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Chapter 3.2
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X and Z matrices from the “stacked” specification (2) of the MLMM to en-
hance the computational speed. The package is freely available from CRAN at
http://CRAN.R-project.org/package=mixAK.
Estimates of individual values of random effects
We now explore the conditional distributions p(bi | y i,ui, θ) and p(bi | y i, θ) resulting
from the Bayesian specification of the model. This will be useful for the developments
in next section. The distribution p(bi | y i,ui = k, θ) (i =1,...,N, k =
1,...,K) is in fact the full conditional distribution for bi and it is used in the MCMC
algorithm to update the values of random effects. It can be shown that it is a (multivariate)
normal distribution with mean and covariance matrix
E(bi | y i,ui = k, θ) =s + SQ −1
i,k η i,k, (17)
var(bi | y i,ui = k, θ) =SQ −1
i,k S′ , (18)
where Q −1
i,k is the covariance matrix and η i,k is the canonical mean of the conditional
distribution p(b ∗ i | y i,ui = k, θ) of shifted and scaled random effects given by
Qi,k = Qi,k(y i, θ) =S ′ Zi ′ Σ −1
η i,k = η i,k(y i, θ) =S ′ Zi ′ Σ −1
i
Note that expression (17) can alternatively be written as
i ZiS + D −1
k , (19)
�
y i − Xiα − Zis � + D −1
k μk. (20)
E(bi | y i,ui = k, θ)
�
= s + S DkS ′ Zi ′ (ZiSDkS ′ Z ′ i +Σi) −1 � y i − Xiα − Zi (s + S μk) � �
+ μk , (21)
which resembles (taking into account the changes due to inclusion of the shift s,
scale S and non-zero mean μ k) a classical expression for empirical Bayes estimation
of a random effect in a standard linear mixed model with normally distributed random
effects (see Verbeke and Molenberghs11 , Section 7.2).
On the other hand, the conditional distribution p(bi | y i, θ) can serve as a basis for
inference on random effects which closely corresponds to classical approaches since
all latent parameters, especially the component allocation ui, are integrated out and
we only condition on observed data and parameters in a classical sense. The mean
of this distribution is given by
�bi =E(bi | y i, θ) =
K�
wi,k(y i, θ) E(bi | y i,ui = k, θ), (22)
k=1