[Rasch] Dimensionality and Correlation

Stephen Humphry shumphry at cyllene.uwa.edu.au
Fri Nov 14 14:22:33 EST 2008


Precisely Andrew. A complementary way of saying the same thing is that  
there may be a correlation between a quantitative property A and  
another quantitative property B that is produced in an experiment for  
a *fixed level* of a property C.

The structure of many physical definitions and laws implies, though,  
that in relevant cases, the measurement itself must be conjoint;  
whereas the description in terms of correlations presupposes the  
properties can be measured separately then correlated. As an example,  
you cannot measure the mass of an object without a force and,  
symmetrically, you cannot measure a force without using an object with  
mass. Acceleration, in some form, is the experimental outcome. Force  
and mass are measured conjointly -- so I agrue at least, and I  
maintain that Poincaré made this point in 'Science and Hypothesis'.

Steve

Quoting Andrew Kyngdon <akyngdon at lexile.com>:

> Anthony,
>
> Those were not my words, they were Steve Humphry's. I was merely   
> supporting them.
>
> Steve was making the point that in the social sciences, the   
> calculation of correlation coefficients has largely supplanted   
> experimental investigation of relationships between quantitative   
> attributes. There is nothing wrong in calculating correlations and   
> they may indeed reveal interesting things from time to time. But   
> they are limited.
>
> The investigation of "trade off" relations is one avenue through   
> which a relationship between two attributes can be ascertained. If   
> the behaviour of the trade offs is consistent with that entailed by   
> the theory of conjoint measurement, then a relationship is   
> discovered without any resort being made to correlation coefficients.
>
> Cheers,
>
> Andrew
>
> Dr Andrew Kyngdon
> Director of Pacific Rim Operations
> MetaMetrics, Inc.
> P.O. Box 754 North Sydney NSW 2059 Australia
> Phone: 0401768090 (Int. +61401768090)
> Email: akyngdon at lexile.com
> Web: www.lexile.com <http://www.lexile.com/>
> Lexile Teacher's Lounge: http://lexile-teachers.blogspot.com/
>
> ________________________________
>
> From: rasch-bounces at acer.edu.au on behalf of Anthony James
> Sent: Fri 14-Nov-08 12:53 AM
> To: rasch at acer.edu.au
> Subject: RE: [Rasch] Dimensionality and Correlation
>
>
>
>
> Andrew,
> Your reply is interesting and raises new questions. What are these
>  "experiments that are required to understand relationships between   
> quantitative properties"?
> Cheers
> Anthony
>
> --- On Tue, 11/11/08, Andrew Kyngdon <akyngdon at lexile.com> wrote:
>
>> From: Andrew Kyngdon <akyngdon at lexile.com>
>> Subject: RE: [Rasch] Dimensionality and Correlation
>> To: "Stephen Humphry" <shumphry at cyllene.uwa.edu.au>, rasch at acer.edu.au
>> Date: Tuesday, November 11, 2008, 4:12 PM
>> 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.
>>
>> Hear, hear!
>>
>> Dr Andrew Kyngdon
>> Director of Pacific Rim Operations
>> MetaMetrics, Inc.
>> P.O. Box 754 North Sydney NSW 2059 Australia
>> Phone: 0401768090 (Int. +61401768090)
>> Email: akyngdon at lexile.com
>> Web: www.lexile.com <http://www.lexile.com/>
>> Lexile Teacher's Lounge:
>> http://lexile-teachers.blogspot.com/
>>
>> ________________________________
>>
>> From: rasch-bounces at acer.edu.au on behalf of Stephen
>> Humphry
>> Sent: Sat 08-Nov-08 7:46 PM
>> To: rasch at acer.edu.au
>> Subject: Re: [Rasch] Dimensionality and Correlation
>>
>>
>>
>>
>> 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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