# Correct conversion of Odds Ratio to Cohen's d

I am trying to convert an Odds Ratio to an Effect Size (ES). I found this article from Cross validated that provides a way of converting a Log Odds Ratio to Cohen's d but am unsure of three things.

1. Is an 'Odds Ratio' the same as a 'Log Odds Ratio'?
2. Does the statistical significance test and heterogeneity tests directly translate to the converted ES or do these need to also be converted?
3. I am also unsure if I am carrying out the calculations correctly. Based on three studies the OR is $$0.52$$ ($$CI = 0.37, 0.74; p = 0.0002$$) Heterogeneity: $$\chi^2 = 3.66; I^2 = 45\%$$. Does my calculated/converted ES below seem accurate? I don't mean to ask someone to do them for me so have tried to figure them out myself but I'm not 100% sure if I've done it right.

Cohen's d = $$0.287$$ ($$CI = 0.204, 0.407$$)

Thank you for any guidance.

1. No, they are not the same thing. A log odds ratio is ... the log of an odds ratio. Hence, if the odds ratio is $$1.5$$, then the log odds ratio is $$\log(1.5) \approx 0.405$$, where $$\log()$$ denotes the natural log often written as $$\ln()$$.
3. Assuming that 0.52 is really an odds ratio (and not a log odds ratio), then this is not correct. You should then do $$\log(0.52) \times \sqrt{3} / \pi \approx -0.36$$. However, if 0.52 is actually a log odds ratio, then your conversion is correct.