## Fundamental Measurement

Fundamental Measurement:
1. Measurement which is not derived from other measurements.
2. Measurement which is produced by an additive (or equivalent) measurement operation.
Rasch analysis operationalizes Fundamental Measurement based on ordered qualitative observations.

Fundamental measures are estimated directly from observation (such as height and weight), rather than derived indirectly by means of other measures (such as density and electrical resistance). Physicist Norman Campbell asserted that fundamental measurement requires a process of concatenation (operational addition), so that measurement units could be seen to be equal.

Difference between those properties which can be measured perfectly, like weight and length, and those which cannot arises from the possibility or impossibility of finding for these properties a physical process of addition. (Campbell, 1920)

In 1964, Duncan Luce and John Tukey perceived that fundamental measurement is not a physical operation, but a theoretical property:

"The essential character of what is classically considered ... the fundamental measurement of extensive quantities is described by an axiomatization for the comparison of effects of (or responses to) arbitrary combinations of 'quantities' of a single specified kind... Measurement on a ratio scale follows from such axioms... The essential character of simultaneous conjoint measurement is described by an axiomatization for the comparison of effects of (or responses to) pairs formed from two specified kinds of 'quantities'... Measurement on interval scales which have a common unit follows from these axioms; usually these scales can be converted in a natural way into ratio scales.

"A close relation exists between conjoint measurement and the establishment of response measures in a two-way table, or other analysis-of-variance situations, for which the 'effects of columns' and the 'effects of rows' are additive. Indeed, the discovery of such measures, which are well known to have important practical advantages, may be viewed as the discovery, via conjoint measurement, of fundamental measures of the row and column variables." (Luce and Tukey 1964 p.1)

"Seeking response measures which make the effects of columns and the effects of rows additive in an analysis-of-variance situation has become increasingly popular as the advantages of such parsimonious descriptions, whether exact or approximate, have become more appreciated. In spite of the practical advantages of such response measures, objections have been raised to their quest, the primary ones being (a) that such 'tampering' with data is somehow unethical, and (b) that one should be interested in fundamental results, not merely empirical ones.

"For those who grant the fundamental character of measurement axiomatized in terms of concatenation [e.g., Campbell], the axioms of simultaneous conjoint measurement overcome both of these objections since their existence shows that qualitatively described 'additivity' over pairs of factors of responses or effects is just as axiomatizable as concatenation. Indeed, the additivity is axiomatizable in terms of axioms that lead to scales of the highest repute: interval and ratio scales.

"Moreover, the axioms of conjoint measurement apply naturally to problems of classical physics and permit the measurement of conventional physical quantities on ratio scales.

"In the various fields, including the behavioral and biological sciences, where factors producing orderable effects and responses deserve both more useful and more fundamental measurement, the moral seems clear: when no natural concatenation operation exists, one should try to discover a way to measure factors and responses such that the 'effects' of different factors are additive." (Luce and Tukey, 1964, p.4)

The Rasch model provides such an additive framework. Furthermore, a 'natural concatenation' also exists! To 'add' two persons' abilities, merely require them to work independently on the same item until they agree on an answer (right or wrong). Then their combined ability (relative to the item's difficulty) is the sum of their individual abilities (also relative to the item). See RMT 2:1 p. 16.

"The Rasch model is a special case of additive conjoint measurement, a form of fundamental measurement... A fit of the Rasch model implies that the cancellation axiom will be satisfied... It then follows that items and persons are measured on an interval scale with a common unit." (Brogden, 1977, p.633)

Since fundamental measurement is accessible by application of Rasch's model, it would seem to follow that social scientists would make every effort to use Rasch's model to construct fundamental measures for the variables they wish to study.

Benjamin D. Wright

Brogden HE. 1977. The Rasch model, the law of comparative judgement and additive conjoint measurement. Psychometrika, 42, 631-634.

Campbell NR. 1920. Physics, the elements. Cambridge: Cambridge University Press

Luce RD & Tukey JW. 1964. Simultaneous conjoint measurement. Journal of Mathematical Psychology, 1, 1-27.

Fundamental Measurement. Wright B. D. … Rasch Measurement Transactions, 1997, 11:2 p. 558.

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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