[Rasch] Dimensionality and Correlation

Trevor Bond trevor.bond at jcu.edu.au
Tue Nov 11 11:20:02 EST 2008


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 (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 
><<mailto:shumphry at cyllene.uwa.edu.au>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 
><<mailto:luckyantonio2003 at yahoo.com>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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