Why are confidence intervals of hazard ratios not symmetric? My understanding is that the confidence interval for a hazard ratio should be symmetrical about the mean (the distance between the lower limit and the mean is the same as the distance between the mean and the upper limit). But often when I see a hazard ratio in the published literature on clinical trials this is not the case.
For example, a study reporting HR, 0.69; 95% CI, 0.54 to 0.89 in mCRC for cetuximab plus FOLFOX-4 vs FOLFOX-4 alone found here: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7044820/pdf/bmjopen-2019-030738.pdf.
The distance between the lower limit and the mean is 0.69 - 0.54 = 0.15, while the distance between the mean and the upper limit is 0.89 - 0.69 = 0.20. Shouldnt the distance for both be 0.15 to be symmetric?

It is just that we are used to symmetric confidence intervals, because this is the most typical/common situation: the case of some sort of linear regression with Gaussian distributed errors.
But... there is not a single unique confidence interval. Any region with 95% confidence will do.
Often one chooses the region that is smallest in some sense. For instance by using a hypothesis test (the confidence interval can be seen a region of hypothesis tests) that computes the p-values based on the region with highest probability density.
Example 1
Below is an example from this question that shows the estimation of the rate parameter $\lambda$ of a population that is exponential distributed. The estimate is based on the observed sample mean $\bar{x}$. You can see that the confidence interval boundaries (thick black lines) are not symmetrical around the estimate (dotted line). This has two reasons.

*

*The estimate for the rate parameter $\hat\lambda$ is not a linear function of the observed $\bar{x}$.

*The sample distribution of $\bar{x}$ for a given parameter $\lambda$ is not symmetric.


Example 2
A very clear example is the estimation of the parameter $p$ in a binomial distribution.
Say you estimate the number of red and blue balls in an urn. If in some sample you observe zero red balls then the estimate for the number of red balls will be zero, but the confidence interval won't be symmetric (or it shouldn't be) or otherwise you include negative values.
Example 3
The answer of Christoph Hanck in the previous linked question shows an example of the Gaussian distribution and how the confidence interval will turn out to be symmetric.
But if you transform the parameter for which you compute the interval then it will again become non-symmetric. Possibly that is the case for the hazard ratio.

