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st: RE: RE: Is there any test to examine disitribution overlap with Stata?


From   "Lachenbruch, Peter" <Peter.Lachenbruch@oregonstate.edu>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: RE: Is there any test to examine disitribution overlap with Stata?
Date   Tue, 3 Jun 2008 08:17:34 -0700

A few years ago, I looked at a way to show equivalence among distributions by fitting with a kernel density and conducting a test based on the distribution with the largest density.  This is essentially like a discriminant function - you pick the population with the highest density.  The error rate is the overlap of the two (or more) distributions.  It was a nice idea, but did not do as well as some other procedures based on parametric assumptions.  I think I would try to find a transformation to normality (won't be possible if there is clumping at a discrete value such as zero) and use a normal based procedure.

Lachenbruch, P. A., Rida, W. N., Kou, J.  (2004) "Lot Consistency as an Equivalence Problem" Journal of Biopharmaceutical Statistics 14(2) 275-290

Tony

Peter A. Lachenbruch
Department of Public Health
Oregon State University
Corvallis, OR 97330
Phone: 541-737-3832
FAX: 541-737-4001

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Verkuilen, Jay
Sent: Monday, June 02, 2008 3:14 PM
To: statalist@hsphsun2.harvard.edu
Subject: st: RE: Is there any test to examine disitribution overlap with Stata?

>>I have 3 correlated probability density functions (pdf), and would like to obtain a simple measure of similarity among them. Cohen´s U statistics do that, but they look at two pdfs at a time. There are also measures of intrinsic discrepancy, but I am not sure if they work with 3 pdf.. could not find any reference. Are you aware of an alternative test to do what I am looking for using Stata?<<<

I'm not entirely sure what you mean by correlated here. I don't understand a measure of one distance though. You have three points in a function space and thus have three distances, dAB, dAC, and dBC. That's just how it is. If you want to define another piont in the function space, e.g., the functional centroid, that's different.  

Anyway, there are many, many possible distance measures, as Nick noted. A natural metric is the max norm, used by the Kolmogorov-Smirnov test. So you could use KS for each. There are lots of other metrics on one dimensional function spaces, with the most obvious one being the Euclidean for functions, i.e.,

           /
d(f,g) = [ | {f(x) - g(x)}^2 dx ]^.5
           / 
           X


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