[Rasch] Unidimensionality
Mark Moulton
MarkM at eddata.com
Wed May 14 08:48:45 EST 2008
Stephen et. al.,
"Physical pace is actually three-dimensional, but nothing else has been
shown to be actually multidimensional in this sense..."
I take a VERY different position. I claim that:
a) There exists an objective, verifiable psychological or mental space
that has the same geometric attributes as physical space, except that it is
not limited to three dimensions;
b) Items and persons can be located in this space as vectors, and data
can be modeled as the multiplication of these vectors;
c) While MDS and other well-known multidimensional models are not yet
"objective" in the Rasch sense, this is a limitation of the methodologies,
not of the geometrical paradigm they attempt to realize.
d) Psychological data and entities are not automatically
multidimensional. They need to be subjected to a measurement model that
forces a common multidimensional metric and flags departures from geometric
assumptions, in a way precisely analogous to Rasch.
e) The distinction between "spatial dimensions" and "dimensions
characterized by different types of quantities or units" is artificial. All
dimensions are different types of quantities (even the three spatial
dimensions) and every different type of quantity that is not a linear
combination of other types of quantities can legitimately be called a
dimension. A common metric is enforced by requiring all data points, in a
multidimensional data set, to be of the same type. Thus, the
correct/incorrect distinction can be used to model educational data sets
that comprise both Math and Language items that erect a demonstrable
two-dimensional space.
The reason for my position is theoretic and pragmatic. I developed several
successful multidimensional algorithms (e.g., see www.eddata.com/resources
<http://www.eddata.com/resources> ), and they rely completely on the
ability to apply the axioms and spatial assumptions of classical geometry to
non-physical spaces. Other multidimensional models, e.g., Singular Value
Decomposition, do as well. And the key point is they work. They make
successful predictions about the world. They are the top algorithms, for
instance, in the Netflix contest for predicting movie ratings.
Such observations make it impossible for me to credit such statements as,
"there is no firm basis for claiming traits are 'multidimensional' in either
sense in psychology, education and so on..." We may or may not succeed in
accurately modeling psychological multidimensionality, but genuine
psychological prediction is nearly impossible without it. Yet it does
occur. Google would collapse without it.
Mark Moulton
Educational Data Systems
-----Original Message-----
From: Stephen Humphry [mailto:stephen.humphry at uwa.edu.au]
Sent: Tuesday, May 13, 2008 12:29 AM
To: rasch at acer.edu.au
Subject: RE: [Rasch] Unidimensionality
Andrew, thanks for the information about that work on multidimensional
scaling, that is interesting.
On the second sense of 'multidimensional', which as you rightly say relates
to a kind of decomposition of dimensions into others, you mention force and
electrical resistance. I would recommend to anyone who is dubious about the
relevance of the anlogy in physics to take even a cursory look at the
expression of electrical resistance, conductance, pressure and so forth in
terms of base SI units. See the last column in Table 3 at the following link
http://physics.nist.gov/cuu/Units/units.html
<blocked::http://physics.nist.gov/cuu/Units/units.html>
Each base unit is a unit of a different dimension; i.e. kind of quantity.
Even from a cursoy look it should be evident that dimensions like electrical
resistance can be expressed in terms of a number of physical dimensions
(length, mass, time etc), each of which has a SI base unit.
The expressions of dimensions in terms of others are not literally
algebraic, as aptly pointed out by Emerson (2008). It's possible to express
a large number (approx 100) of derived units in common use, in terms of just
seven base units, because "some physical measures are related to others by
laws" (Krantz et al, 1971, p. 455). Somtimes one dimension is related to two
others simply by definition, as in the case of density in terms of volume
and mass.
The nature of the relationships is important, however, whether the
relationships are commonly called laws or definitions.
As Andrew says (or at least I take him to be saying), without the
substantive theories regarding dimensions and their relationships, there is
no firm basis for claiming traits are 'multidimensional' in either sense in
psychology, education and so on. And as I said, in my view it is not even
clear in what sense the term is being used.
Steve
Dr Stephen Humphry
Graduate School of Education
University of Western Australia
35 Stirling Highway
CRAWLEY WA 6009
Mailbox M428
P: (08) 6488 7008
F: (08) 6488 1052
CRICOS Provider Code: 00126G
_____
From: rasch-bounces at acer.edu.au [mailto:rasch-bounces at acer.edu.au] On Behalf
Of Andrew Kyngdon
Sent: Tuesday, 13 May 2008 12:05 AM
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
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