Re: Repeated measures analysis of fractions summing to a constant

Posted by Bruce Weaver on Apr 05, 2013; 7:36pm
URL: http://spssx-discussion.165.s1.nabble.com/Repeated-measures-analysis-of-fractions-summing-to-a-constant-tp5719257p5719291.html

The articles by Shaffer (1981) and Greer & Dunlap (1997) say there is no problem.  (I've sent both of them to Rich off-list.)  Meanwhile, here are some relevant excerpts from Greer & Dunlap.

"Periodically, researchers in the behavioral sciences analyze measures that are ipsative. Ipsative measures are those for which the mean for each level of one or more variables (usually the participants) equals the same constant. Data with these constraints are also referred to as allocated observations (Shaffer, 1981) and compositional data (when the scores are proportions; Aitchison, 1986)."  (p. 200)

"The general conclusion is clear: Repeated measures ANOVA with ipsative data works quite well.  Although it is known that techniques such as factor analysis are badly affected by ipsative scores, ANOVA is not, particularly if the epsilon correction for nonuniform variance-covariance matrices is used.  Fortunately, the epsilon correction for repeated measures ANOVA is readily obtainable from most major computer statistical packages.  Therefore, it is hoped that readers will no longer look with suspicion upon ANOVAs with ipsative data, even though the presence of sums of squares equal to zero is disconcerting." (p. 206)

Reference

Greer T, Dunlap WP. (1997). Analysis of variance with ipsative measures. Psychological Methods, 2(2), 200-207.

HTH.

Rich Ulrich-2 wrote

No, you are a bit wrong in concluding that there is no problem.

If you think of the situation of dummy variables, you have provided
an "extra" dummy, like entering dichotomies for both Male and Female.
There is redundancy.  There is over-parameterization.  There is,
somewhere, the loss of one d.f.  for RM when you perform any analysis.  
A "fixed" zero-effect is not the same as a randomly occurring near-zero-effect.

You retain full information (in the statistical sense) if you set up your
model to leave out one of the categories, just as one would for any
dummy coding.  The others will be most "independent" if you omit the
category that has the greatest variance.  The drawback might lie in the
ease of interpreting your results.

--
Rich Ulrich


Date: Fri, 5 Apr 2013 19:36:04 +0400
From: [hidden email]
Subject: Re: Repeated measures analysis of fractions summing to a constant
To: [hidden email]


 

   
 
 
    Thank you for all your answers that came so far. I haven't read them
    carefully yet.

   

    But here is what meanwhile came to my own mind after a little
    meditation.

    It is very simple: I just thought that (PLEASE correct me if I'm
    mistaken!) that there is no problem at all. The constraint that
    repeated-measures sum to a constant within individuals *does not*
    refute using common RM-ANOVA model. If only ANOVA distributional and
    spericity assumptions hold, no need for GEE or other procedures
    arise at all.

   

    Let's have some data: between-subject grouping factor  GROUP and
    within-subject factor RM with 3 levels summing up to a constant
    (100).

   

       group      rm1
      rm2      rm3      sum

       

             1       50       30       20      100

             1       24       42       34      100

             1       34       16       50      100

             1       61       28       11      100

             1       46       46        8      100

             1       23       18       59      100

             2       55       22       23      100

             2       27       39       34      100

             2       44       36       20      100

             2       28       40       32      100

   

    Run usual Repeated-measures ANOVA:

   

      GLM rm1 rm2 rm3 BY group

        /WSFACTOR= rm 3

        /METHOD= SSTYPE(3)

        /WSDESIGN= rm

        /DESIGN= group.

   

    Summing up to a constant just means that upon collapsing the RM
    levels, all respondents appear to be the same: there exist no
    between-subject variation at all, or in other words, the "respondent
    ID" factor's effect is zero. Hence, in the table "Tests of
    Between-Subjects Effects" Error term is zero. Also, the effect of
    GROUP factor is zero too - of course, because the constant sum (100)
    in our data is the same for both groups 1 and 2.

   

    Now, - I'd ask you, - does these results invalid in any way? Do we
    say that ANOVA is misused when an error variation - which is left
    unxplained - is zero? I would not say it, and so RM-ANOVA *is* an
    appropriate method for fractions (i.e values summing up to a
    constant). If I'm wrong, please explain me why.