Survival analysis: Clarification on left censoring

Question

First of all: I know there are some other contributions on stackoverflow around this topic like this, but it seems to me that there is still confusion.

Let's say I am working with failure data of some machine components. Of some of them, I know both when they were installed and when they failed (type I), of others I only know when they were installed but they did not fail until today (type II), of others I don't know when they were installed but when they failed (type III).

• Are both type II and type III right-censored observations, as I only know a lower bound of their lifetime? This would align with the explanation in this comment.
• Or do type III samples represent examples of left censoring, as I only know an upper bond of their possible birth date? This would align with the explanation in this comment, as well as with the explanation of left-censoring given by the author of the lifelines package.

The confusion seems to be about what to look at:

• whether the lower bound of the observed survival time is known
• whether the upper bound of an event time (e.g. installment of a component) is known

When we know the upper bound of the installment date (we know installment was prior to some day), we know the lower bound of the component's survival time. But which of these logics is decisive for us to classify the situation as left- or right-censoring?

Answer 1

I don't know when they were installed but when they failed (type III)..I only know a lower bound of their lifetime...I only know an upper bound of their possible birth date...

It's hard to talk clearly about this unless you specify time origins and the nature of the data collection.

If installation time might have been as early as $-\infty$, you don't have a finite lower bound of the lifetimes of the "Type III" items.

If you know that the installation date had to be after some known earliest possible date and you are evaluating survival from the date of installation, then you have an upper limit to the lifetime: from that known earliest possible date to the date of failure. That's left censoring of the time between installation and failure.

If installation might have been as early as $-\infty$ but you start collecting data on the component at some specific time, then the duration between the time you start collecting data on it and the time it fails is a lower limit to the time between its installation and its failure: right censoring of the time between installation and failure.

Klein and Moeschberger provide examples illustrating the differences among various censoring and truncation schemes.