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Nonparametric Equivalent

  • 1.  Nonparametric Equivalent

    Posted 01-04-2015 21:18
    This message has been cross posted to the following eGroups: Nonparametric Statistics Section and Statistical Consulting Section .
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    Hello and Happy New Year to you all.

    I'm writing to request advice regarding a project I am working on.  

    We are looking to determine if there are differences in Range of Motion scores between 4 groups of rabbits.  Data has been collected 8 weeks pre-operatively, as well as at 8 different time points post-operatively. I recognize this is a repeated measures design (4 x 9) but the sample sizes are quite small for the 4 groups.  (n1= 7, n2=7, n3=7, and n4=6) so I was planning on performing a nonparametric analysis, but cannot find the appropriate one to use in SPSS. The only nonparametric equivalent I'm aware of for a repeated measures ANOVA factorial design is Friedman's but it does not allow me to compare the 4 groups over the 9 time points. I considered computing an "averaged" range of motion score over the 8 post-operative days so as to compare only 2 time point values more efficiently (pre-op at 8 weeks versus averaged post-op days scores), but even that won't work with Friedman as I cannot include the "between groups" comparison component.

    If anyone can suggest the appropriate analysis using SPSS for this dataset, I would be most grateful.

    In advance, thank you for your counsel and discussion.

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    Elizabeth Oddone Paolucci
    Assistant Professor, University of Calgary
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  • 2.  RE: Nonparametric Equivalent

    Posted 01-05-2015 00:03
    As has already been posted, R is free and very powerful.  However, if you don't already know R, there is
    a learning curve.  One alternative is to rank-transform your 27 data points (ranking them with respect
    to the entire data set) and then apply the usual (parametric) ANOVA on the rank-transformed data.

    http://en.wikipedia.org/wiki/ANOVA_on_ranks
    (note the two Conover references in particular)

    >>Kathy

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    Katherine Godfrey
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  • 3.  RE: Nonparametric Equivalent

    Posted 01-05-2015 12:49
    I had a similar thought as Katherine's suggestion to rank transform the data and then perform the usual parametric (presumably repeated measures) anova, except I was going to suggest normal scores instead of rank transformation.

    For comparison, I would also carry out the usual parametric analysis on the raw data as well.
    Mike


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

    As has already been posted, R is free and very powerful.  However, if you don't already know R, there is
    a learning curve.  One alternative is to rank-transform your 27 data points (ranking them with respect
    to the entire data set) and then apply the usual (parametric) ANOVA on the rank-transformed data.

    http://en.wikipedia.org/wiki/ANOVA_on_ranks
    (note the two Conover references in particular)

    >>Kathy

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    Katherine Godfrey
    -------------------------------------------


    Original Message


  • 4.  RE: Nonparametric Equivalent

    Posted 01-05-2015 17:01
    Thank you to all who have already taken the time to reply to my question.  Excellent suggestions and I will look into each of them more closely. It feels great knowing I have some options.

    I welcome any further suggestions from the community and am most grateful to you all for your expertise.

    Sincerely,
    Elizabeth

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    Elizabeth Oddone Paolucci
    Assistant Professor & Associate Director of Office of Surgical Research
    University of Calgary
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    Original Message


  • 5.  RE: Nonparametric Equivalent

    Posted 01-05-2015 11:05
    It's a little data-reductive, but consider the following: Suppose that for each animal you had a single pre-op and a single post-op score, so that this was a pre-post design with four groups; it sounds like you've already considered doing something like this. If so, then you could compare the change scores for each animal, which means that the analysis no longer involves repeated measures. Nonparametric analysis can now proceed using the Kruskal-Wallis test.

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    John Dawson
    Assistant Professor
    Texas Tech University
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  • 6.  RE: Nonparametric Equivalent

    Posted 01-05-2015 21:04
    Hi, Elizabeth, in describing your project, you said "...the sample sizes are quite small...so I was planning on performing a nonparametric analysis...". Um, there is a common misperception out there that one is supposed to do nonparametric analysis when the sample sizes are small. But in fact, nothing could be further from the truth. As sample sizes decrease, nonparametric methods lose power faster than their parametric counterparts, precisely because the nonparametric methods do not make distributional assumptions while their parametric counterparts do. So, um, if small sample size was your motivation for planning to perform a nonparametric repeated-measures analysis, I would instead recommend doing the parametric equivalents...provided, of course, that distributional assumptions are not badly violated.

    If however, you have other reasons for wanting to non-parametric repeated-measures analysis, there is a book I can recommend. It is titled "Nonparametric Analysis of Longitudinal Data in Factorial Experiments", the authors are Edgar Brunner, Sebastian Domhof, and Frank Langer, and it was published in 2002 by Wiley & Sons, Inc. The authors are rather critical of Conover's original suggestion to do repeated-measures analysis on the ranks, and developed their method to be a superior alternative. I used their method recently in a longitudinal experiment where we had two groups of 3 people/group with 8 timepoints/person, where the responses over time strongly violated distributional assumptions. It worked quite well.

    In their book, Brunner, Domhof, and Langer have SAS code, and it is possible to download SAS macros. SAS subsequently incorporated their Wald ChiSquare and ANOVA ChiSquare into Proc Mixed as an option one can specify.  In addition, one of the authors (Brunner) has subsequently assisted in developing an R package for their method, the paper for which can be found at: http://www.jstatsoft.org/v50/i12/paper.


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    Eric Siegel
    Biostatistician
    Univ of Arkansas for Medical Sciences of Biostatistics
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  • 7.  RE: Nonparametric Equivalent

    Posted 01-21-2015 10:30
    Here is my belated reply to this thread:

    Eric is entirely right about the degradation of power of non-parametric statistics when the sample size is low, especially 7 or less for a 2-sample t-test. The conventional wisdom is entirely backwards.  It is in the small sample situation where assumptions such as normality or equal variance is valuable (assuming it is true -- the evidence for which can and should be looked at), as it helps with structure.  One may think about it in a Bayesian way -- the prior is most impactiful when the sample size is small.  On the other hand, the normal scores test (Wilcoxon rank sum test with normal scores) asymptotically has the same power as the t-test whne the data are normal and has greater or equal power for every other distribution.  Moreover, it is MUCH more robust to unequal variances, insofar as one only needs that some monotnice transformation homogenizes the variance of the two groups for the normal scores test, while that transformation needs to be applied to the data in order for equal variance 2-sample t-test to be fully appropriate.

    Eric: thank you for helping to slay that analysis myth.



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    Michael Schell
    Senior Member
    Biostatistics and Bioinformatics Department
    Moffitt Cancer Center
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    Original Message


  • 8.  RE: Nonparametric Equivalent

    Posted 01-21-2015 13:36
    Unfortunately, when sample size is small it is very hard to check the assumptions. Only the grossest deviations will be detected by significance testing and graphics are likely to be inconclusive.

    Peter

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    Peter Flom
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    Original Message


  • 9.  RE: Nonparametric Equivalent

    Posted 01-21-2015 15:58
    I have found that model assumption methods can be used even for n=3 to good advanatge, in this way: use the Anderson-Darling method to assess non-normality (it is is PROC UNIVARIATE in SAS) for both raw and log-transformed data and look at the statistic rather than just the p-value.  One can often see that one of those two distributions of the data is much more normal-like than the other.  Then use the transformation with the more normal-like behavior.  With two groups, one does this separately by group (as the means might be different -- that's why you want to do the test in the first place), adds the scores together and uses the one with the smaller sum, as a lower value reflects a more Gaussian distribution.  If the results suggest that neither transformation is good, then switch to the normal scores (VW), as it is less discrete than the Wilcoxon rank sum test, or give up if both groups have sample sizse of 3 or less, as you can't get a statistically significant p-value.

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    Michael Schell
    Senior Member
    Biostatistics and Bioinformatics Department
    Moffitt Cancer Center
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    Original Message


  • 10.  RE: Nonparametric Equivalent

    Posted 01-06-2015 08:30
    To follow up on Eric's post.  Brunner's suggested nonparametric method for longitudinal ordinal data is well described in their book.  As Eric said, they illustrated their methodology using SAS, and there is now an R package (nparLD).  There is a paper demonstrating several worked examples with SAS, which may be of help:
    http://apsjournals.apsnet.org/doi/abs/10.1094/PHYTO.2004.94.1.33


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    Denis Shah
    Kansas State University
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  • 11.  RE: Nonparametric Equivalent

    Posted 01-06-2015 12:07
      |   view attached
    Dr. Brunner himself apparently saw my post, and contacted me, and asked me to pass along his information to the ASA Statistical Consulting Section. In particular, the book of his that I recommended yesterday is out of print, and Dr. Brunner wishes to recommend articles to take the book's place. In addition, he sent one of the articles he recommends (Brunner E, Puri ML 2001), which I'm hoping I've successfully attached. Below is the text of of Dr. Brunner's email:

    From: Prof. Dr. Edgar Brunner [mailto:ebrunne1@gwdg.de]
    Sent: Tuesday, January 06, 2015 5:17 AM
    To: Siegel, Eric R; eoddone@ucalgary.ca
    Subject: Nonparametric Equivalent

    Dear Eric,

    many thanks for your really sufficient reply to Eilzabeth. Indeed, it is terrible, that Wikipedia still mentions this technique without referencing those papers which have described the topic correctly. Wikipedia does not even mention the papers by Akritas and Arnold (JASA, 1994) or by Akritas, Arnold and Brunner (JASA, 1997) or Akritas and Brunner (JSPI, 1997) where the reason why the rank transform does not work in the majority of the cases is clearly described.

    Unfortunately, our book is out of print and I think that Wiley does not intend to provide a reprint. Therefore, I would like to suggest to refer to the review paper of Madan Puri and me (Brunner, E and Puri M.L. (2001) Nonparametric methods in factorial designs. Statistical Papers, 42, 1-52.) where we give detailed reasons when the rank transform works and where it fails.
    This paper considers theoretical results as well as applications along with some SAS-statements to compute the statistics.

    For your information, I have attached this paper to this mail. I would be happy if you could communicate this idea and refer to that paper. It seems that still the idea of the ranks transform still haunts people's mind without the correct information on when it works and when it fails. In this paper, we suggest to replace the concept of the Rank Transform by the idea of the Rank Transform Property which requires several properties of a statistic.

    @Elizabeth: in case of any questions please do not hesitate to contact me. You may also ask Prof. Arne Bathke (University of Salzburg, Austria:  Arne.Bathke@sbg.ac.at) or Prof. Frank Konietschke (University of Texas at Dallas: fxk141230@utdallas.edu). In particular Frank can give you detailed information on the R-package nparLD to be downloaded from CRAN. Its usage has been described in a paper by Noguchi, Gel, Brunner and Konietschke (Journal of Statistical Software, 2012).

    At present we (Brunner, Bathke, Konietschke) are working on a book on "Rank Procedures in Factorial Designs - Using R and SAS" to appear with Springer.

    I would also appreciate if you could communicate this information to     Katherine Godfrey and to the members of the ASA-Statistical Consulting Section, in particular, if you could draw their attention to our review paper since the book is out of print.

    Kind regards,
    Edgar
     
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    Eric Siegel
    Biostatistician
    Univ of Arkansas for Medical Sciences of Biostatistics
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    Original Message


  • 12.  RE: Nonparametric Equivalent

    Posted 01-06-2015 12:33
      |   view attached
    My first attempt to attach the article failed. Here is a second attempt. If that also fails, try the following URL:
    http://link.springer.com/article/10.1007%2Fs003620000039#page-1

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    Eric Siegel
    Biostatistician
    Univ of Arkansas for Medical Sciences of Biostatistics
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    Original Message


  • 13.  RE: Nonparametric Equivalent

    Posted 01-07-2015 00:45
    Thanks for the Brunner & Puri references.  This may also be of interest:

    http://www.psychologie-aktuell.com/fileadmin/download/PschologyScience/3-2007/02_Hager.pdf

    As Eric noted, nonparametric testing can involve a tradeoff of power for robustness, and power is
    at a premium with small sample sizes. A plot of the data with timelines (the 9 connected points for each rabbit, and for
    the average of each group of rabbits) may tell you as much as an actual test.

    >>Kathy

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    Katherine Godfrey
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    Original Message