Generalized Cohen's d for Multiple Means and Polytomous Settings

Area of Science:

• Statistics
• Psychometrics
• Quantitative Psychology

Background:

• Cohen's d is widely used for two-group mean difference effect size.
• Cohen's f is used for comparing multiple population means.
• Existing methods lack generalization for complex polytomous settings.

Purpose of the Study:

• To derive general formulas for Cohen's d in polytomous settings.
• To evaluate the accuracy of a simplified d = 2f estimator.
• To provide guidance on effect size estimation for multiple populations.

Main Methods:

• Utilized the relationship between Cohen's d and f in dichotomous cases.
• Developed generalized formulas extending Cohen's d to polytomous data.
• Analyzed the performance of the simplified estimator (d = 2f).

Main Results:

• General formulas for Cohen's d in polytomous settings were successfully derived.
• The simplified estimator d = 2f was found to be less accurate.
• Discrepancies in extreme proportions exceeding 0.40 highlight the need for general formulas.

Conclusions:

• Recommended using the derived general formulas for accurate effect size estimation.
• Advised against relying on the simplified d = 2f estimator in most cases.
• Emphasized the importance of accurate effect size measures in multiple population comparisons.