Left censored index date during survival analysis I am trying to do a Time To Event analysis, looking at patients with Multiple Sclerosis, which can lead to wheelchair use.
My intended study is to look at the time from MS diagnosis to first Wheelchair use. I know that everyone in my cohort has MS before the join the cohort.
In most cases, I know the date of diagnosis exactly, in other cases, I know that the diagnosis was some time before they joined the cohort
I believe this means that the index date - the date of diagnosis - is sometimes left censored. How should I account for this in my analysis? Here is some example data:

So for example: We know that Patient 1 was diagnosed in 1995, and we know that Patient 2 was diagnosed some time before 2001
Similarly, Patient 1 started using a wheelchair in 2010, but Patient 2 did not use a wheelchair until at least 2020
For right censored data, I know to differentiate censoring dates from event dates by using the event argument in the survival package.
My question is: Is the situation I am describing above left censoring? As im talking about knowing the index date, not the event of interest date. and secondly, what is the appropriate way to do handle situations like this in survival analysis?
I want to be sure that I'm not confusing the concept of left censoring. the first time I would be able to observe the outcome event in question (wheelchair use) could be thought of as the left_censored date, however I want to account for a left censored index date.


My intended study is to look at the time from MS diagnosis to first Wheelchair use. I know that everyone in my cohort has MS before the join the cohort.

What you want to model thus is the elapsed time between diagnosis and wheelchair use, with time = 0 being the date of diagnosis. With that specific choice of reference time, lack of information about the actual diagnosis date means the associated elapsed time value itself is right censored. For example, the time between diagnosis and wheelchair use for Patient 4 is at least 3 years and 10 months. If you did know the actual diagnosis date, the elapsed time value would be at least that long. That's right censoring of the elapsed time.
Put another way, the left censoring of the diagnosis_date leads to right censoring of the elapsed time between diagnosis_date and wheelchair use--which is what you want to model.
That's somewhat different from the situation in the paper recommended in a comment, where actual age, the time since birth, is of main interest. That paper shows how to take left censoring and left truncation into account in that type of analysis. Unless the actual diagnosis_date is of specific interest in your study, you can take the left censoring of diagnosis_date into account by treating the corresponding elapsed time as right censored. You might have problems, however, in establishing covariate values that held at time = 0 for those individuals--values like age at diagnosis--if such covariates are to be included in your model.
You should also make sure that other hidden biases aren't involved in your study design. For example, say that your study was limited to individuals who had MS but hadn't already started using a wheelchair. Then individuals who start using a wheelchair very soon after they develop MS might be missed in your study design, and you could have left truncation of elapsed times between diagnosis and wheelchair use. That is, you might have no information at all about individuals with very short elapsed times defined that way. So proceed carefully.
