Lost in the Dimensions

The concept of "dimensionality" is bandied about in educational measurement with a looseness that rivals the way anthropologists handle the concept of "culture". A review revealed more than 50 definitions of "culture" in the 1950's. Attempts to implement one standard definition have been resisted vigorously. Each proposed definition expresses the individual preference of an investigator. This ambiguity of definition may enrich the concept of culture in casual conversation because individuals are free to assign their own idiosyncratic meaning to the concept without inhibiting the flow of their ideas. But, for scientific exchange and research, this places a great deal of responsibility on the researcher to be explicit about the precise contextual meaning of the nebulous concept.

Similar considerations apply to the concept of "dimensionality", which, particularly in the context of unidimensionality versus multidimensionality, has become a notorious issue in arguments for and against the use of the Rasch model to construct objective measurement. The apparent importance of "dimensionality" demands that rigor be employed in the way the concept is used. Unfortunately such rigor has been sorely lacking.

Various statistical techniques are supposed to detect item dimensionality, but all are based on the questionable assumption that dimensions can be defined or even caused merely by statistical relationships among variables (i.e. linear or correlational relationships). The tautology is that the supposed dimensions are defined by the same techniques used to detect them. Almost any data set can be subdivided in such a way that subsets have higher internal correlations than their external correlations with one another. Does this mean that any whole is multidimensional and, at the same time, that all its possible subsets are unidimensional? Where does the subdivision stop in order for a final unidimensionality to have been reached? Is statistical unidimensionality achievable or even desirable? To further complicate the issue, there is the effect of data clusters along a single continuum. Most statistical techniques identify such clusters as different dimensions. But if the clustering is along one continuum, aren't the clusters on the same dimension by definition?

What is a "dimension"? Statistical arguments do not seem to provide a useful answer. The question must be approached conceptually. It is the intention of the researcher which defines the dimension. Further, when a construct is developed which defines a dimension, then the potential for establishing hierarchies of dimensions is opened. One researcher's multidimensional space of incongruities may be another researcher's uni-dimensional line of concern. For researchers at a national policy level, there could be a unidimensional operationalization of the entire field of mathematics. At the same time, for researchers focusing on the differences between algebra and geometry, each of these can become a single dimension. It could be argued, from this point of view, that anything can be defined as unidimensional. That is true! Constructing definitions is always the investigator's responsibility.

It is useful to think of unidimensionality, not as an opportune empirical accident, but rather as an intended theoretical construct to be defined. The intention of the researcher must be explicit. What is the construct to be measured? How successful is the instrument designed to implement the desired measurement process? It is the fit of the data to the measurement model that determines the success with which the researcher's intention is implemented through the instrument. If the fit is acceptable, then implementation is successful. If the fit is not acceptable, then an investigator can diagnose the unexpected contradictions and so improve the instrument and refine the intention.

How does this apply to certification testing? If the intention defines the dimension, then it is the intention that must be defined and defensible. The testing agency must outline the exact scope of practice that is being tested. This scope of practice must then be characterized by an itemization of activities. The decision must then be made as to whether these activities can be conceptually related in a way that can be tested using a single measurement instrument or whether a number of qualitatively different instruments are required. The need for such a decision is inherent in the recognition that no field of practice consists of a single activity. Rather, individuals are always required to demonstrate proficiency in a variety of activities before being certified as competent to practice. How the certification agency decides to view the partitioning of these activities determines the "dimensionality" of the certification process.

Lost in the Dimensions, J Stahl … Rasch Measurement Transactions, 1991, 4:4 p. 120

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

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