Where do Dimensions Come from?

Are "dimensions" facts of nature waiting to be discovered, or are they artifacts of our imagination waiting to be invented?

Physicists are in no doubt about how they think of the world about them: "It seems inevitable that we should speak in terms of some definite theoretical model of the world of experience. There appears, however, to be no meaning in supposing there to exist a unique final model that we are trying to discover. We construct a model, we do not discover it" (McCrea 1983, p. 211). The idea of length is a theoretical model for an attribute of an object. Length does not exist on its own in nature, we invented it because it suits our purposes.

The idea of length is operationalized by means of devices, such as rulers. Rulers are always imperfect representations of the idea of length. They are inaccurate and imprecise, but we use them because they are good enough for our purposes. Of course, every length measuring process must be regulated to insure that the resulting number fits with our idea of length. Bending, breaking or otherwise misapplying the ruler still produces numbers, but not numbers that fit our idea of length.

Length is apparent to us, because it is visible, but what about temperature? We want to think about heat in the same way we think about length, as linear quantities. But we don't see heat in this way. Consequently we convert heat to length, or length-like numbers, by thermometers. Now we can think about and manipulate temperature in just the same way that we do length. Representation of abstract ideas requires visualization: rulers meet our need to think in a well-controlled, uniform way.

Educational tests must operate the same way, if we want to make sense of them. We use our imaginations to invent a construct, math ability, that suits our purposes. This construct is our dimension. We express it in terms of an abstract item hierarchy: addition, substraction, multiplication, division. We operationalize it in a math test. But is this dimension useful?

We discover whether our invented dimension, our construct, has any meaning and utility beyond our own imagination by looking for confirmation and contradiction of our intentions. We analyze the responses to our test. Do the item difficulties correspond with our intended hierarchy? Do individual items maintain their locations, i.e., do they fit? Are noticeably different persons positioned at noticeably different locations on the dimension, i.e., separated, in a way that suits our purposes? Contradictions and deficiencies lead us to re-express our construct and revise our operationalization of it. Perhaps it would be more useful for our purposes for addition, substraction, multiplication and division, each to have its own dimension, but that would lead to four measures. We must choose: Are four measures more useful or more confusing than one? Theory can't answer this, only practice can. If we have only one decision to make, then we want only one measure to base it on. If that one decision is actually a series of smaller decisions, then for each of those we want only one measure. Discovering that an individual's height is multi- dimensional with head dimension, torso dimension and leg dimension is of no help, and so ignored, for most purposes involving subject heights.

We know that perfection in conceptualization and operationalization will never be reached. A yardstick is not perfectly unidimensional, nor perfectly precise, nor perfectly accurate. But it is good enough for our purposes. So the ruler we construct from test responses falls short in just the same way. In both cases, we maintain the meaning of our dimension by careful use and maintenance of the ruler. We screen out and investigate errant measurements, misapplications, inconsistent results, and warped test instruments. We insist that only measures in useful accord with our invented dimension have the meaning we impute to the numbers. If no such measures are found, our dimension is useless, however conceptually sound it may be. If such measures are found, then they and only they, suit our purposes and the dimension is useful, however unrefined it may be.

Further thought and investigation will always reveal that our current idea of any dimension, "length", "temperature", "math ability" is deficient, and its operationalization by our ruler is defective. Progress requires that we be prepared to base our actions on what we can usefully achieve now, rather than on the perfection of the infinitely distant future.

Ben Wright

W.H. McCrea (1983) Introductory Remarks. Phil. Trans. R. Soc. Lond. A. 310:211-213.

Where do Dimensions Come from? Wright BD. Rasch Measurement Transactions 1994 7:4 p.325


Where do Dimensions Come from? Wright BD. … Rasch Measurement Transactions, 1994, 7:4 p.325

Please help with Standard Dataset 4: Andrich Rating Scale Model



Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

To be emailed about new material on www.rasch.org
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Rasch.org

www.rasch.org welcomes your comments:

Your email address (if you want us to reply):

 

ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
Sept. 15-16, 2017, Fri.-Sat. IOMC 2017: International Outcome Measurement Conference, Chicago, jampress.org/iomc2017.htm
Oct. 13 - Nov. 10, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Oct. 25-27, 2017, Wed.-Fri. In-person workshop: Applying the Rasch Model hands-on introductory workshop, Melbourne, Australia (T. Bond, B&FSteps), Announcement
Jan. 5 - Feb. 2, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 10-16, 2018, Wed.-Tues. In-person workshop: Advanced Course in Rasch Measurement Theory and the application of RUMM2030, Perth, Australia (D. Andrich), Announcement
Jan. 17-19, 2018, Wed.-Fri. Rasch Conference: Seventh International Conference on Probabilistic Models for Measurement, Matilda Bay Club, Perth, Australia, Website
April 13-17, 2018, Fri.-Tues. AERA, New York, NY, www.aera.net
May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
The HTML to add "Coming Rasch-related Events" to your webpage is:
<script type="text/javascript" src="https://www.rasch.org/events.txt"></script>

 

The URL of this page is www.rasch.org/rmt/rmt74j.htm

Website: www.rasch.org/rmt/contents.htm