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).

enter image description here

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.

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

1 Answer 1


I'd recommend reading the multi-state vignette in the R survival package for guidance. What follows is mostly gleaned from that.

Fine-Gray won't be helpful. First, you don't have a simple competing-risks situation. Second, even if you did, the transitions in Fine-Gray are modeled separately in a way that can lead to the sum of final probabilities over all states being different from 1.

Direct multi-state modeling, as explained in Section 3 of the vignette in the context of a Cox proportional hazards model, can handle all of your data together. That can provide plots of estimated probability in state as a function of time. As you are willing to use a parametric Weibull model with its underlying proportional hazards assumption, that can give you an even more flexible way to proceed.


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