# What is Measurement?

Before proceeding to consider the problem of measurement as applied to psycho-physical problems it is desirable to consider some of the general principles of measurement.

Measurement is primarily a device which enables us to use the laws of arithmetic to solve problems relating to phenomenal events. The laws of arithmetic pertain to numbers and to nothing else: there is nothing inherently numerical in the structure of the phenomenal world. We are, however, so familiar with the description of phenomena in numerical terms (or their formal mathematical equivalents) that the association has become instinctive, and we are apt to imagine that we directly perceive the metrical aspects of nature as inherent constituents of phenomena, existing in their own right, so to speak, and merely observed by us. This induces us to overlook the essentially arbitrary and man-made nature of the association. When this is of an unfamiliar character, as, for instance, when we associate the arithmetical operation x sqrt(-1) with the physical events constituting a phase change of pi/2 in alternating current problems and others of a like nature, we recognize it at once as a mere device of the mathematician. In principle it is no more artificial than the more familiar associations of events with arithmetical concepts which underlie all metrical processes.

The phenomenal world presents itself to us as a complex relation structure. We need not here enter into metaphysical questions concerning the parts played by our sense organs and by things external to us in determining the kind of relations exhibited by phenomena. We will take the phenomenal world as we find it: a structure of related events, which we find it convenient to describe and classify in terms of various concepts.

We have discovered - and this discovery is the foundation stone of physical science - that by employing a simple but ingenious device some aspects of phenomena can be classified so that certain phenomenal relations existing between members of any such class are `similar' to the relations between members of the class of numbers on which the laws of arithmetic are based. In arithmetic, these relations are implicit in the meaning assigned by two important symbols, namely =, the symbol of numerical equality, and +, the symbol of the operation of adding one number to another. All other arithmetical operations, subtraction, multiplication or division, involution or evolution, etc., ultimately derive their significance from the operation of addition.

In any class of phenomenal aspects of the kind we are considering we can perceive various relations. Further, by performing experimental operations on the things which exhibit the aspects in question we can change the actual relations exhibited. But there is nothing inherently numerical in these phenomenal relations: in order to establish a connection we must arbitrarily associate some unique symmetrical transitive phenomenal relation from among those which may have perceptual significance with the arithmetical relation of equality; and, further, we must associate some suitable experimental operation with the arithmetical operation of addition. If, now, we base our phenomenal classification entirely in terms of this symmetrical relation and this experimental operation we obtain a phenomenal series whose members are related to each other in a similar manner to the members of the series of numbers.

We must not confuse `similarity' as here used with identity. Relations are themselves things which can be classified in virtue of certain characteristics irrespective of the kind of things related by them. A relation may be symmetrical or unsymmetrical, transitive or intransitive, etc., and it is these properties of relations themselves, and not any specific properties of the things related by them, which confer relational similarity or dissimilarity on classes defined by relations. However, it is not here necessary to discuss the theory of similar relations, or go into the conditions which must be imposed on our selected criteria of `equality' and `addition ' in order that phenomenal and numerical relations may be similar. The important point to be noted is simply that there is no a priori connection between phenomenal structure and number, and that to make a connection we must artificially associate a phenomenal criterion with numerical equality and a phenomenal operation with numerical addition. When we have done this, but not before, we can predict by arithmetical calculation those phenomenal relations which involve only the stipulated practical criteria of equality and addition.

A phenomenal class defined by two such practical criteria constitutes a measurable magnitude of the type which Dr. Norman Campbell, in his well-known text-book on the principles of measurement, has termed an A magnitude. Any such magnitude can be measured by processes which do not imply the measurability of any other magnitude. It is true that the practical criterion of equality for any A magnitude will always involve the observation of some phenomenal state or condition which may (and usually does) involve other magnitudes; but the observational criterion is always of the null type - no difference, or no observable change, in the prescribed state or condition: no numerical relations have to be determined or even be assumed to exist for these other magnitudes. Familiar examples of A magnitudes are length, volume, mass, electrical resistance, and many others which need not be detailed.

The practical criteria of equality and addition which define these magnitudes for purposes of measurement are sufficiently familiar to require no description. It is their significance which is not so widely recognized as it might be. It is probably usual to regard the experimental processes of determining equality and of adding as something which we have just found to be a convenient method of determining quantitative relations inherent in the nature of the magnitudes, whereas these processes are the necessary connecting links between phenomena and number without which there would be no basis of comparison between the laws of the former and those of the latter. The experimental criteria do not merely enable us to measure a magnitude, they create the magnitude by defining the fundamental relations which are to be the basis of classification.

In Physics the general term measurement is not confined to A magnitudes. By suitable experimental processes we affix numerals to many aspects of phenomena to which no operation can be performed having any similarity in relation structure to the operation of addition. Familiar examples are density, specific heat, electrical resistivity, etc. All those things which we ordinarily regard as properties of substances are magnitudes of this type. They are the B magnitudes of Campbell's classification. We can usually formulate a practical criterion of equality for a B magnitude, but not of addition. ...

Strictly speaking, therefore, the only measurable magnitudes are A magnitudes.

Excerpted from the "Statement by Mr. J. Guild. Are Sensation Intensities Measurable?", Part III of the "Interim Report of the Committee appointed to consider and report upon the possibility of Quantitative Estimates of Sensory Events", Prof. A. Ferguson, Chairman. Report of the 108th Annual Meeting of the British Association for the Advancement of Science, Cambridge, 1938. (Italics Guild's, bold RMT's.)

Rasch measurement constructs measures of A magnitudes. A "phenomenal criterion" associated with numerical equality is 50% success and 50% failure. A "phenomenal operation" associated with numerical addition is the log-odds of joint independent success vs. joint independent failure.

What is Measurement? Guild J. … Rasch Measurement Transactions, 2001, 15:1 p.798-9

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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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