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Is there any way to compare two or more than three using mann-whitney-wilcoxon test? that will also give Z and R values?

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  • $\begingroup$ It's not clear what you mean by "R values" $\endgroup$ – Glen_b Apr 17 '17 at 17:13
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The Wilcoxon-Mann-Whitney test is for two groups. If you want to compare locations of more than two groups with a similar test, then you would usually go to the Kruskal-Wallis test.

The same thing happens with the usual tests under the normality assumption -- for a location comparison (means in that case) on two samples you have a t-test but if you have more than two samples you'd use one-way ANOVA.

The Wilcoxon-Mann-Whitney test doesn't of itself provide Z values (the Mann-Whitney statistic is $U$, while the corresponding Wilcoxon statistic is usually denoted either $W$ or $T$); one can standardize $U$ and call that $Z$, of course (and in large samples it is indeed approximately normal). Some programs will produce such a $Z$ statistic by default, but it's not inherent to the test itself.

[As I mention in comments it's not clear what you mean by "R values". Is it intended to be the proportion of times an observation from sample 1 exceeds one from sample 2?]

When you move to more than two samples, there's no longer a single comparison being made; indeed, it's not even quite the same as performing all possible pairwise comparisons (it's possible with 3 or more groups to have A>B, B>C and C>A on pairwise tests, which the K-W is not sensitive to -- See [1]).

With the Kruskal-Wallis, the usual statistic is in large samples approximately distributed as chi-squared (in the same sense that the ANOVA is F by comparison with the t-statistic in the two sample case).

So a "Z" would not normally be available. You could compute Z's for pairwise comparisons of course, but as outlined above you might have a surprising outcome in that instance (where there is a cycle of differences on pairwise comparisons; you would normally not get a significant Kruskal-Wallis in that instance).

[1] Brown, B.M. and Hettmansperger, T.P. (2002),
"Kruskal–Wallis, Multiple Comparisons and Efron Dice,"
Australian & New Zealand Journal of Statistics, Vol.44 (4), p 427-438

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