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Chapter 3.2
146
The correlation between single observations made on the same subject (between two
measurements of either the same marker or two different markers) is introduced by
specifying a joint distribution for bi =(b ′ i,1,...,b ′ i,R) ′ , the vector of i-th subject
specific random effects pertaining to all R markers. Particularly, the covariance
matrix var(bi) is assumed to be a general (unstructured) positive definite matrix D.
The model is finalized by specifying the distribution of random effects. Morrell et al. 7
or Fieuws et al. 9 assume a normal distribution, i.e., bi
i.i.d.
∼ N(β, D) which together
with the representation (2) allows them to fit model (1) using standard software
like R (R Development Core Team13 ) package lme4 (Bates and Maechler15 )orSAS
PROC MIXED/NLMIXED.
For reasons mentioned in the introduction we replace the normality assumption here
by a heteroscedastic normal mixture. More precisely, we assume that
bi = s + S b ∗ i , b ∗ i
i.i.d.
∼
K�
wk N (μk, Dk), (3)
where s is a fixed shift vector, S a fixed diagonal scale matrix and b ∗ i are shifted and
scaled random effects. Further, w =(w1,...,wK) ′ is a vector of non-negative mixture
weights satisfying �K k=1 wk =1,μ1,...,μK are mixture means and D1,...,DK
are mixture covariance matrices. The number of mixture components K is assumed
to be pre-specified. Note that the shift s and the scale S are included in the specification
of the model (3) only to improve numerical stability of the estimation procedure
described in Section ’Estimation’. However, it is possible to set s = 0 and S = Iq.
The model parameters which need to be estimated are
θ = � α ′ , w ′ , μ ′ 1,...,μ ′ K, vec(D1),...,vec(DK), σ 2 1,...,σ 2 � ′,
R (4)
where vec(Dk) is a vector with the elements of the lower triangle of the matrix Dk.
Note that the means of random effects (β) and the overall covariance matrix of
random effects D are determined by the parameters of the mixture as
D =var(bi) = S
� K�
k=1
wk
�
Dk +
�
μ k −
K�
j=1
k=1
wj μ j
��
μ k −
K�
j=1
wj μ j
� ′ � �
In the model of Verbeke and Lesaffre 10 , maximum-likelihood estimation was used.
Applied to the above heteroscedastic mixture (3) results however in an unbounded
likelihood (McLachlan and Basford16 ). Here, we will use a Bayesian approach where,
with some care, the problem of unbounded likelihood is tackled by using suitable prior
distributions for mixture covariance matrices, see Section ’Prior distributions’.
S ′ .
(5)