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.

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

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

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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