Page 148 - Special Topic Session (STS) - Volume 3

STS535 Edsel A. P. et al.
the World Statistics Congress (WSC) in Kuala Lumpur and in future
manuscripts.

2. Proposed Joint Modeling Approach
Consider a subject or unit in a biomedical, engineering, or socio-economic
setting which is monitored over time. This unit will be monitored over a period
[0, ], where could either be fixed in advance or it could also be random.
Associated with this unit will be a covariate vector, denoted by , representing
relevant demographic features. Of main interest is to determine the health
status of this unit over time. This will be represented by a process = {() ∶
≥ 0} which takes values in a finite state space = ∪ where
1
0
elements of are transient states, whereas those in are absorbing states.
1
0
These absorbing states may correspond to different competing terminal
events, e.g., deaths due to competing causes. The lifetime of this unit will then
be
= { ≥ 0 ∶ () ∈ }.
0

Aside from this health status process, there will also be associated with this
unit a longitudinal marker process, the second component, represented by
= {() ∶ ≥ 0}, which takes values in a finite state space . This
marker process provides information about the health status of the unit and
vice-versa. At any given point in time, the unit will be in one of these states in
. The third component in our setting is the presence of several types of
recurrent events. The occurrences of these recurrent events, which are
competing with each other, will be tracked by a
multivariate counting process = {() ∶ ≥ 0} which takes values in ℤ ,
0,+
where = {0,1,2, . . . }. Similarly to the marker process, the recurrent event
process is also affected by the health status process and vice-versa, and there
will also be synergistic interaction between the marker and the recurrent event
processes. Another important feature governing such systems is the
performance of an intervention at each recurrent event occurrence which
impacts the subsequent rate of occurrences of these recurrent events.
A bio-medical situation where this setting occurs is that where the health
status () of a patient could be in the state space = { = healthy, =
1
2
diseased, = dead} so that 1 = { } and 0 = {, }. Thus, is an absorbing
1 2
0
0
0
state. The blood pressure () marker process () could take values in the
state space = { = Normal , = Low , = High }. The
1
3
2
competing recurrent events could be hospitalizations due to different causes
or ailments.
We shall denote by = { ∶ ≥ 0} the filtration or history governing

this unit. Thus, all the stochastic processes considered, such as , , , etc.,
will be adapted to this filtration. To mathematically simplify our modeling
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