http://spssx-discussion.165.s1.nabble.com/Repeated-measures-analysis-of-fractions-summing-to-a-constant-tp5719257p5719289.html
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.