# How to determine the effect size of a Wilcoxon rank-sum test in R?

I want to determine if there's a difference in mean p-values between two groups. In order to do this I perform a Wilcoxon's rank-sum test (the data is not normally distributed). So far, so good. Finally, I want to calculate the corresponding effect size. Unfortunately, R does not provide this. It also does not provide a z value with which the effect size can easily be calculated using: effect size = z / sqrt(N)

here is some sample R code:

a=rep(0:1,each=20)                            #grouping variable
b=c(rnorm(20, .03,.01), rnorm(20, .02, .009)) #vector of p-values

d=cbind(a,b)

test = wilcox.test(b ~ a, data = d)     #perform Wilcoxon rank-sum test
test


Does anybody know how to obtain the effect size?

• Welcome to SO. I flagged your question for migration to www.crossvalidated.com , as your question is more statistical than anything else. In short : Wilcoxon RANK test works with ranks, so I'm not sure about which effect size you're talking. Obviously it doesn't give you a z value, as that one is linked to parametric testing, not to non-parametric tests like Wilcoxon. Wilcoxon has to be interpreted in terms of location shift. Jun 14 '12 at 14:47
• The Wilcoxon rank sum test does not evaluate a difference in means: it tests for stochastic inequality. The proper measure of an effect size therefore is the probability that $B \gt A$, minus the null value of $1/2$.
– whuber
Jun 14 '12 at 15:07
• @whuber And how is this effect size calculated? From what I've read, it must be a number between 0 and 1 (or between -1 and 1?), but the most voted answer provides a different number (-6 in my case, even divided by sqrt(N) it's still < -1). Nov 13 '20 at 10:30
• Rodrigo, please see stats.stackexchange.com/questions/133077/… and stats.stackexchange.com/questions/342987/….
– whuber
Nov 13 '20 at 14:23

The estimator that corresponds to the Wilcoxon test is the Hodges-Lehmann estimator; it's returned by wilcox.test using the conf.int=TRUE option, under "difference in location".

> wilcox.test(b~a,data=d, conf.int=TRUE)

Wilcoxon rank sum test

data:  b by a
W = 355, p-value = 6.914e-06
alternative hypothesis: true location shift is not equal to 0
95 percent confidence interval:
0.008657301 0.021523993
sample estimates:
difference in location
0.01442617


For more on the Wilcoxon and the assumptions behind it, and what it actually tests for, and other nonparametric estimators, this document is (possibly) helpful: www.stat.umn.edu/geyer/old03/5102/notes/rank.pdf

• So what you say is that the effect size is wilcox.test(b~a,data=d, conf.int=TRUE)$estimate / sqrt(20), correct? – user14650 Jan 9 '15 at 13:29 Get the z for your formula by library(coin) mydf <- as.data.frame(d) wilcoxsign_test(b ~ a, data = mydf, distribution="exact")  and compute the effect size with your formula, setting N to 40 • I suspect this reply would be of higher interest if you could supply additional details on the reasons to use the coin::wilcoxsign_test R function. Also, are you referring to the OP formula,$\text{effect size} = z / \sqrt{N}\$?
– chl
Apr 4 '13 at 19:52
• The reason ti use coin::wilcoxsign_test is that it computes a z value. Apr 5 '13 at 9:43
• Another way to obtain z from standard wilcox.test is the algorithm Andy Fields implemented in his function rFromWilcox. And yes, I was referring to effect size = z / sqrt(N). Apr 5 '13 at 18:17
• Thanks for the clarification although I'm still unsure about that z-value being a proper measure of effect size. (+1)
– chl
Apr 5 '13 at 20:47