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Topic: kolmogov-smirnov, wilcoxon and kruskal tests
Replies: 14   Last Post: Dec 31, 2012 6:38 PM

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Herman Rubin

Posts: 399
Registered: 2/4/10
Re: kolmogov-smirnov, wilcoxon and kruskal tests
Posted: Dec 31, 2012 1:54 PM
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On 2012-12-29, Rich Ulrich <> wrote:
> On Sat, 29 Dec 2012 12:21:31 -0800 (PST), illywhacker
><> wrote:

>>On Sunday, 23 December 2012 01:22:31 UTC, wrote:
>>> Hi Everybody,

>>> I'm trying to do some hypotheses testing in R and I have problems
with interpretation of results.

>>> I have two data sets:

>>> > x<-c(2,1,1,2,2,3,2,2,1,2,2,4,1,2,4,1,1,5,2,1,4,2,2,1,1,2,2,1)

>>> > y<-c(2,2,1,1,2,2,1,4,1,4,2,4,4,4,3,2,4,4,3,2,4,5)

>>> according to the KS test they come from the same distribution:

>>> > ks.test(x,y)

>>> If they come from the same distribution all the characteristics
(mean, median, ... ) should be the same.

>>> However, Wicoxon and Kruskal tests indicate that their null
hypothesis should be rejected:

>>> > wilcox.test(x,y)

>>> > kruskal.test(list(x,y))

>>> Now, I am puzzled with the outcome of the test.

>>> I can simply imagine a situation when Wilcox and Kruskal tests
indicate that their null hypothesis should be accepted but the KS test
can indicate that samples comes from different distributions. Here, it is
the other way round. Does any one has some hints what causes the problems?

> By the way, OP -- WHY could you imagine, at the start,
> without knowing their properties, that the KS test would
> have more power than the other tests for testing a shift?
> - The Wilcox and Kruskal tests do assume that distributions
> have a similar form, for those tests to be valid.
> - That is what the KS tests, if you follow an explicit assumption
> that the distributions are "of the same form", so you are not
> testing for shape.

The Bayes risk efficiency, which is also the "zero" Pitman
efficiency, for the KS test for shift is equivalent to testing the
quanitle for which the density is maximized.

See my paper with Sethuraman in Sankhya 1965, and also my
paper in the Sixth Berkeley Symposium (1970).

>>This just shows the total mess (meaningless concepts, ad hoc tests,
unclarified assumptions, contradictory results with no explanation,
"I would be willing to use...", confusion on the part of the user) that
arises from the nonsensical nature of classical statistical hypothesis
testing. The best advice here is: learn Bayesian methods.

> Are you saying that Bayesian methods are so limited and narrow that
> you, the Bayesian, cannot apply alternate assumptions and tests? Or
> get confused by conflicting results? I've stayed away from Baysian
> because it seemed more confusing, not less.

One CAN apply these Bayesian ideas if one does not insist on
using "optimal" Bayes procedures. One can do this by comparing
the prior Bayes risks, as is done in the above cited papers,
for the various procedures. One can also look at asymptotic
optimality in many cases in which the Bayes procedure cannot be
explicity computed. This "prior Bayes" approach is what can be
used in very many situation, although it is not always easy.

> By the way, Ray showed that the KS test does reject these data, too,
> despite the different test properties.

This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University Phone: (765)494-6054 FAX: (765)494-0558

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