Re: Repeated measures analysis of fractions summing to a constant

Posted by Rich Ulrich on Apr 05, 2013; 11:21pm
URL: http://spssx-discussion.165.s1.nabble.com/Repeated-measures-analysis-of-fractions-summing-to-a-constant-tp5719257p5719292.html

Okay.  I pointed out that there was a loss of d.f.  The cite from
G&D says the analysis is okay if you use the epsilon correction
for repeated measures.  Now, remember that the epsilon correction
makes use of a reduction of d.f. 

I haven't yet looked at what Bruce sent me, but I'm willing to accept
that the epsilon correction reduces the d.f. appropriately, either
100% of what is needed, or nearly 100%.  Epsilon corrections could
be large enough.

It has been a long time since I looked at the epsilon correction, but
I do remember that descriptions mentioned cells with zero or near-zero
for variances.  In my recollection, what was discussed were models
where zeroes were apt to be due to "basement" or "ceiling" scoring
effects.  The d.f. corrections were not always small, so I expect that
they could work here.  (If I were reporting the data, I would be careful
to report the epsilons as evidence that the model has accounted
properly for the d.f.)

--
Rich Ulrich

> Date: Fri, 5 Apr 2013 12:36:24 -0700

> From: [hidden email]
> Subject: Re: Repeated measures analysis of fractions summing to a constant
> To: [hidden email]
>
> 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:
>
> > kior@
>
> > Subject: Re: Repeated measures analysis of fractions summing to a constant
> > To:
>
> > SPSSX-L@.UGA
>
> >
> >
> >
> >
> >
> >
> >
> > 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.
>

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