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In order to calculate an effect size estimator for a non-parametric test such as the Wilcoxon test, the following formula is applied: $ r= \frac{Z}{\sqrt{N}} $

This formula is suggested by several forum contributions and is also used by SPSS for the purpose described above. Originally the formula was developed to my knowledge by Rosenthal (1991, p.19).

Beside just using the formula, I am interested at how to arrive at it as I do not see the connection between the correlation coefficient and the z value. If someone could provide an explanation of how to arrive at the formula or a derivation, I would be glad.

The only hint I have about the formula is in the following table equation 2.1 and 2.2. Rosenthal does not provide unfortunately an explanation about this relationship. enter image description here

Note: There has been already a discussion about this topic but rather in the form how to use and interpret it and not how to arrive at it. Effect size to Wilcoxon signed rank test?

Rosenthal, R. (1991). Applied Social Research Methods: Meta-analytic procedures for social research Thousand Oaks, CA: SAGE Publications Ltd doi: 10.4135/9781412984997

Abar
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  • Minimal references like Rosenthal (1991) can possibly be guessed but are in general a bad idea. Please give full details as expected in an academic thesis or paper. – Nick Cox Jan 16 '18 at 17:29
  • You are absolutely right. Rosenthal, R. (1991). Applied Social Research Methods: Meta-analytic procedures for social research Thousand Oaks, CA: SAGE Publications Ltd doi: 10.4135/9781412984997 – Abar Jan 16 '18 at 17:39
  • Please tell us what "$Z$" might be and how it is related to $r$. – whuber Jan 16 '18 at 18:48
  • Z stands for the z value. How the z value like estimated by the Wilcoxon-Test and the correlation coefficient are related, is exactly my question. As for this formula there is no explanation in the stated paper. However, as I found out SPSS calc. for the Wilcoxon-test an effect size using this formula. – Abar Jan 16 '18 at 19:03
  • @whuber : I hope this rephrasing helps to understand my question. – Abar Jan 16 '18 at 21:22
  • It's unclear--it sounds like you might be referring to an approximation of a test statistic, but maybe you aren't. In your edits you seem to have stopped halfway between asking about correlation coefficients and asking about nonparametric tests, which is further confusing--please check it over again. When you do, providing an explicit formula or example of $Z$ would help. – whuber Jan 16 '18 at 21:26
  • This is the only hint in the original document that I have. I do not know if Rosenthal referred to z or an approximation of it. If this wont help, I will just use the formula like everybody else without questioning how to arrive at it. As nobody seems to know the connection like myself. Thanks anyway @whuber – Abar Jan 16 '18 at 21:46

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