Page 38 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

2 Genetic Covariance Between

Relatives

2.1 Introduction

Of fundamental importance in the prediction of breeding values is the genetic
relationship among individuals. From Chapter 1, it was found that use of the selec-
tion index to predict breeding values requires the genetic covariance between indi-
viduals to construct the genetic covariance matrix. Genetic evaluation using best
linear unbiased prediction (BLUP), the subject of the next chapter, is heavily depend-
ent on the genetic covariance among individuals, both for higher accuracy and for
unbiased results. The genetic covariance among individuals is comprised of three
components: the additive genetic variance, the dominance variance and the epistatic
variance. This chapter addresses the calculation of the additive genetic relationship
among individuals and how to determine the level of inbreeding. Dominance and
epistasis genetic relationships are considered in Chapter 12, which deals with non-
additive models.

2.2 The Numerator Relationship Matrix

The probability of identical genes by descent occurring in two individuals is termed
the coancestry or the coefficient of kinship (Falconer and Mackay, 1996) and the
additive genetic relationship between two individuals is twice their coancestry. The
matrix that indicates the additive genetic relationship among individuals is called
the numerator relationship matrix (A). It is symmetric and its diagonal element for
animal i (a ) is equal to 1 + F , where F is the inbreeding coefficient of animal i
ii i i
(Wright, 1922). The diagonal element represents twice the probability that two
gametes taken at random from animal i will carry identical alleles by descent. The
off-diagonal element, a , equals the numerator of the coefficient of relationship
ij
(Wright, 1922) between animals i and j. When multiplied by the additive genetic
variance (s ), As is the covariance among breeding values. Thus if u is the breeding
2
2
u u i
2
value for animal i, var(u ) = a s = (1 + F )s . The matrix A can be computed using
2
i ii u i u
path coefficients, but a recursive method that is suitable for computerization was
described by Henderson (1976). Initially, animals in the pedigree are coded 1 to n and
ordered such that parents precede their progeny. The following rules are then
employed to compute A.
If both parents (s and d) of animal i are known:
a = a = 0.5(a + a ); j = 1 to (i – 1)
ji ij js jd
a = 1 + 0.5(a )
ii sd
22 © R.A. Mrode 2014. Linear Models for the Prediction of Animal Breeding Values,
3rd Edition (R.A. Mrode)