OUTING INNER FECHNER. Peter R. Killeen Arizona State University asu.edu)

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1 OUTING INNER FECHNER Peter R. Killeen Arizona State University asu.edu) Abstract Fechner bisected psychophysics into the outer, of which he was the master, and the inner, which absorbed error variance and provided notional machinery. This was a category error. A better division of labor allocates outer to techniques that require no assumptions about internal formal operations, and inner to those which exploit them. As a minimalist example of the latter, empirical distribution functions of length productions are used to confirm a square-root psychophysical function for length. Inside, outside. As Caesar divided Gaul, Fechner divided perception. Outer psychophysics was Fechner s terra familiaris, for whose campaign he borrowed methodology from physics. Inner psychophysics was terra incognita. It leans more to physiology and anatomy, particularly of the nervous system (Fechner, 86/965, p. ). That inclination was a bad posture, bent for enlightenment under the wrong lamp. Aristotle distinguished four kinds of explanations becauses required for comprehension of a phenomenon: efficient, material, formal, and final. (He was mistranslated and misconstrued by learned babblers as asserting that each of these types of explanation constituted a version of efficient causality; Hocutt, 974). Efficient causes are triggers: stimuli and instructions. Formal causes are explanatory schema: models and equations. Final causes are goals: purposes and payoff matrices. Material causes are substrates: physiology and anatomy. Understanding material causes inner psychophysics enriches comprehension (for examples, see Norwich, 977, and Ward, 99). But it cannot substitute for the other three causes. One can t mend incomplete formalism with tentative neurophysiology (Uttal, 99; 998), as formal and material constitute different kinds of explanation (Killeen, ). Inner psychophysics conceived as neuroscience is a closet category error.

2 If neuroelectric currents and pet scans can t fill the gap, what can? After 47 pages Fechner came out with a formal cause: we would deem [Weber s Law] to be the unqualified basis of inner psychophysics p. 57). Unfortunately, Fechner s own experiments on lifted weights never demonstrated Weber s law, either on the inside or outside; he posited a generalized Weber s law with additive variance from the weight of the arm. This was a good, but incorrectly executed idea, (he added variances of correlated variables, but the weight is invariant), and not descriptive of the data. He then became too ill to continue experimenting (Stigler, 986). With the blessing of his precedent, however, we can tell the inside story with formal models, leaving the wetware to those whose bent is physiology and anatomy. Local, global. For Fechner, as for others, the local was a stepping stone to the global; he extrapolated from the second moments of psychometric functions to the first moments of psychophysical functions. The relation between the psychometric and the psychophysical has confused tyro and virtuoso alike: Titchener (94, p. ) opined that Fechner s understanding of [Fechner s Law] was wrong; he fell into the very common error of attempting to compose a psychophysical scale from additive operations on psychometric data. Titchener rederived Fechner s law based on his results, that the magnitude of a sensedistance is dependent upon the quotient of the two [stimuli] that limit it (p. ; cf. Ellermeier & Faulhammer, ) that is, based on additive operations on the logarithms of stimuli. We may dispute who was right even today, but we can conclude with confidence that least one of these magisters was wrong. In the present paper, the additivity issue is sidestepped: Local psychometrics analyses of fine discriminations can provide global psychophysical conclusions without invoking either Titchnerian or Fechnerian additivity. There is a curious asymmetry in the tactics of psychophysical researchers. Despite the use of Weber s law to impute the psychophysical function the relation of the physical magnitude to its psychological effect most psychometric studies ignore the psychophysical function. They typically, and pragmatically, use either logarithmic or linear axes for the abscissae of the psychometric function. Why not use the psychological dimension putatively the one on which the discriminations are based? The problem is lack of consensus concerning the psychological scale, its variability among subjects, and the fact that over the small range of psychometric functions, psychophysical functions are almost linear, so it doesn t make much of a difference. But a difference it does make, one that is exploited here. One-way psychophysics. ElGreco s elongated paintings provide no evidence that he was astigmatic: Any warping of the input would be corrected as his hand reproduced the image. Stimuli carried through a psychophysical transformation, and then back through an inverse one, provide no information about those transformations. Nor do introspections and estimations of psychological quantities, for those are not sui generis, but are also subject to output transformations, as often recognized (e.g., Shepard, 98). A one-way flow of information or of noise is necessary to disambiguate the transformations. Most of the variance in perception is due to internal noise, arising from the representation and categorization processes. Noise, the bane of measurement (Lockhead, 99), may also serve it.

3 P = Ψ(S) β =. S Stimulus Input P Distribution over Representation Distribution over Stimulus Figure. Internal noise playing over the perceptual representation warps judgments by the psychophysical function. Discriminations plotted against the stimulus are warped by the inverse transformation. Fechner s use of only the variance of the psychometric functions Weber s law wasted information. Consider the transformation from stimulus S to percept P shown in Figure. Judgmental and memorial fluctuations symmetrically perturb the percepts, which are nonlinear transforms of the stimuli. When translated back to the physical dimension the density suffers an inverse transformation. The resulting density may be found using the transformation technique (Freund, 99). If the psychophysical function is P = S β, and the density of the noise is f(s), then the inverse transformation is S = P /β, and its density is: f' S = f S β βs β. () The absolute factor renormalizes the density. An example of a Gaussian density transformed by a power function is shown in the right panel of Figure. Although more efficient than exploiting only the variance, even this approach wastes information, as the data must be binned for estimates of the density. It is best to consider every datum in its own right. This is possible by comparing the distribution of the data their empirical distribution functions, or EDFs with known functions (D'Agostino & Stephens, 986). It is the most efficient way to use the data. Examples of EDFs are shown in Figure, which displays distributions of 5 lengths produced by 6 participants attempting to measure exactly on a digital tablet (Gilden, Thornton & Mallon, 995).

4 Gilden (995) Length z(p) z(p) z(p) Response/Mean Response/Mean Figure. Data from 6 subjects producing distances of. The data are detrended and reordered from shortest to longest. The ordinates are the Gaussian z-transforms of the cumulative probabilities of the abscissae. The curves are power and logarithmic functions. The abscissae of Figure are the lengths produced, normalized to. The ordinates are their z-scores, Gaussian p-to-z transformations of the cumulative probabilities of producing a length less than or equal to the abscissae. Probabilities were assigned as /5 for the shortest length, /5 for the next, etc. Notice first that all curves are concave, many

5 deviating significantly from linearity: The discriminal process is asymmetric. Two curves are fit to the data: Stevens s z = (x β - )/σ, and Fechner s z = log(x)/σ. They provide equivalent accounts of the data in the top two panels, but in the remainder the power transformation is better. The mean exponent is β =.4, with a substantial inter-subject variability (σ β =.5), and thus consistent with the estimates of β =.46 derived by Parker, Schneider and Kanow (975), β =.55 by Petrusic, Baranski and Kennedy (998), and with a square-root function, which was the maximum likelihood estimate of β for the pooled data. The assumption of a Gaussian distribution on a linear axis may be rejected in of the 6 of the cases (A >.75, p <.5; D'Agostino & Stephens, 986) and of logarithmic curvature in of the 6. Other dispersion functions logistic, Laplace and Weibull were not as good as the normal. Second inning. How can lengths have exponents less than when they are often thought to be represented linearly? The conventional values of the exponents are fixed by assuming that the exponent for numbers is.. But there is evidence from non-metric scaling that the exponent for number magnitude (Schneider, Parker, Ostrosky, Stein & Kanow, 974) is less than. It follows that the exponents for all other continua must be rescaled, and validates the lower exponents found in this study. There are other approaches to one-way psychophysics: Concatenation of psychological quantities generates new internal information that, when inverse transformed into responses, constrains the possible psychophysical functions (e.g., Rule, Curtis & Markley, 97; Ward, 99). Some of these studies also indicate lower exponents for number, and thus length. Will the nonlinearities found in the vicinity of fine discriminations maintain the same form when the range of stimuli under consideration is increased, as in global psychophysical experiments (Ward, Armstrong & Golestani, 996)? Analysis will tell. Fechner was right to separate outer from inner psychophysics, but got the split wrong. His was a gerrymander that permitted him to avoid speculation about unknown material causes. It was in the same spirit as Newton s I feign no hypotheses [about machinery] ; both made space for theory without material correlates. A more useful division is between analytic techniques, with: (a) outer psychophysics comprising traditional unimodal procedures that are agnostic concerning transformations within the psychological stage, and (b) combinatoric inner psychophysics techniques such as multidimensional scaling, concatenation operations, similarity scaling, conjoint and functional measurement, that employ internal operations to add degrees of freedom that constrain psychological structures. The analysis of Figure was an outer technique whose interpretation relied on the simple inner logic of Figure. When extended to cross-modal matching, the internal machinery will play an increasing role in the distributional approach, which has the advantage that global scale properties may be inferred without the complications of bimodal concatenations or additivity assumptions. One hundred and fifty years of psychophysics, the most sedulous and precise of psychology s domains, has left legacies of contentious theories and precise data. Marshaled with both inner (combinatoric) and outer (EDF) techniques, those data may finally realize Fechner s dream of a true psychophysical function.

6 References D'Agostino, R. B., & Stephens, M. A. (Eds.). (986). Goodness-of-fit techniques. New York: Marcel Dekker. Ellermeier, W., & Faulhammer, G. (). Empirical evaluation of axioms fundamental to Stevens's ratio-scaling approach: I. Loudness production. Perception & Psychophysics, 6, 555. Fechner, G. (86/965). Elements of psychophysics (Adler, H. E., Trans., Vol. I). New York: Holt, Rinehart and Winston. Freund, J. E. (99). Mathematical statistics (Fifth ed.). Upper Saddle River: Prentice Hall. Gilden, D. L., Thornton, T., & Mallon, M. W. (995). /f noise in human cognition. Science, 67, Hocutt, M. (974). Aristotle's four becauses. Philosophy, 49, Killeen, P. R. (). Modeling games from the th century. Behavioural Processes, 54, -6. Lockhead, G. R. (99). Psychophysical scaling: Judgments of attributes or objects. Behavioral and Brain Sciences, 5, Norwich, K. H. (977). On the information received by sensory receptors. Bulletin of Mathematical Biology, 9, Parker, S., Schneider, B., & Kanow, G. (975). Ratio scale measurement of the perceived length of lines. Journal of Experimental Psychology: Human Perception and Performance,, 954. Petrusic, W. M., Baranski, J. V., & Kennedy, R. (998). Similarity comparisons with remembered and perceived magnitudes: Memory psychophysics and fundamental measurement. Memory & Cognition, 6, 455. Rule, S. J., Curtis, D. W., & Markley, R. P. (97). Input and output transformations from magnitude estimation. Journal of Experimental Psychology, 86, 449. Schneider, B. A., Parker, S., Ostrosky, D., Stein, D., & Kanow, G. (974). A scale for the psychological magnitude of number. Perception & Psychophysics, 6, Shepard, R. N. (98). Psychological relations and psychophysical scales: On the status of "direct" psychophysical measurement. Journal of Mathematical Psychology, 4, -57. Stigler, S. M. (986). The history of statistics: The measurement of uncertainty before 9. Cambridge, Ma: Harvard University Press. Titchener, E. B. (94). A text-book of psychology. New York: Macmillan. Uttal, W. R. (99). On some two-way barriers between models and mechanisms. Perception & Psychophysics, 48, 88. Uttal, W. R. (998). Toward a new behaviorism: The case against perceptual reductionism. Mahwah, NJ: Lawrence Erlbaum Associates. Ward, L., M. (99). Informational and neural adaptation curves are asynchronous. Perception & Psychophysics, 5, 78. Ward, L. M. (99). Cross-modal additive conjoint structures and psychophysical scale convergence. Journal of Experimental Psychology: General, 9, 675. Ward, L. M., Armstrong, J., & Golestani, N. (996). Intensity resolution and subjective magnitude in psychophysical scaling. Perception & Psychophysics, 58, * I thank David Gilden for the data shown in Figure, and NSF IBN 948 and NIMH K5 MH9 for time to think about them.