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2.1 The general linear mixed model
2.1.6
Gamma parameterization for the linear mixed model
The sigma parameterization of model (2.3) is one possible parameterization of var (y) . In this
parameterization both G(σ g ) and Rv (σ r ) are variance matrices and the variance structure
parameters in σ g and σ r are referred to as sigmas, see above. Other parameterizations
are possible and are sometimes useful. For example, in some of the early development of
REML for the traditional mixed model of (2.5), the variance matrix was parameterized as
the equivalent model
!
b
X
|
(2.6)
var (y) = σe2
γgi Z i Z i + I n
i
for γgi being the ratio of the variance component for the random term ui relative to error
variance, that is, γgi = σu2i /σe2 . In this case ASReml calculated a simple estimate of σe2 and
initial values for the iterative process were specified in terms of the ratios γgi rather than in
terms of the variance components σu2i . It was often easier to specify initial values in terms of
these ratios rather than the variance components which is why this approach was adopted.
Where Rv (σ r ) can be written as a scaled correlation matrix, that is, Rv (σ r ) = σe2 Rc (γ r ),
this suggests the alternative specification of (2.2)
u
0
0
2 G(γ g )
∼N
, σe
(2.7)
e
0
0
Rc (γ r )
where γ g and γ r represent the variance structure parameters associated with scaled (by σe2 )
variance matrices. In this case
|
var (y) = σe2 ZG(γ g )Z + Rc (γ r ) ,
(2.8)
which we will refer to as the gamma parameterization, and the individual variance structure
parameters in γ g and γ r will be referred to as gammas. ASReml switches between the sigma
and gamma parameterizations for estimation. This is discussed in Section 7.6.
2.1.7
Parameter types
Each sigma in σ g and σ r and each gamma in γ g and γ r has a parameter type, for example, variance components, variance component ratios, autocorrelation parameters, factor
loadings. Furthermore, the parameters in σ g , σ r , γ g and γ r can span multiple types. For
example, the spatial analysis of a simple column trial would involve variance components
(sigma parameterization) or variance component ratios (gamma parameterization) and spatial autocorrelation parameters.
2.1.8
Variance structures for the random model terms
The random model terms ui in u define the random effects and associated design matrices,
Z i ∈ Z, but additional information is required before the model can be fitted. This extra
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