# [Rasch] Unidimensionality

Joe Albano joealbano2 at earthlink.net
Tue May 13 07:28:56 EST 2008

```Pardon my ignorance, but I’m trying (1) to learn and (2) to apply some of
this to upcoming work. Does this all translate into a practical observation
that if fix statistics are reasonable (without entering into a debate about
what constitutes reasonable fit stats) and residuals are sufficiently low
(ibid) the measure can be considered unidimensional (even if, in “fact” the
measure characterizes multiple “dimensions” that co-vary in lock step)? Said
another way, can a measure be unidimensional, by Rasch standards, without
the requirement that the underlying phenomenon of interest be
unidimensional?

*IF* this is true I have another question – but I’ll wait to have this
attempt to test my knowledge shot down first, before asking an equally naïve
follow-up J

Best regards,

Joe.

Joe Albano

From: Andrew Kyngdon [mailto:akyngdon at lexile.com]
Sent: Monday, May 12, 2008 12:05 PM
To: rasch at acer.edu.au
Subject: [Rasch] Unidimensionality

To claim that ability or mental attributes are 'multidimensional'
potentially confuses the geometric representation of correlations with the
idea of an actual mental or psychological "space" in which things and
phenomena genuinely exist statically and/or dynamically. Physical pace is
actually three-dimensional, but nothing else has been shown to be actually
multidimensional in this sense
Otherwise, multidimensional might simply mean
test results depend on more than one attribute, each of which could in
principle be measured independently, with relationships contingent on other
factors. In the latter sense, most dimensions in physics could be called
'multidimensional'. The mass of an object depends on its volume and density,
but that doesn't mean it can't be measured.

Well done Steve, you hit the nail right on the head here. There exists
endemic confusion in the behavioural sciences as to what
“multidimensionality” is, and you correctly state that the term can be used
to describe different kinds of structures.

In the case of multidimensional scaling, psychological attributes are
assumed to be multidimensional metric spaces, in that the distance between
any two points in multidimensional space is positive, symmetric and
satisfies the triangle inequality (Beals, Krantz & Tversky, 1968).
Dissimilarities judgments often fail the triangle inequality (Tversky &
Gati, 1982) so it is unwise merely to assume that psychological attributes
form metric spaces. Also, when the Minkowski R metric (or “power” metric) is
used, the properties of interdimensional additivity and intradimensional
subtractivity often fail when more than just one dimension is involved
(Michell, 1990).

In physics and in polynomial conjoint measurement theory more generally,
single quantities can be decomposed into many other variables. Examples in
physics are density, force, and electrical resistance; and possible examples
in psychology are Hull’s (1952) and Spence’s (1956) theories of response
strength. Indeed, most quantities in physics are compositions of other
variables, yet they nonetheless are “unidimensional” and measureable.
Krantz, Luce, Suppes & Tversky (1971) detail several kinds of composition
rules, their attendant cancellation conditions and proofs.

However, as you imply, most behavioural scientists aren’t this specific in
their understanding of “multidimensionality”. Most prefer to entertain an
informal understanding and for guidance will solely rely on the results of
applying a “multidimensional” model to their data. Until they start
developing substantive theories of sufficient depth, there will be no firm
bases for claiming that this or that psychological system is
multidimensional.

Cheers,

Andrew

Andrew Kyngdon, PhD

Senior Research Scientist

MetaMetrics, Inc.

1000 Park Forty Plaza Drive

Durham NC 27713 USA

Tel. 1 919 354 3473

Fax. 1 919 547 3401

MetaMetrics' 2008 Lexile National Conference & Quantile Symposium
Successful Teachers, Successful Students
June 16-19  |  San Antonio Marriott Rivercenter
<http://www.lexile.com/conference2008> www.lexile.com/conference2008

-------------- next part --------------
An HTML attachment was scrubbed...
URL: https://mailinglist.acer.edu.au/pipermail/rasch/attachments/20080512/85dc04d2/attachment.html
```