Page 174 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

With marker information available, the conditional probability that b′ inherits
m
m
Q , given that it has inherited M , is (1 − r), with r being the recombination rate
d′
d′
m
between the ML and the MQTL. Thus if b′ inherits M , the probability in Eqn 10.4
d′
can be calculated recursively as:
p
m
m
m
m
m
P(Q ≡ Q ) = P(Q ≡ Q ) · r + P(Q ≡ Q ) · (1 − r) (10.5)
b b′ b d′ b d′
p
Similarly, given that b′ inherits M , then:
d′
m
m
m
m
p
m
P(Q ≡ Q ) = P(Q ≡ Q ) · (1 − r) + P(Q ≡ Q ) · r (10.6)
b b′ b d′ b d′
m
p
If it is not known whether b′ inherits M or M due to lack of marker information,
d′
d′
p
m
then Q and Q have equal probability of being transmitted to b′. Therefore, r is
d′
d′
replaced by 0.5 in Eqns 10.5 and 10.6.
Using the above information, Fernando and Grossman (1989) developed a
tabular method for constructing G , which is similar to that for calculating A.
v
The rows and columns of G should be such that those for parents precede those
v
for progeny. It should be noted that there are two rows for an individual in G :
v
one each for the paternal and maternal MQTL alleles. Let g be the ij element of
ij
p
G and i , i be the rows of G corresponding to the additive effects of MQTL
m
v o o v
alleles (v , v ) of the oth individual. Similarly, let i , i be the rows for the additive
m
m
p
p
o o s s
p
p
m
effects of the MQTL alleles (v , v ) of its sire (s) additive effects and i , i be the
m
s s d d
m
p
rows for the effects of the MQTL alleles (v , v ) of its dam (d).Then the elements
d d
of the row i below the diagonal, using Eqns 10.4 to 10.6, can be calculated as:
p
o
g p = 1 p ) p g m ; for j = 1 ,..., i -1 (10.7)
p
ij ,
ij , ( - r o g p + r o i s , j o
s
o
p
p
m
p
with r = r if b inherits M or r = (1 − r) if o inherits M . Similarly, elements of row
o s o s
m
i below the diagonal are:
o
g m = 1 m ) m g m ; for j = 1 ,..., i -1 (10.8)
m
ij ,
i o , j ( - r o g p + r o i , j o
d d
where r = r if o inherits M or r = (1 − r) if o inherits M . Since G is symmetric
m
m
p
m
o d o d v
then:
g p = and g m =
ji o , g p o , ji o , g m j
i o ,
i j
m
It is obvious from Eqn 10.4 that, if o = o′, that is, the same individual, then cov(v ,
o
m
m
m
m
v ) = var(v ) as P(Q ≡ Q ) = 1. Therefore, the diagonal elements of G equal unity.
o′
o′
v
o
o
If it is not possible to determine which of the two marker alleles o inherited from its
m
p
sire or dam, then r in Eqn 10.7 and r in Eqn 10.8 are replaced by 0.5.
o o
10.3.1 Numerical application
Example 10.1
Given in the table below are the post-weaning gain data of five calves with the geno-
type at the marker locus given. The aim at this stage is to construct the covariance
matrix G for the MQTL among the five calves.
v
158 Chapter 10