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AI-GEOSTATS: In need of some help.

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  • mercury1@earthnet.net
    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
    Message 1 of 7 , Dec 19, 2000
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      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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    • Eric LEWIN
      ... My thinking is that, on one hand, there is no theoretical lower limit to the minimum number of data, except the evident at least one ! (or two if some
      Message 2 of 7 , Dec 20, 2000
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        > 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 ?

        My thinking is that, on one hand, there is no theoretical lower limit to
        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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      • Hugo PILKINGTON
        You could probably try using the Mann-Whitney U test which is a distribution-free (nonparametric) statistical test (I assume you re comparing two means). I m
        Message 3 of 7 , Dec 20, 2000
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          You could probably try using the Mann-Whitney U test which is a
          distribution-free (nonparametric) statistical test (I assume you're
          comparing two means). I'm not to sure about your very small sample sizes,
          but it should be all right.

          Regards.

          Hugo Pilkington
          ____________________________________________________________
          EuroHIV - European Centre for the Epidemiological Monitoring of AIDS
          Institut de Veille Sanitaire (InVS)
          12, rue du Val d'Osne
          94415 Saint-Maurice cedex
          France

          h.pilkington@...
          Tel: +33 (0)141 79 68 68 http://www.ceses.org
          Fax: +33 (0)141 79 68 02 http://www.invs.sante.fr



          -----Message d'origine-----
          De: mercury1@... [SMTP:mercury1@...]
          Date: mardi 19 décembre 2000 18:08
          À: ai-geostats@...
          Objet: 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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        • Yong Wang
          Mann-Whitney U test does require that the distributions of two populations are similar. It is distribution free but not assumption free. Resampling (or
          Message 4 of 7 , Dec 20, 2000
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            Mann-Whitney U test does require that the distributions of two populations
            are similar. It is distribution free but not assumption free.

            Resampling (or randomization) methods should work. Basically, you will mix
            the two samples, and then randomly separate them into two groups, and
            calculate the parameter you are interested (e.g., average difference);
            repeated the process for a large number of times. You will get a
            distribution of the parameter, compare the parameter estimate from original
            data with this distribution to see how likely you would be able to get the
            original value by chance.

            You can run this through Excel by set up a macro. Or check
            www.resample.com, there is information about the methods and software.

            Best,

            Yong Wang





            At 10:08 AM 12/19/00 -0700, mercury1@... wrote:
            >
            >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
            >
            >--
            >* To post a message to the list, send it to ai-geostats@...
            >* As a general service to the users, please remember to post a summary of
            any useful responses to your questions.
            >* To unsubscribe, send an email to majordomo@... with no subject and
            "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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          • J-F Lenain - L.A.S.E.H.
            You can use nonparametric tests like bootstrap or permutation (randomization) tests: they give better results with nonnormal and small samples. they can be
            Message 5 of 7 , Dec 20, 2000
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              You can use nonparametric tests like bootstrap or permutation
              (randomization) tests: they give better results with nonnormal and small
              samples. they can be used for univariate or multivariate tests.

              The permutation test approximate the distribution of a statistic under the
              H0 hypothesis (no difference):
              - Enumerate all the permutations between the 2 samples (if the samples are
              too large, the permutations are randomly sampled)
              - Evaluate a statistic ( T, F, Manahalobis, nonparametric...) for each
              permutation. The statistic is only used as a distance measure.
              - Order the simulated values to get the permutation distribution, and
              search the position of the true sample.

              The test is significant if
              - the true value is among the 2.5% values at each end (two side test)
              - it is among the 5% upper [lower] values (one side test)

              I hope this may help you
              jf


              -------------------------------------------------------------
              jean-francois LENAIN L.A.S.E.H.
              faculte des Sciences
              e-mail: lenain@... 87060 Limoges CEDEX (France)
              -------------------------------------------------------------


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            • Tom Nolan
              Harland, 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 details are
              Message 6 of 7 , Dec 28, 2000
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                Harland,

                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 details
                are shown on p. 120 of "Statistical Methods in Water Resources," Helsel and
                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
                >
                > --
                > * To post a message to the list, send it to ai-geostats@...
                > * As a general service to the users, please remember to post a
                > summary of any useful responses to your questions.
                > * To unsubscribe, send an email to majordomo@... with no
                > subject and "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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              • Syed Abdul Rahman Shibli
                With five to six samples per population, concluding anything from the tests would really be pushing it. Complementing the results with any deterministic
                Message 7 of 7 , Dec 28, 2000
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                  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,
                  >
                  >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
                  details
                  >are shown on p. 120 of "Statistical Methods in Water Resources," Helsel and
                  >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
                  >>
                  >> --
                  >> * To post a message to the list, send it to ai-geostats@...
                  >> * As a general service to the users, please remember to post a
                  >> summary of any useful responses to your questions.
                  >> * To unsubscribe, send an email to majordomo@... with no
                  >> subject and "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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