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Discriminant analysis using a MLMM with a normal mixture 149
informative prior distribution, we refer to Komárek 14 , where in his notation the
terms yi,j are replaced by reasonable initial values of the random effects. These can
be, for example, equal to their empirical Bayes estimates from separate (for r =
1,...,R) maximum-likelihood fits of models derived from (1). A weakly informative
priorforthefixedeffectsαis obtained by setting ξαr ,l to zero, and c 2 αr ,l (r =
1,...,R,l =1,...,pr ) to a large positive number, e.g., 10 000 (it is necessary to
check that the variance of the posterior distribution is considerably smaller). Finally,
adapting recommendations of Richardson and Green22 , the following values of the
fixed hyperparameters ζε,r ,gε,r ,hε,r (r =1,...,R) related to the prior distribution
of the error terms lead to a weakly informative prior: a small positive number for
ζε,r and gε,r , hε,r =10/R 2 ε,r , where Rε,r is a range of residuals from separate (for
r =1,...,R) initial maximum-likelihood fits of models derived from (1).
Posterior distribution and Markov chain Monte Carlo
Given the parameters ψ, i.e. parameters for which the prior distribution has been
specified in (7), the likelihood corresponding to model (1) takes a relatively simple
form, i.e.,
N�
N� R�
L(ψ) = p(y i | ψ) = p(y i,r | αr , bi,r,σ 2 r )
i=1
∼
i=1 r=1
N�
R�
N (Xi,rαr + Zi,rbi,r, σ
i=1 r=1
2 r Ini,r ).
(16)
Let y be the observed values of all longitudinal markers from the whole data.
Bayesian inference is based on a sample from the posterior distribution p(ψ | y) ∝
L(ψ) p(ψ) obtained using the Markov chain Monte Carlo method with a block Gibbs
sampler.
To improve the numerical properties of the MCMC algorithm, it is useful to choose
the shift vector s and the scale matrix S (see expression (3)) such that the shifted
and scaled random effects b ∗ i have approximately zero mean and unit variances. For
this reason we recommend to set s to the estimated means and diagonal elements of
the S matrix to the estimated standard deviations from separate (for r =1,...,R)
maximum-likelihood fits of models derived from (1). Further, note that a sample
from the posterior distribution p(θ | y) is directly available, simply by ignoring the
γb, γε, b, u parts of sampled values of the vector ψ.
The R package mixAK (Komárek14 ) has been extended to handle the MLMM (1).
Whenever possible, the R implementation exploits the block-diagonal structure of