Prediction Theory 1 Introduction 2 General Linear Mixed Model

17.1 Operational Models
Let
y ijk = µ + C i + S j + e ijk ,
where y ijk are the observations on the trait of interest of individual progeny, assumed to be one
record per progeny only, µ is an overall mean, C i is a random contemporary group effect, S j is
a random sire effect, and e ijk is a random residual error term associated with each observation.
E(y ijk ) = µ,
V ar(e ijk ) = σe
2
V ar(C i ) = σc 2 = σe/6.0
2
V ar(S j ) = σ 2 s = σ 2 e/11.5
The ratio of four times the sire variance to total phenotypic variance (i.e. (σ 2 c + σ 2 s + σ 2 e)), is
known as the heritability of the trait, and in this case is 0.2775. The ratio of the contemporary
group variance to the total phenotypic variance is 0.1329. The important ratios are
σ 2 e/σ 2 c = 6.0
σ 2 e/σ 2 s = 11.5
There are a total of 21 observations, but only five filled subclasses. The individual observations
are not available, only the totals for each subclass. Therefore, an equivalent model is the
“means” model.
ȳ ij = µ + C i + S j + ē ij ,
where ȳ ij is the mean of the progeny of the j th sire in the i th contemporary group, and ē ij is
the mean of the residuals for the (ij) th subclass.
The model assumes that
• Sires were mated randomly to dams within each contemporary group.
• Each dam had only one progeny.
• Sires were not related to each other.
• Progeny were all observed at the same age (or observations are perfectly adjusted for age
effects).
• The contemporary groups were independent from each other.
19