Page 311 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

and:

M = diag[3.0 2.0 3.667 4.0 4.0 4.667 6.0 6.0 5.0 5.0]
The starting values for PCG can now be calculated. Thus:

(0)
−1
(0)
(0)
(0)
b = 0, e = r − Cb = y and d = M r
Thus:
(0)
d ′= [4.333 3.4 0.0 0.0 0.0 0.964 0.483 0.650 0.70 1.0]
ITERATIVE STAGE
Reading through the data and performing the following calculations in each round of
iteration, start the PCG iterative process. Calculations are shown for the first round
of iteration.
The vector v = Cd is accumulated as data are read. For the ith level of fixed effect:
v(i) = v(i) + 1(d(i)) + 1(d(anim ))
k
where anim refers to the animal k associated with the record. Thus for the level 1 of
k
sex of calf effect:
v(1) = 3(4.333) + d(anim ) + d(anim ) + d(anim ) = 15.663
4 7 8
As each record is read, calculate:
z = 4/(2 + number of unknown parents for animal with record)

if either parent is known, otherwise xx = 0
xx = −0.5(z)a
if both parents are known, otherwise xm = 0
xm = 0.25(z)a
If only one parent, p, of animal k is known, then accumulate:

v(anim ) = v(anim ) + 1(d(i)) + M (d(anim )) + xx(d(anim )) (17.14)
k k k,k k p
where d(i) refers to the ith level of the fixed effect and M the diagonal element of
k,k
M for animal k.
Accumulate the contribution to the known parent, p, of k at the same time:
v(anim ) = v(anim ) + xx(d(anim ))
p p k
If both parents p and j of animal k are known, then accumulate for animal k as:

v(anim ) = v(anim ) + 1(d(i)) + M (d(anim )) + xx(d(anim ) + d(anim)) (17.15)
k k k,k k p j
Accumulate for both parents as:

v(anim ) = v(anim ) + xx(d(anim ))
p p k
v(anim ) = v(anim ) + xm(d(anim ))
p p j
v(anim ) = v(anim ) + xx(d(anim ))
j j k
v(anim ) = v(anim ) + xm(d(anim ))
j j p

Solving Linear Equations 295