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?


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?

That depends on how you implement the Cox regression, as you illustrate. As discussed here, tied event times pose problems with Cox regression. The "exact partial likelihood" works OK with small data sets, but it requires "exhaustive enumeration of the possible risk sets at each tied death time, and can require a prohibitive amount of computation time." The Efron approximation to that exact partial likelihood is generally better than the Breslow approximation, but even it performs poorly at times when there is a very high ratio of ties to the size of the risk set.
The odds ratio and the risk ratio are different, although they can be similar if events are rare. See this web page or this Cross Validated page, for example. With a single time point the hazard ratio should match the odds ratio. Your values are so close that I suspect the difference is due to numerical issues from the multiple calculations associated with applying the "exact" method to a risk set of 400 individuals.
