[Rasch] Dimensionality and Correlation

Michael Lamport Commons commons at tiac.net
Tue Nov 11 10:30:33 EST 2008


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.

Michael Lamport Commons

Mark Moulton wrote:
> Stephen,
>
> Great response to Anthony's question!
>
> The only thing I want to add is that dimensions tend to become 
> uncorrelated to the degree the objects (e.g., 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.
>
> This can bite us in Rasch fit analysis, as has been noted in previous 
> discussions.  Items may appear to fit very well (i.e., 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
>
>
>
> On Sat, Nov 8, 2008 at 12:46 AM, Stephen Humphry 
> <shumphry at cyllene.uwa.edu.au <mailto:shumphry at cyllene.uwa.edu.au>> wrote:
>
>
>     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
>     <mailto: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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