Page 178 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where Q = (I − P′) and P is a matrix that relates the effect of the MQTL allele of
an individual to the paternal and maternal MQTL alleles of its parent. Each row
of P contains only two non-zero elements if the parent is known, otherwise only
zeros if the parent is unknown. For instance, for individual i with sire (s) known,
p
p
p
p
row i will have (1 − r ) in the column corresponding to i , and r in the column
o o s o
m
m
m
corresponding to column i . Similarly, if dam (d) is known, row i will contain (1 − r )
s o o
m
m
p
in the column corresponding to i and r in the column corresponding to i . The
d o d
row of P for allele i is equal to s in Eqn 10.12. The matrix P for the pedigree in
i
Example 10.1 is:
1p 1m 2p 2m 3p 3m 4p 4m 5p 5m
1p 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1m 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2p 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2m 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
3p 0.9 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
3m 0.0 0.0 0.1 0.9 0.0 0.0 0.0 0.0 0.0 0.0
4p 0.1 0.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
4m 0.0 0.0 0.0 0.0 0.9 0.1 0.0 0.0 0.0 0.0
5p 0.0 0.0 0.0 0.0 0.0 0.0 0.9 0.1 0.0 0.0
5m 0.0 0.0 0.0 0.0 0.9 0.1 0.0 0.0 0.0 0.0
The matrix H is a diagonal matrix for the covariance of residual effects after
adjusting the effect of the MQTL allele of an individual for the effects of the parent’s
p
paternal and maternal MQTL alleles. For example, the residual effect (e ) for a pater-
o
nal MQTL allele of an individual with sire s known is:
p m
p
p
p
p
e = v − (1 − r )v + r v
o o o s o s
p
and the variance of e is:
o
p
p
p
m
p
p
p 2
m
p 2
p
var(e ) = var(v ) − (1 − r ) · var(v ) − (r ) · var(v ) − 2(1 − r )r · cov(v , (v ))
o o o s o s o o s s
p
m
2
m
p
p
p
p
m
p
2
Since var(v ) = var(v ) = var(v ) = s and cov(v , v ) = var(v ) · P(Q ≡ Q ) = var(v ) · F =s F,
o s s v s s s s s s s v s
the above equation can be written as:
p
p 2
2
p
p
2
p
2
var(e ) = 2s (r ) − 2s (r ) − 2s (1 − r )r F
o v o v o v o o s
p
p
p
2
p
= 2s ((1 − r )r − (1 − r )r F )
v o o o o s
2
p
p
= 2s (1 − r )r (1 − F )
v o o s
p
2
p
p
p
var(e )/s = h = 2(1 − r )r (1 − F ) (10.14)
o v o o o s
p
p
p
where (1 − r )r = (1 − r)r for r = r or (1 − r), F is the inbreeding coefficient at
o o o s
the MQTL of the sire and h is the diagonal element of H for the paternal MQTL
p
o
of individual o. Therefore, if the sire is not inbred, h = 2(1 − r)r with marker infor-
p
o
mation or h = 0.5 with no marker information and h = 1 if the sire is unknown.
p
p
o o
Similarly, for the maternal MQTL of o:
2
m
m
m
var(e )/s = h = 2(1 − r )r (1 − F ) (10.15)
m
o v o o o d
162 Chapter 10