Page 149 - Special Topic Session (STS) - Volume 3

STS535 Edsel A. P. et al.
approach, we shall assume a Markovian property, an assumption typically
made and which is realistic under most situations. We now describe the details
of our proposed joint stochastic model.
The health status process will be a continuous-time Markov Chain
(CTMC). Thus, there is a probability mass function (pmf) over , denoted by
(·), governing the state of (0). This pmf may contain unknown parameters.

There is then a baseline infinitesimal generator matrix = ((, ′) ∶ , ′ ∈
) such that

Since are absorbing states, we have that for all ∈ , (, ′) =
0
0
0. Similarly, the marker process is also a CTMC, with its initial state, (0),
governed by a pmf (·) over . This pmf may also have unknown

parameters. The baseline infinitesimal generate matrix for this marker process
will be = ((, ′): , ′ ∈ ) such that

The holding times and transition probabilities for both of these processes,
which will be governed by these infinitesimal generators, will be further
modulated by the covariate vector and the effects of the other two
components. These will be described below after introducing the model
elements for the recurrent event process.
For the recurrent event process, we take into consideration to aspects
underlying the monitoring of the occurrences of recurrent events. As
articulated in the papers [3, 4], there is a need to take into consideration the
impact of performed interventions at each event occurrence as well as the
impact of the accumulating event occurrences. In addition, for more generality
and to be more applicable in the bio-medical setting, the modeling approach
is usually via a semi-parametric model. Thus, for the ℎ type among the

recurrent event types, we introduce an effective age process ℰ = {ℰ (): ≥

0} which is a dynamically observable nonnegative piecewise continuous -
predictable process, a baseline nonparametric hazard rate function (·), and

a nonnegative function (·; ) defined over ℤ and dependent on an

0,+
unknown parameter vector . This function will encode the impact of the

accumulating event occurrences on the rate of recurrent event occurrences.
Finally, we will have an at-risk process = { () ∶ ≥ 0}, where () =
{ ≥ , ≥ }, where (·) is the indicator function. Thus, () indicates the
subject or unit is still under observation at time s. The process is a bounded,
left-continuous (hence -predictable) process. We are now in position to
describe our proposed joint model. We first introduce the mappings:

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