# How to convert relative risk and confidence intervals/standard error to odds ratio and confidence intervals/standard error

I am conducting a meta-analysis and have one study which reports relative risks, and several studies reporting odds ratios. I would like to convert the relative risks to odds ratios and obtain corresponding confidence intervals and standard errors.

I have data on the prevalence of the outcome in the control group, the relative risk and 95% CIs.

I came across this formula for deriving the odds ratio from the relative risk (from Convert hazards ratio to odds ratio), based on Grant et al, BMJ 2014:

OR = ((1 - p) * RR) / (1 - RR * p)

where RR is the relative risk, OR is the odds ratio, and p is the control event rate.

To calculate the corresponding confidence intervals and standard error, would the below be correct:

OR_lowCI = ((1 - p) * RR_lowCI) / (1 - RR_lowCI * p)

OR_upperCI = ((1 - p) * RR_upperCI) / (1 - RR_upperCI * p)

log_OR_SE = (ln(OR_upperCI) - ln(OR_lowCI))/3.92

Many thanks in advance for any advice.