# How to convert the standard error of the log odds ratio to the odds ratio standard error

I am using the log odds ratio (and its standard error) for meta-analysis. I want to convert back to odds ratio to write up the results... maybe i'm putting in the wrong search terms but can't find info on how to do this.

Obviously the Ln(OR) can be converted by using the exponent, I don't think i can do the same for the standard error of ln(OR)?

• See here or here for the usual approach of exponentiating the end points of a confidence interval on the log odds ratio. – Scortchi - Reinstate Monica Jun 24 '15 at 14:24
• But what if i just wanted to report the standard error of the odds ratio instead of the confidence interval? I mean i guess I could report the CI instead (or aswell!). Oh maybe i can't because its got a weird distribution? – user3084100 Jun 24 '15 at 16:59
• The distribution of the odds ratio estimate is quite asymmetrical, so you wouldn't get an accurate confidence interval by adding/subtracting a multiple of the standard error to the estimate. That's why people usually work with the log odds ratio estimate. – Scortchi - Reinstate Monica Jun 24 '15 at 17:09

With an estimate of the log odds ratio $\hat\omega$ & its standard error $\hat\sigma_{\hat\omega}$ you can use the delta method to get an approximation to the standard error of the odds ratio estimate $\newcommand{\e}{\mathrm{e}}\e^\hat\omega$: \newcommand{\Var}{\operatorname{Var}} \newcommand{\dif}{\mathrm{d}} \begin{align} \sqrt{\Var \e^{\hat\omega}} & \approx \sqrt{\left(\left.\frac{\dif \e^x}{\dif x}\right|_\hat\omega\right)^2 \Var \hat\omega }\\ & = \e^{\hat\omega} \hat\sigma_{\hat\omega} \end{align}