# Multi State Survival Analysis Transition Between States Triggered By different events

I have a state diagram for multistate survival analysis, and between 2 specific states, I have 2 types of transitions triggered by different types of events, i.e, from state 1 to state 2, the transition can be triggered by 2 different events. How can I account for these transitions in multistate survival analysis? Should I add the event type as a covariate for Cox Regression? Or should I fit a separate model for 2 transitions triggered by different events?

The short answer is "it depends."

The medium-length answer: how you model the possibility you describe depends on the assumptions you're willing to make about the data-generating process (DGP) associated with $$1 \rightarrow 2$$ transitions via event 1 vs. $$1 \rightarrow 2$$ transitions occurring via event 2. For a sampling of these assumptions:

• Are you willing to assume event 1's baseline hazard ($$h_0(t)$$) is the same as event 2's $$h_0(t)$$?
• If you have covariates, are you willing to assume each covariate has the same effect on event 1's $$h(t)$$ as event 2's $$h(t)$$?
• Again, if you have covariates: are you willing to assume all covariate effects are unconditional in the same way for both events? (E.g., if $$x$$'s effect is conditional on $$t$$, it would constitute a proportional hazards violation* in any duration model whose hazard can be expressed in proportional hazards form. Are you willing to assume $$x$$ is unconditional on $$t$$ for event 1 AND event 2?)

If you're willing to assume event 1 and event 2 have identical DGPs in every respect, you don't need to do anything extra to distinguish between event 1 vs. event 2. If you think there are differences in the two events' DGPs, then you have additional decisions to make.

If you want to permit the two DGPs to be different (and then test whether differences exist), you can 'split' your true stage 2 into "stage 2 via event 1", with event 1 being the only valid transition event into this new stage, and stage 2 via event 2, with event 2 being the only valid transition into this new stage. You can then check for whether the baseline hazards are the same, covariate effects are the same, etc. in the usual way (discussed here, I know, but they aren't the only authors to discuss these checks).

* There are canned tests for assessing whether the proportional hazards assumption is violated in Cox duration models. The most common is the Schoenfeld-based test of Grambsch and Therneau (1994, Biometrika): estat phtest in Stata, survival::cox.zph in R, lifelines' check_assumptions in Python.