[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.
> At 2:25 PM -0800 11/10/08, Mark Moulton wrote:
>> 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.
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