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

[Nick Cox <njcoxstata@gmail.com>]

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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