Calculate the standard deviation from a hazard ratio's confidence interval I often come across hazard ratios and their confidence intervals in the published literature on clinical trials. I would like to calculate the standard deviation from these confidence intervals for some analysis I'll be doing (generating random draws for this hazard ratio from a log-normal distribution).
Having read on this over the past few days, my thought process is that to convert the confidence intervals of a hazard ratio to the standard deviation of that hazard ratio, I would do the following:

*

*Take the natural log of the upper limit minus the natural log of the lower limit.

*Divide by 2 times the standard error.

*For the 95% confidence interval this would be 2 x 1.96 = 3.92, for the 90% confidence interval this would thus be 2 x 1.645 = 3.29, and for 99% confidence intervals this would be 2 x 2.575 = 5.15.

*If the sample size in either group studied, say a treated group and a control group, is below 100, then I should assume that the authors reporting this hazard ratio calculated this confidence interval using a t distribution, and thus I should replace the numbers 3.92, 3.29 and 5.15 above with numbers specific to the t distribution and the sample size. I do this by going to t distribution tables with degrees of freedom equal to the sample size of both groups summed, minus 2.

This is how I would calculate a standard deviation in the R programming language for an example 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:
(log(0.89) - log(0.54)) / 3.92 = 0.1274623

Is this the right way to calculate the standard deviation from the confidence intervals of a hazard ratio?
EDIT
To more clearly motivate this question, I am a health economist estimating transitions between health states. In my analysis, there is an initial, and well established, transition probability from the stable disease state to the progressive disease state under standard of care treatment.
The literature indicates that this transition probability is decreased by a new medical intervention. The literature describes the hazard ratio for progression with this new intervention vs standard of care based on a clinical trial of cancer patients. Thus, I would like to update the transition probabilities for transitioning from stable disease to progressive disease under standard of care using this hazard ratio, to create transition probabilities for this new intervention as part of a cost-effectiveness analysis of this new medical intervention.
Initially, this will be done just with the hazard ratio reported in the clinical trial. Following this, I would like to conduct a probabilistic sensitivity analysis which reflects the uncertainty in this hazard ratio when creating transition probabilities. To do this, I need to take random draws from the log-normal distribution for the hazard ratio, as hazard ratios are typically skewed unless put on the log scale to normalise.
The following code is used in the R programming language to make these draws:
hr_draws <- rlnorm(nsims, meanlog = log(mean), sdlog = SD). 

This is why I am trying to determine how to create the standard deviation for my hazard ratio as above, in order to create a probabilistic hazard ratio.
My sources are here:
https://handbook-5-1.cochrane.org/chapter_7/7_7_3_2_obtaining_standard_deviations_from_standard_errors_and.htm
https://handbook-5-1.cochrane.org/chapter_7/7_7_3_3_obtaining_standard_deviations_from_standard_errors.htm
https://cran.rstudio.com/web/packages/episensr/vignettes/b_probabilistic.html

The estimated log hazard ratio is assumed to follow a normal distribution. The estimated hazard ratio can't follow a normal distribution because it can't take values less than 0. Confidence intervals for the hazard ratio are calculated by constructing a confidence interval for the log hazard ratio then exponentiating. As you noted, you need to work on the log scale to get your standard errors and do your analysis, then exponentiate after you finish.
Confidence intervals for the log HR are constructed with a normal distribution, not a $t_{df}$ distribution. Further, the sample size in survival analysis refers to the number of events (deaths, progressive disease, etc.), not the total number of patients. Unless there is no censoring, these numbers are not the same.
The standard error of the log hazard ratio can be calculated from the confidence interval with:
$$ SE = \frac{log(Upper CI) - log(lower CI)}{2 * z_{\alpha/2}},$$
where $z_{\alpha/2}$ is the upper $\alpha/2$ quantile of a normal distribution. You can calculate the standard error in R with
(log(upper_ci) - log(lower_ci)) / (2 * qnorm(1 - alpha/2),
replacing alpha with the correct error rate (probably 0.05).
If you want to calculate the standard error of the hazard ratio (not log), you can use the Delta method. The standard error is
$$SE(\text{log HR}) * exp(\text{log HR})$$
EDIT: It seems like OP is a bit confused about the difference between "standard error" and "standard deviation" and how that relates to their simulation. Standard deviation is the square root of the variance. Standard deviation is a measure of variability in an overarching (usually theoretical) population. "Standard error" refers to the standard deviation of a test statistic. These terms are sometimes used interchangeably, but they have different meanings. In OP's question, they are calculating the standard error of the (estimated) log hazard ratio. They are then correctly plugging that into rlnorm() to get random draws from the sampling distribution of the hazard ratio. In less precise terms,  OP is using the reported log HR and the standard error to simulate a distribution of what the true hazard ratio could be.
