# Renewal process vs inhomogeneous Poisson process?

I am analyzing a dataset with recurrent events, and considering two candidate models:

• A model based on a renewal process (time is measured from the previous event)
• A model based on an inhomogeneous Poisson process (all events share the same time reference)

I would like to make a principled decision about which type of model is more suitable, and looking for ideas.

Remarks:

• A similar (but more specific) question was asked here
• In reality, both my models are modified to account for the number of previous events, but I believe that this is not crucial at this point

Update
I was asked to provide additional information about the dataset and the problem. However, I would not want to load the question with inessential details, so I keep it generic:

Dataset: thousands of specimen with dates of events and specimen-specific covariates for each. For some no events has occured (i.e. they are right-censored)

Problem: we are dealing with equipment failures here. After a failure the equipment is repaired and put back in service. However, the risk of failure likely increases, if previous failures took place. Thus, the difference between the two types of models is
a) whether we attribute failures to the fact that this particular piece of equipment is failure prone (fraility?) or b) whether failure is more likely for older equipment.

• Ideally, this decision would be based on knowledge about the data generating process. One important difference is that, for renewal processes, the intervals between events are i.i.d. This is not true for inhomogeneous Poisson processes, where the event rate varies over time. If no decision can be made based on theoretical considerations, you could treat this as a model selection problem. Fit both models and pick one according to some formal criterion. E.g. cross validated likelihood, Bayes factor, etc. Not much more can be said, since you haven't described the data or problem. – user20160 Nov 1 '20 at 0:16
• @user20160 I am not sure what kind of additional information I could give about th problem: I have thousands of specimen, with a set of covariates and event times for each. – Vadim Nov 1 '20 at 16:15
Your situation should be handled well by standard survival analysis methods. The main vignette for the R survival package covers this type of situation in Section 2.2:
That sounds a lot like your situation. With that approach you get the cumulative hazard of event numbers as a function of the time since original installation. An id variable associated with each machine specifically handles machine-specific frailty. The particular model described in the vignette doesn't take into account covariates or increased hazard as a function of prior failures, but those are easily handled. Covariates are straightforward to add to a model, although care needs to be taken with time-dependent covariates. An increased hazard as a function of prior failures could be handled by (a) incorporating the number of prior failures as a covariate or (b) a multi-state model (Section 2.4 of the main vignette), in which each failure puts the machine into a new state.