Loudness functions under inhibition

In both vision and hearing, a masking or inhibiting stimulus increases the slope (exponent) of the power function that relates sensation to stimulus. The power transformation applies only to the inhibited part of the function where the signal is fainter than the masking noise. Where the signal equals the noise, the function shows a discontinuous knee. Experiments were undertaken to see whether the loudness of a tone of 1000 Hz in a white noise would follow a model based on a constant signal-to-noise ratio at two locations, at the effective threshold and at the knee where the inhibited function meets the uninhibited function. The data accord with the slopes (exponents) generated by the model. The same model gives a fairly good account of the recruitment functions for ears suffering from cochlear involvement (e.g., Méniere’s disease). Regardless of degree of hearing loss, loudness recruitment reaches normal when the tone (1000 Hz) is about 30 dB above the affected threshold.     

Binaural summation after learning psychophysical functions for loudness

Do response-related processes affect perceptual processes? Sometimes they may: Algom and Marks (1990) produced different loudness exponents by manipulating stimulus range, and thereby also modified the rules of loudness summation determined by magnitude scaling. The present study manipulated exponents by having a dozen subjects learn prescribed power functions with exponents of 0.3, 0.6, or 1.2 (re sound pressure). Subjects gave magnitude estimates of the loudness of binaural signals during training, and of monaural and binaural signals after training. During training, subjects’ responses followed the nominal functions reasonably well. Immediately following training, subjects applied the numeric response scales uniformly to binaural and monaural signals alike; the implicit monaural-binaural loudness matches, and thus the basic rules underlying binaural summation, were unaffected by the exponent learned. Comparison of these results with those of Algom and Marks leads us to conclude that changing stimulus range likely influences underlying perceptual events, whereas “calibrating” a loudness scale through pretraining leaves the perceptual processes unaffected.     

Empirical evaluation of a model of global psychophysical judgments: IV. Forms for the weighting function

Understanding the psychological interpretation of numerals is of both practical and theoretical interest. In classical magnitude estimation, respondents match numerals to sensations and in magnitude production they select sensations that stand in a prescribed numerical ratio to a given standard. The present work focusses on evaluating several possible, and related, forms for the function W formulating the distortion of numerals. The main form, of which a power function is a special case, is the Prelec exponential/power representation. Behavioral equivalents to power and to Prelec functions are formulated, tested, and rejected. It is argued that either the mathematical form or the assumption W(1)=1 is wrong. Whereas, the axiomatic literature has focussed exclusively on the former inference, we explore the alternate that W(1)≠1. Behavioral axioms are formulated in each case and experimentally tested. We conclude that most respondents satisfy a general power function and that those who do not, satisfy the general Prelec function.     

Stevens' power law and the problem of meaningfulness

Frequently, it is postulated that the results of a ratio production (resp., ratio estimation) experiment can be summarized by Stevens' power law psi=alphaphi(beta). In the present article, it is argued that the power law parameters depend, among other things, on the standard stimulus presented as a reference point, and the physical stimulus scale by which the physical intensities are measured. To formalize this idea, a new formulation of Stevens' power law is presented. We show that the exponent in Stevens' power law can only be interpreted in a meaningful way if the stimulus scale is a ratio scale. Furthermore, we present empirically testable axioms (termed invertibility and weak multiplicativity) which are both necessary and sufficient for the power law exponent to be invariant under changes of the standard stimulus. Finally, invertibility and weak multiplicativity are evaluated in a ratio production experiment. Ten participants were required to adjust the area of variable circles to prescribed ratio production factors. Both axioms are violated for all participants. The results cast doubts on the well-established practice of comparing power law exponents across different modalities.     

Evaluating a model of global psychophysical judgments—II: Behavioral properties linking summations and productions

Steingrimsson and Luce [Journal of Mathematical Psychology, in press] outlined the second author's proposed psychophysical theory [Luce (2002), Psychological Review, 109, 520-532; Luce (2004a) Psychological Review, 111, 446-454] and tested behavioral attributes that, separately, gave rise to two psychophysical functions, Ψ_⊕ and Ψ_°p. The function Ψ_⊕ maps pairs of physical intensities onto the positive real numbers and represents subjective summation, and the function Ψ_°p represents a form of ratio production. This article evaluates properties linking summation and production in such a way as to force Ψ_°p=Ψ_⊕=Ψ. These properties, which are a form of distributivity, are subjected to an empirical evaluation in three experiments. The testing strategy is carried out in the auditory domain and concerns the subjective perception of loudness. Considerable support is provided for the existence of a single function Ψ for both summation and ratio production.     

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.     

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.     

Power functions of loudness magnitude estimations and auditory brainstem evoked responses

The correspondence between subjective and neural response to change in acoustic intensity was considered by deriving power functions from subjective loudness estimations and from the amplitude and latency of auditory brainstem evoked response components (BER). Thirty-six subjects provided loudness magnitude estimations of 2-sec trains of positive polarity click stimuli, 20/sec, at intensity levels ranging from 55 to 90 dB in 5-dB steps. The loudness power function yielded an exponent of .48. With longer trains of the same click stimuli, the exponents of BER latency measures ranged from -.14 for wave I to -.03 for later waves. The exponents of BER amplitude-intensity functions ranged from .40 to .19. Although these exponents tended to be larger than exponents previously reported, they were all lower than the exponent derived from the subjective loudness estimates, and a clear correspondence between the exponents of the loudness and BER component intensity functions was not found.     

On the psychophysical law and estimation procedures in psychophysical scaling

A general formulation of the power law is presented which has two special features: (1) negative exponents are admissible; and (2) the log law is a special limiting case. Estimation procedures, which provide joint estimates of the exponent and the absolute threshold, are derived for the direct ratio scaling methods. A solution is provided for theaveraging problem for ratio production and bisection scaling, two methods generating observations on the physical scale, and Monte Carlo methods are used to evaluate the resulting estimators.     

On predicting exponents for cross-modality matches

Mashhour and Hosman used magnitude estimations to scale seven continua: line length, time duration, finger span, loudness of noise, weight, gray reflectance, and surface area. The first four continua also served as the adjusted stimuli in 17 cross-modality matches among the various continua. Contrary to the view expressed by Mashhour and Hosman, the results appear to support the psychophysical power law. A reanalysis of the data shows that the exponents of the power functions obtained in cross-modality matches agree with the exponents of the power functions produced by magnitude estimations, provided correction is made for the regression effect. The measured discrepancies between the exponents predicted and those actually obtained show scatter that is consistent with that of other experiments. In particular, the scatter accords well with the distribution of 68 exponents predicted by Moskowitz from experiments in which Os matched both number and loudness to various taste concentrations.     

Dichotic, diotic, and monaural summation of loudness: a comprehensive analysis of composition and psychophysical functions

In a series of six experiments, the method of magnitude estimation, constrained by a multivariate model, was used to assess the rules that govern the summation of the loudness of two-tone complexes. This methodology enabled us to specify the amounts of summation and simultaneously to construct the corresponding loudness scales. The components had different frequency separations and in the different experiments were presented (1) dichotically, a different frequency to each ear; (2) diotically, to both ears; and (3) monaurally. Results replicated and in some conditions extended known features of multiple signal processing by the auditory system. Thus, qualitatively different rules of loudness integration appeared. For monaural and diotic modes of stimulation, overall loudness depended on total sound energy within the critical band, but on the simple sum of component loudnesses beyond the critical band. For dichotic presentations, a fully additive rule of loudness summation appeared, regardless of frequency spacing. For the latter (but not the former), loudness summation was perfect, with the underlying loudness scales closely approximating Stevens's sone scale.     

Bias effects on magnitude and ratio estimation power function exponents

A bias model of relative judgment was used to derive a ratio estimation (RE) power function, and its effectiveness in providing estimates of exponents free of the effects of standards was evaluated. The RE bias model was compared with the simple RE power function that ignores bias. Results showed that when bias was not taken into account, estimates of exponents exhibited the usual effects of standards observed in previous research. However, the introduction of bias parameters into the RE power function virtually eliminated these effects. Exponents calculated from "equal-range segments" (e.g., low stimulus range vs. high stimulus range) judged by magnitude estimation (ME) were examined: the effects of equal-range segments on exponents were much stronger for ME than standards were for RE, using the bias model.     

Individual differences in power functions for a 1-week intersession interval

Two experiments examined correlations of the power function exponents of individual Ss obtained in each of two sessions. Half the Ss for any task performed second sessions immediately after the first, the other half after a week’s delay. In Experiment I, groups of 16 Ss gave magnitude estimations of apparent area, or else of area and loudness. In Experiment II, groups of 16 Ss made cross-modality matches of apparent time duration to area. Significant correlations in all cases indicated consistent and persisting S differences in exponents. The results are related to the findings of other studies of such individual differences.     

Individual psychophysical functions for 28 odorants

Individual scales of odor intensity were obtained for 28 different chemical compounds by the method of magnitude estimation. Eleven Ss participated in an experiment with 196 olfactory stimuli which differed in both quality and intensity. It was found (1) that power functions described the relationship between partial vapor pressure of the odorants and their subjective odor intensity for all Ss, (2) that all exponents were less than one but varied greatly between Ss, (3) that consistent intraindividual differences in the exponents of different odorants exist, and (4) that these are attributable to perceptual differences rather than to response bias.     

Constrained scaling: the effect of learned psychophysical scales on idiosyncratic response bias

We report seven experiments in which subjects were trained to respond with numbers to the loudness of 1000-Hz pure tones according to power functions with exponents of 0.60, 0.30, and 0.90. Subjects were then presented with stimuli from other continua (65-Hz pure tones or 565-nm lights varying in amplitude) and were asked to judge the subjective magnitude of these stimuli on the same numerical scale. Stimuli from the training continuum were presented, with feedback, on every other trial in order to maintain the trained scale. Except for the 0.90 scale, subjects readily learned the predetermined scales and were able to use them to judge the non-training stimuli with group results consistent with those usually reported. Also, in contrast to the usual magnitude estimation results, these results produced extremely low levels of intersubject variability. We argue that such learned scales can be used as "rulers" for measuring perceived magnitudes, according to a common unit.     

Variability and sequential effects in cross-modality matching of area and loudness

Individual subjects' performance was examined for cross-modality matching (CMM) of loudness to visual area, as well as for magnitude estimation (ME) of the component continua. Average exponents of power functions relating response magnitude to stimulus intensity were .73 for area, .20 for loudness, and 2.44 for CMM. Predictions of the CMM exponent based on ME were higher than the empirical values, whereas more accurate predictions were made from magnitude production exponents obtained in a previous study. Sequential dependencies were assessed by comparing the response on trial n to the response on trial n--1. The coefficient of variation of the response ratio Rn/Rn-1 was systematically related to the stimulus ratio Sn/Sn-1 for both area and loudness. The coefficient was lowest for ratios near 1 and increased for larger or smaller values. For CMM, the coefficient of variation appeared to be independent of stimulus ratios. The correlation between log Rn and log Rn-1 was also related to Sn/Sn-1 for both ME and CMM. The correlation was highest when Sn/Sn-1 was 1 and dropped to 0 with increasing stimulus separation, but CMM yielded a shallower function than ME.     

Binaural and temporal integration of the loudness of tones and noises

Subjects judged the loudness of tones (Experiment 1) and of bursts of noise (Experiment 2) that varied in intensity and duration as well as in mode of presentation (monaural vs. binaural). Both monaural and binaural loudness, for both types of signals, obeyed the bilinear-interaction prediction of the classic temporal integration model. The loudness of short tones grows as a power function of both intensity and duration with different exponents for the two factors (.2 and .3, respectively). The loudness of wide-band noises grows as a power function of duration (with an exponent of approximately .6) but not of sound pressure. For tones, binaural summation was constant but fell short of full additivity. For noises, summation changed across level and duration. Temporal summation followed the same course for monaural and binaural tonal stimuli but not for noise stimuli. Notwithstanding these differences between tone and noise, we concluded that binaural and temporal summation are independently operating integrative networks within the auditory system. The usefulness of establishing the underlying metric structure for temporal summation is emphasized.     

Memory mechanisms and the psychophysical scaling of duration

Two experiments were performed to examine the suggestion that underlying memory mechanisms may be revealed in the form of the psychophysical function for duration. In experiment 1 a broad range of durations, with fine spacing at the lower end, was employed to bring out any transition in function that might reflect a change from 'ionic' memory to short-term memory. Estimation in conventional time-units (Verbal Estimation) was also compared with unit-free estimation (Magnitude Estimation). In experiment 2 Verbal Estimation was compared with the Production method, for a different range of stimulus values, and with varying interval content. Contrary to earlier claims, memory mechanisms were not found to be reflected in the values of power exponents for subjective duration. The value of the search for such functions is questioned, as simple linear plots fit the data at least as well.     

Perceived vibration and the loudness of low-frequency tones

Vibration and low-frequency tones were scaled for loudness by two numerical estimation procedures and by cross-modality matching. The same ranges of frequencies, from 30 to 250 Hz, were delivered to the ear and to the fingertip. For vibratory loudness, two sets of power functions were obtained, of which the low-frequency set was somewhat steeper. Tonal loudness gave a family of power functions of approximately the same slope at all the frequencies tested. For frequencies above 100 Hz, the growth of loudness is about the same for both modalities. Below this frequency, vibratory loudness grows more rapidly than tonal loudness.     

Consistency of individual exponents in cross-modal matching

An important question about individual differences in the exponent of the psychophysical power law is how they should be interpreted. The differences may reflect permanent characteristics of individuals, and it has been argued that, if this is so, the range of these differences is so great as to identify the class of data as exceptional among the physical and biological sciences. Cited as evidence of such permanence has been the correlation between individual exponents obtained on two separate occasions. In a previous paper, we showed that increasing the time interval between occasions reduced the correlation to a nonsignificant level; we argued, therefore, that obtained individual differences in exponents, even though large, depended upon the operation of factors only incidentally associated with the particular observer. In a series of new studies of session-to-session correlation between individual exponents, we provide evidence that: (1) our original finding for magnitude estimates of visual size is repeatable, with the correlation dropping to nearly zero after 1 week; (2) when judged line length is matched to brightness, a delay of I week is sufficient to produce a nonsignificant correlation; (3) in contrast, magnitude estimates of loudness yield significant correlations after a week’s delay; (4) but, when moduli are arbitrarily changed between sessions by the experimenter, these correlations for magnitude estimates of loudness drop to a nonsignificant level, even for a zero-delay condition. We conclude that, whereas in some scaling tasks the passage of time alone between sessions is sufficient to disrupt what appears to be the mnemonic basis for session-to-session correlation, in other (less familiar) tasks, more positive interference (in the form of a modulus change) is needed to achieve the same end. The evidence is consistent with the belief that enduring characteristics of the observer contribute only a small portion of the variability in individual power law exponents.