Date: Feb 2, 2013 11:22 AM
Author: Luis A. Afonso
Subject: Intra-permutations to test two different mean values
Intra-permutations to test two different mean values
Among a number of no-Parametric tests the Permutation Test is so prominent that his inventor, Ronald A. Fisher, named it as *exact*. In fact, from its own nature, if complete data permutations (CDP) are put under consideration, one cannot find a sound argument against its validity. if some conditions are met, namely, the two Populations are unimodal, have the same shape and supposedly differ only by their locations, i.e., are shifted by an unknown amount d. It´s clear that these demands leads to a practical drawback, such as the impossibility to test, by Fisher PT samples originated from two normal Populations with different variances. It is simple to see why: the data switching between samples does mix values with not equal dispersions and therefore each simulated sample, globally, have different features according to the not own included items. In this instance I propose that item permutations are only allowed concerned each sample (intra-permutations).
The Intra-Permutation Method. The arrival coefficients W( )
Let be X= X1, ?, Xm and chose at random and exhaustively all m without replacement, affecting each one by the index Wx(j)= j/(m*(m+1)/2 where j is the order the item is chosen. The same for Y= Y1, . . .,Yn, Wy(j)= j/(n*(n+1)/2. Noting that
_____mmX = E(Sum (Wx(j)*X(j))) = E(Xhat)
_____mmY = E(Sum (Wy(j)*Y(j))) = E(Yhat)
_______Xhat= (X1 + ?+ Xn)/m
_______Yhat= (Y1 + ? +Ym)/n
we view mmX - mmY as an element that can sample the r.v. D = E(X) - E(Y). So, contrarily to the Fisher´s Test, which gives evidence if the two Populations have similar mean values (H0 condition, unrestricted mixing) this intra-permutation method provides a way to see how the difference on Population means are not distinct from the source samples difference Xhat - Yhat. This is a fundamental difference that could surprise, as it happens to me, a careless person: an example of two similar statistics from different universes, if is allowed to say so. . .
Luis A. Afonso