# Survival analysis: How to code 'left censored' data?

I have data on the closure of brands of stores and I would like to model the factors that influence the risk of closure. I know all the stores that are open on a certain date, 1st January 2015 (but they may have actually opened for example 5, 10, 20 or 30 years ago). I know the month of closure up until 21st December 2021. For covariates I am using information such as the catchment population, distances to competitor stores and distance to stores of the same brand. These vary by year. To model the outcome of store closure I am proposing to use Cox proportional hazard models in R using package survival. Because all stores are open on the 1st January 2015 I have an unusual form of right censoring in that the survival time I observe for each store is at least the time to the month in which they closed or 84 months (7*12) (see this post : Misconception about left censoring).

My question is that in models of such a form, information is used on the survival time and an indication of censoring (traditionally 0=censored, survived; 1=non-censored, eg died). Since all my observations are right censored in some sense, does this mean that this indication of censoring is the same for all stores?

• Please edit your question to say more about what covariates you will be evaluating in your model and whether any are time-varying. Cox models assume that the hazard at any time is a function of instantaneous values of covariates. If you are treating time = 0 as the original date of opening, then you won't have covariate values at all until 1st January 2015. Also, is there some reason why you can't just use 1st January 2015 as your time= 0 reference for the analysis? See this thread for an example.
– EdM
Jan 11, 2022 at 17:36

For covariates I am using information such as the catchment population, distances to competitor stores and distance to stores of the same brand. These vary by year.

The simplest solution is to choose 1st January 2015 as your time = 0 for all stores. That might not seem as satisfying as modeling from the original store-opening date, but it might be the best you can do.

With those time-varying covariates you don't have information about values prior to that date, so you couldn't properly model survival at prior times anyway. A Cox proportional hazards model has an advantage here in that the risk of an event is assumed to be a function of the instantaneous covariate values. Thus even with a 1st January 2015 time reference you will still get useful information about the hazards associated with your covariates since that date, given that a store was open on that date.

This thread discusses a similar situation in modeling customer churn in the insurance industry. Following on from that discussion, one thing that might help here would be to include the length of time that a store had been open prior to 1st January 2015 as a fixed-time covariate in your model, if you could get just that single piece of information for each store. If you can't, the best you can do is to work with the information that you have.

• Thanks. Unfortunately I do not know for how long the store had been open before 1st January, 2015. I have found this thread stats.stackexchange.com/questions/144037/… where Allen Wang on Nov 15 '16 at 23:05 asks about the coding of the event. Firstly I understand that all my data are right-censored, even those that closed during the observation period to December 2021. In R there is an event field that is normally coded as 0=alive, 1=dead, but this is really denoting those that are right censored, BUT all my data is right censored! So all event=0? Jan 13, 2022 at 14:53
• @StephenClark the problem is that without events you can't analyze survival with a Cox model. Cox regressions are based on comparing covariate values for those who have an event at a given time against all those still in the at-risk set. Without events, there's nothing to analyze in a Cox model. So probably the best you can do is to start with 1st January 2015 as time = 0 for all cases, use the information that you do have about hazards since then, and recognize that length of operation prior to that date is something you can't control for.
– EdM
Jan 13, 2022 at 15:00
• Thanks for taking the time to engage and sharing your thoughts. Jan 13, 2022 at 20:54