Competing hazards for event that makes the event of interest more likely

Question

Suppose I'm Netflix. I'm using survival analysis methods (kaplan-meier curves) to study when my customers decide to cancel their subscription. However, I've noticed that customers that experience problems and have to contact support decide to cancel their subscriptions at a higher rate than those that never contact support.

I'd like to know what my cancellation rate (survival curve) would look like if nobody had problems that required them to contact support. So, I model a competing hazard for first-touch with support. When customers contact support the first time, they are modeled out as a second type of event. My assumption is that this is a more accurate methodology than simply censoring those people at that time point.

Is this an appropriate methodology? Why or why not?

Disclaimer: I don't work for Netflix.

Edit: Another possible methodology

Another way of looking at this would be to model the competing event not as first touch with support, but instead as cancellation after touching support. This makes the events of equivalent types (much like if you had death by cancer vs. death by getting hit by a bus).

Answer 1

I think you might look for modeling frameworks where clients are in a few different states (perhaps: new user, established-user, established-new-problem, frustrated-user) and use survival analysis to estimate time dependency of transitioning to other states. The survival package in R (which is what your citation was using) also has a multi-state outcome type, but it's description seems to imply that various states are either censored or "absorbing" (in the language of Markov modeling). That might be useful if one of the states was "contract terminated". I would look for citations to "inhomogeneous Markov models". Homogeneous Markov models have time-invariant transition rates. You should be able to use some of the data that you analyzed in your (hypothetical) Netflix contract to specify the nature of the time-dependency of the various transition rates. You could then refine the model by adding in client descriptors.

Answer 2

Neither of the methodologies I proposed was correct. The proper way to do this is to fit a cox proportional hazards model, with a time-dependent covariate representing whether or not somebody experienced a problem. You can use the results of the model to build a hypothetical survival curve based on any value of the covariate.

See here for reference: http://www.econ.uiuc.edu/~roger/courses/478/lectures/L6.pdf