Power laws for the perception of rotation and the oculogyral illusion

Magnitude estimation was used to measure subjective motion for two indicators of vestibular function. Twelve as made estimates of 5-sec pulses of angular acceleration across the range of angular acceleration × time (at) =10-150 deg/sec. Results were: (1) the power law describes subjective motion for all individual as, (2) the power function exponent (1.41) for the perception of rotation is slightly greater than the exponent (1.25) for the oculogyral illusion, (3) a significant number of as gave higher exponents for the perception of rotation, and (4) the magnitude estimates of the oculogyral illusion and perception of rotation were highly correlated within and across as.     

The range effect as a function of stimulus set,presence of a standard,and modulus

The intramodal range effect (an inverse relationship between stimulus range and exponent in Stevens’s power law) has been well documented, but its conditions have not been tested. Both the estimates of stimulus magnitudes and their exponents are affected by context, stimulus location, and different standards and moduli, but how these variables might interact with the variable of stimulus range has not been studied. In the present research, exponents were derived from magnitude estimates of line length for each of three different stimulus: ranges at two different locations on the scale of length, with or without a modulus. Moduli of 50 and 500 permitted an analysis of the effect of response magnitude on the range effect. Because different ranges had stimulus values in common, the effect of range and location on exponents from those common values could be determined. Exponents decreased as stimulus range increased, but only in the free-modulus condition. For that condition, exponents derived from magnitude estimates of only the common stimuli also showed the range effect and response magnitude did not influence the range effect. Exponents were higher for stimulus ranges at the lower location, but location does not appear to contribute to the range effect. Although the range effect is not explained, the conditions under which it holds and some factors that may influence it are considered.     

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.     

Shift in stimulus range and the exponent of the power function for loudness

The exponent of the power function for loudness was tracked over the course of 60 trials with one stimulus range and compared to the exponent over the course of 60 subsequent trials with a different stimulus range. Three stimulus sets were used: (1) weak, a short range of relatively soft tones (45-55 dBA); (2) strong, a short range of relatively loud tones (64-74 dBA); and (3) complete, a longer range of soft to loud tones (40-90 dBA). All pairs of stimulus sets were tested, together with three control conditions in which no shift in range occurred. Ten subjects were run in each of the nine groups. For preshift trials, the mean exponent was lowest for the strong stimulus series, highest for the weak series, and at an intermediate value for the complete series. These differences were all significant. Following a shift in stimulus range, the weak series still yielded the highest exponent, but the exponents were not reliably different for the complete and strong series. Postshift exponents also depended significantly on the preshift range experienced by the subjects. These effects were not confined to the period immediately following the shift in range, but persisted for up to 60 trials.     

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.     

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.     

Neutral vs ego-orienting instructions: Effects on judgments of magnitude estimation

Three experiments were performed to examine the relative constancy of the exponent in the psychophysical power law under varying motivating conditions. The method of magnitude estimation was used to obtain judgments of apparent tactual roughness or of apparent area size of squares. Patterns of the qualitative observations of the three Es and of the various exponents for the six groups of Ss indicated that neutral instructions and “ego-orienting” instructions, which were perceived as unbelievable coming from an equal fellow student, both yielded exponents identical to those reported in the literature. Believable ego-orienting instructions given by an E of clearly perceived higher social status produced a statistically significantly lower exponent than neutral. Intermediate conditions, wherein Ss apparently disbelieved both types of instructions, but assumed that the superior-status E was “analyzing” them, yielded exponents of intermediate size. Results and supplementary trend analyses are discussed as possible, highly sensitive indicators of motivational impacts on sensory judgments.     

Scales for perceived egocentric distance in a large open field: comparison of three psychophysical methods

Scales for perceived egocentric distance produced by three psychophysical methods in four ranges of distances were compared. It was found that (a) the exponents produced by ratio and fractionation methods are in good agreement; (b) the exponents of both these methods were larger than those produced by magnitude estimation; (c) an increase in range of distance was associated with a decrease in exponent, but this diminution seems to interact with the method used; (d) for all the psychophysical methods used, there was large variability in the individual exponent; (e) the exponent was smaller than 1.0 for approximately 78% of the pooled sample, with all adult observers (N = 612) considered; and (f) an arithmetic mean exponent equal to 0.90 represents fairly all the results obtained.     

The perception of rate in spoken and sign languages

The methods of magnitude production and estimation were used to scale the perception of signing rate by signers and observers. As in the case of voice level and speech rate, the autophonic scale of signing rate has a slope greater than unity and is steeper than the corresponding extraphonic scale; the obtained exponents of the two power functions for signing are, respectively, 2.66 and 1.56. When English-speaking subjects estimated their own rate of reading of the translated version of the signed passage, they produced an autophonic reading scale quite similar to that for signing (exponent of 2.51), but when they made magnitude estimations of English rates covering the same range of rates as the signed passage, the exponent of the extraphonic reading scale was significantly larger (1.89). This was also the case when French subjects estimated French reading rates. The difference between extraphonic signing and reading scales was confirmed by subjects who knew no Sign Language or French; their results appear to indicate, in addition, that the processes involved in extraphonic perception of rate are purely acoustic (speech) or visual (sign) and do not require, as one could have thought, deeper linguistic operations.     

On the relation between stimulus intensity and processing time: Piéron’s law and choice reaction time

Piéron (1914, 1920, 1952) demonstrated that simple reaction time (SRT) decays as a hyperbolic function of luminance in detection tasks. However, whether such a relationship holds equally for choice reaction time (CRT) has been questioned (Luce, 1986; Nissen, 1977), at least when the task is not brightness discrimination. In two SRT and three CRT experiments, we investigated the function that relates reaction time (RT) to stimulus intensity for five levels of luminance covering the entire mesopic range. The psychophysical experiments consisted of simple detection, two-alternative forced choice (2 AFC) with spatial uncertainty, 2 AFC with semantic categorization, and 2 AFC with orientation discrimination. The results of the experiments showed that mean RT increases with task complexity. However, the exponents of the functions relating RT to stimulus intensity were found to be similar in the different experiments. This finding indicates that Piéron’s law holds for CRT as well as for SRT. It describes RT as a power function of stimulus intensity, with similar exponents, regardless of the complexity of the task.     

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.     

Variations in the relationship between radius of curvature and velocity as a function of joint motion

A power law describes the relationship between the geometric properties of a trajectory (radius of curvature) and movement kinematics (tangential velocity) in curved drawing movements. Although the power law is a general law of motion, there are conditions under which it degrades. In particular, the power law may be less explanatory of movements around certain joints. The present study considered how varying motion around different joints influenced the fit of the power law. Motions associated with finger and wrist, or elicited by an isometric force production task, were compared. The power law was most explanatory of finger motion and isometric production and least explanatory of wrist motion. The fit of the power law for finger and wrist motion suggested separate laws for each joint system. Since the fit of the power law was better for finger than for wrist motion, there is some suggestion that the power law better explains motion around fewer or simpler joint systems.     

Single-parameter power law psychophysics of auditory numerosity and the psychological moment hypothesis

In the present study, numerosity estimation was investigated. A two-parameter Stevens power law analysis was performed on a total of944 subjects in six experiments. Two pulse ranges (2–17 or 17–253 pulses) and six pulse rates (either constant or randomly varied within trial blocks) were used, variously, in an unsuccessful attempt to find evidence for a psychological moment, under the supposition that the exponent (or, possibly, the measure constant) would become smaller as increasing numbers of pulses fell within the interval determined by each psychological moment. A single-parameter reanalysis of these six experiments under the initial value condition that a (standard) stimulus of one pulse be assigned a theoretical response (modulus) of one yielded single-parameter equations whose exponents were reliably less varied than those for conventional two-parameter equations in Experiments 1–4 (with randomly varying pulse rates from trial to trial) but not less varied in Experiments 5 and 6 (in which pulse rates were constant within trial blocks). It was concluded that the variable pulse rate condition, with its reduced exponent variability and presumed reduced temporal confounding, provides a more valid estimate of the single-parameter power law exponent for numerosity, which was found to be 0.80.     

Single-parameter power law psychophysics of auditory numerosity and the psychological moment hypothesis.

In the present study, numerosity estimation was investigated. A two-parameter Stevens power law analysis was performed on a total of 944 subjects in six experiments. Two pulse ranges (2-17 or 17-253 pulses) and six pulse rates (either constant or randomly varied within trial blocks) were used, variously, in an unsuccessful attempt to find evidence for a psychological moment, under the supposition that the exponent (or, possibly, the measure constant) would become smaller as increasing numbers of pulses fell within the interval determined by each psychological moment. A single-parameter reanalysis of these six experiments under the initial value condition that a (standard) stimulus of one pulse be assigned a theoretical response (modulus) of one yielded single-parameter equations whose exponents were reliably less varied than those for conventional two-parameter equations in Experiments 1-4 (with randomly varying pulse rates from trial to trial) but not less varied in Experiments 5 and 6 (in which pulse rates were constant within trial blocks). It was concluded that the variable pulse rate condition, with its reduced exponent variability and presumed reduced temporal confounding, provides a more valid estimate of the single-parameter power law exponent for numerosity, which was found to be 0.80.     

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.     

Invariance tests for bisection and fractionation scaling

If we are to accept the power function as the psychophysical law, then estimates of the exponent of a particular form of the power law should be independent of changes in basic independent variables. In the present study, various power law hypotheses were tested for the bisection and fractionation scaling of brightness. The results indicated that for bisection, estimates of exponents were dependent on the particular interval bisected, suggesting rejection of the simple power law for bisection. For fractionation, two additional forms of the power law were tested, each form involving a threshold parameter. One was the Φ-law (involving a translation on the intensity axis), and the other was the ψ-law (involving a translation on the psychological axis). The Φ-law provided a poor fit to the data, whereas the ψ-law appeared to fit well for at least one S. Analysis for individuals showed that for all five Ss, the variance due to standards was appreciably larger for the Φ-law.     

Reaction times reveal the contribution of the different receptor components in luminance perception

Piéron (1914, 1920, 1952) demonstrated that simple reaction time decays as a hyperbolic function of luminance. Similarities between cell latencies and reaction time (RT) to luminance suggest that this relationship may be determined by retinal processes. If the exponent of the Piéron function is specific to a given sensory modality, as assumed by some authors (e.g., Bonnet, 1992a, 1992b; Norwich, 1987), it should reflect receptor activities. Consequently, functions with different exponents should fit data for different luminance ranges. In a contrast-discrimination experiment, we investigated this question with a large range of luminance levels in a two-alternative spatial forced-choice task. The results of the experiment show that three functions with different exponents fit RT to the three luminance ranges (scotopic, mesopic, and photopic). The exponent decreases with increasing luminance. The findings indicate that the exponent and the asymptotic latency of the RT function reflect receptor activities of the visual system.     

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.     

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.     

The effects of size,clutter, and complexity on vanishing-point distances in visual imagery

Summary The portrayal of vanishing-point distances in visual imagery was examined in six experiments. In all experiments, subjects formed visual images of squares, and the squares were to be oriented orthogonally to subjects' line of sight. The squares differed in their level of surface complexity, and were either undivided, divided into 4 equally sized smaller squares, or divided into 16 equally sized smaller squares. Squares also differed in stated referent size, and ranged from 3 in. to 128 ft along each side. After subjects had formed an image of a specified square, they transformed their image so that the square was portrayed to move away from them. Eventually, the imaged square was portrayed to be so far away that if it were any further away, it could not be identified. Subjects estimated the distance to the square that was portrayed in their image at that time, the vanishing-point distance, and the relationship between stated referent size and imaged vanishing-point distance was best described by a power function with an exponent less than 1. In general, there were trends for exponents (slopes on log axes) to increase slightly and for multiplicative constants (y intercepts on log axes) to decrease as surface complexity increased. No differences in exponents or in multiplicative constants were found when the vanishing-point was approached from either sub-threshold or suprathreshold directions. When clutter in the form of additional imaged objects located to either side of the primary imaged object was added to the image, the exponent of the vanishing-point function increased slightly and the multiplicative constant decreased. The success of a power function (and the failure of the size-distance invariance hypothesis) in describing the vanishing-point distance function calls into question the notions (a) that a constant grain size exists in the, imaginal visual field at a given location and (b) that grain size specifies a lower limit in the storage of information in visual images.