Question
Asked 8th Mar, 2014

# How can I calculate the effect sizes of small samples (<20) for non-parametric tests Wilcoxon and Mann-Whitney U?

I want to calculate the effect sizes of Wilcoxon and Mann-Whitney tests. Can I use the formula r=Z/sqrt(N) when the samples are smaller than 20 (where N is all the observations) ?

## Most recent answer

10th Sep, 2022
Ponmathi Paramanandam
Sri Ramachandra Institute of Higher Education and Research
very useful thread,thanks to all who shared their ideas here

## Popular answers (1)

20th Mar, 2014
Ronán Michael Conroy
Royal College of Surgeons in Ireland
To add to Jochen's reply, effect size is indeed defined for the Wilcoxon Mann-Whitney test. The underlying test statistic, U, can easily be rescaled to give the probability that an observation from one group will be higher than an observation from the other. This measure of effect size is variously known as the common language effect size, the area under the ROC curve, Harrel's c (though C is a special case) etc.
But it is a very useful measure of effect size indeed - in a clinical trial, for example, it corresponds to the probability that a person on the new treatment will do better than a person on the comparison treatment.
To calculate it, simply divide U by its maximum value, which is the product of the Ns for the two groups
I've a little paper on this, which is no longer behind a paywall, but will pretty soon go free. http://www.stata-journal.com/article.html?article=st0253
11 Recommendations

## All Answers (19)

8th Mar, 2014
Jochen Wilhelm
Justus-Liebig-Universität Gießen
"effect size" is not defined when you abandon the data and use the ranks. If you need this for detemining the sample size required to achieve a power for the test (of a location shift), then you can use bootstap methods (there cannnot be a general solution since there is no general distributional model that could be used!). Thus you have to have pilot data for the distribution under H0.
Possibly read:
Biometrics. 1988 Sep;44(3):847-60.
Estimating the power of the two-sample Wilcoxon test for location shift.
Collings BJ1, Hamilton MA.
7 Recommendations
8th Mar, 2014
Menelaos Emmanouel Sarris
University of Patras
You may calculate effect size via r = z/√N (r: effect size; z: z value; N: Observation number). You should divide z value to square root of observation number for getting effect size. You can find z value on the output case -at the end of the Wilcoxon and Mann-Whitney tests-.
2 Recommendations
20th Mar, 2014
Ronán Michael Conroy
Royal College of Surgeons in Ireland
To add to Jochen's reply, effect size is indeed defined for the Wilcoxon Mann-Whitney test. The underlying test statistic, U, can easily be rescaled to give the probability that an observation from one group will be higher than an observation from the other. This measure of effect size is variously known as the common language effect size, the area under the ROC curve, Harrel's c (though C is a special case) etc.
But it is a very useful measure of effect size indeed - in a clinical trial, for example, it corresponds to the probability that a person on the new treatment will do better than a person on the comparison treatment.
To calculate it, simply divide U by its maximum value, which is the product of the Ns for the two groups
I've a little paper on this, which is no longer behind a paywall, but will pretty soon go free. http://www.stata-journal.com/article.html?article=st0253
11 Recommendations
10th Jun, 2014
Bruce Weaver
Lakehead University Thunder Bay Campus
Although I've not read them yet, I see that Robert Newcombe's book on confidence intervals for proportions & related measures (http://www.crcpress.com/product/isbn/9781439812785) has a chapter called "Generalised Mann-Whitney Measure" and another called "Generalised Wilcoxon Measure" (chapters 15 & 16 respectively). The original poster may find chapter 15 useful.
1 Recommendation
10th Jun, 2014
Jochen Wilhelm
Justus-Liebig-Universität Gießen
Thank you! It is an entirely different kind of "effect". Mathematically this is perfectly fine (I think), I am only a little itchy about the fact that probabilities itself are used as effect-size measure (this is related to the meaning of probability, what is not so clear, though).
27th Jun, 2014
Mei Teng Woo
Republic Polytechnic
May I know the meaning of the "negative sign" of the effect size? e.g. r = -0.57
27th Jun, 2014
Jochen Wilhelm
Justus-Liebig-Universität Gießen
it means that the expected value for the response decreases for higher vaues of the predictor.
27th Jun, 2014
Mei Teng Woo
Republic Polytechnic
e.g.
I'm comparing between two independent groups (control and Treatment), the treatment has higher mean ranks compared to control group. Can I say a significant effect of Group (The mean ranks of control and treatment group were 5.2 and 9.3, respectively)", p < 0.05, r= -.57. There is a medium effect observed of the intervention on treatment group.
26th Nov, 2015
Reneh Karamians
The Ohio State University
r=Z/sq.root of N abs(r) works fine. Note Small=.1 moderate=.3 large=.5
4 Recommendations
12th Oct, 2016
Trudy Mallinson
George Washington University
I realize this thread is a little dated but so glad I stumbled upon it. Many thanks Dr. Conroy, the article is really helpful.
1 Recommendation
18th Oct, 2016
Ronán Michael Conroy
Royal College of Surgeons in Ireland
Glad you found it useful, Trudy
4th Dec, 2016
AmirAli Jafarnezhadgero
University of Mohaghegh Ardabili
According to Fritz, Morris, and Richler (2011) we can calculate effect size for the Mann-Whitney U-test using the the mentioned formula. The following website is realy helpful:
19th Feb, 2017
Dido Green
Brunel University London
Thank you for this thread.  Do you think this formula could be used for within subject repeated measures?  Or should a factor be added to consider the extent of estimated progress. Without an a priori estimation of the expected treatment effect, the instruments’ ability to measure change cannot be disentangled from the treatment effect.  I am concerned about the overstatement of treatment results with the use of the Goal Attainment Scale when one would expect some progress in attainment of goals that were the focus of the intervention.
3rd Jul, 2017
Sharina Windmuller
Hogeschool Fontys
can anyone tell me iof the effect size in the calculation of the Wilcoxcon Signed Rank Test can be a negative outcome?
i've got (-2,005/ sqare of 146) and the outcome is -0.17)
how do I report this APA?
thnx!
You can find a formula proposal here, in addition to the interpretation:
24th Mar, 2021
Azam Zarneshan
Azarbaijan Shahid Madani University
use the G power analysis ,tests, independent two groups (Mann‐Whitney test), the Determine Effect size section Or the following formula that has similar results:
Mean difference between the two groups (M2-M1) / mean standard deviation
24th Mar, 2021
Ronán Michael Conroy
Royal College of Surgeons in Ireland
Azam Zarneshan The effect sizes in G*Power depend on the data following a known distribution. They are based on Cohen's d, which measures effect based on the difference between means. However, the Wilcoxon Mann-Whitney test does not test a difference between means (or medians), so the effect size is inappropriate to the test.
4 Recommendations
10th Sep, 2022
Ponmathi Paramanandam
Sri Ramachandra Institute of Higher Education and Research
very useful thread,thanks to all who shared their ideas here

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