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).
