What question did this study set out to answer?

To develop general formulas for Cohen’s d applicable in polytomous settings and compare them with simplified estimators.

January 22, 2026Open Access

Generalized Cohen’s d for Multiple Means and Polytomous Settings

Key Points

To develop general formulas for Cohen’s d applicable in polytomous settings and compare them with simplified estimators.
Derived formulas for Cohen’s d based on its relationship with Cohen’s f for multiple populations.
Evaluated the performance of traditional shortcut estimator d = 2f.
Analyzed scenarios where subpopulation case proportions differ significantly.
Generalized formulas provide more accurate effect size estimates than the simplified estimator.
Reliability of effect size estimation improves as discrepancies between extreme proportions exceed 0.40.

Abstract

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 = 2 f 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.

Generalized Cohen’s d for Multiple Means and Polytomous Settings

Key Points

Abstract

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