Page 163 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Animal Reliability
1 0.09
2 0.04
3 0.07
4 0.12
5 0.15
6 0.06
7 0.10
8 0.05
In practice, calculating the inverse of the MME is not feasible for large populations
and PEV has to be approximated. As indicated earlier, EBV from RR models are linear
functions of the random regressions; therefore, methods to approximate reliabilities should
simultaneously approximate PEV and the prediction error covariance (PEC) among the
individual random regressions (Liu et al., 2002; Meyer and Tier, 2003). Such an approxima-
tion method presented by Meyer and Tier (2003) is outlined in Appendix D, Section D.2.

9.3.5 Random regression models with spline function

Random regression models with Legendre polynomials have been considered to have
better convergence properties as the regressions are orthogonal. However, some stud-
ies have reported high genetic variances at the extremes of the lactation and negative
correlations between the most distant test days. In order to overcome this limitation,
some workers have fitted RRM using splines (Misztal, 2006; Bohmanova et al., 2008).
Splines are piece-wise functions consisting of independent segments that are connected
in knots. The segments are described by lower-order polynomials. Linear splines are
the simplest spline function where the segments are fitted by linear polynomials
between two knots adjacent to the record and zero between all other knots. Thus the
system of equations is sparse as only two coefficients are non-zero for a given record.
The use of cubic splines for the modelling of the lactation curve has also been pre-
sented by White et al. (1999). However, the linear spline is considered in this section.
Let T be a vector of n knots, then the covariables of the linear spline for DIM
t (F (t)) located between knots T and T can be calculated as:
i i i+1
F (t) = (t − T)/(T − T)
i i i+1 i
F (t) = (T − t)/(T − T)
i+1 i+1 i+1 i
= 0
i 1...i−1, i+2...n
= 1 − F (t) and F
= 0.
i
i
If t = T, F (t) = 1 and F 1...i−1, i+1...n
Thus the vector F for DIM t has at most two non-zero elements, which sum up to
one. The above formula assumes that T £ t < T . If, however, t < T or t > T , the fol-
i n i n
lowing can be used and the sum of the elements of the vector will not sum up to one:
= 0
1
1
1
if t < T , F (t) = t/T and F 1+ i...n
= 0
n
n
5
if t > T , F (t) = T /t and F 1...n−1
Using the data in Example 9.1, assume that the four knots are fitted for the fixed
lactation curve and knots are placed at days 4, 106, 208 and 310, the covariables for
the spine function for particular DIM are as follows:
Analysis of Longitudinal Data 147