Average exponent from magnitude production

When magnitude production is used to obtain a psychophysical power function for a group ofSs, the group exponent is shown to be the harmonic mean of individual-S exponents under the conditions usually employed in empirical research. The conditions are (a) the same numbers are presented to eachS, (b) responses to each number are pooled overSs by taking geometric means, and (c) the function is fit by the method of least squares in which regression is of the logarithm ofSs' responses upon the logarithm of numbers presented byE.This paper was supported in part by grant no. APA-151 from the National Research Council of Canada. The author wishes to thank William W. Rozeboom for his helpful comments on this paper.     

Evaluation of three sensitivity measures in magnitude estimation tasks

Taking into account the studies about the measure of sensitivity in magnitude estimation tasks, we analyze the three most common measures used in this topic: Pearson's product-moment correlation between the logarithm of the stimulus and the logarithm of the response (R), the exponent of Stevens' power function (K), and the measure "M" proposed by Garriga-Trillo. Using a sample of participants greater than usual in psychophysical studies (180 participants), we designed an experiment with two sets of stimuli with different stimulus ranges. In each of these sets, we used two kinds of stimuli (line segments and squares). Our conclusions were: (1) we rejected the use of K as a sensitivity measure because the results provided by this index were the opposite of those expected when we compared the two stimulus ranges. (2) We also rejected the use of M because this measure is a linear transformation of Kendall's coefficient of concordance. (3) Lastly, we suggest the mathematical transformation proposed by Fisher to achieve a normal distribution, and recommend this transformation as the best sensitivity measure.     

The bisection of a brightness interval by pigeons

Pigeons were trained on a discrete trials, successive discrimination procedure, in which the stimuli were two luminance values on the center key. Behavior was maintained by 25% reinforcement of correct responses on two side-keys. During occasional test trials the luminance of the center key was maintained at one of a number of values, intermediate to those of the two training stimuli, and a function relating the relative frequency of responses on the two side keys to stimulus intensity was obtained. The intersection of this function with the 50% line provided an estimate of the bisection point. Since no bisection point occurred below the geometric mean of the interval, the results were not consistent with a logarithmic scale of brightness but fitted the general mean theorem with an exponent of 0.24. With continued testing, the performance of individual subjects oscillated in an irregular manner about the mean bisection point. The relative stability of the test behavior and the absence of context effects indicated that the method was suitable as a general procedure for measuring stimulus distances.     

Odor intensity: Differences in the exponent of the psychophysical function

A series of five experiments showed that there are reliable differences among the exponents of the psychophysical power functions for odorants. There was virtually a perfect rank-order correlation between the size of the exponent and the water-solubility of the odorants. The exponents for odorants that are completely soluble in water (n-propanol and acetone) were approximately 2.5 times the size of the exponents for odorants that are insoluble in water (n-octanol and geraniol). For n-aliphatic alcohols, the size of the exponent and solubility in water decrease as a function of carbon chain-length. Although the exponents were higher when the stimuli were delivered with an air-dilution olfactometer than when they were sniffed from cotton swabs, the relative values among odorants were independent of the method of stimulus presentation.     

Stimulus sequence and the exponent of the power function for loudness

In two experiments, 15 and 13 subjects estimated the loudness of 12 sound-pressure levels (38-104 dB; 6-dB intervals) of a 1000-Hz tone by the method of magnitude estimation with a modulus assigned to the first stimulus presented. The tone duration was 1 sec. and the interstimulus interval was 6 sec. The presentation order was systematically ascending-descending in one experiment and balanced-irregular in the other. The results indicate that (1) loudness is a power function of sound pressure with an exponent of 0.60 for the systematic order and 0.29 for the irregular order. (2) For both the irregular and systematic orders, a large step-size (12 or 18 dB) between the stimulus on Trial n and on Trial n-1 (or n-3) results in a slight assimilation effect. This also occurs for the small step-size (6 dB) in the irregular order. (3) The size of momentary exponents (based on two points, Trials n and n-1 or n-3) depends on the sound pressures of successive stimuli, whether the steps are positive or negative, and whether the stimuli have been presented in systematic or irregular order. For positive steps, the momentary exponent is lower for a soft tone (Trial n) than for a loud tone, whereas for negative steps the momentary exponent is lower for a loud tone than for a soft tone. These effects ar more pronounced when these stimuli are presented in an irregular order. A relative judgment model is offered for magnitude estimation. It assumes that subjects judge the loudness of a stimulus in terms of three reference markers: the minimum and maximum sound pressures as well as the sound pressure of the previous stimulus.     

A test of a two-stage model of magnitude judgment

It has been suggested that the power law J = aⁿ, describing the relationship between numerical magnitude judgments and physical magnitudes, confounds a sensory or input function with an output function flawing to do with O’s use of numbers. Judged magnitudes of differences between stimuli offer some opportunity for separating these functions. We obtained magnitude judgments of differences between paired weights, as well as magnitude judgments of the weights making up the pairs. From the former we calculated simultaneously an input exponent and an output exponent, working upon Attneave’s assumption that both transformations are describable as power functions. The inferred input and output functions, in combination, closely predict the judgments of individual weights by the same Os. Although pooled data (geometric means of judgments) conform fairly well to a linear output function, individual data do not; i.e., individual Os deviate quite significantly fromlinearity and from one another in their use of numbers. Individual values of the inferred sensory exponent, k, show significantly better uniformity over Os than do values of the phenotypica! magnitude exponent previously found to describe interval judgments of weight.     

Random intercept item factor analysis

The common factor model assumes that the linear coefficients (intercepts and factor loadings) linking the observed variables to the latent factors are fixed coefficients (i.e., common for all participants). When the observed variables are participants' observed responses to stimuli, such as their responses to the items of a questionnaire, the assumption of common linear coefficients may be too restrictive. For instance, this may occur if participants consistently use the response scale idiosyncratically. To account for this phenomenon, the authors partially relax the fixed coefficients assumption by allowing the intercepts in the factor model to change across participants. The model is attractive when m factors are expected on the basis of substantive theory but m + 1 factors are needed in practice to adequately reproduce the data. Also, this model for single-level data can be fitted with conventional software for structural equation modeling. The authors demonstrate the use of this model with an empirical data set on optimism in which they compare it with competing models such as the bifactor and the correlated trait-correlated method minus 1 models.     

The influence of object familiarity on magnitude estimates of apparent size.

Three magnitude-estimation experiments were used to determine the exponents of the power function relating size judgments and physical size for two-dimensional familiar and unfamiliar stimuli. The exponent of the power function was used to index the effect of familiar size on perceived size under a variety of conditions, from full-cue to reduced-cue viewing conditions. Although the value of the exponents varied across the three experiments, within each experiment the exponent of the familiar stimulus was not significantly different from that of the unfamiliar stimulus, indicating that familiar size does not influence the rate of growth of perceived size. The results of a fourth experiment excluded a possible explanation of the findings of experiments 1-3 in terms of subjects responding to relative angular size as a consequence of the successive presentation of the different-sized representations of the familiar stimulus. Taken together, the present findings are consistent with the hypothesis that the influence of familiar size on estimates of size mainly reflects the intrusion of nonperceptual processes in spatial responses.     

Intensity resolution and subjective magnitude in psychophysical scaling

Several successful theories of psychophysical judgment imply that exponents of power functions in scaling tasks should covary with measures of intensity resolution such asd’ in the same tasks, whereas the prevailing metatheory of ideal psychophysical scaling asserts the independence of the two. In a direct test of this relationship, three prominent psychophysical scaling paradigms were studied: category judgment without an identification function, absolute magnitude estimation, and cross-modality matching with light intensity as the response continuum. Separate groups of subjects for each scaling paradigm made repeated judgments of the loudnesses of the pure tones that constituted each of two stimulus ensembles. The narrow- and wide-range ensembles shared six identical stimulus intensities in the middle of each set. Intensity resolution, as measured byd’-like distances, of these physically identical stimuli was significantly worse for the wide-range set for all three methods. Exponents of power functions fitted to geometric mean responses, and in magnitude estimation and cross-modality matching the geometric mean responses themselves, were also significantly smaller in the wide-range condition. The variation of power function exponents, and of psychophysical scale values, for stimulus intensities that were identical in the two stimulus sets with the intensities of other members of the ensembles is inconsistent with the metatheory on which modern psychophysical scaling practice is based, although it is consistent with other useful approaches to measurement of psychological magnitudes.     

Interresponse-time analysis of stimulus control in multiple schedules

Interresponse-time distributions were recorded in two components of multiple variable-interval schedules that were varied over several conditions. Values of the exponent for power functions relating ratios of interresponse times emitted per opportunity to ratios of reinforcers obtained in the two components varied with interresponse-time class interval. The exponent (sensitivity to reinforcement) afforded a measure of stimulus control exerted by the discriminative stimuli. Exponents were near zero for short interresponse times, consistent with previous conclusions that responses following short interresponse times are controlled by response-produced or proprioceptive stimuli. Values of exponents increased with longer interresponse times, indicating strong control by exteroceptive stimuli over responses following interresponse times of approximately one second or longer.     

Interpersonal and intrapersonal variance of exponent of power function observed in grip strength task.

For grip strength there is a power function with an exponent of 1.7 between the subjective magnitude and the actual force exerted by a subject, but large variabilities among and within individuals were found. We focused on these variabilities and investigated the relationship between them by conducting a ratio production procedure requiring trials of maximum effort and half of maximum effort. For 30 adults we conducted four measurement trials, two on the same day, and the remaining two trials on a day or two later. The mean value of the exponent, the standard deviation, and the coefficient of variation of the four trials for each subject were calculated. The mean value of the exponent of the power function for all subjects was 1.6. This value approximated the value of 1.7 reported by Stevens and Mack. The values ranged from .50 to 5.39. The correlation between subjects' mean exponent value and standard deviation was .90, and the correlation between the mean value of the exponent and the coefficient of variation was .50. There was a close relationship between interpersonal and intrapersonal variance.     

Size contrast as a function of conceptual similarity between test and inducers

In four experiments, the effect of the semantic relationship between test and inducing stimuli on the magnitude of size contrast in an Ebbinghaus-type illusion was explored. In Experiments 1 and 2, the greatest illusion was found when test and inducing stimuli were identical in shape and differed only in size. Decreased size contrast was found when inducing stimuli were drawn from the same category as the test stimulus, but were not visually identical. Even less size contrast was found when inducing stimuli were from a near conceptual category, with the least effect when they were drawn from a completely different category. In Experiment 3, it was demonstrated that even if test and inducing stimuli are drawn with identical geometric elements, the size contrast illusion is greatly reduced if they represent apparently different conceptual categories (through the manipulation of orientation and perceptual set). In Experiment 4, any geometric or spatial confounds were ruled out. These results suggest that size contrast is strongly influenced by the conceptual similarity between test and inducing stimuli.     

Quantitative functions for size and distance judgments

A psychophysical approach was used to obtain judgments of visual extent under three conditions. In tuvo conditions a comparison stimulus at each of two distances was matched in size to a standard which varied in distance. Stimuli were presented on a well-lighted table and were judged by two observers under Objective instructions. Both the standard and comparison were located in either a frontal or longitudinal plane. In a third condition relative distance estimates were given of two stimuli which varied in their relative positions along the table. The mean results for all conditions were described as a power function of physical stimulus measures. The exponent was greater than 1.0 for frontal size and usually less than 1.0 for flat size and distance. The position of the comparison affected the magnitude of the exponents to a lesser degree. These findings have relevance for interpretations of size and distance judgments.     

Haptic processing of the location of a known property: does knowing what you've touched tell you where it is?

The relationship between knowing where a haptic property is located and knowing what it is was investigated using a haptic-search paradigm. Across trials, from one to six stimuli were presented simultaneously to varying combinations of the middle three fingertips of both hands. Participants reported the presence/absence of a target or its location for four perceptual dimensions: rough/smooth, edge/no edge, relative position (right/left), and relative orientation (right/left). Reaction time data were plotted as a function of set size. The slope data indicated no difference in processing load for location as compared to identity processing. However, the intercept data did reveal a cost associated with processing location information. Location information was not obtained for "free" when identity was processed. The data also supported a critical distinction between material and edge dimensions versus geometric dimensions, as the size of the cost associated with processing location was larger for spatial than for intensive stimuli.     

Apparent combined length of two-line and four-line sets

Recent studies, using single-line stimuli, show apparent length to be a power function, with exponent 1.0, of objective or physical length. If apparent lengths are additive, then, given the 1.0 exponent, two lines should appear to S to have the same total length as the single line they would form if physically joined. When S adjusted the length of one line to match the combined length of two other lines, however, he generally made the variable line much longer than the actual combined length of the two lines. Dividing the total length equally between the two lines, so that each had 50%, represents the greatest departure from a single line, yet the largest overestimation occurred when one of the two lines had 65% to 75% of the total length. The overestimation was greater when the lines were spaced farther apart, suggesting that the amount of area occupied by the lines affected apparent combined length.     

Magnitude estimation of manually assessed elastic stiffness: stability of the exponent

Magnitude estimations were obtained for manual assessments of pure elastic stiffness stimuli (metal springs). 20 subjects of varied experience in manual assessment of spinal stiffness volunteered to participate. The mean exponent of the power function relating perceived magnitude of elastic stiffness to measured physical magnitude was 1.65. Exponents varied across the 20 individuals but were stable across testing sessions held at least 2 weeks apart, and the size of the exponent was not related to prior experience.     

Stimulus Range Effects in Temporal Bisection by Humans

Two experiments with human subjects, using short-duration tones as stimuli to be judged, investigated the effect of the range of the stimulus set on temporal bisection performance. In Experiment 1, six groups of subjects were tested on a temporal bisection task, where each stimulus had to be classified as "short" or "long". For three groups, the difference between the longest (L) and shortest (S) durations in the to-be-bisected stimulus set was kept constant at 400 msec, and the L / S ratio was varied over values of 5:1 and 2:1. For three other groups, the L/S ratio was kept constant at 4:1 but the L-S difference varied from 300 to 600 msec. The bisection point (the stimulus value resulting in 50% 'long' responses) was located closer to the arithmetic mean of L and S than the geometric mean for all groups except that for which the L / S ratio was 2:1, in which case geometric mean bisection was found. In Experiment 2, stimuli were spaced between L and S either linearly or logarithmically, and the L / S ratio took values of either 2:1 or 19:1. Geometric mean bisection was found in both cases when the L / S ratio was 2:1, but effects of stimulus spacing were found only when the L / S ratio was 19:1. Overall, the results supported a previous conjecture that the L / S ratio used in a bisection task played a critical role in determining the behaviour obtained. A theoretical model of bisection advanced by Wearden (1991) dealt appropriately with bisection point shifts discussed above but encountered difficulties with stimulus spacing effects.     

Determinants of cumulative successive contrast in saltiness intensity judgments

When all stimuli elicit the same taste quality, solutions preceded by a high concentration level are judged to be significantly less intense than solutions preceded by a low concentration level. After repetitious stimulation with a different tasting stimulus, the intensity of the present stimulus is overestimated. This phenomenon is called “successive contrast.” In the present study, the cumulative effects of three identical stimuli on the saltiness ratings for a test stimulus are investigated. The preceding stimuli are manipulated with regard to taste quality, saltiness intensity, total taste intensity, and complexity. Whether the size of the cumulative contrast effect is associated with the degree of dissimilarity between preceding stimuli and test stimulus, or with the saltiness or total taste intensity of the preceding stimuli, is investigated. The size of the contrast effect depends on the type of preceding stimulus, its intensity, and the type of test stimulus. No association was found with judgments of the degree of dissimilarity between the preceding stimuli and the test stimulus. For nonsalty preceding stimuli, the contrast effects are independent of concentration level. When the preceding stimuli taste at least partly salty, the total intensity appears to determine the size of the contrast for an unmixed salty test stimulus.     

Psychological analyses of haptic and haptic-visual judgements of extent

Magnitude estimates of haptic extent resulted in positively accelerated psychophysical power function with an exponent of 1.18. However, in two further experiments right-handed male subjects made rating-scale judgements of the combined width of two stimulus blocks. Six widths were used and five replications of the 36 factorial combinations were presented to each subject. In Experiment II both stimuli were out of view and one was held between the thumb and index finger of each hand. In Experiment III one stimulus was held out of view between thumb and finger of the right hand and the second was shown to the subject. Mean ratings in both experiments were fit by a model which assumes that responses are a weighted average of the scale values of the two stimuli (Anderson, 1974a).     

Free-operant forgetting: Delayed stimulus control of multiple-schedule performance

Pigeons' responses were reinforced in two components of several multiple variable-interval variable-interval schedules of food reinforcement. In one component, the key was illuminated green for 15 seconds and white for 45 seconds. In the other component, the key was illuminated red for 15 seconds and white for 45 seconds. Values for the exponent of the power functions relating response ratios to reinforcement ratios were higher in the presence of the discriminative stimuli (green or red) than in the presence of white. Sensitivity of response ratios to changes in reinforcement ratios provided an index of the extent to which responding was under delayed stimulus control by prior discriminative stimuli.