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Re: AI-GEOSTATS: In need of some help.

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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 1 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 2 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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      • 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 3 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 4 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 5 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 6 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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