Range of acceptable stimulus intensities: An estimator of dynamic range for intensive perceptual continua

The dynamic range (DR) of a sensory system is the span (usually given in log units) from the lowest to highest intensities over which a continuously graded response is evoked, and may be a distinctive feature of each such system. Teghtsoonian (1971) proposed that, although DR varies widely over sensory systems, itssubjective size (SDR) is invariant. Assuming the psychophysical power law, the exponent for any continuum is given by the ratio of subjective span to DR, both quantities expressed logarithmically. Thus, exponents are inversely related to DR and may be interpreted as indexes of it. Because DR can be difficult or even dangerous to measure directly, we sought to define a smaller range representing some fixed proportion of DR that could be used in its place to test the hypothesis of an invariant subjective range. Observers manipulated the intensities of five target continua to produce the broadest range they found acceptable and reasonably comfortable, a range of acceptable stimulus intensities (RASIN). Combined with an assumed constant SDR (derived from previous research), RASINs accurately predicted exponents obtained by magnitude production from the same observers on the five continua, as well as exponents reported in the literature.     

The limits of perceived magnitude: Comparison among individuals and among perceptual continua

Two models of perceived magnitude that focus on the concept of dynamic range (DR) for sensory systems are summarized and compared. The first asserts that, whereas the size of DR varies physically from one perceptual continuum to another, all are subjectively equal. The other asserts that, for any given continuum, whereas the size of DR varies over individual observers, the subjective range does not.A test of the second model is reported for perceived effort for a range of loads on a bicycle ergometer. Both DR and judgmental (magnitude estimation) range estimates were obtained for 30 subjects. The former exhibited the expected variability over subjects, but there was no support for the predicted invariance of the latter. However, it is concluded, first, that the finding constitutes disconfirming evidence for the second model only if it is assumed that magnitude estimates are proportional to perceived magnitude, an issue which itself remains in dispute, and second, that the testability of the second model is doubtful unless a measure can be identified that is a stateable function of perceived magnitude.     

Electrocutaneous psychophysical input-output functions and temporal integration

Electrocutaneous magnitude estimation functions were generated by stimuli ranging from 1.0 to 5.0 mA and from 100 to 6,400 msec in duration. The results indicate that when these functions are fitted by a two-parameter power function (ME = aI^b), the values of the constant, a, and the exponent, b, are altered by increases in stimulus duration, with a increasing and b decreasing. The exponent decreases from around 1.4 to 0.93 as duration increases from 100 to 6,400 sec. Equal magnitude estimation contours drawn for estimates ranging from “2“ to “50“ can be fitted by an equation representing partial integration, I × t^a = K. The exponent a decreases as a function of the level of the magnitude estimation, indicating less partial integration at higher than at lower levels of estimated magnitude. The electrocutaneous data are compared to data in other sensory modalities.     

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.     

THE DEPENDENCE OF VIBROTACTILE THRESHOLD AND MAGNITUDE FUNCTIONS ON STIMULATION FREQUENCY AND SIGNAL LEVEL: A perceptual and neural comparison

F ranzén , O. The dependence of vibrotactile threshold and magnitude functions on stimulation frequency and signal level. A perceptual and neural comparison. Scand. J. Psychol ., 1969, 10 , 289–298.—Apparent intensity of vibrotactile signals was examined as a function of stimulation frequency and displacement. Two sorts of power transformations were observed. (1) The exponent decreased as frequency was increased. (2) At low signal levels perceived intensity was linearly related to the physical input. At a certain point on the intensity continuum an abrupt shift to a power function with a slope of 0.58 occurred. The successive change of the exponent of the magnitude functions is tentatively interpreted as a kind of successively increasing demultiplication in the rate of firing frequency in the sensory pathway. A dual mechanism of mechanoreception is discussed.     

Sequential dependencies and regression in psychophysical judgments

A tendency for judgments of stimulus magnitude to be biased in the direction of the value of the immediately preceding stimulus is found in magnitude estimations of loudness. This produces a bias in the empirical psychophysical function that results in underestimation of the exponent of the unbiased function presumed to relate number and stimulus intensity, N = aSⁿ. The biased judgment can be represented as a power product of focal and preceding stimulus intensity, N_ij= aS ^m S_j ^b. A bias-free estimate of the correct exponent, n, can be obtained from the relation n = m + b.     

Cross-modality matching functions generated by magnitude estimation

It is possible to generate cross-modality matching functions by having subjects make magnitude estimates of sets of stimuli appropriate to different modalities. The sets are interspersed among each other in the same test session and judged on a common absolute scale of sensory magnitude. An appropriate statistical device locates stimulus levels that appear, on the average, to match. The method is fast, efficient, circumvents the need for continuous stimulus adjustment, and holds promise for the study of the individual as well as the average psychophysical function. To illustrate its potential uses, advantages, and limitations, we used the method to generate cross-modality matching functions relating loudness and brightness. Compared to the scales of loudness and brightness generated by the magnitude estimations of the same stimuli, the matching functions (1) conform better to power functions, (2) may show less variation in slope (exponent), and (3) show far less variation in absolute magnitude (position).     

Range and regression effects in magnitude scaling

The relation between power law exponents obtained by magnitude estimation and magnitude production was studied for both loudness and perceived distance. While the results confirm the usual finding of higher values for production for relatively large stimulus ranges, just the opposite occurs when the stimulus range is short, necessitating a revision of the Stevens-Greenbaum regression principle. The relation between range and exponent was explored, both for the case in which several intensities are presented for judgment and for the simpler case of only two intensities. In both cases, a power relation was described relating stimulus ratios to judgmental ratios, with exponents containing both range-dependent and range-independent components.     

Ratio scales of acid sourness

Experiments were conducted to assess the relation between concentration, or pH, and the perceived sourness of 24 acids. The psychophysical functions for sourness conform to the power relation S = kCⁿ which relates sensory intensity, S, to physical concentration, C. Averaged across the 24 acids, the exponent for sourness was 0.85 for both molar and percentage concentrations, and about ?1.70 for pH concentration. The intercept, k, which is a measure of relative sourness, differed across acids. The particular measure used to designate the concentration of an acid markedly influenced its magnitude and rank order of sourness.     

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.     

Perceived warmth and skin temperature as functions of the duration and level of thermal irradiation!

This study examined the correlations among the degree of perceived warmth. the level and duration of irradiant flux. and the thermal response of the skin. For any constant duration. perceived warmth grew as a power function of the difference between the irradiant flux of the stimulus and the flux that approximates the absolute threshold for warmth. The exponent of the power function was about 0.87 for the shortest durations (2—6 sec). but rose to 1.04 for the longest duration of exposure (12 sec). For any constant level of flux. perceived warmth changed only slightly with duration. In contrast, superficial skin temperature. and inferred temperatures of deeper layers of the skin, rose continuously and markedly with duration. Neither the change in tissue temperature. nor the rate of change of tissue temperature, nor thermal gradient correlated consistently with level of perceived warmth. The change in the difference between the temperature 0.2 mm and that 1.0 mm below the skin surface provided a fairly good but not perfect correlate to perceived warmth. The findings suggest the possibility that sensory adaptation at the site of the receptor system mediating warmth could act in such a way as nearly to offset the effect of rising skin temperature with increased duration ofstimulation.     

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.     

Response bias in category and magnitude estimation of difference and similarity for loudness and pitch

Interval scales of sensory magnitude were derived from magnitude and category estimates of loudness differences, loudness similarities, pitch differences, and pitch similarities. In each of the four loudness experiments, a loudness scale was constructed from a nonmetric analysis of the rank order of the judgments. The four loudness scales so constructed were found to be equivalent to one another and indicated that loudness was a power function of sound pressure with an exponent of .29. A similar analysis for the four pitch experiments found the pitch scales derived in each case to be equivalent to one another and linear with the mel scale of pitch. Thus the same sensory and similarities for two distinct perceptual continua. For both pitch and loudness, these sensory scales were used to generate scales of sensory differences. A comparison of the category and magnitude estimates of sensory differences with the scale of sensory differences derived from the nonmetric analyses indicated the presence of significant response biases in both category and magnitude estimation procedures.     

Pitch strength and Stevens’s power law

Recent studies have suggested that the saliency or the strength of pitch of complex sounds can be accounted for on the basis of the temporal properties in the stimulus waveform as measured by the height of the first peak in the waveform autocorrelation function. We used a scaling procedure to measure the pitch strength from 15 listeners for four different pitches of complex sounds in which the height of the first peak in the autocorrelation function systematically varied. Pitch strength judgments were evaluated in terms of a modification of Stevens’s power law in which temporal information was used from both the waveform fine structure and the envelope. Best fits of this modified power law to the judged pitch strengths indicate that the exponent in Stevens’s power law is greater than 1. The results suggest that pitch strength is primarily determined by the waveform fine structure, but the stimulus envelope can also contribute to the pitch strength.     

Pitch strength and Stevens's power law

Recent studies have suggested that the saliency or the strength of pitch of complex sounds can be accounted for on the basis of the temporal properties in the stimulus waveform as measured by the height of the first peak in the waveform autocorrelation function. We used a scaling procedure to measure the pitch strength from 15 listeners for four different pitches of complex sounds in which the height of the first peak in the autocorrelation function systematically varied. Pitch strength judgments were evaluated in terms of a modification of Stevens's power law in which temporal information was used from both the waveform fine structure and the envelope. Best fits of this modified power law to the judged pitch strengths indicate that the exponent in Stevens's power law is greater than 1. The results suggest that pitch strength is primarily determined by the waveform fine structure, but the stimulus envelope can also contribute to the pitch strength.     

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.     

Identification variability as a measure of loudness: an application to gender differences.

It is well known that discrimination response variability increases with stimulus intensity, closely related to Weber's Law. It is also an axiom that sensation magnitude increases with stimulus intensity. Following earlier researchers such as Thurstone, Garner, and Durlach and Braida, we explored a new method of exploiting these relationships to estimate the power function exponent relating sound pressure level to loudness, using the accuracy with which listeners could identify the intensity of pure tones. The log standard deviation of the normally distributed identification errors increases linearly with stimulus range in decibels, and the slope, a, of the regression is proportional to the loudness exponent, n. Interestingly, in a demonstration experiment, the loudness exponent estimated in this way is greater for females than for males.     

Stability of individual loudness functions obtained by magnitude estimation and production

A correlational analysis of individual magnitude estimation and production exponents at the same frequency was perfor.med, as well as an analysis of individual exponents produced in different sessions by the same procedure across frequency(250, 1, 000, and 3, 000 Hz). Taken together, results show, first, that individual exponent differences do not decrease by counterbalancing magnitude estimation with magnitude production, and, second, that individual exponent differences remain stable over time despite changes in stimulus frequency. Further results disclose that although individual magnitude estimation and production exponents do not necessarily obey the .6 power law, it is possible to predict the slope (exponent) of an equal-sensation function averaged for a group of listeners from individual magnitude estimation and production data. Assuming that individual listeners with sensorineural hearing loss also produce stable and reliable magnitude functions, it is also shown that the slope of the loudness-recruitment function measured by magnitude estimation and production can be predicted for individuals with bilateral losses of long duration. Thus, results obtained in normal and in pathological ears suggest that individual listeners can produce loudness judgments that reveal, albeit indirectly, the input-output characteristic of the auditory system.