Multi-trait linear mixed model

Formula: Y = (1 (x) I_T) mu + X beta + Z b + eps, b ~ N(0, G (x) Sigma_T), eps ~ N(0, I_J (x) R_T)

Usage

Mtlmmfit(Y, Z, G, Sigma_T, R_T, X = NULL)

Arguments

Y

Lines by traits.

Z

Design matrix of lines.

G

Genomic relationship matrix.

Sigma_T

Genetic covariance between traits.

R_T

Residual covariance between traits.

X

Extra fixed-effect columns; None uses only the trait intercepts.

Value

List with mu, beta, b, J, T.

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. (5.5)/(5.5a) p. 153: the traits of each line are stacked, giving a mixed model with Kronecker-structured covariances. The book notes on p. 153 that when Sigma_T and R_T are diagonal this is equivalent to fitting each trait separately. The solution is in the stacked ordering (line 1 traits, line 2 traits, ...).

Examples

set.seed(9)
J <- 6; nT <- 2
G <- crossprod(matrix(rnorm(J * J), J)) / J
Y <- matrix(rnorm(J * nT, 5), J, nT)
Mtlmmfit(Y, Z = diag(J), G = G, Sigma_T = diag(nT), R_T = diag(nT))$mu
#> [1] 4.459407 4.465805