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Cohen's d and Hedges'g effect sizes are used when two distributions have equal variances, which is a requirement and assumption to use these statistical tests.

When two distributions have unequal variances, Glass'delta should be used instead, putting in the equation only the standard deviation of the control/pre-measurement group.

What happens when two distributions have unequal variances but none of them is a control or pre-measurement group? For example, two sets of measurements are made on the same participants on different body sides. For some reason, the two data distributions have unequal variances, so I cannot use Cohen's d or Hedges'g effect sizes, because the equal-variance assumption is violated. Therefore, I have to use Glass'delta, but none of the two distributions is a control or pre-measurement group, compared to the other. What to do in this case?

Thanks.

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You can also compute a standardized mean difference that does not assume homoscedasticity but that uses the square-root of the average of the two variances in the denominator. Then you don't have to choose which group's SD is used in the denominator. See Bonett (2008, 2009) for further details.

Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means. Psychological Methods, 13(2), 99–109. https://doi.org/10.1037/1082-989X.13.2.99

Bonett, D. G. (2009). Meta-analytic interval estimation for standardized and unstandardized mean differences. Psychological Methods, 14(3), 225–238. https://doi.org/10.1037/a0016619

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  • $\begingroup$ I will close this question as resolved, but there is an important caveat. While Bonett's ES does not assume equal variance, its parametric CI depends on the correlation between the data sets, and thus on the study design. The CC between the data sets is used to calculate the standard error, which is used to calculate the parametric CI around the ES. Bonett confirmed this point. It should be noted that standardized mean differences are used as ES because they are independent of the experiment design, and thus allow for a comparison among different studies, for example in meta-analyses. Thanks. $\endgroup$
    – Federico
    Apr 5, 2023 at 10:15
  • $\begingroup$ Another option could be to use Glass' delta, considering as control/pre-measurement group that one with the smallest standard deviation, which is generally the case in comparison to the intervention/post-measurement group, where the effect can vary among participants. However, this choice should be explained in the publication. $\endgroup$
    – Federico
    Apr 5, 2023 at 10:23

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