Measurement, Scaling, and Psychophysics

Chapter 1 MEASUREMENT, SCALING, AND PSYCHOPHYSICS R. Duncan Luce, Harvard University Carol L. Krumhansl, Cornell University Introduction Aximatic Measurement Scaling Psychophysics INTRODUCTION kinetic and potential energy, acceleration, impedance, and charge. To most physical scientists, the term measure- Psychological measurement is in a compar- ment brings to mind issues of precision, reliabil- able, floundering state with informal concepts ity, bias, and technique. One takes for granted that seem to admit of degree or amount, but that the attribute in question-be it the velocity whose means of measurement remain crude, of light, the mass of an electron, or the diameter controversial, and uncertain. Subjectively, our of the sun-is measurable, that it can be quan- sensations of hunger, loudness, and sweetness tified in a consistent and meaningful way. In the are no less real than our sensations of force, physical sciences very precise systems of meas- energy, and acceleration. Moreover, for both urement have been developed that permit largely psychological and physical attributes, our observer-free descriptions of physical objects sensations suggest a measurement system that and events. But think back to the early history codes the degree or amount of the attribute or of physics, when the very nature of many meas- property in question. However, it is important to ures was a matter of doubt and debate, and most recognize an essential difference between the measurement systems in use resulted from the objectives of physical and psychological meas- needs of trade and commerce, not physical urement. Physics studies certain properties of theory. True, there were strong intuitive space, matter, and radiation, not the sensations notions of what we now call force, weight, mass, they engender, although there is no doubt that time, velocity, momentum, work, heat, and tem- in its early development physics was strongly perature, but it took a great deal of effort to guided, and sometimes misguided, by these develop methods for measuring these attributes sensations. In psychology, we are concerned of physical systems and to understand the rela- with the sensations themselves, and this dif- tions among them. In the process, it became ference poses a challenge to develop measures necessary to introduce some far less intuitive that are appropriate for our special purposes. but equally important measures, such as entropy, The system of measurement that develops in 4 MEASUREMENT, SCALING, AND PSYCHOPHYSICS psychology will undoubtedly turn out to sensations themselves are attributes of the resemble that of physics in certain fundamental organism, comparable in many ways to the ways, but new problems requiring novel solu- properties of physical objects. As such, these tions are also likely to emerge. attributes are assumed to be highly stable and Physics and its applications exhibit three dis- regular, and thus they can be subjected to care- tinct levels of theory. First, there are the pri- ful analytical study and represented numeric- mitive relations among attributes that deter- ally in a manner analogous to physical measure- mine both the measurement representations and ment. This approach is concerned not only with the interconnections among measures, as the assignment of numerical values to the reflected in the structure of physical units. psychological states, but also with the way in Second, there are the various laws of physics which the observed psychological measui-es that are expressed entirely in terms of these possess various formal properties of the number measures, such as the basic equations of electro- system into which they are being mapped. For magnetic theory (Maxwell's equations), of hydro- the purpose of this chapter, we will refer to this dynamics, of kinetics, and of statistical mechan- as the measurement approach. ics. And third, certain particular systems are If this approach is followed and succeeds, one modeled and the equations are solved for those anticipates the discovery of general laws relat- cases. An example is the force exerted on a par- ing the sensations to physical attributes. That ticular airfoil in a particular fluid flow, or the is, the measured sensations are expected to cor- movement of the needle of a voltmeter when respond systematically to the physical quan- placed across a particular circuit under par- tities that give rise to them. This is the point of ticular conditions. Keep in mind that, at each of view, more or less explicit, that has informed the the three levels, important regularities are tradition of psychophysical measurement that found, all of which are called laws; those of began with Fechner (1860) and received major measurement, those of general physical pro- impetus in this century by Thurstone (1927), cesses, and those of particular physical systems. Stevens (1951, 1975; see the latter for numerous Although they all express regularities, they are references), and many others. For example, obviously not all of the same type, since they sound intensity, that is, the amplitude of the differ in generality. pressure wave impinging on the eardrum, is a If three distinct levels really can be distin- well-understood physical attribute that is highly guished, at which level does the psychophysicist correlated with the sensation called loudness. It operate when studying loudness? If a scale of is, however, by no means identical to it. For one loudness is developed, is it (1) an example of thing, the relationship is not linear; twice the measurement, (2) a part of some psychophysical amplitude does not give rise to twice the per- theory, or (3) the manifestation of the operation ceived loudness. Moreover loudness is affected of a mechanism? In the first case, it must eventu- by the frequency of the sound as well as by its ally be shown to arise as the representation of a intensity. A more striking example, perhaps, is lawful qualitative structure. In the second, it is color vision. Light stimuli are described at the best thought of as a construct whose ultimate physical level by the energy at each of an infin- status is not necessarily clear; often constructs ity of wavelengths, but perceptually this infi- of theories eventually become basic measures, nite-dimensional space is mapped onto the much once the situation is better understood. And in simpler, three-dimensional structure of the the third, it may best be thought of as a random psychological color space. Thus, the relation variable, reflecting the probabilistic character between the physical stimulus and the corre- of a mechanism, but not necessarily having any- sponding psychological response may be quite thing to do with fundamental measurement. complex, although this approach assumes that We discuss the first separately and then lump the relation is lawful. the other two. For the measurement view to prevail in psychophysics or, more generally, in psychol- The Strategy of Measurement and ogy, or even more generally, in biology, it Scaling in Psychophysics appears necessary for some structure of inter- connected scales to arise, much as exists in In following this strategy we suppose that the physics. This would mean a complex pattern of INTRODUCTION 5 reductions of some psychological scales to are for the most part fallible in the sense of others, with a resulting simple pattern of units, exhibiting a certain amount of inconsistency and quite possibly some simple connection with and irregularity, into some familiar numerical the scales of physics. This appears to be the representation. For example, the question may program that Stevens was pursuing; we discuss be how best to locate points in some geometri- it below. To anticipate, Stevens attempted to cal, often metric, space so as to represent the argue that the psychological scales of intensity psychological relationships among stimuli? are all power functions of the primary physical This approach accepts the fallible nature of the scales that manipulate each attribute. Moreover data, and is, therefore, not so much concerned he attempted to show that a consistent pattern with verifying that the data satisfy specific of power functions holds among the scales of properties entailed by the chosen representa- intensity, suggesting a single underlying attri- tion as with searching for the best possible fit of bute of psychological intensity. Were that true, the data within the chosen class of representa- then psychophysical measurement would simply tions. The fit between the original data and the enlarge the system of physical measures, just as solution is evaluated in some global, statistical electromagnetic theory did in the second half of sense. the last century, and one could anticipate the In the short run, this scaling approach has development of psychological theories involv- proved to be very directly usable. In the long ing these sensory (and perhaps other) variables run, solving some of the deeper questions posed on the model established in physics. by the axiomatic approach to measurement, Within this tradition of measurement there such as defining the possibilities for measure- are a number of subschools and points of view. ment, discovering how complexes of structures Perhaps most notable is the division between interrelate (as, for example, those underlying axiomatizers and scalers. The axiomatizers the system of physical units), and determining (whose approach is exemplified in Krantz, Luce, what kinds of statements are meaningful in a Suppes & Tversky, 1971, Narens, 1985, and measurement structure, may have more pro- Roberts, 1979) tend to treat the following type of found implications for progress in psychology. problem: If a body of (potential) qualitative observations satisfies certain primitive laws- The Strategy of Mechanisms in axioms that capture properties of these observa- Psychophysics tions-then is it possible to find a numerical structure that accurately summarizes these The second major strategy for describing psycho- observations? In technical terms, the question physical attributes is to focus on sensory and is: To which numerical structures is the set of other mechanisms in the organism.

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