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

["JVerkuilen (Gmail)" <jvverkuilen@gmail.com>]

On Sat, Mar 9, 2013 at 8:34 AM, David Hoaglin <dchoaglin@gmail.com> wrote:

> It seems odd to use a two-sample test when you actually have only one
> sample.

100% agreed, as the sampling variability in the second sample is
illusory and costly.



> 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.
>

Yeah that's the problem. It seems to me that the KS test isn't really
reflecting the hypothesis because of its vague alternative. I think
the Mann-Whitney came from the notion of using an ROC type approach to
the problem, as per:

     Bamber, D. 1975. The area above the ordinal dominance graph and
the area below the receiver operating characteristic graph. J. Math.
Psych., 12, 387-415.

but I'm not sure.

Bamber shows that the ordinal dominance plot test of the area under
the curve is equivalent to the Mann-Whitney statistic, but that's for
a two sample problem. The reason I thought -rocgold- might be tricked
into giving the right answer is because in that sense the gold
standard ROC curve is the uniform here, and Teodora wants to know if
her data differ from it. ROC procedures also give confidence
intervals, which will likely be very helpful. I'm guessing that it's
time to consult a good text book (for which see the Stata
documentation as a starting point as they usually have several good
suggestions).



> 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.

That's certainly one method, as it would allow the use of qnorm.
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