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.
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