Ch7, pg16

Hi Shashi

Are you referring to the paragraph on left truncation vs. left censoring?

As an example of left truncation, consider trying to estimate the survival function since onset of a particular disease. Say we are observing a particular hospital and for the relevant, say 30, patients that started treatment in that hospital, we know when they contracted the disease and we started observing them from that point on. Let's say our timeline for these patients looks like the following:

\( event \hspace{1cm} 30E \hspace{1.5cm} D \hspace{2.5cm} C \)
\(\hspace{2cm} |-----|------|----- \hspace{1cm} etc \)
\( time \hspace{1cm} 0 \hspace{2.3cm} 5 \hspace{2.5cm} 12 \)

Where 30E is 30 entrants (we started observing them at time 0, which is the time they contracted the disease). D is a death and C is a censoring.

Now say that at time 20 the hospital also receives a transfer from another hospital during our study, consisting of ten patients who have the disease. Say that we know when they contracted the disease but we didn't start observing them from that point (as they were in the other hospital). This is an example of left truncated data.

Now, we could say, well we know that these patients survived from time 0 to time 20, so let's update our timeline as follows:

\( event \hspace{1cm} 40E \hspace{1.5cm} D \hspace{2.5cm} C \)
\(\hspace{2cm} |-----|------|----- \hspace{1cm} etc \)
\( time \hspace{1cm} 0 \hspace{2.3cm} 5 \hspace{2.5cm} 12 \)

So, just adding the 10 patients to the start, time 0 and then we'll record their deaths / censorings past time 20 as they happen.

However, this is not appropriate. This is because although we received 10 new patients who we know survived from time 0 to time 20, we don't have information about other patients from that hospital who may have died in that time. So, we have a biased group of people over this time period, which leads to an underestimation of the death probabilities.

To perhaps see this more clearly, imagine the extreme of receiving a transfer of alive patients at time 20 from every other hospital that exists and adding these to our group at time 0. This is going to make the proportion of deaths up to time 20 look smaller and smaller because we're ignoring the deaths up to that point from the other hospitals and only taking their survivors.

So, the appropriate thing to do in this case is to add these 10 patients to our timeline at time 20:

\( event \hspace{1cm} 30E \hspace{1.5cm} D \hspace{2.5cm} C \hspace{2.6cm} 10E \)
\(\hspace{2cm} |-----|------|-------| \hspace{1cm} etc \)
\( time \hspace{1cm} 0 \hspace{2.3cm} 5 \hspace{2.5cm} 12 \hspace{2.6cm} 20 \)

Then we can record death or censoring results from time 20 onwards for this group, treating them just like the other patients in our study from this point on.

Hope this helps!

Andy