How to compare right-censored discrete data?

Question

I have 2 groups of 24 members, let's call the groups x and y. In some sense, x is "untreated" and y is "treated". For each member of x and y, there is a discrete value which refers to a time (1-10 days) at which an event occurred (a binary state change). The data is right-censored, meaning there is no data after 10 days. Each member of x has a value, something like [2, 4, 3, 3, 2, 2, 4, 2, 3, 5, etc.]. On the other hand, y is incomplete, looking something like: [9, 6, 5, NA, NA, 8, NA, 9, NA, 5, NA, NA, NA, etc.]

The NA value may mean to "the event might have occurred on some day after day 10" or "the event will never occur".

Two questions:

1. What test can I use to assess a difference in time-to-event for groups x and y, where time is a discrete variable?
2. How can I address the right-censored nature of group y?

This is a lot like survival analysis, but it's slightly different because in survival analysis everyone dies at some point (even if the death was right-censored). We hypothesized that the "treatment" would result in a group that looks a lot like y: time-to-event is much longer than x or does not occur at all.

The differences between the 2 groups are as extreme as presented here, but I still need to put a p-value on this to please the scientific reviewers. Also, in some cases, only 2-3 members of y have a value, and the other 21-22 members are NA.

I would appreciate some guidance on this one, thanks!

Answer 1

Your statement

This is a lot like survival analysis, but it's slightly different because in survival analysis everyone dies at some point (even if the death was right-censored).

isn't strictly true. There are, for example, cure models in survival analysis that explicitly model time-to-event along with the probability of no event at all. And in practice, even though everyone dies at some point in standard overall-survival studies, the time to death is often well past any practical upper limit of observation time. Your data aren't really different from standard survival data taken at discrete times.

Thus there's no reason why you can't analyze these data with a discrete-time survival model. It's essentially a set of binomial regressions at each discrete time value. The data have to be formatted properly and you need to make a choice about the link function for the regression (e.g., a complementary log-log link is most closely related to proportional hazards models in continuous time). There are over pages on this site on discrete-time survival.

If the presence/absence of an event is of more interest than time to event and there is no loss to follow up before 10 days (that is, all NA values represent times beyond 10 days), you might accomplish what you need with a simple binomial regression of no-event/event at 10 days.