[Rasch] Dimensionality and Correlation

MarkHMoulton at gmail.com MarkHMoulton at gmail.com
Tue Nov 11 11:42:11 EST 2008


Trevor,

Point duly noted and accepted. My observation was along the lines that the  
(relatively) high correlation between Math and Language, and the high  
correlations between many mental and social constructs, can be an artifact  
of how we sample the persons. The sampling constraints of educational data  
especially, are highly non-random and may lead to artificially high  
correlations even in traits that have little to do with each other, such as  
physical development and standardized test performance. That said, Math and  
Language, being related to a common cognitive development construct, are  
bound to have significant non-zero correlations regardless of how we sample.

Mark

On Nov 10, 2008 4:20pm, Trevor Bond <trevor.bond at jcu.edu.au> wrote:
>
>
> Well, Mark,
>
> That might be your view. But there is another alternative
> competing hypothesis: School achievement is driven by cognitive
> development- the necessary but not sufficient pre-cursor of all
> meaningful school learning.
>
> Tendentiously
>
> Trevor
>
>
>
>
>
>
>
> At 2:25 PM -0800 11/10/08, 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
> (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.
>
>
> 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
>
>
>
>
>
>
> On Sat, Nov 8, 2008 at 12:46 AM, Stephen
> Humphry 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 (eg 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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> --
>
> Professor Trevor G
> BOND Ph D
>
> Visiting
> Scholar
>
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