[Rasch] Dimensionality and Correlation

MarkHMoulton at gmail.com MarkHMoulton at gmail.com
Tue Nov 11 11:45:14 EST 2008


Michael,

I had forgotten how relevant and useful an analytically derived  
hierarchical complexity construct can be in this context.

Mark

On Nov 10, 2008 3:30pm, Michael Lamport Commons <commons at tiac.net> wrote:
> That is why we always regress Rasch Score performance on the Items  
analytically determined order of hierarchical complexity, a truly  
independent variable. This insures that the scales are working. Still,  
development and learning are pushed by similar environments as twin studies  
show.
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> Michael Lamport Commons
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> Mark Moulton wrote:
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> Stephen,
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> Great response to Anthony's question!
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> The only thing I want to add is that dimensions tend to become  
uncorrelated to the degree the objects (eg, persons) are randomly drawn  
from the population of all possible objects., However, the objects studied  
in education and social science, and in science at large, tend to be  
sampled in highly non-random ways. When kids go to school they study Math  
and Language in tandem, causing the subject areas to have high correlations  
artifactually caused by the students having school in common. However, in a  
hypothetical society without school where people learn things at random,  
Math and Language scores would have a lower correlation.
>
>
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> This can bite us in Rasch fit analysis, as has been noted in previous  
discussions. Items may appear to fit very well (ie, be highly correlated)  
simply because all the students experience a similar curriculum. When a  
subsample of students experiences a different kind of curriculum, the same  
items may end up with different relative difficulties. In these cases,  
Rasch is not giving us sample free statistics and we don't know it. The  
only remedy is either to constrain ourselves to students who have  
experienced a similar curriculum, or to use items whose difficulties are  
robust across different curricula.
>
> Mark Moulton
>
> Educational Data Systems
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> On Sat, Nov 8, 2008 at 12:46 AM, Stephen Humphry  
shumphry at cyllene.uwa.edu.au shumphry at cyllene.uwa.edu.au>> wrote:
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> Hi Anthony. In my view there's a lot of confusion surrounding this
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> in the social sciences.
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> Think of volume and mass. If the densities of a set of objects is
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> uniform (eg they're all made of uranium), then the correlation
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> between measurements of their mass and volume will be near 1.
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> Does that make mass and volume the same quantitative property?
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> Clearly not.
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> Suppose on the other hand, objects are made of a range of
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> substances -- styrofoam, rubbers, metals (even gases). Then
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> measurements of mass and volume will not be highly correlated and
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> may have a very low correlation.
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> It is possible to have a perfect correlation between two sets of
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> measurements yet for those measurements to be of different kinds
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> of quantities (dimensions).
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> So if two sets of measurements (i) are genuinely measurements and
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> (ii) have a low correlation, then they must be different kinds of
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> quantities (dimensions).
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> On the other hand, though, the fact two sets of measurements has a
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> high (even perfect) correlation is not sufficient to demonstrate
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> there is only one kind of quantity (kind of dimension). In my view
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> it's theoretically instructive to consider whether it is
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> conceivable that levels of two properties could be uncorrelated
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> among any group of individuals under any conditions. It's an
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> experimental question whether there are conditions under which two
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> things are not correlated. Also in my view, a lot of people try to
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> use correlations as a substitute for experiments that are required
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> to understand relationships between quantitative properties.
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> I also personally think the word unidimensionality is bit frought
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> with traps. Hope that helps.
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> Steve Humphry
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> Quoting Anthony James luckyantonio2003 at yahoo.com
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> luckyantonio2003 at yahoo.com>>:
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> Dear all,
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> I have difficulty understanding the difference between
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> dimensionality and correlation. I have seen several times in
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> the literature that people talk about correlated dimensions
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> and uncorrelated dimensions. I was always under the
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> impression that if two dimensions are correlated then they
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> are not two separate dimensions. They are one. But
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> apparently, this is not true and there can be two separate,
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> and at the same time, correlated dimensions. Is that right?
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> I'd be grateful for any comments on the relationship between
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> correlation and dimensionality. Apparantly corrlation doesn't
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> have much to do with unidimensionality.
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> Cheers
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> Anthony
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