Determining the probability that a competing event will eventually occur

How does one determine the probability that an event will eventually occur given competing risks?

Given the following state machine, I am trying to determine the probability that an observation in state A at time $$t_i$$ eventually ends up in state C (as opposed to state D).

I am considering using Weibull-AFT to estimate the hazard functions $$h_{AB}(t)$$, $$h_{BC}(t)$$, $$h_{AD}(t)$$, $$h_{BD}(t)$$ for each of the transitions. This would give me the shape $$k$$ and scale $$\lambda$$ parameters for each of the underlying Weibull distributions.

What I would like to do is get probabilities for the AB vs. AD transitions and probabilities for the BC vs. BD transitions, then treat the state model like a Markov chain.

From there I get stuck. I have looked into Fine and Gray methods, but these seem to be more focused on the likelihood of an event happening during the next time period. I have also considered taking the ratio of the integral of the competing hazard functions, but it seems like I am making things up at this point.

• Are all the state transitions irreversible, as implied by the single-headed arrows?
– EdM
Commented Mar 14 at 20:11
• @EdM that is correct Commented Mar 14 at 21:06