# Difference between left censoring and interval censoring

I'm dealing with some medical data which is made up of all admissions to hospital in a certain area, where each row is one patient admission alongside the various tests and demographic data done. Also some patients have multiple admissions.

To model the data I am using a joint model for survival analysis, with the outcome being a patient being diagnosed with dementia. I have two questions regarding the correct use of censoring in mainly Cox regression. For each patient I class $$time = 0$$ as their first admission to hospital.

1. If a patient turns up on their first admission, and already has dementia, I currently just exclude them from the analysis. Does this however count as left censoring, and can I include them if I change the way I measure time? Currently this would be assuming that they got dementia at negative time. I'm only considering old age patients, so I thought I could maybe choose time = 0 to be 65 years of age etc. That would avoid the negative time problem but feels wrong or just a quick fix.

2. What is the difference between interval and left censoring? As far as I understand, interval censoring assumes the event happens within an interval $$[t_{1},t_{2}]$$ but we don't know exactly when. Whereas left censoring assumes we just know the event happens before a time $$t$$. However isn't this just the same as knowing the event happened within the interval $$[0,t]$$?

Most sources seem to say the same things about left and interval censoring, so any advice would be great.

With respect to Question 2, Klein and Moeschberger explain in Section 3.3:

Of course, interval censoring is a generalization of left and right censoring because, when the left end point is 0 and the right end point is $$C_l$$ we have left censoring and, when the left end point is $$C_r$$ and the right end point is infinite, we have right censoring.

With respect to Question 1, when you omit individuals who first present with the dementia diagnosis you are effectively left truncating your observations. Your analysis is conditional on an individual not already having experienced the event. You are including no information about those who already had the event. From the perspective of that analysis, it's like such individuals didn't even exist.

From your description, using a fixed chronological age* as the time reference instead of the first hospital admission would seem to make a lot of sense. Then you can include those who first present with dementia as having left-censored event times and include them in the analysis. There's nothing wrong with that, and a strong argument could be made that it would be preferable in a situation like this. There's a good deal of interest in the age at which dementia is diagnosed, while I don't see a reason to care so much about how soon it occurs after a first hospital admission (of possibly several), an admission that might be for an unrelated condition.

Note, however, that even then you are analyzing the dementia diagnosis time conditional upon an individual being admitted to the hospital. You are omitting anyone with dementia in the study area who doesn't happen to be admitted to a hospital.

*Even a reference of time = 0 at birth would be OK for a Cox model, as a Cox model only evaluates the order of event times, not the event times per se. Then you don't have to worry about someone with early-onset dementia forcing you to re-set the time reference later on.

• I actually went to that exact book and found said section 3.3, so thank you for that. The issue with my dataset, is that it is only made up of hospital admissions. I get rid of 10,000+ patients by my current pipeline, so it's good to know I can include them. I think I'll try out a fixed age, and also birth as a reference time and see what happens. Thank you! Commented Aug 20, 2023 at 10:12
• If I used time from birth, would that mean I am no longer able to use age as a predictor anywhere? Although would this be fine, since that is all now contained in the survival time? Commented Aug 20, 2023 at 10:20
• @JPiekos I haven't thought this through completely, but you might still include age at first admission as a predictor. You probably should include the calendar date/year of first admission as a predictor, to allow for changes in incidence/diagnosis rates over calendar time. Whenever I get confused about censoring/truncation, I find that Klein and Moeschberger point me in the correct direction. Read their sections on left truncated survival times for further guidance.
– EdM
Commented Aug 20, 2023 at 15:14
• thanks a lot, and yes the Klein and Moeschberge book I'm realising is very good. Commented Aug 24, 2023 at 13:34