How do I model state transitions, but conditioned on extra features?

Question

I'm trying to model a dataset in which the goal is to predict the path of users through a series of states. The state space is very simple, with just three states, and I have plenty of sequence observations (~O(10^6)).

I further know some structure about the transition matrix, as there are only 3 transitions possible:

$$ Y_0 -> Y_0\\ Y_0 -> Y_1\\ Y_0 -> Y_2 $$ and all sequences start in state 0 and end in either state 1 (success) or 2 (failure), so each observed sequence is just a series of $$[Y_0,...Y_0,Y_m] \text{ where } m \in {1,2}$$. Furthermore, there is a fixed maximum sequence length, after which all sequences transition to failure by default.

Where things start to get complicated is that each user has a number of other attributes/features $\mathbf{x} = x_0, …, x_K$, and I would like to include these in the model. I suppose I am looking to learn something like $$ \mathbb{P}[S_{T+1}| S_t,...S_0, X=\mathbf{x}] $$

Put in words, the question I really want to answer is "What is the probability that a user moves to state 1 or state 2, and for how many time steps does it stay in state 0 until then?".

It could be that I'm over-complicating what might look like a simple classification problem, but a key insight we are trying to generate is the second part of the question above:

How long does a user take to convert to success/failure, and how does this depend on their user specific attributes $\mathbf{x}$"

rather than just the simple binary prediction of the end state.

Alternatively, is there anything from survival modelling that might be a better approach to answering this question?

Answer 1

With your interest in the time to event, this seems for the most part to be compatible with a standard competing-risks survival model. You have 2 mutually exclusive and final event types (ending either in $Y1$ or $Y2$), while those remaining in state $Y0$ would be considered "censored" in the sense that the time to an event for those cases hasn't yet been reached. See this vignette, for example.

One potentially complicating factor is the granularity of your sequence times. Standard survival analysis works best if time is effectively continuous. Otherwise discrete-time survival analysis might be better, which is essentially a set of logistic regressions (multinomial rather than binomial in this case) at each point of your sequence. I think that's what @kjetil b halvorsen was getting at in a comment on the question.

Another thing to consider is the "fixed maximum sequence length, after which all sequences transition to failure by default." With that handling of the final transition, the proportional hazards (PH) assumption underlying approaches like Cox regression wouldn't hold at the end. If you were willing to treat those as censored instead of transitioning explicitly to $Y2$ then you could still get the probability of being such a case (time-to-event greater than final time point) from a Cox or parametric survival model if the PH assumption holds up to that point. Alternatively, such a transition could be handled in discrete-time survival analysis with a separate logistic model for that last time point.