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where
wi,k(y i, θ) =
(k =1,...,K).
wk |Dk| −1/2 |Qi,k| −1/2 �
exp − 1
K�
wj |Dj|
j=1
−1/2 |Qi,j| −1/2 exp
Discriminant analysis using a MLMM with a normal mixture 151
�
′ −1
μk Dk μ ′ −1
k − ηi,k Qi,kη �
i,k
�
2
�
− 1
�
′ −1
μj Dj 2
μ ′ −1
j − ηi,j Qi,j η �
i,j
�
(23)
The Bayesian estimate of bi integrating out the uncertainty with which the pa-
rameters θ are estimated is the posterior mean of � bi. That is, E � E(bi | y i, θ) � � y � ,
where the second expectation is done over the different possible θ values. For
θ (1) ,...,θ (M) , the (MCMC) sample from the posterior distribution p(θ | y), � b (m)
i =
E(bi | y i, θ (m) )(m =1,...,M)andE � E(bi | y i, θ) � �
� y is estimated as
�bi = � E � E(bi | y i, θ) � � y � = 1
M
M�
E � �
bi
� y i, θ (m)� = 1
M
m=1
M�
m=1
�b (m)
i , (24)
leading to the Bayesian estimate of the i-th individual value of random effects.
Discrimination procedure
To develop a discrimination procedure, we assume that a training data set is available
for which we know the allocation of the involved subjects (patients, longitudinal
profiles) to prognostic groups (g =0,...,G − 1). Let y g = {y i : i ∈
prognostic group g} be the observed values of the longitudinal markers for subjects
in the training data set belonging to prognostic group g. Each prognostic group is
characterized by MLMM written as (1) or written in a condensed way as (2), with
the random effects distribution specified by the mixture (3), i.e.,
Y i = X g
i αg + Z g
i bi
⎫
+ εi,
bi = s g + S g b ∗ i ,
b ∗ i
i.i.d.
∼ � K
k=1
w g
k
εi ∼N(0, diag(σ g
1
N (μg k , Dg
k ),
2 ,...,σ g
1
(i ∈ prognostic group g).
2 ,...,σ g
R
2 g 2
,...,σR ))
⎪⎬
⎪⎭
(25)
Let θ g = � αg′ , w g′ , μ g′
g ′ g
1 ,...,μK , vec(D1 ),...,vec(Dg K ),σg
2 g 2
1 ,...,σR � ′
be the model
parameters for group g. Note that not only the values of parameters θ g but also
the structure of the MLMM (25), for example the structure of matrices X g
i and Zg
i ,
may differ between prognostic groups.