Page 171 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where D contains the k largest eigenvalues of C . However, Mantysaari (1999) indi-
p p
cated that with several biological traits, Eqn 9.11 could easily lead to a non-positive definite
C and the decomposition may not be possible. He used an expectation maximization
p
(EM) algorithm to fit the CF to the environmental covariance matrix. However, if C
p
has been estimated directly using REML (Meyer and Hill, 1997), the EM algorithm
would not be necessary and the covariance matrix for pe can be approximated as QD Q′.
p
In addition to reducing the number of equations to k per animal in the MME with this
method, the system of equations is very sparse since D or D are diagonal.
a p

9.5 Equivalence of the Random Regression Model
to the Covariance Function

Meyer and Hill (1997) indicated that the RR model is equivalent to a covariance
function model. The equivalence of the RR model fitting either a parametric curve or
Legendre polynomials to the CF model is presented below. Similar to the model in
Section 9.3, the RR model with a parametric curve can be represented as:
f -1 k-1 k-1
jt å
+
y = F + zt ()b m å zt ()a jm å zt ()l jm + e jt (9.12)
+
jt
m
m
m
m=0 m=0 m=0
where y is the test day record of cow j made on day t; b are fixed regressions coef-
jt m
ficients; a and l are the additive genetic and permanent environmental random
jm jm
regressions for cow j; F represents the remaining fixed effects in the model; z (t) is
jt m
the mth parameter of a parametric function of days in milk; and e is the random
jt
error term. For example, in the model of Jamrozik et al. (1997), z was a function of
2
2
days in milk with five parameters: z = (1 cc dd ), where c = t/305 and d = ln(1/c),
with ln being the natural logarithm. Then the covariance between breeding values u
i
and u on an animal recorded at DIM t and t is:
l i l
k−1 k−1
uu =
t
t
t
cov(, ) f ( , ) = ∑ ∑ z m ( ) ( ) cov(a m ,a r ) (9.13)
z
t
l
i
l
r
i
l
i
m=0 r=0
However, instead of a parametric curve, assume that orthogonal polynomials such
as Legendre polynomials were fitted in an RR model as described in Section 9.3. Let
a and a represent TD records on days t and t of animal j standardized to the interval
i l i l
−1 to 1 as outlined in Appendix G. Furthermore, assume that the mth Legendre poly-
nomial of a be f (a ), for m = 0,...,k − 1. The covariance between breeding values u
i m l i
and u on an animal recorded at DIM a and a could then be represented as:
l i l
k−1 k−1
cov(, ) = f ( , ) = ∑ ∑ ( ) ( ) )
uu
a
a
a
i
i l i l f m a f r l cov(a m ,a r (9.14)
m=0 r=0
The right-hand sides of Eqns 9.13 and 9.14 are clearly equivalent to the right-
hand side of Eqn 9.6, with cov(a , a ) equal to C , the ijth element of the coefficient
m r ij
matrix of the covariance function. This equivalence of the RR model with the covari-
ance function is useful when analysing data observed at many ages or time periods,
as only k regression coefficients and their k(k + 1)/2 covariances need to be estimated
for each source of variation in a univariate model.
Analysis of Longitudinal Data 155