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Re: st: Using Wilcoxon rank-sum (Mann-Whitney) test to compare an emipirical and a uniform distribution


From   Nick Cox <njcoxstata@gmail.com>
To   "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Subject   Re: st: Using Wilcoxon rank-sum (Mann-Whitney) test to compare an emipirical and a uniform distribution
Date   Sat, 9 Mar 2013 17:31:49 +0000

In addition to other suggestions, note that -quantile- is already a built-in graphical test of a uniform distribution.

Nick


On 9 Mar 2013, at 13:34, David Hoaglin <dchoaglin@gmail.com> wrote:

Teodora,

It seems odd to use a two-sample test when you actually have only one
sample.  What was the basis for the advice to use the
Wilcoxon-Mann-Whitney test?

A one-sided KS test would be all right.  Some people might be more
comfortable making the test two-sided, unless you would not have any
interest in a situation where the data departed from the null
hypothesis in the other direction.  I don't know what the literature
says about whether any other test has greater power for the type of
alternative that you are interested in.

With only 15 observations, the departure would have to be substantial
to reject the uniform null hypothesis.

I have an offbeat suggestion.  Transform the sample to normal deviates
by applying the inverse of the standard normal cumulative distribution
function to each observation, and test whether the transformed sample
departs from the standard normal distribution.  You can also make a
normal probability plot of the transformed sample.

What do you mean by "a constant markup of 1/14"?

David Hoaglin

On Thu, Mar 7, 2013 at 3:11 PM, Tsankova, Teodora <TsankovT@ebrd.com> wrote:
Some time ago I posted on statlist with a question regarding the use of a one-sided KS test and I was advised that for my purpose I can use the Wilcoxon-Mann-Whitney test (ranksum command in Stata).

I basically have 15 observations that go from 0 to 1 and constitute my empirical distribution and I want to prove that those take higher values than a uniform distribution would suggest. I have three questions related to the test:

1) I generate myself 15 more observation which take values from 0 to 1 with a constant markup of 1/14 (I simulate a uniform distribution of 15 variables in the same interval). Has anyone else used this method for creating uniform distribution and do you see any problems with it?

2) I use the ponder option to compute the p-value for the one sided test and I get the following output:

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

ObservedOr~m |      obs    rank sum    expected
-------------+---------------------------------
   Observed |       15         236       232.5
    Uniform |       15         229       232.5
-------------+---------------------------------
   combined |       30         465         465

unadjusted variance      581.25
adjustment for ties        0.00
                    ----------
adjusted variance        581.25

Ho: ktaub_~m(Observ~m==Observed) = ktaub_~m(Observ~m==Uniform)
            z =   0.145
   Prob > |z| =   0.8846

P{ktaub_~m(Observ~m==Observed) > ktaub_~m(Observ~m==Uniform)} = 0.516
999996
(15 real changes made)
(0 real changes made)
(0 real changes made)

I would interpret it in the following way: In 51.6% of the cases you would draw a random number from Observed that would be higher than a random draw from Uniform. Is this the correct interpretation?

3) My last question is related to the fact that Wilcoxon Mann- Whitney test is used to analyse ordinal data. My data has an ordinal meaning in the sense higher values represent more homogenous group lending villages in my case. However, the values the variable takes are not interval but continuous ones. Can I still use this test?

Thank you,

Teodora

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