How to account for right-censored data in a hypothesis test

Question

Right censoring was not a major issue in my prior research on education because I was modeling outcomes like grades and graduation rates where all participant results occur on the same day.

But now I am seeing loads of hypothesis tests where data arrive sequentially such as when patients enroll in a medical study or online customers accumulate in an industry study. In both cases right censoring of the outcomes is a given: some patient and customer outcome events happen after data collection has ended.

We have to truncate the outcomes in order to conduct our analysis, and I have found little formal guidance on how to handle these truncation issues:

• If we stop data collection on a fixed date, then the earlier enrollees have had a much longer window in which to complete their event (e.g., return for lab work or check out their shopping cart).
• If we apply a standard baking period for the metric (e.g., 7-day return rate for for lab work or 7-day checkout rate), what are the implications for truncating events that occur after day 7?

What factors can introduce bias when modeling these kinds of truncated outcomes?

Answer 1

What you describe isn't right truncation, it's right censoring. That might seem like a trivial difference in a choice of words, but it's a critical distinction in statistical analysis.

Right truncation means something different from your situations. It means that, if the time to an event is beyond some value, then you have no observation at all. For example, as Klein and Moeschberger explain (Section 1.19, Second Edition), some work on the time between HIV infection and development of AIDS only included individuals who had developed the disease by the end of the study. Others, whose time since infection hadn't yet been long enough to lead to AIDS, were omitted. Such data are considered right-truncated, as the data set provides no information about potentially longer times to events.

The scenarios you describe are different. You have at least some information for all cases, but for some you only have a lower limit for the time until the event. Such times to events are considered right-censored. This review discusses many ways that censoring can occur and how to deal with it.

Right censoring, as in your scenarios, is readily handled by survival analysis methods. In the economics literature, regression involving censored observations might be called "tobit regression" instead, but the principles are the same and implementation can simply be behind-the-scenes invocation of survival analysis routines.

This web site currently has almost 3000 pages tagged survival. The R survival package provides many useful tools and vignettes that help you learn how to use them.