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.     

ON EISLER'S GENERAL PSYCHOPHYSICAL DIFFERENTIAL EQUATION AND HIS FECHNERIAN INTEGRATION

The general psychophysical differential equation (GPDE) for calculating 'intrasubjective' relations, is reviewed and it is argued that: ( a ) the use of calculus and Weber functions are unnecessary for the determination of such relations: simple algebraic methods are available, ( b ) Taylor's expansion in the derivation of GPDE is not necessary: a short cut method is suggested and applied instead, ( c ) GPDE is untenable as an empirical model, for it is based on one definition and one assumption which are both arbitrary and purely mathematical, ( d ) The sources of error in the validation procedures used by Eisler are pointed out; ( e ) by means of a validity test and the due treatment of a dozen sets of experimental data, the Fechnerian integration as well as GPDE are invalidated.     

ON PSYCHOPHYSICS IN GENERAL AND THE GENERAL PSYCHOPHYSICAL DIFFERENTIAL EQUATION IN PARTICULAR

The first part of the paper discusses theoretical issues regarding the problem of measurement in psychophysics and the suitable choice of a subjective scale, with special reference to the views of Ekman, Ross, Suppes & Zinnes, and Treisman. The theory of the GPDE is discussed more fully than before. In the second part, further empirical confirmation of the GPDE is given for the following combinations of Weber functions: constant-constant, linear-linear, linear-parabolic, parabolic-parabolic.     

Using contextual effects to derive psychophysical scales

This paper presents a functional measurement analysis of Parducci’s range-frequency theory. The theory provides a basis for (1)finding context-invariant psychophysical scales, (2)establishing the validity of rating scales, and (3) explaining contextual effects in judgment. In an experimental illustration, Ss judged the magnitude of numerals in nine different distributions. All data were used to obtain the psychophysical function for numerals and also to test the model. The large contextual effects were consistent with the model. Quantitative tests of fit supported the equal-interval assumption of the category scale. The psychophysical function appeared nearly linear but with a significant negative acceleration. The data were shown to be qualitatively inconsistent with generalizations of Helson’s theory of adaptation level and Johnson’s correlation-regression theory. They supported Parducci’s range-frequency theory and illustrated how it could be used to factor out contextual effects from effects of stimulus magnitude. Extensions of the range-frequency approach are also discussed.     

The connection between magnitude and discrimination scales and direct and indirect scaling methods

(1) An analysis of the method of paired comparisons shows that Case V and Case VI, the latter characterized by log-normal distributions and Weber's law for subjective continua, are fundamentally indistinguishable. Case VI produces a log-arithmetic interval scale of subjective magnitude. (2) It is demonstrated that the difference between discrimination scales according to Case V and from category rating is due to the difference between intra- and interindividual variability yielding different Weber functions.This research was supported by the Humanistic Faculty of the University of Stockholm and by the Swedish Social Science Research Council. I am indebted to Mr. U. Forsberg for computational assistance.     

Magnitude scales for cold pressor pain

Subjects in two experiments immersed a hand and forearm in ice water (cold pressor test) for 60 sec and gave magnitude estimates of pain every 5 sec. Forty subjects immersed an arm for three trials in Experiment 1, and 60 subjects immersed an arm for two trials in Experiment 2. Linear, power, or additive constant functions fitted the data almost equally well. Exponents of the two-constant power function differed from those of the additive constant function. Individual differences in exponents were marked, but various psychological manipulations aimed at reducing pain failed to affect the exponents. The utility of power functions to describe cold pressor pain has yet to be determined.     

Sensory scales of taste intensity

The subjective intensity of taste was scaled by the method of magnitude estimation in which Os assigned numbers to designate the apparent strength ofstimulus concentrations. Substances used were sucrose, dextrose, maltose, fructose, saccharin, Sucaryl, sodium chloride, and quinine sulfate. For aqueous solutions of each substance, taste intensity was found to increase as a power function of concentration by weight. Some approximate exponents were: sucrose, 1.3; sodium chloride, 1.4; quinine sulfate. 1.0. The magnitude scale for sucrose was compared with the category scale obtained by a commonly used rating procedure. The category scale turned out to be highly nonlinear.     

The response criterion,the stimulus configuration,and the relationship between brightness contrast and brightness constancy

Four experiments were designed to investigate the nature of the relationship between brightness contrast and brightness constancy while controlling the response criterion, the area of the surround, the stimulus configuration, and the mode of appearance of the modulus target. Ten Os in each of the four experiments estimated the apparent whiteness or brightness of targets with different contrast ratios. All targets were viewed at several illumination levels. Most constancy (whiteness and brightness) functions displayed shallow slopes that reflected a good approximation to constancy. The functions within Experiments I, III, and IV were vertically displaced and parallel; those in Experiment II were vertically displaced and increased in slope. This suggests that decreasing the contrast ratio had no effect on the tendency towards constancy when the area of the surround was greater than that of the target but resulted in a decrease in constancy when the area of the surround was equal to that of the target.     

Two subjective scales of number

The way in which the apparent magnitude of numbers grows with their absolute magnitude was measured with a modified version of the direct technique Marks and Slawson (1966) used to determine the psychophysical exponent for loudness. This modified technique required subjects to estimate how evenly and randomly a sequence of integers appeared to sample the numerical continuum. The results indicate that the apparent magnitude of numbers increases with a decelerated power function of their arithmetic magnitude when a series samples from an open-ended range. However, when an upper boundary of the range is specified, the subjective scale seems to be linear. Random productions of numbers parallel the results found with judgments of presented sequences. The two scales of number provide the basis for an interpretation of the difference between magnitude and category scales: that subjects use numbers differently when the response scale is open-ended Imagnitude estimation than when it has a fixed upper limit tcategory scale. Given the assumption that subjects use numbers in this way in the two tasks, the qualitative relation between magnitude and category scales is predicted.     

Weber laws,the Weber law,and psychophysical analysis

In failing to define the units in which the stimulus is to be measured, the Weber law might seem to make no definite assertion, and indeed, it is shown that any single empirical function, supposed to relate a given stimulus intensity with that intensity which is just noticeably greater, can be put into the Weber form by a suitable change of scale in which the stimulus intensity is to be measured. Nevertheless, it turns out that if different individuals have different Weber functions, when the intensities are measured on a given scale, then it is by no means always possible to transform the scale so that all of the functions can take on the Weber form. Some necessary conditions are given for the possibility of such a transformation when there is at hand a finite number of functions, and when the functions depend upon a single parameter the necessary and sufficient condition is easily derived. The same discussion leads to a generalization of Thurstone's psychophysical scale and shows that such a scale is always possible.     

When are semantic targets detected faster than visual or acoustic ones?

Duration-discrimination data from an experiment using empty auditory intervals in a two-alternative forced-choice paradigm are presented. The observed functional relationship between standard deviation of the psychometric density function and stimulus duration is shown to be fit significantly better by a Weber’s law model of duration discrimination than by Creelman’s counter model. Both models fail to predict the rapid rise in the Weber fraction observed for durations longer than about 2 sec. However, the Weber’s law model, based on a generalization of Weber’s law, accurately predicts the initial drop in the Weber fraction for very short durations and the observed constancy of the Weber fraction for durations up to 2 sec.     

Neither strong nor weak space constancy is coded in striate cortex

Space constancy (the perception that the world remains stable despite eye movements) might be mediated by receptive fields with constant locations in the world rather than constant locations on the retina. In a strong form the hypothesis, receptive fields of single neurons would compensate for all eye movements, whereas in a weak form they would compensate only for the small movements of fixation that occur without retargeting. Early receptive field mapping confounded world-centered and retina-centered organizations by fixing the retina relative to the world. This paper links the strong and weak forms of constancy by relating previously isolated data, and illustrates the issues with a previously unpublished figure showing that striate receptive field have constant retinal locations. The first tests of space constancy in striate cortex contradicted the strong hypothesis; recent tests contradict the weak hypothesis, also. The results are consistent with modern theories of space constancy, differentiating between motor-oriented and perceptual functions of vision, and differentiating between constancy during pursuit and constancy across saccades. Received: 5 January 1998 / Accepted: 16 July 1998     

Direct estimation: effect of methodological factors on scale tupe

Abstract.— An experiment was performed to investigate the importance of methodological differences between magnitude estimation and category rating. It was found that the form of a scale varied with (a) the range of responses used by the subject and (b) with his freedom of choosing a highest number. Other factors investigated played a minor role. A narrow range and fixed upper scale value yielded the typical category scale, a wide range with no restriction on the highest value the typical magnitude scale. The same factors that determined the form of the scale also affected the Weber functions.     

Direct and indirect tests of memory for category exemplars in young and older adults

Young and older adults were compared on direct (cued recall) and indirect (exemplar generation) tests of memory for category members. Because category names served as cues in both tasks, amount of retrieval support was constant across tasks. Although older adults produced fewer category members in cued recall, priming of category exemplars in the generation task did not vary with age. These results suggest that age constancy in priming tasks does not depend on physical similarity between study materials and retrieval cues provided at test and point to the importance of deliberate recollection as a factor in determining the extent of age differences in memory.     

Psychophysical functions for active tactual discrimination of line width by blind children

Two psychophysical experiments were conducted on active tactual discrimination of line width using the method of constant stimuli with blind subjects. In Experiment I, 93 subjects made single judgments over each of five standards between 0.04 and 0.12 in. In Experiment II, 6 subjects were tested repeatedly over seven standards between 0.04 and 0.50 in. Over the range of standards common to both experiments, the Weber fractions decreased as the width of the standard increased. For two larger standards (0.25 and .50 in.), the Weber fractions approached asymptote and then increased respectively suggesting a U-shaped function. No differences in accuracy of discrimination were found between males vs. females, age groupings, or grade groupings.     

The effects of duration,area, and background intensity on the visual intensity difference threshold given by the forced-choice procedure: Derivations from a statistical decision model for sensory discrimination

A model for visual discrimination is outlined. This assumes that detection is based on separating signaland noise by means of a statistical decision procedure, and that the noise arises from physical and biological sources. When the model is applied to such traditional detection procedures as the method of constant stimuli, it successfully predicts the laws of spatial and temporal summation and the form of the Weber function. The model is extended to the forced-choice procedures. It is found that it predicts departures from the classical laws for spatial and temporal summation and an increase in the slope of the Weber function when the pulsed forced-choice procedure is used, and data are presented that agree with the predictions. It is shown that data obtained with forced choice against a continuous background are consistent with the assumption that the Seither covertly detects or fails to detect an increment at each ofthe two alternative locations and this determines his final overt response, rather than that he directly compares the inputs at the two locations. A comparison between results obtained with the forced-choice procedure with a pedestal, or with the continuous forced-choice procedure, provides further evidence that Ss employ a double detection strategy in the latter case.     

Ratio scales of sugar sweetness

In a series of 10 experiments, groups of Os judged the sweetness of 16 sugars. The results suggest that, for all sugars except mannose, the intensity of sweetness grows as a power function of concentration, with an exponent of about 1.3. The relative sweetness of sugars was determined using both molarity and per cent by weight. With both measures, sucrose and fructose were the sweetest sugars. The order of the remaining sugars in the sweetness hierarchy was partiy a function of the measure of concentration. The variability of the magnitude estimates of sweetness was roughly proportional to the stimulus concentration, supporting Weber’s law.     

Direct quantitative judgments of sums and a two-stage model for psychophysical judgments

Judged magnitudes of differences between stimuli have previously been shown to support a two-stage interpretation of magnitude estimation, in which input transformations and output transformations are each describable as power functions. In an effort to provide support for the model independent of the difference estimation procedure. the present investigation employed two additional judgment tasks. We obtained magnitude judgments and category judgments of the combined magnitudes (sums) of paired weights from two groups of Ss. Values of the inferred input exponent k calculated from the two sets of data were very similar and were also remarkably similar to the exponent previously calculated from magnitude estimations of differences between weights. The output exponent calculated from magnitude judgments of sums described a concave upward function; however. the similar function describing category judgments was essentially linear. These results show that the inferred input exponent is not the result of the difference estimation task, and in addition provides support for the contention that the interval scale may be a less biased sensory measure than the magnitude scale. The introduction of an additive constant to the model improved its fit to the data but the rule by which it was introduced made very little difference.     

Psychophysical scales of apparent heaviness and the size-weight illusion

The apparent heaviness of a set of 40 cylindrical objects was scaled by the method of magnitude estimation. The objects varied in weight, volume. and density. There were three main conclusions: (1) For any constant volume, heaviness grows as a power function of weight; the larger the volume. the larger the exponent of the power function. The family of such power functions converge at a common point in the vicinity of the heaviest weight that can be lifted. (2) For any constant density (i:e., weight proportional to volume), heaviness does not grow as a power function of weight. (3) For any constant weight, heaviness decreases approximately as a logarithmic function of volume; the constants of the log function depend systematically on the weight of the object. The outcome furnishes a broad quantitative picture of apparent heaviness and of the size-weight illusion (Charpentier’s illusion).     

F tests for the absolute invariance of dominance and composition scales

Significance tests are developed for evaluating the absolute invariance of the dominance and composition scales presented in a previous paper [Bechtel, 1967]. These tests derive from the multivariate normality of the estimated scale values—this multinormality stemming from that of the observations upon which these estimated scale parameters are based. The scales compared by means of the present techniques will typically be constructed under distinct experimental conditions, which may take the concrete form of treatments, occasions, individuals, or groups. Thus the present tests possess a wide range of applicability including various experimental designs in which scale configuration is the dependent variable. Illustrative data are used in statistically comparing the scale structures of two different individuals.This research was supported in part by National Institutes of Health Grant MH-04439-06. The author would like to express his appreciation to Fred C. Andrews of the University of Oregon and J. E. Keith Smith of the University of Michigan for their helpful comments concerning aspects of this work. Computing assistance was obtained from the Health Sciences Computing Facility, UCLA, sponsored by NIH Grant FR-3.