Hazard ratio and relative risk (or odds ratio) in case of very few event/censoring times As far as I understand relative risks and hazard ratios are very similar concepts (although this paper seems to disagree). The advantage of hazard ratios is the inclusion of temporal information. I always thought that the Cox model at least theoretically requires that events and censoring can take place at any given moment so that time is necessarily a continuous variable. Obviously, time is usually discretised but this I considered okay as long as there are many (preferably equidistant) possible intervals at which an event or censoring can occur. The other day someone told me that there is no problem even if there is just one point in time in Cox regression. So my first question is: Is that true?
And if so shouldn't hazard ratio and relative risk be equal as there would be no additional temporal information? Here, I have an example with 150/200 events for placebo and 130/200 for verum with event and censoring time respectively the same for all patients.
> library(survival)
> d=data.frame(trt=(c(rep(1,200),rep(0,200))),event=c(rep(1,130),rep(0,70),rep(1,150),rep(0,50)),time=2)
> (cox.model=(coxph(Surv(time,event)~trt,data=d,ties="exact")))
Call:
coxph(formula = Surv(time, event) ~ trt, data = d, ties = "exact")

       coef exp(coef) se(coef)      z      p
trt -0.4784    0.6198   0.2203 -2.172 0.0299
> exp(confint(cox.model))
        2.5 %    97.5 %
trt 0.4024838 0.9544346


Likelihood ratio test=4.77  on 1 df, p=0.02901
n= 400, number of events= 280

(The "exact" method for ties is correct right? "Efron" gives quite a different result of 0.7820 and "Breslow" is far from significant.) The relative risk on the other hand is this:
> meta::metabin(130,200,150,200,sm="RR")
Number of observations: o = 400
Number of events: e = 280

     RR           95%-CI     z p-value
 0.8667 [0.7615; 0.9864] -2.17  0.0302

Interestingly, the odds ratio is much closer but also not equal:
     OR           95%-CI     z p-value
 0.6190 [0.4018; 0.9538] -2.17  0.0297

My second question is: Should HR and RR (or OR) be the same under these conditions?

These terms and calculations are essentially convenient ways to bias results of studies in various directions to support the hoped-for outcomes.
Example - Relative risk...
Pharma Co. X produces its latest/greatest FLU vaccine, and needs some data to bring it to market.
So - it recruits 2000 healthy volunteers.  1000 are given vax.  1000 are not.
Company exposes all to FLU virus.
1 of vaxed group tests positive for flu.  2 of non-vaxed test positive.
THE ABSOLUTE difference is OBVIOUSLY 1/1000 (.001).  In other words - zero statistical difference.
This sells ZERO vaccine doses.  Data must be spun.
SO -- Enter RELATIVE RISK!!
ONE vaxed person has flu. TWO non-vaxed...  OH -- that's a 50% difference.
VOILA.. - 50% RELATIVE RISK reduction with the vax.
Totally corrupt - but totally acceptable for their marketing.
