Page 190 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

either var(e ) or var(e ) is similar to that for var(e) in Section 10.5. For instance, for
12 21
the oth progeny:
= v − q(v ) − p(v )
o12 o12 s11 s22
e
2
2
) = var(v ) − q var(v ) − p var(v ) − 2qpcov(v , v )
o12 o12 s11 s22 s11 s22
var(e
2
2
2
2
2
2
= s − (1 − p) s + p s − 2(1 − p)ps F
v v v v s
2
= 2s ((1 − p)p − (1 − p)pF )
v s
2
2
= 2s (1 − p)p(1 − F ) = 2s pq(1 − F )
v s v s
2
)/s = H = 2pq(1 − F )
o12 v s
var(e
Therefore, if the sire is not inbred, the diagonal element of H for progeny o(h )
oo
with the allele v equals 2pq. If the sire is unknown, h = 1. Similarly, for a
o12 oo
progeny o with allele v , h = 2qp if the sire is known, otherwise 0 if the sire is
o21 oo
unknown.
The matrix G can be calculated using rules similar to those defined in Section
10.3. The relationship of the MQTL paternal allele of a progeny o with MQTL
p
alleles of individuals 1 to (o – 1) can be calculated using Eqn 10.7, with r = p
o
p
when o inherits marker haplotype v or r = (1 − p) when o inherits marker
s12 o
haplotype v . Similarly, for the maternal MQTL allele, Eqn 10.8 can be used with
s21
m
m
r = p when o inherits marker haplotype v or r = (1 − p) when o inherits
o m12 o
marker haplotype v .
m21
Using Eqn 10.26, Goddard (1992) derived the following rules for calculating G .
−1
v
1. Replace v with v in all equations and then delete the row and column for v
o11 s11 o11
−1
in G . Similarly, replace v with v . Set G to zero.
−1
o22 s22
2. For progeny allele v , add:
o12
q/2p to the element corresponding to (v , v )
s11 s11
p/2q to the element corresponding to (v , v )
s22 s22
1/2pq to the element corresponding to (v , v )
o12 o12
−1/2p to the element corresponding to (v , v ) and (v , v )
s11 o12 o12 s11
−1/2q to the element corresponding to (v , v ) and (v , v )
s22 o12 o12 s22
1/2 to the element corresponding to (v , v ) and (v , v )
s11 s22 s22 s11
3. For a progeny allele v , replace p with q and v with v in the rules above.
o21 o12 o21
4. For an allele v without known parents, add 1 to element corresponding to (v , v )
s11 s11 s11
Goddard (1992) indicated that the use of the approximate means to calculate P
implies that v and v are forced to be identical even if double crossover occurs.
s11 o11
Therefore, it might be desirable to use a correlation (r) slightly less than unity between
v and v . This is achieved by using:
s11 o11
v 2 2
s11
o11 = (1 − r /4)v + r /4v + e 11
s22
−1
Then the row and column for v are retained in G , and, in the above rules, v is
o11 v o12
2
replaced by v and p by r /4.
o11
10.9.3 An illustration
Example 10.5
Consider that the four calves in the following data set have the following genotype at
two linked loci.
174 Chapter 10