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?
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: