Livestock Breeding and Genomics - Excercise 7

Pedigree

nr_animal <- 6
tbl_pedigree <- tibble::tibble(Calf = c(1:nr_animal),
                               Sire = c(NA, NA, NA, 1 ,3, 4),
                               Dam = c(NA, NA, NA, 2, 2, 5))
tbl_pedigree

Numerator Relationship Matrix

We call the numerator relationship matrix \(A\). The computation of the elements of \(A\) are done separately for

1. the diagnoal elements \((A)_{ii}\) and
2. the off-diagonal elements \((A)_{ij}\) for \(i \ne j\)

First all elements of the matrix \(A\) are initialized to \(0\)

A = matrix(0, nrow = nr_animal, ncol = nr_animal)
A

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0
[5,]    0    0    0    0    0    0
[6,]    0    0    0    0    0    0

Diagonal Elements

Computation: \((A)_{ii} = (1 + F_i)\) and \(F_i = 1/2 (A)_{sd}\)

i <- 1
s <- tbl_pedigree$Sire[i]
d <- tbl_pedigree$Dam[i]

Fi <- ifelse((is.na(s) | is.na(d)), 0, 0.5 * A[s,d])
A[i,i] <- 1+Fi
A

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    0    0    0    0    0
[2,]    0    0    0    0    0    0
[3,]    0    0    0    0    0    0
[4,]    0    0    0    0    0    0
[5,]    0    0    0    0    0    0
[6,]    0    0    0    0    0    0

Off-diagonal Elements

Off-diagonal \((A)_{ij} = 1/2 (A)_{io} + 1/2 (A)_{iq}\) where \(o\) and \(q\) are parents of \(j\)

for (j in (i+1):6){
  o <- tbl_pedigree$Sire[j]
  q <- tbl_pedigree$Dam[j]
  Aio <- ifelse(is.na(o), 0, A[i,o])
  Aiq <- ifelse(is.na(q), 0, A[i,q])
  A[i,j] <- 0.5 * Aio + 0.5 * Aiq
}
A[(i+1):6,i] <- A[i,(i+1):6]
A

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,] 1.00    0    0  0.5    0 0.25
[2,] 0.00    0    0  0.0    0 0.00
[3,] 0.00    0    0  0.0    0 0.00
[4,] 0.50    0    0  0.0    0 0.00
[5,] 0.00    0    0  0.0    0 0.00
[6,] 0.25    0    0  0.0    0 0.00

Problem 1: Numerator Relationship Matrix

Use the above steps of computation for the complete matrix.

Hint

• Construct a loop with loop-variable \(i\) that runs over all the rows of the matrix \(A\).

Solution

Check Result

The function getA() of the pedigreemm package can be used to check the result