Page 196 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The equations for obtaining the solutions for SNP and polygenic effects are:
⎛ XX XR X X Z ⎞ ⎛ ⎞ ˆ ⎛ Xy ′ ⎞
′
′
′
−1
b
⎜ ′ ′ ′ ⎟ ⎜ ⎟ ⎜ ⎟
g ˆ = Zy ′
⎜ ZX ZX Z Z ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ (11.4)
⎜ ⎝ WX WX WW + A a ⎟ ⎜ u ˆ ⎟ ⎝ Wy ′ ⎠
′
′
⎠ ⎝ ⎠
−1
where:
2 2
e u
a = s /s
If the vector of observations, y in Eqn 11.3, are de-regressed breeding values of bulls
(see Section 5.5.2), then each observation may be associated with differing reliabilities.
Thus a weighted analysis may be required to account for these differences in bull reli-
abilities. The weight (wt ) for each observation could be the reciprocal of the effective
i
daughter contribution (see Section 5.5.2) or wt = (1/rel ) – 1, where rel is the bull’s
i dtr dtr
reliability from daughters with parent information excluded (VanRaden, 2008). Then the
MME are:
æ æ X¢RX X¢RX X¢R Z öæ ö ˆ b æ X¢ Ry ö
-1
-1
-1
-1
ç ÷ç ÷ ç ÷ ÷
-1
ç Z¢RX Z¢RX Z¢R Z ÷ç ÷ = çZ¢ Ry ÷ (11.5)
-1
-1
-1
ˆ g
ç ç -1 -1 -1 -1 ÷ç ÷ ç -1 ÷
øè ø
è W¢RX W¢RX WR W + A a ÷ç ÷ ˆ u ç W¢ Ry ÷ ø
è
where R = D and D is a diagonal matrix with diagonal element i = wt .
i
Example 11.1
Given below is the real genotype for the first ten SNPs of a popular dairy bull and
those of his sons and some other unrelated bulls genotyped using the 50K Illumin chip.
The genotypes of animals are coded as described in Section 11.3. The observations are
the DYDs for fat yield, and the effective daughter contribution (EDC) for each bull is
also given. The EDC can be used as weights in the analysis. It is assumed the genetic
2
2
variance for fat yield is 35.241 kg and residual variance of 245 kg , and animals 13
to 20 are assumed as the reference population and 21 to 26 as selection candidates.
Assuming that the first three SNPs have been identified as having the most significant
effect, the aim is to fit Eqn 11.3 with and without weights using these three SNPs:
Fat
Animal Sire Dam Mean EDC DYD SNP Genotype
13 0 0 1 558 9.0 2 0 1 1 0 0 0 2 1 2
14 0 0 1 722 13.4 1 0 0 0 0 2 0 2 1 0
15 13 4 1 300 12.7 1 1 2 1 1 0 0 2 1 2
16 15 2 1 73 15.4 0 0 2 1 0 1 0 2 2 1
17 15 5 1 52 5.9 0 1 1 2 0 0 0 2 1 2
18 14 6 1 87 7.7 1 1 0 1 0 2 0 2 2 1
19 14 9 1 64 10.2 0 0 1 1 0 2 0 2 2 0
20 14 9 1 103 4.8 0 1 1 0 0 1 0 2 2 0
21 1 3 1 13 7.6 2 0 0 0 0 1 2 2 1 2
22 14 8 1 125 8.8 0 0 0 1 1 2 0 2 0 0
23 14 11 1 93 9.8 0 1 1 0 0 1 0 2 2 1
Continued
180 Chapter 11