Page 179 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

m
m
m
where (1 − r )r = (1 − r)r for r = r or (1 − r), F is the inbreeding coefficient at
o o o d
m
the MQTL of the dam and h is the diagonal element of D for the paternal MQTL
o
m
of individual o. Therefore, if the dam is not inbred, h = 2(1 − r)r with marker
o
2
m
m
information or h = 0.5 with no marker information and h = s if the dam is
o o v
unknown.
Equation 10.13 may be written as:
n
G -1 = å qqh -1
v j j j
j=1
where n is the number of individuals in the pedigree, q is the column of Q and h is
j j
the jth diagonal element of H. Since Q = (1 − P′), the jth element of q (i.e. the dia-
j
gonal element) is unity and q has at most only two other non-zero elements. If the
j
p
p
p
m
p
sire of o is known, j = i , element i = −(1 − r ) and element i = −r . Similarly, if
o s o s o
m
the dam is known, then for j = i , element i = −(1 − r ) and element i = −r .
m
p
m
m
o d o d o
Therefore, the contribution corresponding to the paternal and maternal MQTL
alleles of an individual to G can easily be calculated from parent and marker
−1
v
information.
Fernando and Grossman (1989) gave the following rules for obtaining G . First,
−1
v
calculate the diagonals of H using Eqns 10.14 and 10.15 and its inverse. Second, set
G to zero and for each offspring o, with sire s and dam d, add the following to the
−1
v
indicated elements of G :
−1
v
If the sire is known, add:
p 2 − 1 p p
s s
(1 − r o ) h p to diagonal element i i ;
i o
p p
p p
(1 p ) h p to elemeents ii and ii ;
−
1
−− r o s o o s
i o
p p − 1 p m m m p
)
o
s s
s s
(1 − rr o h p to elements i i and i i ;
i o
p 2 − 1 m m
s s
(r o ) h p to diagonal element i i ;
i o
− p h p to elemeents ii and i i
m p
p m
−
1
o s o o s
r
i o
If the dam is known, add:
m 2 − 1 p p
d d
(1 − r o ) h m to diagonal element i i ;
i o
p m
m p
(1 m ) h m to elemeents ii and i i ;
−
1
−− r o d o o d
i o
m m − 1 p m m m p
)
d d
o
d d
(1 − rr o h m to elements i i and i i ;
i o
m 2 − 1 m m
d d
o ) h m to diagonal element i i ;
(r
i o
− m h m to elemeents ii and ii
m m
m m
−
1
o
r
i o d o o d
And always add:
p
m
p
h p to element ii and h m to element i , i m
,
- 1
- 1
i o o o i o o o
Use of Genetic Markers in Breeding Value Prediction 163