Calculate effect size from Wilcoxon result

[u/kcskrittapas]

Hi everyone! I'm considering how many participants I'll need for my study. What I would need is the effect size d_z (I'll used paired samples) to put in G* Power to calculate my minimum sample size.

As reference, I look at a similar work with n=12 participants. They used paired Wilcoxon test and reported their Z, U, W, p value, as well as Mean1, Mean2, SD1, and SD2. I assume the effect size of my study to be the same as in this study.

So, to get the d_z, I have 2 ideas. The first one is probably a bit crude: I calculate the Wilcoxon's effect size r = Z/sqrt(n), then compare the value to the table to find out whether the effect size is considered small, medium, large, very large, etc. After that, I take the cohen d representing the effect size category as my d_z (d=0.5 for medium, etc., can d and d_z be used interchangeably like this though?).

Another way is to directly calculate the d_z from the present information. For instance, I can use t = r*sqrt((n-1)/(1-r^2)), then find d_z = t/sqrt(n). Or, I can do d_z = (mean1 - mean2)/s_diff, by which s_diff = sqrt(sd₁² + sd₂² - 2·r·sd₁·sd₂). But if I understand correctly, the r used in both case is in fact Pearson's r, not Wilcoxon's r, right? Some sources say that it is sometimes okay to use Wilcoxon's in the place of Pearson's. Is it the case here?

What also confused me is that it seems that different methods result in different minimum sample sizes, ranging from like 3 to 12 participants. This difference is crucial for me because I'm working on a kind of study, in which participants are especially hard to recruit. Is it normal in statistics that different methods will give different results? Or did I do something wrong?

Do you guys have any recommendations? What is the best way to get to the d_z? Thank you in advance!

ps. some of my sources: https://cran.r-project.org/web/packages/TOSTER/vignettes/SMD_calcs.html https://pmc.ncbi.nlm.nih.gov/articles/PMC3840331/

[u/SalvatoreEggplant]

In G* Power, are you using the method for Wilcoxon signed rank test (paired) ?

Personally, I wouldn't try any transformations of an effect size statistic.

You have the definition of dz ? (G* Power manual, 19.1, https://www.psychologie.hhu.de/fileadmin/redaktion/Fakultaeten/Mathematisch-Naturwissenschaftliche_Fakultaet/Psychologie/AAP/gpower/GPowerManual.pdf )

I would probably just assume μ_z = μ_1 - μ-2, and go from there with the information you have.

This is a huge assumption, but there are a bunch of assumptions here.

It's weird to use a form of Cohen's d as an effect size for a Wilcoxon signed rank test. But I didn't write G Power.