[R] Genstat into R - Randomisation test

[Meyners, Michael, LAUSANNE, AppliedMathematics]

Robert, Tom, Peter and all, 

If I remember correctly (don't have my copy at hand right now),
Edgington and Onghena differentiate between randomization tests and
permutation tests along the following lines:

Randomization test: Apply only to randomized experiments, for which we
consider the theoretically possible re-randomizations given the
constraints for the original randomization. In this setting, inference
is valid for the observed population, and does not require random
sampling (while, of course, this assumption is needed if we want to
generalize the findings). I think they also call it re-randomization
tests at least once, while the prefix is usually omitted for simplicity,
but this makes it clearer that an initial randomization is required.

Permutation test: Can be applied even to data from non-randomized
experiments, requires, however, random sampling. Hence, one additional
assumption is needed (one that is common to inference using classical
tests). This is hardly addressed in Edgington & Onghena. 

In both cases, the "answer" can be approximated by random draws if full
randomizations/permutations are difficult or impossible to perform, but
this applies to both methods. The distinction in Edginton (1980) as
mentioned by Tom is not used in the latest edition. 

In this regard, the comparison with "exact vs. approximate integral"
seems inappropriate to me, as the distinction is conceptual, not
computational. Neither is one the (non-)exhaustive variant of the other.
I doubt that "randomization" test is named according to "_random_ choice
of permutations" as implied by Peter, it's rather based on
"randomization" (of an experiment) in its best statistical meaning. 

Not sure whether the distinction is needed, but it might be helpful in
some instances, at least if used consistently. I have never seen another
distinction between these two, and the literature is inconsistent in its
use as well.

Just my two cents worth.
Michael

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
On Behalf Of Robert A LaBudde
Sent: Donnerstag, 9. April 2009 17:38
To: Tom Backer Johnsen
Cc: r-help at r-project.org
Subject: Re: [R] Genstat into R - Randomisation test

At 04:43 AM 4/9/2009, Tom Backer Johnsen wrote:
>Peter Dalgaard wrote:
> > Mike Lawrence wrote:
> >> Looks like that code implements a non-exhaustive variant of the 
> >> randomization test, sometimes called a permutation test.
> >
> > Isn't it the other way around? (Permutation tests can be
> exhaustive by looking at all permutations, if a randomization test did

> that, then it wouldn't be random.)
>
>Eugene Edgington wrote an early book (1980) on this subject called 
>"Randomization tests", published by Dekker.  As far as I remember, he 
>differentiates between "Systematic permutation" tests where one looks 
>at all possible permutations.  This is of course prohibitive for 
>anything beyond trivially small samples.  For larger samples he uses 
>what he calls "Random permutations", where a random sample of the 
>possible permutations is used.
>
>Tom
>
>Peter Dalgaard wrote:
>>Mike Lawrence wrote:
>>>Looks like that code implements a non-exhaustive variant of the 
>>>randomization test, sometimes called a permutation test.
>>Isn't it the other way around? (Permutation tests can be exhaustive by

>>looking at all permutations, if a randomization test did that, then it

>>wouldn't be random.)

Edginton and Onghena make a similar distinction in their book, but I
think such a distinction is without merit.

Do we distinguish between "exact" definite integrals and "approximate"
ones obtained by numerical integration, of which Monte Carlo sampling is
just one class of algorithms? Don't we just say: 
"The integral was evaluated numerically by the [whatever] method to an
accuracy of [whatever], and the value was found to be [whatever]." 
Ditto for optimization problems.

A randomization test has one correct answer based upon theory. We are
simply trying to calculate that answer's value when it is difficult to
do so. Any approximate method that is used must be performed such that
the error of approximation is trivial with respect to the contemplated
use.

Doing Monte Carlo sampling to find an approximate answer to a
randomization test, or to an optimization problem, or to a bootstrap
distribution should be carried out with enough realizations to make sure
the residual error is trivial.

As Monte Carlo sampling is a "random" sampling-based approximate method.
The name does create confusion in terminology for "randomization" tests
for bootstrapping.

================================================================
Robert A. LaBudde, PhD, PAS, Dpl. ACAFS  e-mail: ral at lcfltd.com
Least Cost Formulations, Ltd.            URL: http://lcfltd.com/
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