Page 151 - Special Topic Session (STS) - Volume 3

STS535 Edsel A. P. et al.
where the first form results from performing imperfect repairs or
interventions at event occurrences, while the latter form results from
performing a perfect repair or intervention after the last event occurrence
prior to time , and is the time of the last event occurrence prior to time

. The effective age processes are determined dynamically since the repairs
or interventions performed after each recurrent event occurrence are
usually not determined at time zero but decided upon after the event
occurrence. The functions on the other hand could for instance be of

form

(: ) = exp {[log(1 + )] }, ∈ ℤ 0,+

4. The final requirement to completely specify the joint stochastic model is
the assumption that, given − , then {(), ≥ }, {(), ≥ } , and
{(), ≥ } are conditionally independent. This conditional independence
assumption enables the construction, in a dynamic fashion, of the full
likelihood function or process.

3. Some Aspects of the Class of Joint Models
The distinctive trait of this joint model is the interplay among the three
components: the health status, the marker, and the recurrent events. Each of
these affect the others in the sense that the future occurrences of transitions
or events, given the present, for each of the components are affected by the
current state of the other two components. As such dependencies of the
random paths are induced and there is a synergistic dynamicity to the paths
of the different processes. Each of them have some baseline behavior which
are encoded in the baseline parameters: the infinitesimal generator for the
-process; the infinitesimal generator for the -process; and the baseline
hazard rate functions for the -process. Some form of proportionality is
0
then imposed to model the modulation induced by the other components and
the covariate vector through the exponential link functions.
There are many model parameters in this joint model, which implies that in
order to perform reasonable inference, a sufficient number of subjects or units
over reasonable monitoring periods will be required. The model parameters
are:

• Parameter of (·) and parameter of (·).

• Baseline infinitesimal generators (, ′), , ′ ∈ .

• Baseline infinitesimal generators (, ′), , ′ ∈ .

• Baseline hazard rate functions (·), = 1,2, . . . , , which are specified
0
nonparametrically.

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