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Discriminant analysis using a MLMM with a normal mixture 155
diction uses only the latent characteristics of the subjects with the between subjects
variability.
Application to PBC data
As an illustration, we have applied our methodology to the Dutch PBC study data
described in the introduction. Of the 375 patients, 178 patients are known to be alive
at T = 10 years without needing a liver transplantation. They will be further referred
to as prognostic group 0. A total of 41 patients died from a liver related cause or
needed a liver transplantation during the first 10 years and will be further referred as
prognostic group 1. The remaining 156 patients can be divided in three categories
for whom the prognostic group is unknown because loss of follow-up (14 patients),
or is unknown because the follow-up time was less than 10 years (112 patients).
The third category consists of 30 patients who died in the first T = 10 years
from another than a liver related cause. These 156 patients could be considered
as the third prognostic group in our methodology. However, the results of our
discrimination procedure can be better exemplified with G = 2 groups by means of
sensitivity, specificity and receiver operating curves (ROC). Since the main objective
of this section is to illustrate and explore the performance of our methodology, we
will therefore consider only the two prognostic groups mentioned above. Hence
our training data consist of 178 + 41 patients. The markers used to predict the
prognostic group include values of: bilirubin, albumin, alkaline phosphatase (AP).
Hence, R = 3 in model (1). To better satisfy mixed model (1), the natural logarithm
of AP was used as the marker instead of the original AP value. Figure 1 shows the
observed longitudinal profiles of the markers, separately for Group 0 and Group 1.
We can observe that the bilirubin levels in Group 0 are rather low and stable over
time whereas they start to increase dramatically from a specific time point for many
patients in Group 1. Albumin levels are in general higher in Group 0 than in Group 1
while the reverse is true for AP. For practically all patients in Group 1, log(AP)
is almost never negative whereas in Group 0 negative values of log(AP) are quite
frequent.
Although not required with our approach, the mixed models (1) for the PBC data
will have the same structure in both prognostic groups. Namely, for each of three
markers (r =1, 2, 3) and both groups (g =0, 1), the X g
i,r matrix contains two
columns: age of the patient at start of treatment (median 54.7) and a dose of
ursodeoxycholic acid (UDCA) in mg/day (median 750). Consequently, the vector
αg of fixed effects has a length of 6 for g = 0, 1. Matrices Z correspond to