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?
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.