Page 162 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

M are of the order nr. Actual daughter yield deviation for any DIM can be generated
3
using Eqn 9.2.
As indicated in Section 5.2, for ease of computation, W in Eqn 5.12 is pre-
2prog
−1
multiplied with G , such that the equation for DYD becomes:
-
-1
DYD = å G W 2prog a prog ( 2YD u ˆ mate ) / G W 2prog a prog
-1

9.3.4 Reliability of breeding values

The reliability of an EBV depends on its prediction error variance (PEV) relative to
the genetic variance. It can therefore be regarded as a statistic summarizing the value
of information available in calculating the EBV. The published EBV from an RR
model is usually a linear function of the random regression coefficients obtained by
solving the MME. The principles for calculating PEV and reliability under this situa-
tion are presented using the diagonal elements of the inverse of the coefficient matrix
of the MME for Example 9.2.
Let k′u define the EBV for the trait of interest for animal i from the RR model.
The vector k = w t, where w might be the weighting factor for the ith age or lactation
i i
if the study was on body weight at several ages or fat yield in different lactations
analysed as different traits. For instance, if fat yields in lactations 1 and 2 were ana-
lysed as different traits, w′ might be [0.70 0.3], indicating a weight of 0.7 and 0.3,
i
respectively, for first and second lactation EBV. The vector t defines how within lacta-
tion EBV was calculated and is the same as in Eqn 9.2. For Example 9.2, k is a scalar
with a value of 1. Given that G is the additive genetic covariance matrix for random
regression effect for animal effects and P is the covariance matrix for pe effects, then
the additive genetic variance of k′u = g = k′Gk and the variance for the pe effect for
the trait of interest = p = k′Pk. The heritability of k′u can therefore be calculated as
2
2
(g/(g + p + e) and a = (4 − h )/h .
ii
Let C be the subset of the inverse of MME corresponding to the genetic
effect for the ith animal. Then for animal i, prediction error variance (PEV ) =
i
ii
k′C k. The reliability of k′u can therefore be calculated as 1 − PEV /g. As an illus-
i
tration, in Example 9.2, k′ = wT = [215.6655 2.4414 −1.5561], g = k′Gk =
2
2
2
154896.766 kg , p = k′Pk = 323462.969 kg and h = 0.32. For animal 1, the
11
matrix C is:
é 2 9911 0 5159 - 1 2295ù
.
.
.
.
.
C 11 = ê ê 0 5159 0 8683 - 0 2480 ú ú
.
.
ë - ê 0 2295 - 0 2480 0.99183ú û
.
and:
PEV = k′C k = 140499.97
11
1
Therefore, reliability for animal 1 equals 1 − 140499.97/154896.766 = 0.09. The relia-
bilities for the animals in Example 9.2 are:
146 Chapter 9