design matrix of lines.

Z_EL

Design matrix of the line-by-environment interaction.

G

Genomic relationship matrix.

sigma2_g

Genomic variance component.

Sigma_E

Genetic covariance between environments.

sigma2_e

Residual variance component.

Value

List with beta, b_lines, b_gxe, n.

References

Montesinos Lopez, Montesinos Lopez and Crossa (2022), Multivariate Statistical Machine Learning Methods for Genomic Prediction, Springer, doi:10.1007/978-3-030-89010-0. Chapter 5, Eq. (5.4) p. 150. The two random terms are stacked into one Z = [Z_L Z_EL] with a block-diagonal Sigma and solved as Eq. (5.1).

Examples

set.seed(5)
nl <- 4; ne <- 2
G <- crossprod(matrix(rnorm(16), 4)) / 4 + diag(0.1, 4)
Z_L <- kronecker(matrix(1, ne, 1), diag(nl)); X_E <- kronecker(diag(ne), matrix(1, nl, 1))
Z_EL <- diag(nl * ne)
y <- rnorm(nl * ne, 10)
Gxeblup(y, X_E, Z_L, Z_EL, G, sigma2_g = 1, Sigma_E = diag(ne))$beta
#> [1] 6.795224 2.535797 4.259427