Correct conversion of Odds Ratio to Cohen's d

Question

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.

Answer 1

An odds ratio is an effect size. A lot of people use the term effect size to mean standardized mean difference (i.e., Cohen's d), but this is not correct terminology. Imagine the term effect size stands for car and things like odds ratio, risk ratio, standardized mean difference, and so on are brands/types of cars. Right now, you are asking how to transform your BMW into a car ... not a very sensible question (although that might depend on how you feel about BMWs, but that's another issue). Instead, you should ask how to transform your BMW into a Mercedes (which is kind of silly as well, but at least not completely nonsensical). Anyway, to answer your questions:

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 $\ln()$.

2. One can either meta-analyze the log odds ratio of the three studies and then convert the summary estimate to a standardized mean difference or one could convert the three log odds ratios into standardized mean differences and meta-analyze those. The result will actually be identical. And the test for heterogeneity and the test of significance for the summary estimate will also be identical. So there is no need to apply any kind of conversion to those.

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.