# How to calculate CI for Median Odds Ratio?

According to Austin et al (2016) the median odds ratio (MOR) is defined by

$$\exp(\sqrt{2\sigma^2} \times \Phi^{-1}(0.75))$$

, where $$\Phi^{-1}$$ denotes the inverse of the standard normal cumulative distribution function and $$\sigma^2$$ is the variance of the random effects.

I am doing a multilevel logistic regression with random intercepts. My data is copy-pastable from here and the following code should run everything.

library(lme4)

m1 <- glmer("error ~ 1 + year + categorisation + statistic + (1 | journalID)",
data = data,
v <- as.data.frame(VarCorr(m1))[4]
MOR = exp(sqrt(2*v)*qnorm(0.75))


I get a MOR of 1.233264.

However, I don't understand how to compute a confidence interval on this MOR.

Edit: @user2957945 kindly points out a question that is similar. Although it refers to a 95% Credible Interval rather than a Confidence Interval, a bootstrapping approach is suggested. However, aside from not being competent to assess whether the approach there is valid, I also wasn't sure how to adapt the code for my data.

Austin, P. C., Wagner, P., & Merlo, J. (2017). The median hazard ratio: a useful measure of variance and general contextual effects in multilevel survival analysis. Statistics in Medicine, 36(6), 928-938.