Generalized Cohen's d for Multiple Means and Polytomous Settings

Cohen's d is the most commonly used estimator to quantify the magnitude of the difference between the means of two subpopulations. When comparing multiple populations simultaneously, Cohen's f can be used for the same purpose. Using their relationship in the dichotomous setting, several general formulas for d are derived that generalize d to the polytomous setting. The traditional simplified estimator d = 2f is studied as a shortcut estimator. It is strongly recommended to use the general formulas instead of the simplified ones when assessing the magnitude of the effect size, especially when the discrepancy of the extreme proportions of cases in the subpopulations exceeds 0.40.