Page 119 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

with terms defined as in Eqn 5.1 and MME as in Eqn 5.2. If G is represented by an
FA structure (Eqn 6.8), then an equivalent model to Eqn 6.9 is:
*
y = Xb + Z(Iq × F)c + Ws + e = Xb + Z c + Ws + e (6.10)
with q being the number of individuals, c is a vector of common factor effects of
*
order m, Z = Z(Iq × F), and s is the vector for the specific factor effects. In some
contexts, application of Eqn 6.10, i.e. with elements of S ≠ 0, is referred to as the
extended factor analysis (XFA) compared with models with no specific effects (S = 0),
which is simply referred to as factor analysis (FA). The MME for XFA then are:

⎡ X′RX X′R Z ∗ X′R W⎤ ⎛ ⎞ ˆ ⎡ X′ Ry⎤
−
−1
1
−1
−1
b
⎢ ⎥ ⎜ ⎟ ⎢ ⎥
c = ⎢
∗
∗
∗
∗
⎢ ZR X Z R Z + I ⊗ A −1 ZR W⎥ ⎜ ⎟ ZR y⎥ (6.11)
′
∗
′
′
′
1
−1
−1
−1
−1
ˆ
⎢ m ⎥ ⎢ ⎥
−
−
−
1
s
−1
1
1 ⎜ ⎟ ˆ
1
−1
⎢ ⎣ W′RX W′R Z ∗ W′ R W + S − 1 ⊗ A ⎥ ⎝ ⎠ ⎢ ⎣ W′ Ry⎥ ⎦
⎦
The vector a of solutions for animal i can be obtained as:
ˆ
i
a = Fcˆ + sˆ (6.12)
ˆ
i i i
The number of equations in the MME (Eqn 5.2) for the usual multivariate model
are equal to the number of equations for b and s in Eqn 6.11. However, there are an
*
additional mq equations for the common effects and Z , which is a vector of order m
with elements j , is denser than Z in Eqn 5.2, which contains a single element of
ij
unity in a row or column. However, the section of the coefficient matrix for random
−1
effects is much sparser as effects are genetically uncorrelated and A contributes only
2
(m + n) non-zero elements compared to n for Eqn 5.1. For the estimation of covari-
ance estimates using REML, Thompson et al. (2003) showed that the sparsity of the
MME with an XFA structure imposed dramatically reduced computational require-
ments compared to the standard multivariate model. Note that fitting an FA structure
to G with no specific effects, the MME are similar to Eqn 6.11 but with the row of
*
equations for s omitted, and the Z will be a vector of order n.
An illustration
Example 6.3
The data on pre-weaning weight gain (WWG) and post-weaning gain (PWG) in
Example 5.1 is extended to include two additional traits of muscle score (MS) and
backfat thickness (BFAT), and data is presented below. The objective is to undertake
multi-trait analysis imposing an XFA on G and the results obtained compared to
those from full MBLUP or FA structure on G with no specific factors.
Calf Sex WWG PWG MS BFAT
4 Male 4.5 6.8 5.0 0.226
5 Female 2.9 5.0 3.0 0.573
6 Female 3.9 6.8 12.0 0.386
7 Male 3.5 6.0 8.0 0.290
8 Male 5.0 7.5 15.0 0.175
Methods to Reduce the Dimension of Multivariate Models 103