Page 165 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

9.4 Covariance Functions

Kirkpatrick et al. (1990, 1994) introduced the concept of analysing repeated records
taken along a trajectory such as time or age by means of covariance functions. In
view of the fact that such a trait can take on a value at each of an infinite number of
ages and its value at each age can be regarded as a distinct trait, the trajectory for such
a trait could be regarded as an infinite-dimensional trait. Thus the growth trajectory
or milk yield trajectory of an individual could be represented by a continuous function.
Covariance function describes the covariance structure of an infinite-dimensional
character as a function of time. Therefore, the covariance function is the infinite-
dimensional equivalent of a covariance matrix for a given number of records taken
over time at different ages. The value of the phenotypic covariance function, þ(t , t ),
i j
gives the phenotypic covariance between the value of the trait at ages t and t .
i j
Similarly, the value of the additive genetic covariance function, f(t , t ), gives the addi-
i j
tive genetic covariance between the value of the trait at ages t and t . In mathematical
i j
terms, given t ages, the covariance between breeding values u and u on an animal at
l m
ages a and a could be written as:
l m
k-1 k-1
) =
a
cov( ,uu ) = f ( ,a m å å ( ) (a )C (9.6)
l
l m l f i a f j m ij
i=0 j=0
k 1 −1
− k
= ∑ ∑ ij l m (9.7)
i
j
t aa
= i 0 = j 0
where f with factors t is the covariance function (CF), C is the coefficient matrix
ij
associated with the CF with elements C , a is the lth age standardized to the intervals
ij l
for which the polynomials are defined and k is the order of fit. Kirkpatrick et al.
(1990, 1994) used Legendre polynomials that span the interval −1 to + 1. The ages
can be standardized as described in Appendix G.
Given that G is the observed genetic covariance matrix of order t, and
assuming a full order polynomial fit (k = t), Eqn 9.6 can be written in matrix
notation as:
ˆ
G = F C F′ (9.8)
ˆ
ˆ
and C can be estimated as:
ˆ
−1 ˆ
C = F G (F ) (9.9)
−1
where F is the matrix of Legendre polynomials of order t by k with element f = f(a ) =
ij j t
the jth polynomial evaluated at standardized age t.
As an illustration, assume body weight measurements in beef cattle have been
taken at three different ages, 90, 160 and 240 months old, and that the genetic
ˆ
covariance matrix (G) estimated was:
. ⎤
⎡132 .3 127 .0 136 6
ˆ
G = ⎢ ⎢ 127 .0 172 .8 200 .8 ⎥ ⎥
⎣ ⎢136 .6 200 .8 288 . ⎥ 0 ⎦
Using the method described in Appendix G, the vector of standardized ages is:
a′ = [−1.0 −0.0667 1.000]
Analysis of Longitudinal Data 149