# R: st: R: the sign of z statatistic after -ranksum-? (flag: Stata 9/2 SE)

 From "Carlo Lazzaro" To Subject R: st: R: the sign of z statatistic after -ranksum-? (flag: Stata 9/2 SE) Date Sun, 25 Oct 2009 17:00:38 +0100

```Dear Rose,
The formula used in Stata (release 9/2 SE) is reported in:
[R] Stata Base Reference Manual. Volume 3. R-Z. Release 9: 1-5.

Kind Regards,
Carlo

-----Messaggio originale-----
Da: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] Per conto di gjhxmu@sina.com
Inviato: domenica 25 ottobre 2009 15.00
A: statalist
Oggetto: Re: st: R: the sign of z statatistic after -ranksum-? (flag: Stata
9/2 SE)

Dear Carlo and Martin,

thank you very much for your help, especially for Carlo's detailed
illustration.

Just as Carlo's guess, I followed the formula from textbooks as follows to
get the z-statistic.

Assuming the number for the two independent sample  is m and n respectively.

Firstly to get the U statistic,

U=w-1/2*k*(k+1) // where w is the rank sum for the bigger number and k is
the bigger number. If m equals n, k takes the number for the group first
appearing in the rank.

Then to get the Z statistic,

Z=(U-1/2*m*n)/sqrt(1/12*m*n*(m+n+1))

Following the formula, I think no matter which group is compared with which
group, the Z statistic is the same. I am sorry for not reading the concrete
formula used in the stata.

Thank you for your kind help.

Best regards,

Rose.

----- Original Message -----
From: Carlo Lazzaro <carlo.lazzaro@tin.it>
To: <statalist@hsphsun2.harvard.edu>
Subject: st: R: the sign of z statatistic after -ranksum-? (flag: Stata 9/2
SE)
Date: 2009-10-25 19:18:29

Dear Rose,
the following examples may shed light on the oddity you came across in
performing -ranksum-.

webuse fuel2,clear
ranksum mpg, by(treat)

Two-sample Wilcoxon rank-sum (Mann-Whitney) test

treat | obs rank sum expected
-------------+---------------------------------
0 | 12 128 150
1 | 12 172 150
-------------+---------------------------------
combined | 24 300 300

----------

Ho: mpg(treat==0) = mpg(treat==1)
z = -1.279
Prob > |z| = 0.2010

return list

scalars:
r(N_2) = 12
r(N_1) = 12
r(Var_a) = 295.9565217391304
r(z) = -1.278817949868369
r(sum_exp) = 150
r(sum_obs) = 128
r(group1) = 0

As you can see, after -return list-, Stata consider - r(sum_obs) = 128 -.

z can be obtained via the following formula (see also: Pagano M, Gauvreau K.
Principles of Biostatistics. 2nd edition. Brooks/Cole, 2000)

di (128-150)/295.96^.5
-1.2788104

When you decide to invert the samples rank-sum:

ranksum mpg, by(treat2)

Two-sample Wilcoxon rank-sum (Mann-Whitney) test

treat2 | obs rank sum expected
-------------+---------------------------------
0 | 12 172 150
1 | 12 128 150
-------------+---------------------------------
combined | 24 300 300

----------

Ho: mpg(treat2==0) = mpg(treat2==1)
z = 1.279
Prob > |z| = 0.2010

. return list

scalars:
r(N_2) = 12
r(N_1) = 12
r(Var_a) = 295.9565217391304
r(z) = 1.278817949868369
r(sum_exp) = 150
r(sum_obs) = 172
r(group1) = 0

As you can see after -return list-, Stata consider - r(sum_obs) = 172 -.

Again, z can be obtained via the following formula (see also: Pagano M,
Gauvreau K. Principles of Biostatistics. 2nd edition. Brooks/Cole, 2000)

di (172-150)/295.96^.5
1.2788104.

As Martin said, usually <the test is against a two-sided alternative, so the
sign hardly matters...>

However, your concern is probably driven by the awareness that some
textbooks on statistics report the following formula for calculating
Wilkoxon rank sum test (see again: Pagano M, Gauvreau K. Principles of
Biostatistics. 2nd edition. Brooks/Cole, 2000):

zw = (W-mw)/sw

Where the z-statistic is obtained by subtracting the mean of the ranks sum
from W (the smallest of the two rank sums)and ; in this way, as you stated
in your thread, <z statistic the result should be the same no matter which
group is compared with which group>.

HTH and Kind Regards,
Carlo
-----Messaggio originale-----
Da: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] Per conto di gjhxmu@sina.com
Inviato: domenica 25 ottobre 2009 4.29
A: statalist
Oggetto: st: the sign of z statatistic after -ranksum-?

Dear statalists,

I use -ranksum- in the stata to do the Wilcoxon rank-sum test, which is
also known as the Mann-Whitney two-sample statistic.

My typing is as followings,

webuse fuel2,clear
ranksum mpg, by(treat) // the z statistic equals -1.279
replace treat=treat==0
ranksum mpg, by(treat) // the z statistic equals 1.279

My doubt is why the sign of z statistic is opposite. As far as I know, in
the computation of z statistic the result should be the same no matter which
group is compared with which group.

Any help will be appreciated!

Best regards,

Rose.

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