# How do right censoring and right truncation interact?

I am working with data related to time of event for some physical product. the data is current status data, also sometimes called interval censored type II data, which mean i only have one inspection per data point where I know if an event has happened or not. This means all my data points are either left censored (event has happened) at time of inspection, or right censored (event has not happened) at time of inspection. So if time to event is $$T_i$$ and the censoring time corresponding to inspection is $$C_i$$, then the observed data is a sequence of intervals, either $$T_i \in [0,C_i]$$ or $$T_i \in [C_i,\infty)$$ depending of event has happened or not.

As an attempt to make it reasonable that the data points are comparable due to possibly different quality of material over time, i am only considering data points created younger than some fixed time $$A$$. This means that my data is also truncated since it is not possible to observe event times larger than $$A$$ so all my datapoints are from a right truncated distribution lets say $$T_i \sim F(\, \cdot \,|\, T_i \leq A)$$. So far i have not considered this truncation, but i want to make sure that it does not make too large a difference.

The question is how to make the censoring and truncation interact. It seems intuitive to me that the censoring interval must be contained in the truncation interval, meaning that either I limit the right censoring $$T_i \in [C_i,A] \subset [0,A]$$, or I ignore the right truncation for the right censored observations $$T_i|T_i> C_i \sim F$$ and $$T_i|T_i\leq C_i \sim F(\, \cdot \,|\, T_i \leq A)$$. Of course the third possibility is my intuition is wrong and i simply keep both right censoring and right truncation, and the censoring interval is then not contained in the truncation interval.

## 1 Answer

If I understand the situation correctly, the statement:

This means that my data is also truncated since it is not possible to observe event times larger than $$A$$

is incorrect. That would be right censoring: you have information about those items, but you only have a lower limit for the time to event. It's similar, for example, to a clinical study that ends administratively at a time of, say, 5 years. It "is not possible to observe event times larger than" 5 years in that case, but you do have a known lower limit of 5 years (or less) for the times of those individuals' events.

Truncation, whether right or left, is something else. It represents a situation in which you have no information about cases that have an event time shorter or longer than some limit. Klein and Moeschberger present an example of right-truncated data in Section 1.19:

... data on the infection and induction times for 258 adults and 37 children who were infected with the AIDS virus and developed AIDS by June 30, 1986... In this sampling scheme, only individuals who have developed AIDS prior to the end of the study period are included in the study. Infected individuals who have yet to develop AIDS are not included in the sample.

That's different from your situation: you have included all the items in your study, but some only have a lower limits for the time to event. You have right censoring, not right truncation.

• Perhaps you are right, or perhaps i have not explained it well. So all data points from objects created more than 100 years ago i do not consider, meaning i am looking at a sample from a distribution that cannot exceed 100 years, but of course the time to event could be longer than 100 years. Is this not a right truncation? Mar 7 at 6:24
• @NikolajPedersen the time to event for an object created more than 100 years ago also could be a good deal less than 100 years. You simply have no information on objects created more than 100 years ago, regardless of their event times. Right truncation, following Klein and Moeschberger, is based on a limit to observing event times. Your situation is best described as a current-status analysis conditional on an object's being less than 100 years old. My question would be why you limited the study to objects less than 100 years old, but I suppose you had a good reason for that.
– EdM
Mar 7 at 17:11
• I see. Thank you so much! Mar 7 at 20:07