Discrete vs. Continuous Survival regression - for business case / subscription churn

Question

I'm trying apply a survival analysis to a churn problem - customer subscriptions.

There's nothing particularly unusual about these subscription - customers either pay, or leave, monthly, or annually, and can start at any time during the year.

I was initially looking at some Cox regression models but I'm a bit confused on the difference between continuous and discrete time survival models -- it sounds like the latter (discrete) might lose some potential information but allows more additional methods to be applied.

Anyway from my reading so far -- I wonder if various models using the same inputs would really vary that dramatically. But -- which is the better selection?

I mean -- yes, customers only have relevant data points either once a month, or once a year (live or die) -- so it at least "seems" like discrete makes sense, but not sure I fully understand the difference here.

Answer 1

A Cox model can be considered a type of discrete time survival model, as the model is based only on the discrete number of event times included in the data set and there is no explicit modeling of time. It's certainly possible to analyze monthly data on churn with Cox models, as in this thesis (Erik Gustafsson, "Customer duration in non-life insurance industry," Mathematical Statistics Stockholm University, 2009). One warning: there are difference among implementations in handling tied event times, of which you will have many in this type of data set.

As you note, a discrete time model does seem more natural for this type of data, if you have a series of fixed time intervals shared among all individuals. A discrete time model is just a series of binomial regressions for each of the time intervals in question. You don't even have to choose between a discrete time model and a proportional hazards model, as a complementary log-log link in the binomial regression is equivalent to a proportional hazards assumption. You aren't stuck with a proportional hazards assumption with a discrete time model, however, as you can choose from a set of link functions for the binomial regressions that don't implicitly make that assumption. That might provide more flexibility in describing your data.