Unidimensional Models in a Multidimensional World

Question: "Unidimensionality is one of the assumptions underlying most Rasch models. But everything we encounter is multidimensional. Why aren't all Rasch models multidimensional?"

Reply: The world is multidimensional and confusing. A fundamental activity of physical science is to decompose the world around us into unidimensional variables (weight, height, temperature, pressure, ...). Using these unidimensional variables, physicists can think clearly and make strong inferences. The history of the thermometer is an illustrative example of this process. Early thermometers (around 1600 A.D.) combined temperature with atmospheric pressure. They were "multidimensional". It was a major advance when scientists discovered how to separate those two dimensions in order to make both temperature and atmospheric pressure into unidimensional variables.

In Rasch measurement, we are attempting to perform the same process of splitting a multidimensional world into unidimensional variables, but now with social science. Asserting and then building unidimensional variables has been very useful in physical science. We expect it will also be in social science.

What if there are two dimensions?

Question: When we know there are two dimensions in the data, what is the next step - two separate analyses? Then, how can we make it sense out of the two analysis when we only want to report one number?

Answer: Under these circumstances, we need to consider:

1. How big is the difference between the dimensions?

2. How many people, and which people, does it impact?

3. Is it important enough to merit reporting two numbers?

This may require a separate analysis of each dimension. For instance, in an elementary-arithmetic test, we will probably find there is an "addition" dimension and a "subtraction" dimension. Unless the test is intended to identify learning difficulties, it is unlikely we will want to report two numbers. But the dimensionality may have useful information for instruction. In one situation, relatively bad performance on "subtraction" was discovered to be related to poverty. Children in poverty did not like the thought of something being "taken away" (subtracted). This suggests that teaching "subtraction" to impoverished children should avoid using emotive words or personal implications.

It is unusual in a carefully-constructed test that two dimensions are different enough inferentially to merit reporting two numbers. But secondary dimensions may indicate that care should be taken in test-construction in order to balance items between dimensions. For instance, aim for 50% addition items and 50% subtraction items, not 80% addition items and 20% subtraction items.

Linacre J.M. (2009) Unidimensional Models in a Multidimensional World, Rasch Measurement Transactions, 2009, 23:2, 1209

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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