Page 197 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

(Continued)
Fat
Animal Sire Dam Mean EDC DYD SNP Genotype
24 14 10 1 66 9.2 1 0 0 0 1 1 0 2 0 0
25 14 7 1 75 11.5 0 0 0 1 1 2 0 2 1 0
26 14 12 1 33 13.3 1 0 1 1 0 2 0 1 0 0
EDC, effective daughter contribution; DYD, daughter yield deviation.
The prediction of marker effects and polygenic effects for the reference population and
−1
selection candidates can be done simultaneously by including A for all animals but using only
the fat yield records for the reference animals. Thus y′ = (9.0 13.4 12.7 15.4 5.9 7.7 10.2 4.8).
The incidence matrix X = I , with q = 8 (the number of animals in the reference population).
q
COMPUTING THE MATRIX Z
The computation of Z requires calculating the allele frequency for each SNP. The
allele frequency for the ith SNP was computed as:
n
∑ m
j ij
2* n
where n = 14, the number of animals with genotypes, and m are elements of M. The
ij
allele frequencies for the ten SNPs were 0.312, 0.179, 0.357, 0.357, 0.143, 0.607, 0.071,
0.964, 0.571 and 0.393, respectively. However, only the first three SNPs are needed for
this example, therefore Z is of order 8 by 3 with elements z = m – p , with j =1, 3. Thus:
i,j i,j i,j
⎛ 1.357 − 0.357 0.286 ⎞
⎜ 0.357 − 0.357 − 0.714 ⎟
⎜ ⎟
⎜ 0.357 0.643 1.286 ⎟
⎜ − − ⎟
Z = ⎜ 0.643 0..357 1.286 ⎟
⎜ − 0.643 0.643 0.286 ⎟
⎜ ⎟
⎜ 0.357 0.643 − 0.714 ⎟
⎜ − 0.643 − 0.357 0.286 ⎟
6
⎜ ⎟
⎝ − 0.643 0.643 0.286 ⎠
The W matrix is a diagonal matrix for the eight reference animals with records. This is aug-
mented with 12 columns of zeros to account for ancestors 1 to 12. For the weighted analysis,
the R was a diagonal matrix with the diagonal elements equal to the EDC of the first eight
animals in the data set. The matrix A is computed using the usual rules for all 26 animals
−1
and a = 245/35.241 = 6.952. Solving the system of equations gives the following results:
Unweighted analysis Weighted analysis

Mean effect
9.895 9.178
SNP effect
1 0.607 2.655
2 −4.080 −4.640
3 1.934 2.951
Continued

Computation of Genomic Breeding Values and Genomic Selection 181