Mathematical Models and Measurement

Mathematical models have often fallen short in providing useful representations of social science problems. Modelling involves "a heroic simplification of a problem using the minimum possible number of basic variables in order to come to grips with the essentials" (Saaty & Alexander, p. 11). Mathematical models enable abstraction based on logical formulations using the convenient language of mathematics. Useful models are simple enough to allow data collection and analysis. They are also practical in the sense that they serve as an aid to implement their own application. Models enable better visualization of the main elements of a problem. They also form a basis for communication, decreasing ambiguity and improving the chances of agreement on results.

Choosing a model, however, can present some difficulty since each model, good or otherwise, is an alternate reality. Casti notes four features of good models that can help in making an effective choice: 1) simplicity, 2) agreement with known facts, 3) explanatory power and 4) predictive capability. The point of making models is to bring order to our experiences and observations, and to make predictions. Models are tools for organizing reality, for ordering experiences rather than simply describing them. A description is only an account of what is. It does not motivate any investigation into the reasons why things are the way they are. Models motivate explanations and it's clear that explanation is preferable to description. Too often, however, the data are allowed to become the model, so that there is a different model, a new and confusing alternate reality, for every set of data collected.

The Cyclical Nature of Mathematical Models The use of a mathematical model involves several steps: data collection, hypothesis construction, analysis, prediction and conclusion. The flow chart (F.S. Roberts, 1976) provides an idealization of modelling. When the predictions provided by a model agree with what is observed, the system can be declared a model of the process under investigation. But, as Casti states, "is this all it takes to be deemed a model of a natural phenomenon? Shouldn't there be something more than just agreement with the data?" The model must go beyond the data: it must be based on theory.

The Rasch model coordinates data with the requirements of a useful definition of measurement. The Rasch model specifies what must dominate when a person responds to a test item in order for linear measurement to be constructed. As with all real-life problems, the interaction between person and item is approximate, interconnected and occasionally misleading. Nevertheless the constructed measures must dominate the observed data if serviceable conclusions are to be drawn from the results of person/item interactions. The Rasch model constructs measurement and not merely description. The theory dominates the data.

Estimating with Models:
Mathematical models involve equations containing variables and constants. But the equations themselves must be conducive to solution, a process which may take the theoretician through four steps (Saaty & Alexander, p. 7):

1. A priori bounds - Establish that there is a limit to the number of solutions the problem has.

2. Existence and uniqueness - Prove that there is exactly one solution.

3. Convergence - If an iterative instead of a closed form method is used, do the iterations converge to a meaningful solution?

4. Approximations - How good an approximation to the ideal solution does the converged solution provide? How precise is the approximation?

Rasch model estimates routinely satisfy these four criteria. Since Rasch estimates are based on sufficient statistics, a unique solution to the estimation equation always exists provided all persons and items are connected through a sufficient network of successes and failures. Whenever divergence does occur with a particular set of data, it is not due to failure in the model but to deficiency in the data (and occasionally to inadequacy in the estimation algorithm employed). When data are good enough to be used for the construction of measurement, which, in practice, is most of the time, no artificial parameter bounds or specially selected starting values are needed in order for a functional converged solution to be obtained.

Rasch measurement models meet the requirements for useful models as well as the requirements for processes of solution. Rasch analysis demonstrates that good model building is possible in the social sciences.

Casti JL 1989. Alternate Realities. New York: Wiley

Roberts FS 1976. Discrete Mathematical Models. Englewood Cliffs, NJ: Prentice-Hall.

Saaty TL & Alexander JM 1981. Thinking with Models. Oxford: Pergamon Press.

                          Prediction
                          Deduction
      Mathematical Model  ------>  Mathematical Predications
          Translation ^                 Interpretation
          Induction                  v
      Real-world Data     <------  Real-world Predictions
                          Testing

Mathematical models and measurement. Wisniewski DR. … Rasch Measurement Transactions, 1992, 5:4 p.184

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

Forum

Rasch 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.

The URL of this page is www.rasch.org/rmt/rmt54f.htm

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