Perceptual Ratios, Differences, and the Underlying Scale

Torgerson’s (1961) conjecture that subjects cannot meaningfully distinguish perceptual ratios and differences has triggered a continuing debate in psychophysics on what constitutes adequate scaling methodology. It is shown that an axiomatic theory of magnitude estimation proposed by Narens (1996, 1997) sheds new light on this problem by dropping the assumption that subjects can report sensation magnitudes veridically. Furthermore, this theory provides qualitative predictions suited to test Torgerson’s conjecture, by checking whether ratio and difference productions may be combined commutatively, which through the theory implies that the same psychological operation underlies both tasks. A loudness production experiment in which six subjects were instructed to double, or triple loudness, and to add a small, or a large loudness difference to a standard revealed that the two kinds of instructions were indeed commutative in most instances, consistent with Torgerson (1961) and other empirical results. Only two subjects showed small, but significant violations of commutativity. These results, when interpreted in the context of Narens’ (1997) axiomatic theory, suggest that most individuals’ ratio and difference adjustments can be simultaneously represented as ratios. This conclusion, which implies Torgerson’s conjecture, is derived using methods that are more rigorous than those employed previously to tackle the problem. In psychophysics, scaling methods supposedly based on the evaluation of sensory differences, such as equisection or categorisation, appear to be incompatible with methods supposedly based on the evaluation of sensory ratios, such as fractionation, or magnitude estimation. As early as 1961, however, Torgerson explicitly refering to these two classes of scaling methods proposed a provocative hypothesis about the relation between perceptual ratios and differences that initiated an ongoing research tradition of its own. Based on his work on scaling the lightness of Munsell grays, Torgerson observed that category ratings (on a scale from 0 to 10) and ratio estimates (magnitude estimation with a fixed standard and modulus) of the same stimuli were non-linearly related, suggesting that subjects do not evaluate differences and ratios on the same underlying scale “in the manner required by the number system” (Torgerson, 1961, p. 203). Rather, the difference scale exhibited a linear relationship with the logarithm of the ratio scale, suggesting that stimuli judged to be separated by equal subjective distances under one instruction were also separated by equal subjective ratios under the other instruction. This lead Torgerson to hypothesize that “the subject perceives or appreciates but a single quantitative relation between a pair of stimuli” (p. 203), which he/she reports as a ratio or a difference, depending on what the instructions require. This claim has been known as Torgerson’s conjecture in the research literature (e.g. Schneider, 1980; Birnbaum, 1982). Given the many assumptions implicit in Torgerson’s conjecture, it appears desirable (a) to have a more formalized theory of judgments of ratios and differences, and (b) to derive tests from it that do not require more than qualitative comparisons. Precisely that has been accomplished based on an axiomatic theory of magnitude scaling proposed by Narens (1996) and later extended to Torgerson’s problem (Narens, 1997). Narens’ theory of subjective intensity Narens’ (1996) “Theory of Ratio Magnitude Estimation” formulates the assumptions inherent in S.S. Stevens’ direct scaling approach by specifying them in the form of an axiomatic theory. This theory takes care to treat the numerals uttered by the subject in a magnitude estimation task as distinct from (scientific) numbers, of which the subject may or may not have a sound understanding. Two of the axioms (commutativity and multiplicativity) formulated by Narens (1996) are crucial to both an empirical test of the theory and the interpretation of subjects’ scaling behavior. These axioms were empirically evaluated by Ellermeier & Faulhammer (2000) by having subjects produce loudness ratios. They showed that subjects’ loudness adjustments were consistent with the weaker commutativity axiom (e.g., making a reference sound twice as loud and tripling the loudness of the resultant yields the same sound-pressure level as first tripling and then doubling the loudness). However, the multiplicativity axiom (e.g., the fact that consecutive doubling and tripling of loudness should be equivalent to making the starting intensity six times as loud), was violated in a significant number of cases. According to Narens’ axiomatization, this outcome implies that though in principle a ratio scale of loudness exists, the numerals used by subjects in order to describe sensation ratios may not be taken at face value. Narens’ (1997) extension of the theory is based on the following idea: A person’s intensity judgments on a continuum of stimuli result from “calculations” (e.g., algorithms) based on an inner psychological structure that is ratio scalable. Using this, and results from Narens theory of meaningfulness (published later in Narens, 2002), it follows that for functions f and g on the stimuli that are produced using ratio production or difference production or both kinds of productions the following Generalized Commutativity Principle holds: f ∗ g = g ∗ f, where, of course, ∗ is the operation of functional composition. Because it has been empirically established by Ellermeier and Faulhammer (2000) that ratio production functions commute, Torgerson’s conjecture that subjective differences are subjective ratios suggests that difference production functions should commute with ratio production functions. Note, however, that Togerson’s conjecture was based on a theory of measurement due to S. S. Stevens which identifies subjects’ responses to ratio and difference instructions veridically; that is, in the ratio condition, the “numeral ratio 3” was interpreted by the measurement procedure as the numerical ratio represented by the real number 3, and the “numeral difference 2” was interpreted as the numerical difference represented by the real number 2. Such veridicality is not part of Narens (1997) theory nor its deduction of the Generalized Commutativity Principle. Operationalization of Ratio and Difference Production While ratio productions are easily implemented by asking the subject to ”make the second tone p times as loud” as the first one (denoted by Rp(x) = y, with x being the level of the first (reference) tone, and y being the result of the adjustment), we cannot simply say for difference productions “adjust the second tone so that the loudness difference is p” without providing a unit. Therefore, the following difference matching instruction Da,b was implemented where Da,b(x) = y holds if and only if the subject adjusts a stimulus y such that “the difference in loudness between y and x is the same as the difference between b and a.” The above observation leads to the following specification of the General Commutativity Principle (GCP): For behavioral functions Rp and Da,b and all stimuli x in X: Da,b[Rp(x)] = Rp[Da,b(x)]. (1) For each such test of the GCP, the subject has to perform two ratio productions (here specified as making the comparison tone three times as loud), and two interval matches as illustrated in Figure 1. If the GCP holds, the two orders of chaining these operations should not make a difference, and multiple productions of y and v (as specified in Figure 1) should be statistically indistinguishable. Figure 1: Illustration of the paradigm used to investigate the validity of the General Commutativity Principle (GCP). For an explanation, see text.