A GENERAL DIFFERENTIAL EQUATION IN PSYCHO-PHYSICS: DERIVATION AND EMPIRICAL TEST

By a reasoning similar to Fechner's a general psychophysical differential equation is developed. Its integration allows the calculation of intrasubjective relations, provided the Weber functions of the two pertinent variables are known. The method is empirically validated for the relation between category and magnitude scales and for the similarity function. Theoretical implications are discussed.     

The Fechnerian idea

From the principle that subjective dissimilarity between 2 stimuli is determined by their ratio, Fechner derives his logarithmic law in 2 ways. In one derivation, ignored and forgotten in modern accounts of Fechner's theory, he formulates the principle in question as a functional equation and reduces it to one with a known solution. In the other derivation, well known and often criticized, he solves the same functional equation by differentiation. Both derivations are mathematically valid (the much-derided "expedient principle" mentioned by Fechner can be viewed as merely an inept way of pointing at a certain property of the differentiation he uses). Neither derivation uses the notion of just-noticeable differences. But if Weber's law is accepted in addition to the principle in question, then the dissimilarity between 2 stimuli is approximately proportional to the number of just-noticeable differences that fit between these stimuli: The smaller Weber's fraction the better the approximation, and Weber's fraction can always be made arbitrarily small by an appropriate convention. We argue, however, that neither the 2 derivations of Fechner's law nor the relation of this law to thresholds constitutes the essence of Fechner's approach. We see this essence in the idea of additive cumulation of sensitivity values. Fechner's work contains a surprisingly modern definition of sensitivity at a given stimulus: the rate of growth of the probability-of-greater function, with this stimulus serving as a standard. The idea of additive cumulation of sensitivity values lends itself to sweeping generalizations of Fechnerian scaling.     

Analytical methods in the theory of psychophysical discrimination I: Inequalities, convexity and integration of just noticeable differences

We re-examine the theoretical status of Fechner's Mathematical Auxiliary Principle [Fechner, G. T. (1889). Elemente der psychophysik. Leipzig: Breitkopf und Härtel] which underlies Fechner's method of constructing a sensory scale by integrating just noticeable differences (jnds). That “Principle” has been roundly criticized [Luce, R. D., & Edwards, W. (1958). The derivation of subjective scales from just noticeable differences. Psychological Review, 65, 227-237] as being inconsistent with the very basis of Fechner's psychophysical theory, and indeed this is the case. In important papers Pfanzagl [(1962). Über die stochastische Fundierung des psychophysischen Gesetzes (On the stochastic foundations of the psychophysical law). Biometrische Zeitschrift, 4, 1-14] and Krantz [(1971). Integration of just noticeable differences. Journal of Mathematical Psychology, 8, 591-599] resurrected Fechner's method; their analysis showed that the sensory scale could be written as the limit of a sequence of integrals, each of the form suggested by the auxiliary principle. In this work, we investigate the properties of a typical member of the Krantz-Pfanzagl sequence of integrals; we do so with the view to obtaining useful approximations to the true scale. Weber's inequality [Falmagne, J.-Cl. (1977). Note: Weber's inequality and Fechner's problem. Journal of Mathematical Psychology, 16, 267-271] plays an important role in our developments. That inequality, and other inequalities of a similar nature, allows us to place bounds on the error incurred by approximating a true scale u by an integral of jnds. Under appropriate conditions these bounds are sufficiently tight that the relative error is very small over the entire stimulus domain. We illustrate our theoretical results with a number of examples.     

A note on cross-modality matching

”Stevens' law”, that the psychophysical law is a power function, is often taken to be confirmed by results of cross-modality matching experiments, for it predicts both the fact that cross-modality matching experiments yield power functions, and the exponents of these power functions. Both these predictions, however, follow from a more general form of the psychophysical law, of which Fechner's law is a special case. In view of this, an interpretation of cross-modality matching based on stimulus discriminability rather than sensation magnitude is proposed.     

The psychophysics of mental test difficulty

By the use of the Seashore tests of Pitch Discrimination, Intensity Discrimination, Time Discrimination, and the test of Tonal Memory, it is shown that the easiness of an item, as determined by absolute scaling methods, is proportional to the logarithm of the magnitude of the stimulus. It is proposed that this is a case of Fechner's psychophysical law and that the unit of absolute scaling as applied to test items may become a satisfactory unit of allS-scales in the more traditional psychophysical problem.     

Antecedents to Fechner's law: the astronomers J. Herschel, W. R. Dawes, and N. R. Pogson

The history of the stellar magnitude scale is briefly traced from the second century b.c. until the middle of the nineteenth century. It becomes clear that astronomers formulated "Fechner's Law" by about 1850. While Fechner is credited with the grander view of things, the contention is made that the astronomers John Herschel, W. R. Dawes, and N. R. Pogson have not been given their due by historians of psychology.     

Is there an auditory analogue to Fechner's paradox in binaural loudness perception?

In binocular brightness perception a phenomenon called Fechner's paradox can be observed. This paradox implies non-monotonicities in the psychometric functions of binocular brightness. Lehky (1983) proposed a model that describes such non-monotonicities. He suggested that Fechner's paradox also exists in binaural loudness perception. However, until now no sufficient data have been collected to test this hypothesis. Therefore, an experiment was conducted in which 36 psychometric functions were obtained using binaural stimuli in the range of intensities in which Fechner's paradox supposedly occurs. As a result, no significant non-monotonicities were found. However, it is shown that jnds derived from the psychometric functions contradict predictions derived from the limited binaural additivity model of Gigerenzer and Strube (1983).     

Magnitude scales,category scales,and the general psychophysical differential equation

The general psychophysical differential equation (GPDE) which relates scale values and Weber functions for two subjective variables was tested statistically with data from a number of category rating and magnitude estimation experiments. In all tests. it was investigated whether the GPDE is compatible with the assumption that the Weber function of the category scale is constant. This assumption implies that the category scale may be regarded as a discrimination scale. The tests were carried out by estimating the Weber function of the category scale by means of the GPDE and testing these estimates for constancy, For group data, there were fairly small systematic deviations from constancy across different experiments in the Weber functions that were estimated by the GPDE, and the hypothesis of a constant estimated Weber function could not be rejected statistically (p>.05) in most experiments. For individual data the estimated Weber functions deviated from constancy in idiosyncratic ways, and these deviations were statistically significant (p<.05)for most subjects.     

ON THE PROBLEM OF CATEGORY SCALES IN PSYCHOPHYSICS

E isler , H. On the problem of category scales in psychophysics. Scand. J. Psychol ., 1962, 3 , 81–87.—In the first Part of the paper it is shown that, if three assumptions are granted, the category scale must be logarithmically related to the magnitude scale: K =α log φ +β . In the second part of the paper, the relation K =α log ( kφ+ q) +β is derived, starting from the assumptions that the category scale is a pure function of discrimination and that discrimination is appropriately described by the linear generalization of Weber's law for prothetic continua. We can reconcile these two formulas by defining the zero-point of the magnitude scale as the point where variability vanishes.     

Psychophysical judgment: Wundt's theory revisited

Summary Wundt's psychological interpretation of Weber's and Fechner's laws rests on two main postulates. First, the magnitude of sensory excitation is proportional to the intensity of the stimulus that produces it. Second, in apperception, sensory comparisons and judgments are made according to psychological relationships: Two sensations are just-noticeably different when they fall in a constant psychological ratio. Wundt's theory implies a hierarchical organization, in which sensory-perceptual processes are embedded within higher-level, apperceptive processes. Such an organization is compatible with recent psychophysical research on loudness and with a model that goes toward resolving some controversies in loudness scaling, notably the seeming incompatibility of Stevens's sone scale and Garner's lambda scale.     

Fechner's "inner psychophysics"

Though psychologists are generally aware that Gustav Fechner introduced psychophysics and set down its essential methodology, most of them only know about the part that Fechner called "outer psychophysics." In his classic publication of 1860, Fechner insisted that "inner psychophysics" was more important, yet this aspect of Fechner's work failed to receive any attention. The article reviews Fechner's presentation of inner psychophysics and suggests reasons why that part of his work was neglected and has been forgotten.     

Ratio scale measurement of the perceived lengths of lines.

In Experiments 1 and 2, magnitude and category estimates of perceived line length difference were used to rank order 36 pairs of lines with respect to the psychological length difference between the lines forming a pair. In Experiment 3 magnitude estimates of line length ratios were used to order the same 36 pairs with respect to the psychological length ratio between the lines comprising a pair. A nonmetric analysis of the rank-order of the difference and ratio judgments was used to construct a ratio scale of perceived line length. Perceived line length was found to be a power function of physical line length with an exponent of .46. In Experiments 4 and 5, magnitude and category estimates of line length similarity were obtained. A nonmetric analysis of these data indicated that similarity judgments were monotonic inverses of judgments of psychological ratios.     

On theoretical and realizable ideal conditions in psychophysics: Magnitude and category scales and their relation

By spacing 10 stimuli (white noise) between 40 and 110 dB according to two criteria [equal response ambiguity (ERA) and equal discriminability (ED)], an attempt was made to construct an “ideal” case for magnitude estimation and category rating. The “ideal” case is defined by linear and constant Weber functions (SDs as a function of scale values) for the two scales, respectively. Altogether, three group and two individual magnitude and category rating experiments were run with these two spacings. It was found that the ERA spacing approximated the ideal case well for both Weber functions and the ED spacing only for the Weber function of the category scale. The general psychophysical differential equation that relates scale values and Weber functions for the two scales allowed good prediction of the category scales from the magnitude scales and the Weber functions. The data suggested a distinction between phenotypic (empirical) and genotypic Weber functions, analogous to “real” and “ideal” cases in physics.     

Inference using categories

How do people use category membership and similarity for making inductive inferences? The authors addressed this question by examining the impact of category labels and category features on inference and classification tasks that were designed to be comparable. In the inference task, participants predicted the value of a missing feature of an item given its category label and other feature values. In the classification task, participants predicted the category label of an item given its feature values. The results from 4 experiments suggest that category membership influences inference even when similarity information contradicts the category label. This tendency was stronger when the category label conveyed class inclusion information than when the label reflected a feature of the category. These findings suggest that category membership affects inference beyond similarity and that category labels and category features are 2 different things.     

The contribution of category relationships to similarity ratings.

Two experiments produced evidence that category relationships contribute to similarity ratings. In Experiment 1, participants gave similarity ratings with respect to a semantic category (VEGETABLE) and a set of exemplars, some of which were members of the category (e.g., BROCCOLI) and some of which were not (e.g., BANANA). A regression analysis was used to predict the similarity ratings in terms of numbers of common and distinctive features, as reported by other participants. Perceived similarity was greater for examplars that were members of the category, independently of feature overlap. Experiment 2 examined similarity ratings with respect to pairs of exemplars. In some cases, both exemplars were members of the same category (e.g., BROCCOLI/CUCUMBER). In other cases, one exemplar was a member of the category and the other was not (e.g., BROCCOLI/BANANA). A regression analysis was used to predict the similarity ratings in terms of numbers of common and distinctive features. Perceived similarity was greater when both exemplars were members of the same category, independently of feature overlap.     

Similar as well as dissimilar contextual stimuli increase rated similarity

Contextual stimuli from one category increase the rated similarity of stimuli from a second category. In Experiment 1, associates of one of the members of pairs of target words increased the rated similarity of the members slightly more than did much less similar contextual stimuli. In Experiment 2, contextual category names and good exemplars increased the rated similarity of pairs of poor to medium target exemplars and were also rated as more similar to these targets than the targets were to each other. In both cases, the category names resulted in the stronger effect. This ability of contextually similar stimuli to increase rated similarity opposes the distance-density and averaging-judgmental theories of the effect of context on similarity. However, the theory that a positive association exists between the perceived similarity of stimuli and the strength of the category to which they belong explains how both contextually dissimilar and contextually similar stimuli can increase rated similarity.     

特征相似性、类别标签与因果关系在归纳推理中的作用

采用人工材料,比较类别标签、特征相似性与因果关系对归纳推理强度的影响。包括两个实验,实验1比较类别标签与特征相似性对归纳推理的影响,结果表明,当类别标签对归纳推理的影响显著强于特征相似性时,不能将类别标签等同于一个相似性特征。实验2进一步探讨类别标签与因果关系对归纳推理的作用,结果表明,因果关系作用明显强于类别标签的作用。综合两个实验的结果并整合前人相关研究,提出了不同关系影响归纳推理的强度假想。     

The effect of semantic relatedness on immediate serial recall and serial recognition

Several researchers have reported that learning a particular categorization leads to compatible changes in the similarity structure of the categorized stimuli. The purpose of this study is to examine whether different category structures may lead to greater or less corresponding similarity change. We created six category structures and examined changes in similarity within categories or between categories, as a result of categorization, in between-participant conditions. The best supported hypothesis was that the ease of learning a categorization affects change in within-categories similarity, so that greater (within-categories) similarity change was observed for category structures that were harder to learn.     

Relative olfactory intensity perception as mediated by ratio-range category scale responses

Scales whose categories are labeled with ranges of ratio values are compared with verbal category scales and magnitude estimation. Relative perceived intensities of Eugenol odor were scaled by power law methods, using 102 Ss, five scaling methods-one verbal, two numerical, and two magnitude estimation-and making comparisons against two alternate odor reference standards. Variations in the psychophysical exponent values derived under each condition were examined. Comparisons between scale types were made: numerical ratio-range category scales may behave as magnitude estimations or as category scales depending on the way responses are scored by the experimenter.     

Categories and particulars: prototype effects in estimating spatial location

A model of category effects on reports from memory is presented. The model holds that stimuli are represented at 2 levels of detail: a fine-grain value and a category. When memory is inexact but people must report an exact value, they use estimation processes that combine the remembered stimulus value with category information. The proposed estimation processes include truncation at category boundaries and weighting with a central (prototypic) category value. These processes introduce bias in reporting even when memory is unbiased, but nevertheless may improve overall accuracy (by decreasing the variability of reports). Four experiments are presented in which people report the location of a dot in a circle. Subjects spontaneously impose horizontal and vertical boundaries that divide the circle into quadrants. They misplace dots toward a central (prototypic) location in each quadrant, as predicted by the model. The proposed model has broad implications; notably, it has the potential to explain biases of the sort described in psychophysics (contraction bias and the bias captured by Weber's law) as well as symmetries in similarity judgments, without positing distorted representations of physical scales.