# [R] R-help Digest, Vol 217, Issue 20

John Maindonald john@m@|ndon@|d @end|ng |rom @nu@edu@@u
Sat Mar 20 21:02:41 CET 2021

```No, it is not distribution free.  Independent random sampling is assumed.
That is a non-trivial assumption, and one that is very often not true or not
strictly true.

John Maindonald             email: john.maindonald using anu.edu.au<mailto:john.maindonald using anu.edu.au>

On 21/03/2021, at 00:00, r-help-request using r-project.org<mailto:r-help-request using r-project.org> wrote:

From: Jiefei Wang <szwjf08 using gmail.com<mailto:szwjf08 using gmail.com>>
Subject: Re: [R] about a p-value < 2.2e-16
Date: 20 March 2021 at 04:41:33 NZDT
To: Spencer Graves <spencer.graves using effectivedefense.org<mailto:spencer.graves using effectivedefense.org>>
Cc: Bogdan Tanasa <tanasa using gmail.com<mailto:tanasa using gmail.com>>, Vivek Das <vd4mmind using gmail.com<mailto:vd4mmind using gmail.com>>, r-help <r-help using r-project.org<mailto:r-help using r-project.org>>

Hi Spencer,

Thanks for your test results, I do not know the answer as I haven't
used wilcox.test for many years. I do not know if it is possible to compute
the exact distribution of the Wilcoxon rank sum statistic, but I think it
is very likely, as the document of `Wilcoxon` says:

This distribution is obtained as follows. Let x and y be two random,
independent samples of size m and n. Then the Wilcoxon rank sum statistic
is the number of all pairs (x[i], y[j]) for which y[j] is not greater than
x[i]. This statistic takes values between 0 and m * n, and its mean and
variance are m * n / 2 and m * n * (m + n + 1) / 12, respectively.

As a nice feature of the non-parametric statistic, it is usually
distribution-free so you can pick any distribution you like to compute the
same statistic. I wonder if this is the case, but I might be wrong.

Cheers,
Jiefei

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