I am working on meta-analysis, 90% of my data is continuous and expressed in same units, some 10 % studies use different unit of measure.
What should be the appropriate effect size?
Mean difference?
Or SMD,
At the end I need effect size % decrease or increase
As far as I know, two scales are usually used in these transformations: Cohen's d scale or r scale. I guess, the most used common scale in the meta-analytic settings is the r scale. I just derived new formulae for transforming r to de or d to r in any kind of scale. See Preprint R effect size and Generalized Cohen d 2024 with name
Is it feasible to transform Standardized Mean Differences (SMD) into percentage units? I'm curious if such a conversion is possible and, if so, what methods or approaches might be utilized for this purpose.
Cohen's d in its original form is SMD (although we usually see it in the sample-based form, also given by Cohen). r^2 or R^2 are usually taken as percents/proportions ("percentage/proportion of the variance explained by the grouping variable"). Then, if you convert d to the scale of r by using the traditional formula by Cohen and, if there are more categories than two, you can use the new formulae given in the previous link. When you have transform Cohen's d to the scale of r, you can take the second power and express SMD in the scale of percentages (of some kind).
The interpretation may be make more complex the fact that, when the scales are not equal, two things affect the result: the number of categories in the variable with the narrower scale and the dicrepancy of the cases in different categories. They both affect the magnitude of the deflation or attenuation in the correlation. This causes that, if the proportions of the cases in different categories radically deviate from each other, the scale of r (and R) is no more -1 to +1, but it may range -0.6 to +0.6 (see simulations and publiced datasets inArticle The effect of various simultaneous sources of mechanical err...
. Then, the challenge is that what means the "percent" of explaining power or r^2 = 0.4? It means "the proportion of what we CAN predict by the given condition of variables". For this you could try the deflation- or attenuation corrected estimates of r (rDC or rAC; see https://link.springer.com/content/pdf/10.1007/s41237-022-00162-2.pdf andArticle Directional nature of the product–moment correlation coeffic...
). They strictly reflect the proportion of the observed correlation of the maximum possible correlation. The relation of these latter correlations with Cohen's d is not studied.Good luck!
JMe
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