# How to predict survival curves with step changes over time

I'm trying to predict the survival curves for customers, knowing that for some rare customers after a certain time the survival probability has a jump. Those jumps are due to endings of minimum contract duration. E.g., some customer commit to stay for 6 month. We know that those customers (usually) stay for 6 month. After that they can cancel there subscription on a monthly basis (thanks @EdM for the advice to include that information in my question). As far as I understand Cox Proportional Hazard and Accelerated Failure Models, they, in in layman's terms, take the average survival over time of the whole population and than multiply that with a constant, which is dependent on covariates. As those jumps are really rare and happen at different points of the lifetime, the average survival curves don't have any jumps in my case. With multiplying those curves with a constant, I can't model those jumps. Have I understood that correctly? If so, is there a way to predict remaining survival, knowing, that the survival probability will make a jump at a certain time?

• If you do not know when these probability jumps might occur and they are rare, it seems unlikely you could ever model them. Compare that to for example people buying car insurance: you may lose a few during a year if they dispose of their car, but many more at the end of every 12 months when their annual policies come up for renewal - that should be possible to model. Commented Feb 8, 2023 at 16:57
• But we know when the jumps occurs. We just don't know how strong it effects the survival probability
– TiTo
Commented Feb 9, 2023 at 8:33
• It would help if you could give an example of how you determine that "for some rare customers after a certain time the survival probability has a jump." That makes it sound like you already have some type of survival model implemented. Or is your "survival" model some type of repeated-events model, and some customers have a big increase or decrease in something like the rate of placing orders? Please provide that information by editing the question, as comments are easy to overlook and can be deleted.
– EdM
Commented Feb 9, 2023 at 17:02
• just did, thanks @EdM
– TiTo
Commented Feb 10, 2023 at 9:10

A simple way to deal with this is to use the end of the initial minimum contract as time = 0 for your survival model instead of the start of the initial contract. For the few who end before that time, you could use a binomial regression (stayed the duration/didn't) unless the particular ending time is of interest. You could use the duration of the initial minimum contract as a covariate under my suggested time = 0.
Another choice if you want to keep time=0 as the start date of the initial contract, as you seem to be doing now, would be to include underInitialContract as a binary time-varying covariate in a Cox model. That would allow for different hazards depending on whether an individual was under that initial contract. Interaction terms of other covariates with that would allow for different associations with outcome depending on whether or not an individual was underInitialContract. I'm often skeptical of using time-varying covariates, but in this case it won't lead to the potential circular reasoning that can arise in general.