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Discriminant analysis using a MLMM with a normal mixture 153
profiles and fitted mixed model leading to marginal, conditional and random effects
prediction.
Marginal prediction
For marginal prediction, the predictive density fg,new is equal to the marginal density
of Y new where the term marginal reflects the fact that the random effects are
integrated out. That is, for our model
f marg
g,new = f marg (y new; θ g ) ≡ p � �
y � g
new θ � =
K�
k=1
w g
k pk
� �
y � g
new θ � , (28)
� �
where pk y � g
new θ � �
is the density of N Xg newαg + Zg new(sg + Sgμk), V g
�
new,k
V g
new,k = ZgnewSg D g
kSg′ Zg new ′ +Σ g new.
Conditional prediction
with
For conditional prediction, the predictive density fg,new is equal to the conditional
density of Y new given the estimated values of individual random effects. That is,
for our model
f cond
g,new = f cond (y new; θ g ) ≡ p � �
y �
new bnew = � b g
new, θ g�
(29)
�
which is a density of N Xg newαg + Zg new � b g
new, Σ g �
new . As explained in Section
’Estimates of individual values of random effects’, a suitable estimate of the individual
random effects, denoted by � b g
new, is the mean of the conditional distribution
p(bnew | y new, θ g ) which is computed using an expression analogous to (22), with
y i, bi, θ replaced by y new, bnew, θ g , respectively.
Random effects prediction
Random effects prediction is based on the distribution of the individual random
effects. The predictive density fg,new is then equal to the density of bnew evaluated
at the estimated value of the random effect, i.e., at � b g
new. Hence, in our case,
f rand
g,new = f rand (y new; θ g ) ≡ p � g �
b� � g
new θ � =
K�
k=1
� g �
where pk
�b � g
new θ � is the density of N � sg + Sgμ g
k , SgD g
kSg′� .
w g
k pk
� g �
�b � g
new θ � , (30)