In 2015 I wrote a post about the causal interpretation of hazard ratios estimated in randomised trials, following a paper by Aalen and colleagues. One of the arguments made in that paper was that the hazard ratio does not have a valid interpretation as a causal effect in this setting, even when the proportional hazards assumption holds:
This makes it unclear what the hazard ratio computed for a randomized survival study really means. Note, that this has nothing to do with the fit of the Cox model. The model may fit perfectly in the marginal case with X as the only covariate, but the present problem remains.
With recent discussions on estimands in light of the estimand addendum to ICH E9, I have been thinking more on the argument/claim by Aalen et al.
Suppose we have a randomised trial where we know (somehow) that the ratio of the hazards in the two groups is constant over time – that is the proportional hazards assumption holds. Suppose that the true hazard ratio comparing treatment 1 to treatment 0 is . Then due to the relationship between the survival function and hazard, as is well known, it follows that:
where and are the marginal survival functions in the two treatment groups. This can then be re-expressed as:
Thus under proportional hazards, at any time, the hazard ratio is equal to the ratio of the log of the survival probabilities under the two treatments to this time. The survival functions can be expressed as the population means of where denotes the potential failure time under treatment 1 for a randomly selected individual from the population (and similarly for treatment 0). Thus the hazard ratio can be expressed as a functional of the potential outcomes under the two treatments.
Admittedly the interpretation is not that nice, but I would argue the hazard ratio nevertheless does have a causal interpretation (assuming proportional hazards holds). If at a particular time t the survival probabilities are both close to 1, the preceding expression can be approximated to show the hazard ratio is approximately the relative risk of failure under the two treatments, something which has also been written about before.
I am certainly not a causal inference expert, so I do not know what is required to claim that a quantity is a valid causal effect measure. But it would seem to me that if one views the hazard ratio as the previously described ratio of (logged) marginal probabilities, at least when proportional hazards holds, it is a valid causal effect measure. If any causal inference or survival analysis people can help, or point out that I am missing something obvious, please add a comment!