Cox model with different follow-up times in groups We have a dataset where the non-exposed group has follow up to 5 years but the exposed group has follow up only to 1 year (>1 year not possible in the dataset).  Analysis is with Cox regression.  
The question is whether we should censor the non-exposed patients at 1 year to match the maximum follow up in the exposed group, or not.  Would the coefficient for the exposure be different using the full follow-up time vs. the 1-year censored follow-up time for the non-exposed?
Any advice much appreciated.
 A: If you only have exposure and no other covariates it makes no difference.  The Cox partial likelihood compares observations at the same time, so when you  have no observations still at risk in one group, those in the other group provide no information.
In R
> library(survival)
> set.seed(2020-6-28)
> z<-rep(1:2,each=100)
> x<-rexp(200,z/2)
> c<-ifelse(z==1,5,1)
> t<-pmin(c,x)
> d<-x<=c
> table(z,d)
   d
z   FALSE TRUE
  1     3   97
  2    37   63
> coxph(Surv(t,d)~factor(z))
Call:
coxph(formula = Surv(t, d) ~ factor(z))

             coef exp(coef) se(coef)     z       p
factor(z)2 0.5748    1.7768   0.2011 2.859 0.00425

Likelihood ratio test=8.4  on 1 df, p=0.003757
n= 200, number of events= 160 

Now re-do the censoring at 1 for both groups
> c.early<-rep(1,200)
> t.early<-pmin(c.early,x)
> d.early<-x<c.early
> table(z,d.early)
   d.early
z   FALSE TRUE
  1    59   41
  2    37   63
> coxph(Surv(t.early,d.early)~factor(z))
Call:
coxph(formula = Surv(t.early, d.early) ~ factor(z))

             coef exp(coef) se(coef)     z       p
factor(z)2 0.5748    1.7768   0.2011 2.859 0.00425

Likelihood ratio test=8.4  on 1 df, p=0.003757
n= 200, number of events= 104 

Precisely no change in the Cox model, as claimed.
If you have other covariates the results will not be identical. The question then is whether you expect the relationship between the other covariates and survival to stay the same after 1 year or not.  If it does stay about the same, you'll get a better estimate of it (and so potentially better adjustment) by using the whole data.  If it changes too much, you will get an estimate that's averaged over the whole time and so is biased for the one-year period where you have information on exposure.
The censoring itself won't introduce a bias (or rather, it's a basic assumption of survival analysis that it doesn't and there's no fix if it does).
A: You should censor the non-exposed patients at 1 year to match the maximum follow up in the exposed group to minimize biases.
