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Chapter 3.2
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multiple markers was considered by Morrell et al. 7 and Marshall et al. 8 who based
their discrimination rule on a multivariate (non)linear mixed model (M(N)LMM).
Finally, Fieuws et al. 9 describe a discrimination procedure based on a multivariate
generalized linear mixed model (MGLMM) which allowed them to include discrete
markers as well.
All of the above approaches assume a normal distribution for the random effects in
the underlying mixed model. Nevertheless, it is known that it is difficult to check this
assumption which cannot be evaluated using commonly used empirical Bayes estimates
of individual random effects due to their shrinkage (Verbeke and Lesaffre10 ).
Consequently, Verbeke and Molenberghs11 , Chapter 7 conclude that non-normality
of the random effects can only be detected by comparing the results obtained under
the normality assumption with results obtained from fitting a mixed model with
relaxed distributional assumptions for the random effects. Moreover, according to
Komárek et al. 12 , the most promising approach for discrimination based on longitudinal
profiles is based on predictors of individual random effects and the distribution
of these random effects. It is therefore natural that the correct specification of the
random effects distribution plays an important role.
For these reasons, we are targeting a method in which the normality assumption
of random effects is relaxed. A suitable semi-parametric model for an unknown
distribution is a normal mixture. Verbeke and Lesaffre10 used the homoscedastic
version (variances of the mixture components are equal) as a model for random
effects in LMM for a single marker. This approach relaxes the strong parametric
assumption of the normal random effects distribution and also allows to cluster the
longitudinal profiles in the absence of a training data set. The first objective of
this paper is to generalize the model of Verbeke and Lesaffre10 to (a) allow for
multiple longitudinal markers in a computationally tractable manner; (b) consider
more general heteroscedastic normal mixtures (variances of the mixture components
are unequal) in the random effect distribution. The second objective of this paper
is to apply the developed model to the training (Dutch PBC) data set and to
discriminate future patients using their multivariate longitudinal profiles.
The paper proceeds as follows. The first Section describes the multivariate linear
mixed model with a normal mixture in the random effects distribution. This approach
will be used to model in each prognostic group the longitudinal evolution of the
markers and their dependence on possible covariates. The estimation procedure for
the proposed model is based on the Markov chain Monte Carlo methodology and
is given in Section ’Estimation’. Section ’Discrimination procedure’ explains how
the fitted mixed models can be used to discriminate future patients into prognostic