# Understanding left-truncated data in survival analysis

I am having a hard time grasping the concept of left truncation.

According to what I understand, left truncation occurs when we observe individuals only if their event of interest takes place after some specified time $$Y_L$$. We could say that we only observe an individual if their event of interest occurs in the interval $$(Y_L, \infty)$$.

This document provides an example where some psychiatric patients are followed in order to see how their death times compare with the general population. Data includes gender, age at first admission to the hospital, and the time that they were followed (whether it be death or censoring that ended this time). It goes on to say how the data is left truncated since the patients admitted into the hospital have already survived until that point (age) in their life. But then, as I see it, wouldn't every study that measures lifetimes have left truncation? By this I mean every study conducted on lifetimes where individuals were not observed since birth.

My doubts on this came when I wanted to create some (basic) Kaplan-Meier and Cox Proportional-Hazard models for data on patients with covid-19. I am aware that not considering left truncation may cause a bias in the model. The data set is specific to Mexico and contains information on people who were tested for this virus. It contains:

• General details about patients (age, gender, state where they live)
• Date on which the patients first started having symptoms and date on which patients where tested
• Date for death times (if they occurred)
• Result for the test (positive, negative or pending)
• Other variables taking into account underlying diseases.

My aim is to create a model for the survival probability for covid-19 patients (Kaplan-Meier), and then try to study which covariates are associated with the hazard function (Cox).

I am only considering a subset of the data (patients that have tested positive for the virus) in order to create the models, but I am unsure if am dealing with left truncation or not. It makes sense to me that the individuals considered in the data are left truncated (since in order to be considered they would have to survive until the age they were tested), but it also makes sense to me that we are just considering a subset of the population, so I don't know exactly why we would be considering left truncation. Does left truncation always occur in biological or medical studies?

If anyone could clarify the general concept and/or explain how it applies in my specific example I would appreciate it very much.

wouldn't every study that measures lifetimes have left truncation? By this I mean every study conducted on lifetimes where individuals were not observed since birth.

That's basically correct. But many models don't follow lifetimes since birth. Rather, they consider survival after some other defined reference time.

My aim is to create a model for the survival probability for covid-19 patients (Kaplan-Meier), and then try to study which covariates are associated with the hazard function (Cox).

In that case, then the reference time would seem best to be taken as the time either of first symptoms or of Covid-19 diagnosis, and survival would be calculated from that reference date. You aren't modeling survival up to a certain age, so the type of left censoring you worry about isn't a problem. Instead, you could include age at first symptoms/diagnosis as a covariate in your model.

You model would include some bias in terms of the broad population, in that asymptomatic individuals or those with minimal symptoms and thus never tested for Covid won't be in your data set. Your data on disease severity are left truncated, in that you have no data on disease severity below a certain level. But that's not the outcome you're modeling; it's left truncation of outcome that needs this particular type of attention in survival analysis. Your outcome data, in terms of survival since initial symptom/diagnosis, aren't left censored. You would, however, have to be careful about just what class of patients is covered by your model.

Note that a standard Cox survival model might not be best for these data. The assumption is then that all individuals have the same basic shape of survival curve, with relative hazards proportional to values of predictors. Death from this disease might better be evaluated with a cure model if the time to death is of interest, or a simple binary-outcome model otherwise.

• Cure models do indeed seem like a better option! Thank you for your complete answer. Commented Sep 13, 2021 at 15:59