Sabiha Khatoon, the following site might be helpful:

https://www.psychometrica.de/effect_size.html

There are other sites on the web, also.

You can calculate the effect size r from the z-score of the U-test.

r = z/sqrt(N)

Robert Trevethan Thank You

Rainer Duesing Thank You!

The effect size is dependent on what you are measuring, not necessarily how you are testing any particular hypothesis, which in the M-Whitney case is often not the same. What is the null hypothesis that you ideally want to test. You said you used M-W because of distribution assumptions, so you might be interested in the difference of means. Related, why are you using M-W rather than say a bootstrap or turning the percentiles into normal scores. Lots of people recommend against using it: e.g., https://notstatschat.tumblr.com/post/63237480043/rock-paper-scissors-wilcoxon-test

Hello Sabiha Khatoon. You can find some good info in this discussion from a few years ago:

https://www.researchgate.net/post/How_can_I_calculate_the_effect_sizes_of_small_samples_20_for_non-parametric_tests_Wilcoxon_and_Mann-Whitney_U

In the (currently) most popular answer, Ronán Michael Conroy said that "[t]he 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." That rescaling entails dividing U by the product of the two sample sizes:

p(X > Y) = U/(n1*n2)

The Stata Journal article mentioned in that answer is no longer behind a paywall, and is an excellent resource.

• https://www.stata-journal.com/article.html?article=st0253

Salvatore S. Mangiafico describes some other options on this web-page:

• https://rcompanion.org/handbook/F_04.html

As Sal notes in the introduction to the Effect Size section, all of these measures essentially rescale p(X > Y):

"Statistics of effect size for the Mann–Whitney test report the degree to which one group has data with higher ranks than the other group.  They are related to the probability that a value from one group will be greater than a value from the other group."

HTH.

As Daniel Wright noted, it seems you wanted to use some other procedure initially. I'll hazard a guess that you wanted to carry out an unpaired t-test. I decided to risk opening the Word document attached to the original question, and see that the sample sizes are quite large:

• n1 = 392
• n2 = 166

With those sample sizes, an unequal variances version of the t-test should be quite robust. What are the means and SDs that go along with those sample sizes?

Bruce Weaver by using K-S Test i checked the normality. As the data was nonnormal so i applied WMM Test.

For n1, SD= .799, Mean= 3.631

& N2, SD= .907, M= 3.215

Bruce Weaver 'unequal variances version of the t-test' you mean Welch t-test but this is parametric test which carries the assumption of normality though it is robust for unequal sample distribution variance. My data violates the assumptions of normality and homogeneity, that's why WMM test was applied, inspite of the fact that the sample size is large.

Parametric test means you are estimating a parameter. It is not about the normality assumption. Given your sample size, it is not surprising that you can reject that the data were not drawn from an exactly normal distribution. Depending how anormal the distribution is, but don't you think the central limit theorem helps here? It is important to realize the WMW is testing something different than either of these t-tests (or doing a mean difference test with a bootstrap). Do you want to investigate a parameter for the difference in means?

Daniel Wright thanks!!!

I'll just two more points. First, you say you are interested in the impact of gender. Some folks would find this problematic. Here is an extract from my teaching notes:

According to Holland, one of Jerry Garcia's banjo pupils (really! see, Wainer, 2005, p. 160): "As an example, the schooling a student receives can be a cause, in our sense, of the student's performance on a test, whereas the student's race or gender cannot" Holland (1986, p. 946).

The Wainer quote is a book of his and the Holland quote his big paper on causation and statistics. Other people (e.g., Pearl) don't have this worry. I'm just throwing it out there (and checking if there are any banjo fans out there ... my notes continue with references to Reuben and Cherise and mandolins).

My second point is that if you are worried about the normality and heterogeniety issues, here is a bootstrap code in R for a couple of packages for this with some made up data (though is you use bootstrapping much a package like boot is recommended):

set.seed(300)

g1

Daniel Wright A big thank U!!! I have a question... i cant understand why there is unnecessary recommendations for converting the non normal data into normal, so as to use parametric tests. Why there is little focus on non parametric tests? Why cant we go with non normal data?

Without having read above replies, I want to share my proposal that the effect size should be AUC in this case.

To rephrase, we can certainly define AUC as the effect size and it seems to fit very well here.

Hi Sabiha Khatoon , I think the first two problems are cause by people often being taught that there are parametric tests that assume normal distributions of the residuals and then there are rank based tests that do not and these based test often do not estimate any parameters of the original unranked data. So, problem 1 is over-applying the phrase non-parametric. Second, is people not considering other alternatives. These alternatives include having the assumed distribution of the residuals something other than normal (like generalized linear models), using estimation procedures that lessen the impact of far outliers (robust regression), transformations to fit the data to be closer to the assumptions allowing the parameters to be back-transformed to the original units (e.g., the geometric mean), and bootstrapping. I view rank based procedures as a last resort because you lose information and cannot because the results in terms of the original units. This is why in my original comment I questioned if you want an effect size for a rank based test if you are conceptualizing things like investment habits in money spent. I would want the effect size in those monetary units. The MW could be used as a last resort for testing the hypothesis because the t-test assumptions are not meet, but you may still be interest in shifts in means.

Also, remember the assumptions are seldom meet, but may still yield useful results. Here is what I have in my teaching notes:

"But consider the statistician John Tukey's (\citeyear{Tukey1986}, p.~72) argument that researchers should reject the idea that ``all assumptions are right,'' and instead work with the idea that ``all assumptions are wrong.'' And then try to figure out how wrong these assumptions likely are and how much this affects the value of the information that statistical procedures extract. This may affect how much this information should be relied upon when making decisions. If there are alternatives that will make better use of the data without these errant assumptions, consider them. It is important to stress that with real-world problems assumptions are often necessary and the associated procedures can still yield useful information.

"

Glass rank bi-serial correlation coefficient (rg) is the appropriate method of obtaining effect size for Mann-Whitney U test. The formulae and interpretation for rg is given in the book of King, Rasopa and Minium (7th Ed) book. In fact effect sizes for all the major non-parametric statistics are given in this book. Pls use this book. Thanks

r = z/√N There you have!

Ghazi Shahnawaz Thank You Sir!!

hi. use G Power analysis and the mean scores of investment habit or buying habit for estimated effect size d.