[Rasch] Dimensionality and Correlation

Michael Lamport Commons commons at tiac.net
Sat Sep 12 04:40:50 EST 2009

This is a two way street.  Education also is the best predictor of stage.


Trevor Bond 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 (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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> -- 
> Professor Trevor G BOND Ph D
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> Faculty of Education Studies & Graduate Programmes Office
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