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I am familiar with the generic formula to convert OR to Cohen's d. However, I get the very strong impression from a relatively recent paper (https://www.tandfonline.com/doi/abs/10.1080/03610911003650383) that the rate of an outcome in the unexposed group can matter a great deal in this conversion. Depending on the OR value, and the rate, the converted d can vary. Specifically, as the rate increases, the d increases for a given OR.

The authors of that paper did not bother to give their method. How does one take this into account?

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The referenced article is using the probit method of computing Cohen's $d$ from the dichotomous outcome data. Two other common methods are based on converting the logged odds ratio, the first method by the standard deviation of the logistic distribution ($\frac{\pi}{\sqrt{3}} = 1.8138$) or by dividing by 1.65. The latter is called the Cox method and better approximates the probit values. However, it should be obvious that these odds ratio based methods will produce an identical $d$ for a given OR, independent of the underlying base rate. The reason that the probit method does depend, at least to a small degree, on the base rate is that the standard normal distribution and the logistic distribution, while both symmetrical, have a somewhat different shape with the latter being more heavy-tailed. See:Sánchez-Meca, J., Marín-Martínez, F., & Chacón-Moscoso, S. (2003). Effect-size indices for dichotomized outcomes in meta-analysis. Psychological methods, 8(4), 448, for a comparison of the various methods of estimating Cohen's $d$ from dichotomous outcome data.

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