In charlotte-ngs/lbgfs2021: Livestock Breeding and Genomics (ETHZ) - Fall Semester 2021

knitr::opts_chunk$set(echo = TRUE)

Data

nr_animal <- 5
tbl_decomp <- tibble::tibble(Animal = c(1:nr_animal),
                             Sire = c(NA, NA, NA, 1, 4),
                             Dam = c(NA, NA, NA, 2, 3),
                             Trait = c(4.5, 2.9, 3.9, 3.5, 5.0))
tbl_decomp

Model

$$y = X\mu + Zu + e$$

Components

X = matrix(1, nrow = nr_animal, ncol = 1)
X

Z = diag(1, nrow = nr_animal)
Z

Pedigree

ped = pedigreemm::pedigree(sire = tbl_decomp$Sire, 
                           dam = tbl_decomp$Dam,
                           label = as.character(1:nr_animal))
Ainv <- as.matrix(pedigreemm::getAInv(ped = ped))
Ainv

MME

xtx <- crossprod(X)
xtx

xtz <- crossprod(X, Z)
xtz

ztx <- crossprod(Z, X)
ztx

ztz <- crossprod(Z)
ztz

sigmae2 <- 40
sigmau2 <- 20
lambda <- sigmae2/sigmau2
lambda

ztzainvlambda <- ztz + Ainv * lambda
ztzainvlambda

coef_mat <- rbind(cbind(xtx, xtz), cbind(ztx, ztzainvlambda))
coef_mat

y <- tbl_decomp$Trait
xty <- crossprod(X,y)
zty <- crossprod(Z,y)
rhs <- rbind(xty, zty)
rhs 

Solutions

sol <- solve(coef_mat, rhs)
sol

Problem 1: Decomposition

From the lecture, we know how the predicted breeding value of animal $4 can be decomposed into different components. Verify this decomposition based on the solutions obtained above.

Solution

The decomposition is given as follows:

$$\hat{u}_4 = {1\over 6}\left[ y_4 - \hat{\mu} + 2 * (\hat{u}_1 + \hat{u}_2) - \hat{u}_3 + 2 \hat{u}_5\right] $$

This can be verified as follows

(tbl_decomp$Trait[4] - sol[1] + 2 * (sol[2]+sol[3]) - sol[4] + 2*sol[6])/6

The breeding value of animal 4 is

sol[5]

Problem 2: Reliabilities

Compute the reliabilities of all the predicted breeding values. The reliability $B_i$ of the predicted breeding value for animal $i$ is computed as

$$B_i = 1 - \frac{(C^{22})_{ii}}{var(u_i)} $$

C <- solve(coef_mat) * sigmae2
C

The lower right corner $C^{22}$ is

nr_fixed_effect <- ncol(X)
nr_sol <- nrow(sol)
C22 <- C[(nr_fixed_effect+1):nr_sol, (nr_fixed_effect+1):nr_sol]
C22

The diagonal elements of $C^{22}$

C22ii <- diag(C22)
C22ii

The reliabilities are

B <- 1 - C22ii / sigmau2
B

charlotte-ngs/lbgfs2021 documentation built on Dec. 19, 2021, 3:01 p.m.