In charlotte-ngs/lbgfs2020: Livestock Breeding and Genomics (Fall 2020)

Disclaimer

When working with multi-trait BLUP models Kronecker products are important.

Data Set

tbl_si_dat <- tibble::tibble(Animal = c(1:5),
                             Sire   = c(NA, NA, NA, 1, 4),
                             Dam    = c(NA, NA, NA, 2, 3),
                             Sex    = c("M", "F", "F", "M", "F"),
                             WWG    = c(4.5, 2.9, 3.9, 3.5, 5.0),
                             PWG    = c(7, 4.2, 5.9, 6.1, 6.9))

tbl_si_dat

Observations

The vector of observations is now the columns 'WWG' and 'PWG' appended to each other

(vec_y <- c(tbl_si_dat$WWG, tbl_si_dat$PWG))

In vector notation

cat(paste(rmdhelp::bcolumn_vector(pvec = vec_y, ps_name = 'y', ps_env = '$$')))

Covariances

Between traits

(mat_G0 <- matrix(c(20, 18,
                    18, 40), nrow = 2, byrow = TRUE))

(mat_R0 <- matrix(c(40, 11,
                    11, 30), nrow = 2, byrow = TRUE))

Error Term

Assume that errors between animals are not correlated. Hence

$$var(e) = R = R_0 \otimes I_n$$ where $n$ is the number of observations

nobs <- nrow(tbl_si_dat)
mat_In <- diag(1, nrow = nobs)
(mat_R <- mat_R0 %x% mat_In)

Pedigree

The variance ($G = var(u)$) of all breeding values ($u$) is

$$G = var(u) = G_0 \otimes A$$

(ped <- pedigreemm::pedigree(sire = tbl_si_dat$Sire, dam = tbl_si_dat$Dam, label = as.character(tbl_si_dat$Animal)))

The numerator relationship matrix $A$

(mat_A <- as.matrix(pedigreemm::getA(ped = ped)))

Then the matrix $G$ is

(mat_G <- mat_G0 %x% mat_A)

charlotte-ngs/lbgfs2020 documentation built on Dec. 20, 2020, 5:39 p.m.