# Survival analysis when event cannot happen after timepoint

People are contacted for a survey for a fixed number of days. For simplicity, let's say this happens over 5 days. People can complete their survey on day 1, day 2, ..., up to the end of day 5. After day 5, anyone who hasn't completed their survey is considered unresponsive. The goal is to predict who will not respond, i.e. anyone who didn't complete their survey by end of day 5.

If I interpret this as a time-to-event analysis with $$T$$ denoting the time to survey completion, I see a potential problem. No one can have an event after day 5. This makes the survival function discontinuous since it immediately drops to 0 at T > 5.

I'm not sure how to fit a survival model like this. It seems to fall outside of the usual survival analysis models. Are there any methods to handle this situation?

The goal is to predict who will not respond, i.e. anyone who didn't complete their survey by end of day 5.

For this goal, instead of using the response as the event, use the failure to "complete their survey by end of day 5" as the event. That then becomes a simple logistic regression: didn't complete versus completed within 5 days.

If the time to respond isn't important, then you don't need to think about this as a time-to-event model.

If the time to respond is important and you want to use response as the event, you have a cure model. If the time points are as discrete as your example implies, then a binomial regression model with the time points included as predictors (a discrete-time survival model) will do all that you want together, as you get probabilities of response as a function of time point, and the probability of no response at all is just 1 minus the sum of probabilities of responding at earlier tune points.

• Time to respond isn't important to predict, but time is an important factor. If someone hasn't responded by day 4, you would expect a small chance of completion than someone who hasn't responded by day 2. I guess time can be put into a logistic regression model though.
– Eli
Aug 5, 2022 at 18:02
• @Eli if you have just a few time points, then look into discrete-time survival analysis, which is just a logistic regression model with data structured appropriately and the time points included as predictors. See this page and its links, among others. As you don't care about the possibility of response after 5 days, if you need to work in continuous time you could do standard survival analysis with "responded" as event, and just use the proportion "surviving" after 5 days as your estimate of the fraction not responding.
– EdM
Aug 5, 2022 at 18:17