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Is there a way to transform a coefficient from a logistic regression into a cohen's d value?

I am trying to calculate the cohen's d for a study with a binary DV while controlling for another variable. I can calculate cohen's d from the log odd ratio, but that does not control for the other variables, which I have included as controls in the logistic regression. This is for an internal meta-analysis where we combine across 12 studies with binary and scale DVs. Let me know if you have any suggestions.

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You could fit the logistic regression model with the IV of interest and the covariate that you want to control for. The coefficient for the IV of interest is the 'adjusted' log odds ratio (let's call this $\mbox{lnor}$), which you could convert to a d-value, either using $$y = \sqrt{3} / \pi \times \mbox{lnor}$$ or $$y = \mbox{lnor} / 1.65.$$ The former assumes that the dichotomous outcome is a dichotomized version of a continuous variable that follows a logistic distribution, while the latter assumes a normal distribution. The corresponding sampling variance of $y$ is $$\mbox{Var}[y] = 3 / \pi^2 \times \mbox{SE}[\mbox{lnor}]^2$$ or $$\mbox{Var}[y] = \mbox{SE}[\mbox{lnor}]^2 / 1.65^2,$$ where $\mbox{SE}[\mbox{lnor}]$ is the standard error of $\mbox{lnor}$ (which you can extract from the logistic regression model).

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