[Rasch] Dimensionality and Correlation

Stephen Humphry shumphry at cyllene.uwa.edu.au
Sat Nov 8 19:46:58 EST 2008

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  

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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