What to do with Cox proportional hazards regression, when exposure and outcome does not have the same follow up

I'm working on a cox regression to analyse the association between an exposure and outcome. I also have a start time (time start for follow-up) and end time (time when getting the outcome or censored). So the model is written like this in R:

model <- coxph(Surv(start, end, outcome) ~ 1 + exposure + strata(gender), robust = TRUE, data=my_data)

I have a big sample 10.000+ for people born between 1997 and 2011. The follow-up is also starting from 1997 to 2011.

• The exposure is registered in the system from 1997 (it's a diagnosis that you cold get from birth).
• The outcome is registered in the system from 2004 (it's a diagnosis that can only be detected later in life).

Meaning I would have fx, individuals in my sample that were born in 1997 with the exposure, but their risk for getting the outcome, would start from 2004 as it was not possible to register that in the system before 2004, (meaning they would be 7 years old before they get the outcome registered, even though they may got the outcome before the age of 7). At the same time I would have individuals born fx in 2003 with the exposure, and their risk of getting the outcome start from when they are 1 year old.

What should I do with that kind of data? Can I minimise bias, by adjusting for age in the model ?

Update: As a response to EdM comment on what time = 0 is in my case. My data is a type of data where the individuals have multiple row pr. person. Meaning those who got the outcome has two rows. Here is an example for an individual with two rows in my dataset:

• First row for an individual: start time is when they are able to get the outcome (from the age of 1), meaning that is where there follow-up start time = 1 , and the end time is when they got the outcome during follow up, fx the person got the outcome at age 4 --> this row indicates the contributing time interval for that individual to the study outcome = 0
• second row for the same individual: start time is 4 and end time would still be 4 --> this row indicates that the person got the outcome at age of 4 outcome = 1

An example for those individuals that did not get the outcome, they have one row each

• Fx start time for an individual would still be 1 (as the outcome can only be observed from the age of 1) and the end time will be end of follow-up or due to other competing risks --> this row indicates that the person did not get the outcome outcome = 0 as well as the contributing time interval to the study.

IMPORTANT It just occurred to me that I have written some misleading information at the begning of my Q. The correct information is the following:

• The follow-up starts at 2004 (where the outcome is able to be registered in the system). The follow-up ends at 2011.
• The people in my sample are born between 1997 and 2009.
• The exposure information is able to be registered from 1997.
• Just to recap, my Q is, people that are born in 1997 with the exposure, and if they also get the outcome, they would first be registered as getting the outcome in 2004, even though they potentially got it before 2004. So there would be a 6 years gab before they can be reported as getting the outcome (as the outcome can be observed from the age at 1).
• The people born at 2004 with the exposure, are able to get the outcome registered already a year after birth. So they would contribute with a lower time interval in the analysis.

What I am confused about is: would that be a problem in the analysis? That some contributes to a longer time interval - not due to natural causes, but due to the issue of reporting the outcome in the system? and some are able to contributing to a lower time interval due to natural causes. Would my HR estimate be to biased?

• I haven't time right now to give a full answer, but look up left-censoring, which this seems to be a case of. Unless I am misunderstanding Jan 27 at 14:57
• Thanks for answering. It is left-censoring. The potentiel of the outcome occurring before the follow-up period. Even though the follow-up starts from 1997 in my sample, the follow-up for the specific outcome starts from 2004. But is there a solution to it? Like adjusting? Or should I just report that the estimate is influenced by left-censoring? Jan 27 at 15:01
• There should be ways to deal with this, but they would depend on your choice of time = 0 for the survival analysis. Is that the date of birth or some other time reference? Please provide that information by editing the question, as comments are easy to overlook and can be deleted.
– EdM
Jan 27 at 23:10
• EdM, could you please read my Q again, thanks! I have updated the information at the end of the Q. I have also added some extra information at the IMPORTANT title. Jan 28 at 11:45
• There are much more serious epidemiologic issues to deal with, touched upon above, but from just the title, and ignoring time-dependent covariates and all other complexities, the Cox model like Kaplan-Meier estimates do not assume equal follow-up for exposed and non-exposed but assume that censoring time is random conditioning on X, e.g., censoring has nothing to do with impending failure. The censoring process has to be independent of the failure process. Jan 28 at 14:31

Based on the edited question, I'll assume that once exposure = 1 is found the exposure continues in that state thereafter and that the time scale is age since birth (date of birth is time = 0 for each individual). I'll also assume that outcome = 1 can occur no more than once in an individual.

If everyone in the sample was followed from birth, then there is no problem with left truncation. All individuals provide information starting at time = 0 (birth).

There is one potential problem, depending on the nature of the exposure. If someone was born with exposure = 0, switched to exposure = 1 prior to 2004, and was found to have the outcome in 2004 when the outcomes were first recorded, then you won't know which exposure value was in place at the time of the outcome event. If the exposure is something that you are born with, however, then that won't be a problem.

I'll assume that the exposure is something you are born with, like an inherited metabolic defect or some other genotype. Unless exposure (or some other predictor variable) changes over time, you don't need multiple lines per individual. You simply use the outcome indicator to record the status at the age of last observation.

The outcome here requires special handling, however, because (under the above assumptions) the event times for those born prior to 2004 but found to have the outcome at the first recording would be left censored, as Peter Flom suggested in a comment. That is, you have only an upper limit for the time to event (the age at recording the outcome). The help page for the Surv() function says:

For interval censored data, the status indicator is 0=right censored, 1=event at time, 2=left censored, 3=interval censored.

Unless you have interval censored event times or the exposure (or some other modeled predictor) changes over time, then you might not need the counting process data format Surv(start, end, outcome). A simple Surv(end, outcome) format will do, with end being the age when outcome = 1 or at the last observation.

I'm not sure that coxph(), however, can handle left censored event times. (It certainly can't handle interval censored event times.) There's no way for the usual Cox model to know whether individuals with left-censored event times should be in the risk sets for event times found to be earlier. For a regression model you probably need to use tools like those provided by the icenReg package. A regression model could include date of birth as an additional predictor, if you expect that the hazard of developing the outcome changed with calendar time.

There are a few matters that you would need to discuss with a statistician willing to take responsibility for the reported results. First, if the simplifying assumptions above don't hold, the modeling will be more complicated. Second is how to handle individuals who were born in 1997 or after but were lost to follow up before 2004. You won't know whether, at the last follow up, they had outcome = 1 (left censored event time) or outcome = 0 (right censored). Third is how to deal with the implications of not being able to record outcome = 1 until age = 1. Fourth, if you do want to include date of birth as a predictor, is whether that inclusion will pose problems for modeling individuals born prior to 2004.