Page 267 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

15Estimation of Genetic

Parameters

ROBIN THOMPSON
Rothamsted Research, Harpenden, UK

15.1 Introduction

In order to carry out prediction of breeding values, estimates of variance components
are usually needed. In this chapter the estimation of variance parameters is considered
using univariate sire and animal models.

15.2 Univariate Sire Model

To motivate this work, the mixed effect sire model introduced in Chapter 3 is used.
This model (Eqn 3.15) has:

y = Xb + Zs + e
and:
2
var(s) = As s
2
var(y) = ZAZ ′ s + R
s
2
2
2
where A is the numerator relationship matrix for sires, s = 0.25s and R = Is . The
e
s
a
2
2
aim is to estimate s and s . The simplest case with this sire model is when X is a n × 1
s e
matrix with elements 1, b having one element representing an overall effect and the q
sires being unrelated, so that A = I.
An analysis of variance can be constructed by fitting: (i) a model with the overall
effect b; and (ii) a model with sire effects, these models giving residual sums of
squares that can be put into an analysis of variance of the form:
Source Degrees of freedom Sums of squares
Overall Rank (X) = 1 y ′ X(X ′ X) X ′ y = F
−1
−1
−1
Sires Rank (Z) – rank (X) = q − 1 y ′ Z(Z ′ Z) Z ′ y − y ′ X(X ′ X) X ′ y = S
−1
Residual n – rank (Z) = n − q y ′ y − y ′ Z(Z ′ Z) Z ′ y = R
Essentially, the effects b and s are thought of as fixed effects to construct an
2
2
unweighted analysis. If estimates of s and s are required, then the sums of squares
s
e
S and R can be equated to their expectation E(R) = (n − q)s and E(S) = (q − 1)s +
2
2
e e
2
−1
trace(Z ′ SZ)s where S = I − X(X ′ X) X ′ .
s
© R.A. Mrode 2014. Linear Models for the Prediction of Animal Breeding Values, 251
3rd Edition (R.A. Mrode)