Misconception about left censoring

Question

While trying to learn about censoring, I stumbled across a sentence on Wikipedia which I do not understand:

A common misconception with time interval data is to class as left censored intervals where the start time is unknown. In these cases we have a lower bound on the time interval, thus the data is right censored (despite the fact that the missing start point is to the left of the known interval when viewed as a timeline!).

I know what right and left censoring means but I have trouble following this reasoning. Maybe somebody here could help me and explain what they mean!

Answer 1

This is a statement about subtraction and absolute value of real numbers.

In the context we must understand a "time interval" as being an interval of the form $[A,B]$ where one or both of the endpoints is a random time. The object of study is the duration $X=B-A.$

When the start of the interval $A$ is not known, but only the fact that $A\le a$ is known (with $a$ marking the beginning of the study period), the variable $A$ is left censored. However, the value of the duration $X$ when the end of the interval $B=b$ is observed is

$$X = b-A \ge b-a,$$

which is right censoring.

Answer 2

This can be very confusing.

Start with a simple situation, when time = 0 is both the time of study entry and the time of starting some therapy, for example in a cancer outcome study in which you want to evaluate time to death after therapy starts. If Participant A both enters the study and starts therapy at time = 0, and is still alive at time = 2 years, then you know that the time between starting therapy and death will be at least 2 years for Participant A. That's standard right-censoring at 2 years.

Now say that Participant B enters your study after having started treatment at some prior unknown time. You don't know exactly when treatment began, so you call time = 0 for Participant B the time of study entry. If Participant B dies at time = 2 years, you know that the time between starting therapy and death was at least 2 years. That's logically the same as for Participant A: in both cases, you have a lower limit on the time between starting therapy and death. That's right censoring on the survival time of interest in both cases.

Potential confusion with Participant B can come from losing the more typical association of lack of an event with right censoring, which you do have with Participant A. With Participant B you observe an event but you don't know the actual elapsed time from starting therapy to death, so you have to treat that event time (starting from your time = 0 at study entry) as a right-censored observation.