
150 NORMAL LINEAR MODELS
8.3.3 Marginal IM for the variance components
For simplicity, here we will assume that λ
L
= 0; this assumption often holds, as in
the three examples discussed in Section 8.3.4.4 below, but there are models, such as
the full animal model [132], where it may fail. To start, for a given S = s and σ
2
, we
can solve for the auxiliary variable v in the above baseline association:
v
s,σ
2
,`
=
s
`
λ
`
σ
2
α
+ σ
2
ε
, ` = 1,... ,L −1, v
s,σ
2
,L
=
s
L
σ
2
ε
.
Differentiating this expression with respect to both components of σ
2
gives an L ×2
matrix ∂ v
s,σ
2
/∂ σ
2
= diag{v
s,σ
2
}W (σ
2
), where the rows of W (σ
2
) are given by
w
`
(σ
2
) =
−
λ
`
λ
`
σ
2
α
+ σ
2
ε
, −
1
λ
`
σ
2
α
+ σ
2
ε
, `