> I need a way to compare two small populations (very small sample sizes...

My thinking is that, on one hand, there is no theoretical lower limit to

>5 and 6... both of which lack normality). I would like to compare them

>based on 3-5 parameters. Because of the above limitations I have given up

>on the validity of a t-test (which assumes a normal distribution and

>larger sample sizes). My basic question is this: are these two small

>populations statistically different or do they belong to the same

>population ?

the minimum number of data, except the evident "at least one" ! (or two if

some spreadth information is required). What changes with such very small

samples is the 2nd kind risk value, if calculable, which becomes high. Said

differently, to a 1st kind risk given, the decision criteron becomes so

"wide", that it has a very low power of discreminating between statistical

coherence, which is the question given, and bad luck coincidence of the

data (for instance, on one scales plate, five or six realisations of a

random variate defined between 0 and 1 and having a bimodal distribution

with modes at 0.25 ans 0.75, and on the other plate, not more much data

from a normal random variate of mean = 0.5 and standard deviation = 0.25 --

hue ! just a guess of a counter-example...).

On the other hand, practically, having no restriction on the class of

plausible probability laws implies the non-parametric test, which decision

intervals can not necessarily be calculated to a known precision. More

precisely, in the present case, the test I am thinking of, to compare two

samples for being from the same parent distribution with no other

assumption, is the Kolmogorov-Smirnov test, which is based on the

distribution of the maximum absolute difference between the two empirical

cumulative functions (CDFs), a distribution which pdf expression is only

known _assymptotical_ (as far as I have learned in stat books... ; more

other, the assymptotic function is an infinite serie, which may show in

some cases a poor numerical convergence -- but this is another story). By

assymptotic is meant that the approximation becomes more and more valid

when the sample size increases. However, no idea is given to the quality of

this approximation ! And in the present case, as the question relates to

very small samples, I have found nothing on the validity of this

criteria... So if some theoretical statistitian can confirm, or invalidate

and complete this, I'll be happy to learn more.

A last word : in case you (I mean, anyone on the list) is interested, I

have written a Matlab script (v.4.2) that does the job : asking for two

samples files, drawing these samples and their two associated empirical

CDFs, calculating the max difference, and evaluating the corresponding

probability according the (assymptotic) K-S law.

--E'ric Lewin

PS: I am not fully sure of the exact statistical english terminology (1st

or 2nd "kind risk", etc.); if I am wrong, thanks for correcting me.

+=[ Éric LEWIN <mailto:eric.lewin@...> Tél: (33/0)4 76 63 59 13 ]=+

+===[ LGCA (Labo. de Géodynamique des Chaînes Alpines), Grenoble (France) ]===+

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* Support to the list is provided at http://www.ai-geostats.org- With five to six samples per population, concluding

anything from the tests would really be pushing it.

Complementing the results with any deterministic

knowledge of the underlying population (genesis,

noteworthy features, prior experience, etc) could lend

some measure of validity to what you will eventually

conclude from such tests (i.e. do they make sense).

Unfortunately, doing that often leaves one in the

unsavory position of realizing that there is more

uncertainty than first thought of. Somewhat counter-intuitive,

but so true in my personal experience.

Syed

-----Original Message-----

From: Tom Nolan <btnolan@...>

To: <ai-geostats@...>

Date: Thursday, December 28, 2000 10:41 PM

Subject: RE: AI-GEOSTATS: In need of some help.

>Harland,

details

>

>You could use the exact form of the Wilcoxon Rank Sum test, which is

>appropriate for sample sizes of 10 or less per group. Computational

>are shown on p. 120 of "Statistical Methods in Water Resources," Helsel and

any useful responses to your questions.

>Hirsch, 1992, Elsevier. The test is commonly used to determine whether two

>groups are from the same population (i.e. have the same median and other

>percentiles), or alternatively whether the medians are different.

>

>Tom Nolan

>

>> -----Original Message-----

>> From: ai-geostats-list@... [mailto:ai-geostats-list@...]On

>> Behalf Of mercury1@...

>> Sent: Tuesday, December 19, 2000 12:08 PM

>> To: ai-geostats@...

>> Subject: AI-GEOSTATS: In need of some help.

>>

>>

>>

>> Hi Folks!

>> This is my first post to this list. Hope it is not out of place.

>> I need a

>> way to compare two small populations (very small sample sizes..5

>> and 6....both

>> of which lack normality). I would like to compare them based on

>> 3-5 parameters.

>> Because of the above limitations I have given up on the validity

>> of a t-test

>> (which assumes a normal distribution and larger sample sizes).

>> My basic question

>> is this: are these two small populations statistically different

>> or do they

>> belong to the same population? I have asked many elementary

>> level stats folks

>> and have not been entirely satisfied with their solutions. So, I

>> pose this

>> 'problem' to you.

>> Thanks for your time.

>> Happy Holidays!

>> -Harland

>>

>> --

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>> * As a general service to the users, please remember to post a

>> summary of any useful responses to your questions.

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>> requests to the list

>> * Support to the list is provided at http://www.ai-geostats.org

>

>

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"unsubscribe ai-geostats" followed by "end" on the next line in the message

body. DO NOT SEND Subscribe/Unsubscribe requests to the list>* Support to the list is provided at http://www.ai-geostats.org

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