Hello Dear All
I have a SPSS data set of pre and post treatment of a group of patient and the data are not-normally distributed, for that I used (mann whittey U test) for p-value and for effect size (r) but the are result is (-) and I want to ask.
1- is the (r) is appropriate for non-parametric test effect size?
2- what dose (negative r) mean?
3- and is there any other effect size can I use for (mann whittey U test)?
4- and what if the (r) result was bigger than (0.5) which is (large)?
and I will attach the SPSS data set file, hopefully I could find any help.
regards
r is calculated from the z-statistic, which is calculated from the U-statistic. U is neccesarily positive, as it is a standardized sum of (positive) ranks. z is the standardized difference between the observed U and the U expected under H0. If U is smaller that the expected U under H0, this difference is negative, and thus the z-value can be negative. r is the z-value divided by the sqrare-root of the total sample size. If z is negative, r will be negative, too. However, there are also formulas that take the absolute of this ratio, like the positive square-root from (z²/n). Following this definition of r, r must be positive.
Your data:
n1 = n2 = 38
rankSum1 = 741
rankSum2 = 2185
U(obs.) = n1*n2 + (n2*(n2+1))/2 - rankSum2 = 0
U(exp.) = n1*n2 / 2 = 722
SE(U) = sqrt(n1*n2*(n1+n2+1)/12) = 96.3
z = (U(obs.) - U(exp.)) / SE(U) = (0 - 722) / 96.3 = -7.5
r = z/sqrt(n1+n2) = -7.5 / sqrt(38+38) = -0.86
r = +sqrt(z²/n) =|z/sqrt(n1+n2)| = |-7.5 / sqrt(38+38)| = 0.86
That brings me to your concrete questions, which I like to answer with a question:
What does r actually tell you? What is the practical implication of this effect size measure? From my point-of-view, this is difficult if not impossible to interpret in any practical way. There is a better approach to judge an "effect" in rank tests, which is the probability that of two randomly selected values from each distribution, the value in group B will be larger (or smaller) than than that from group A. However, I am not sure if this interpretation makes sense if there is more than just a shift between the distributions (that is, the groups show different variance and/or different skew).
This paper from @Ronán M. Conroy is a valuable read:
https://www.stata-journal.com/sjpdf.html?articlenum=st0253
Dear Jochen Wilhelm you done a great job, thank you so much, but I did not understand how to interpret the effect size, because my supervisor insist to interpret effect size.
regards
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