Antw: Re: [Rasch] Dimensionality and Correlation
Moritz Heene
heene at psy.lmu.de
Wed Nov 12 09:12:03 EST 2008
As for me this discussion sounds very much like "to 'g' or not to 'g' ".
I agree with Stephen and just want to illustrate that point a bit more. Funnily enough, the question whether correlational methods can tell us anything about the structure of the mind was raised almost 100 years ago by Godfrey Thomson (s. reference bellow) in his sampling theory.
Let us say we have 50 items in ten tests and 700 uncorrelated but randomly (!) overlapping group factors (which he called the neural bonds) and 10 uncorrelated noise factors. Simulating this kind of data structure, running a factor analysis and extracting only one factor yields the following results:
Factor loadings:
0.509
0.796
0.634
0.446
0.602
0.919
0.621
0.568
0.640
0.836
And now the best: test of the hypothesis that one factor is sufficient: chi square = 28.84 df=35. p-value = .759.
So does correlation/correlational methods tell us anything about the "true" dimensionality? As Thomson puts it: "Hierarchical order [i.e., rank one which implies unidimensionality] will arise among correlation coefficients unless we take pains to suppress it. It does not point to the presence of a general factor, nor can it be the touchstone for any particular form of hypothesis, for it occurs even if we make only the negative assumption that we don't know how the correlations are caused, if we assume only that the connexions are random".
Of course, one might still argue that this travesty of "g" is the most parsimonious explanations of test intercorrelations but note this is not how -especially factor analytic results- are interpreted. A factor is usually interpreted as the "common cause". This implies that the latent variable is a constituent of natural reality and factor analysis is like a detector being able to detect this cause. Given the results presented above this sounds really ridiculous to me. Furthermore, think about the following: "If I make the discovery that the angles of a quadrilateral are equal in sums to four right angles, I may not conclude that it is a square" (Thomson, 1920). But this is exactly what happens in factor analysis and dimensionality analysis.
So it appears that factor analysis (and the correlational approach on which FA is based) is notoriously unreliable in detecting the true underlying structure. Note that I didn't say that FA is always wrong but in most of the cases we simply don't know. Perhaps sometimes FA reveals the true dimensionality, perhaps it fails to do so (see example above). So FA is simply a variable generator aggregating observed variables or, to be more precise, is the best linear combination of the common parts of variables.
Best,
Moritz.
References:
Thomson, G. H. (1916). A hierachy without a general factor. British Journal of Psychology, 8, 271-281.
Thomson, G. H. (1920). The general factor fallacy in psychology. British Journal of Psychology, 10, 319-326.
********************************************
Dr. Moritz Heene
Ludwig Maximilian University Munich
Department Psychology
Methodology and Psychological Assessment
Leopoldstrasse 13
80802 Munich, Germany
Tel.: +49 89 / 2180 5192
Fax: +49 89 / 2180 3000
Web: http://www.psy.lmu.de/pm/Mitarbeiter/Moritz-Heene.html
********************************************
>>> Stephen Humphry <shumphry at cyllene.uwa.edu.au> 08.11.08 9.48 Uhr >>>
Hi Anthony. In my view there's a lot of confusion surrounding this in
the social sciences.
Think of volume and mass. If the densities of a set of objects is
uniform (e.g. they're all made of uranium), then the correlation
between measurements of their mass and volume will be near 1.
Does that make mass and volume the same quantitative property? Clearly not.
Suppose on the other hand, objects are made of a range of substances
-- styrofoam, rubbers, metals (even gases). Then measurements of mass
and volume will not be highly correlated and may have a very low
correlation.
It is possible to have a perfect correlation between two sets of
measurements yet for those measurements to be of different kinds of
quantities (dimensions).
So if two sets of measurements (i) are genuinely measurements and (ii)
have a low correlation, then they must be different kinds of
quantities (dimensions).
On the other hand, though, the fact two sets of measurements has a
high (even perfect) correlation is not sufficient to demonstrate there
is only one kind of quantity (kind of dimension). In my view it's
theoretically instructive to consider whether it is conceivable that
levels of two properties could be uncorrelated among any group of
individuals under any conditions. It's an experimental question
whether there are conditions under which two things are not
correlated. Also in my view, a lot of people try to use correlations
as a substitute for experiments that are required to understand
relationships between quantitative properties.
I also personally think the word unidimensionality is bit frought with
traps. Hope that helps.
Steve Humphry
Quoting Anthony James <luckyantonio2003 at yahoo.com>:
> Dear all,
> I have difficulty understanding the difference between
> dimensionality and correlation. I have seen several times in the
> literature that people talk about correlated dimensions and
> uncorrelated dimensions. I was always under the impression that if
> two dimensions are correlated then they are not two separate
> dimensions. They are one. But apparently, this is not true and there
> can be two separate, and at the same time, correlated dimensions.
> Is that right? I'd be grateful for any comments on the relationship
> between correlation and dimensionality. Apparantly corrlation
> doesn't have much to do with unidimensionality.
> Cheers
> Anthony
>
>
>
>
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