# 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?

• What time (calender time, age) is used for the analysis, and how is the follow-up time of the two groups distributed on that time scale?
– swmo
Mar 10, 2015 at 21:46
• We're using days since initiation of treatment. Mar 11, 2015 at 5:06

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).

You should censor the non-exposed patients at 1 year to match the maximum follow up in the exposed group to minimize biases.

• I will not agree with this, I would like to treat short follow up group as censored, since they were really "censored' in term of longer follow up time. Aug 30, 2017 at 23:00
• What do you mean by "treat short follow up group as censored"? Both groups should be censored by event (i.e. death), not by follow up time. If the groups were censored by death, by censoring the exposed patients at 5 years and non-exposed patients at 1 year, we allow the exposed patients more time to die, thus creating the bias.
– Meo
Aug 31, 2017 at 14:45
• censor means no events. If a subject is dead (event) , he is not censored. And you should use follow up time not arbitrary 5 years. Aug 31, 2017 at 23:19
• If you treat the short follow up group as censored, then how would you consider the event?
– Meo
Sep 6, 2017 at 22:35
• Censored means you did not observe the event because you finished the follow up earlier or lost follow up or people dead. Sep 6, 2017 at 23:16