Page 225 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

12.4 Method for Rapid Inversion of the Dominance Matrix

Hoeschele and VanRaden (1991) developed a method for computing directly the
inverse of the dominance relationship matrix for populations that are not inbred,
by including sire and dam or sire and maternal grandsire subclass effects in the
model. However, only the inclusion of sire and dam subclasses is considered in
this text. Dominance effects result from interaction of pairs of genes and are not
inherited through individuals. Since animals receive half of their genes from the
sire and half from the dam, the dominance effect of an individual could be
expressed as:
d = f (12.5)
S,D + e
where f represents the average dominance effect of many hypothetical full-sibs pro-
duced by sire (S) and dam (D) and e is the Mendelian sampling deviation of the
2
individual from the S by D subclass effect. Variance of S by D subclass effects, s , is equal
f
2
2
to the covariance among full-sibs due to dominance, i.e. s = 0.25s ; therefore,
f
d
2
var(e) = 0.75s . On the basis of Eqn 12.5, Hoeschele and VanRaden developed sim-
d
ple recurrence formulae for dominance effects using pairs of animals (sire and dam)
and interaction between their parents.
For a particular sire and dam subclass (f ), the combination effect results
SD
from the interactions between the sire and the parents of D, interactions of the
dam with the parents of S and interactions of the parents of S with the parents of D.
Thus:
f = 0.5(f + f + f + f )
SD S,SD S,DD SS,D DS,D
− 0.25(f + f + f + f ) + e (12.6)
SS,SD SS,DD DS,SD DS,DD
where SS and DS denote sire and dam of sire, respectively, and SD and DD corre-
sponding parents for the dam. Equation 12.6 can also be obtained by regressing f on
SD
its parent subclasses effects as:
f = b′f + e
SD par
where f is a vector of eight parent subclasses in Eqn 12.6 and b is a vector of
par
corresponding partial regression coefficients with:
b′ = cov(f , f )/var(f ) (12.7)
SD par par
and:
2
var(e) = s − b′var(f )b (12.8)
f par
The covariance between subclasses in Eqn 12.7, for instance between f and
SD
f , is:
PM
cov(f , f ) = (a a + a 2 (12.9)
SD PM SP DM SM DP f
a )s
with a being the additive relationship between i and j. Thus:
ij
cov(f , f ) = (a a + a a 2 2
SD SS,DD S,SS D,DD S,DD D,SS )s = (0.5(0.5)) + (0(0)) = 0.25s f
f
and:
cov(f , f ) = (a a + a a 2 2
f
SD S,SD SS D,SD S,SD D,S )s = (1(0.5)) + (0(0)) = 0.5s f
Non-additive Animal Models 209