[Rasch] Dimensionality and Correlation

Mark Moulton markhmoulton at gmail.com
Tue Nov 11 09:25:26 EST 2008

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> 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>:
>  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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