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The z-statistic is based on an approximation of the normal distribution.

When n samples are equal or greater than 25, it could be argued that a sample is asymptotically normal, and that a z-statistic can be used to report the Wilcoxon signed-rank test output.

For n < 25, where the summed rank value may not be approximated by the normal distribution, what would you report as the test statistic for the Wilcoxon signed-rank test?

To note: I am working in R and using the wilcox.test function, which reports the test statistic V.

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  • $\begingroup$ Is your question really about reporting, or about which one to use for testing in small samples? $\endgroup$
    – jbowman
    Dec 29, 2019 at 23:07
  • $\begingroup$ Thanks for your reply. To clarify - I'm not clear whether the z-statistic can be used to report a Wilcoxon signed-rank test with sample sizes less than 25. And, if it is not, which test statistic should be reported (in general), and why. $\endgroup$
    – emperorgoose
    Dec 30, 2019 at 11:47

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As long as you make it plain which statistic you use it shouldn't matter which of several equivalent statistics you use - including a standardized statistic. They're each giving the same information about the sample, and the exact p-value for each would be identical.

[Using a normal approximation for any of the versions of the statistic would give a slightly different p-value to the exact p-value, but this is a separate issue. The normal is pretty good down a fair bit smaller than n=25 - with a continuity correction.]

I'd suggest choosing to report whichever version of the statistic you expect that your intended audience will be most familiar with.

Unless there was a clear reason to do otherwise, I'd probably just report the statistic given by the package I was using (after specifying the statistic sufficiently clearly that it wasn't ambiguous).

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  • $\begingroup$ That's very helpful. I have a further question: is there a good justification for using the z-statistic for sample sizes as low as 10? (would reporting a t statistic instead of a z-statistic be better here?). $\endgroup$
    – emperorgoose
    Dec 30, 2019 at 12:22
  • $\begingroup$ Whichever sample size you consider doing it at, you're dealing with an approximation, one which is (overall) worse the smaller the sample size. There's no universal standard of "close enough"; I might think it's just fine at 10 and you might not. (Or vice-versa.) And we could both be 100% correct, since our purposes and tolerance for inexactness of significance level would differ. $\endgroup$
    – Glen_b
    Dec 30, 2019 at 14:29
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As a software note, the wilcox.test function in R does not return the Z value. If the sample size is less than 50 and there are no ties, by default the software computes the p value with an "exact" method, and doesn't compute a Z value at all. In other cases, the function computes the Z value but doesn't report it. 

A = c(1,3,5,7,9)
B = c(2,4,6,8,10)

wilcox.test(A, B, exact=TRUE)

   ### Wilcoxon rank sum test
   ###
   ### W = 10, p-value = 0.6905

wilcox.test(A, B, exact=FALSE, correct=FALSE)

   ### Wilcoxon rank sum test
   ### 
   ###  W = 10, p-value = 0.6015

If one were interested in the Z value, I know a couple of methods to extract it in R. One is to use the coin package. Another is to use the rcompanion package (with the caveat that I am the author of that package.)

if(!require(coin)){install.packages("coin")}
if(!require(rcompanion)){install.packages("rcompanion")}

Y = c(A, B)
Group = c(rep("A", length(A)), rep("B", length(B)))
Data=data.frame(Group, Y)

library(coin)

wilcox_test(Y ~ Group, data=Data)

   ### Asymptotic Wilcoxon-Mann-Whitney Test
   ###
   ### Z = -0.52223, p-value = 0.6015

library(rcompanion)

wilcoxonZ(A, B, exact=FALSE, correct=FALSE)

   ###      z 
   ### -0.522

wilcoxonZ(A,B, exact=TRUE)

   ###      z 
   ###     NA

The Z value itself doesn't give any more information than the p value does. However, sometimes Z / sqrt(N) is used as an effect size statistic, often called r.

-0.522 / sqrt((length(A) + length(B)))

   ###  -0.1650709

library(rcompanion)

wilcoxonR(x=Y, g=Group)

   ###      r 
   ### -0.165
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  • $\begingroup$ Thanks very much. I have been using z <- qnorm(w$p.value/2) to calculate the z statistic, where w is the output of the wilcoxon signed rank test. I have not come across many papers which report the wilcoxon.test V-statistic, which is why I am converting to Z. $\endgroup$
    – emperorgoose
    Dec 30, 2019 at 17:47
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    $\begingroup$ That's fine. Though note that z is often reported with a sign, where it is negative if the second group has higher values than the first. Likewise, if the effect size statistic Z / sqrt(N) is used, it is also often reported as signed. $\endgroup$
    – Sal Mangiafico
    Dec 30, 2019 at 17:52

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