[Rasch] Dimensionality and Correlation

Michael Lamport Commons commons at tiac.net
Wed Nov 12 03:43:53 EST 2008

I thought I had answered Theo good point when I pointed out how common 
environments lead to very similar IQ.  Clearly, similar environments 
will cause correlations.  That is why family estimates of the 
heritability of IQ are better than identical twin estimates (.77).  
Identical twins get treated too similarly.  But there is something to 
regressing Rasch Stage scores on order of hierarchical complexity.  We 
find that within a test, the r is slightly higher than the r is if all 
the tests are thrown into a single Rasch, math/logic/physics/chemistry.  
I can send a presentation to whomever would like it.  As the domain gets 
fuzzier, the r drops much more, such as in social/moral domain.  That 
does not mean that there is nothing called overall stage, which shows up 
in Factor Analysis.  In fact, in the math/logic/physics/chemistry 
domain, the loadings are as high as .95 and only the first factor 
matters at all.  But what it does say is that content also contributes 
to Rasch scaled scores along with some "general" stage term.  There 
should not be such high correlations when we test adolescents and adults 
who do not have courses in all of math/logic/physics/chemistry.  So how 
context contributes should be an issue.  Also common experience would 
contribute to there being a general factor as Theo points out.

So to sum it up, Rasch performance, although appearing to yield a single 
dimension, always reflects multiple factors.  Sometimes the single 
dimension contributes huge amounts as shown by the r's of .95 between 
Rasch scores and orders of hierarchical complexity.  But we also showed 
small contributions by other variables affecting difficulty. number 
size, number of calculation place in order within a stimulus. raising 
the r to .991 from ..982  Lots of things affect performance as we all 

So we need to pay attention to whether or not we are interested in the 
items or the participants.  I have focused on the items and the scales 
they generate.  Then hierarchical complexity helps a lot.  It gives real 
validity to a scale.  When it comes to participants, there are all sorts 
of things going on as Theo points out.

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