# Constructing Scientific Measurement Models

The aim of the scientific process is, in some sense, to predict the future. It may be a future-in-the-past, for instance an eclipse of the sun that occurred in Ireland in 688 A.D., or it may a future-yet-to-happen. Scientific models deliberately embody simplified, but manageable, versions of reality. Henry David Thoreau wrote a universal truth in another context: "Our life is frittered away by detail ... Simplify , simplify." If we attempt to include every possible detail into our analysis, we exhaust ourselves and obtain results that are so specific as to become merely restatements of the original details.

Thus the scientific challenge is to formulate models general enough to encompass the scope of situations usually encountered, but specific enough to give practical and useable guidance in the outcomes to be expected in those situations. Thus the scientific model embodies a theory about the relationships that generate the data. Of course, the predicted outcomes only approximate the actual ones. "Empirical problems are frequently solved because, for problem solving purposes, we do not require an exact, but only an approximate, resemblance between theoretical results and experimental ones." (Laudan, 1977). Indeed "in many aspects of statistics it is necessary to assume a mathematical model to make progress." (Draper and Smith, 1966).

There are an infinity of possible models that generate outcomes which approximate the data, so which ones to choose? There is no absolute or correct answer, but there is the answer of utility. "All science is only a refinement of everyday thinking" (Einstein, 1936). The more generally applicable the model, and the more useable the results, the more it is likely to meet practical needs and form the basis for scientific progress. William of Ockham suggests that "What can be accounted for by fewer assumptions is explained in vain by more." Scientists are also generally comfortable performing arithmetical operations. "Measurement is primarily a device which enables us to use the laws of arithmetic to solve problems relating to phenomenal events" (Guild, 1938). Accordingly, a good starting point would be to look for models with as few parameters as possible within a framework that can be manipulated by arithmetical operations.

Classical test theory (CTT) appears to meet these requirements. In fact, it is almost ubiquitously used for summarizing and reporting the results of scoreable tests. Its strength is that the outcome of a test for an examinee can be expressed as one number which has at least the arithmetical properties of rank order, and often approximates linearity. CTT fails when results must be compared across tests, or there is missing data, or score differences within a test need to be compared, or when ...

Rasch's insight was that a simple logistic transformation overcomes the obvious predictive flaws of CTT. The logistic transformation is mathematically tractable, and yet, as Derek de Solla Price observed, it underlies a multitude of natural process.

Under many circumstances, merely replacing a reported percent with
Measure = 50 + 25 * Log10 ( %Right / %Wrong )
will approximate linearity will enough.

John Michael Linacre

Draper, N. R., & Smith, H., Jr. (1966) Applied Regression Analysis. New York: Wiley.

Einstein, A. (1936) Physics and reality. Journal of the Franklin Institute, 221. Translated by Syllabus Division, University of Chicago.

Guild, J. (1938) Are Sensation Intensities Measurable? Report of the 108th Annual Meeting of the British Association for the Advancement of Science, Cambridge.

Laudan, L. (1977) Progress and its Problems. Berkeley, CA: University of California Press.

Price, D. J. de Solla (1986) Little Science, Big Science ... and Beyond. New York: Columbia University Press.

Constructing Scientific Measurement Models. J.M. Linacre … Rasch Measurement Transactions, 2003, 17:1, 907

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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
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