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Chapter 3.2
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effects correct the mean structure to get the patient specific profile. Nevertheless,
a further improvement of the mean structure may lead to a further improvement of
the discriminant procedure. Further, we have also fitted models with K = 3 mixture
components and compared the resulting discriminant rule to these described above.
However, the solution with K = 3 performed worse than the above procedures based
on K =1orK = 2 implying that the structure of the models with K = 3 is already
overparametrized. Note that with K = 3, the dimension of the parameter space
increases with 1 + 6 + 21 = 28 in each prognostic group.
Discussion
In this paper, we have generalized the discriminant analysis of multivariate longitudinal
profiles by assuming a normal mixture in the random effects distribution in
the mixed model. The application of our approach to the PBC Dutch Study data
showed some improvements compared to the methodology based on mixed models
with normal random effects. Due to the fact that the normal mixture serves as
a semi-parametric model for the unknown random effects distribution, the first obvious
question is how to choose K, the number of mixture components. In general,
models with a different number of components can be fitted and then compared
by means of a suitable measure of model complexity and fit like the deviance information
criterion (DIC, Spiegelhalter et al. 25 ), or the penalized expected deviance
(PED, Plummer27 ). Alternatively, posterior distributions of deviances under different
models can be compared (Aitkin, Liu and Chadwick28 ). Nevertheless, when
discrimination is of primary interest and a training data set is available then it is
preferable to choose the optimal model by evaluating the resulting discrimination
rule by, e.g. means of cross-validation, as was done in Section ’Application to PBC
data’.
In our specification of the MLMM, we assumed that the errors εi,r,j (i =1,...,N, r =
1,...,R, j =1,...,ni,r) are independent and hence the markers Yi,r,j are conditionally
independent given the random effects. Hence, one can further generalize
the proposed model by using a more general covariance structure for the vectors
of errors εi,r (i =1,...,N, r =1,...,R as was done, e.g., by Shah, Laird and
Schoenfeld 29 or Morrell et al. 7 With such generalization, the results of Section ’Ap-
plication to PBC data’ can even improve. Further, it is certainly possible to relax
the normality assumption on random effects in several other directions than was
done in this paper. For example, a multivariate t-distribution (see, e.g., Pinheiro,
Liu and Wu 30 ) or a mixture of multivariate t-distributions (Lin, Lee and Ni 31 )for