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.     

On the development of stimulus control

Pigeons were trained to peck one key when presented with white noise at any of five intensities lower than a reference intensity, and to peck another key when presented with white noise at any of five intensities greater than the reference intensity. The shape of the stimulus control curves (proportion of responses to one key versus stimulus intensity) changed from a horizontal line at the beginning of training to the sigmoid form of typical psychometric functions at the end of training. The development of stimulus control is described in terms of a model based on the theory of signal recognition and a concept of attention.     

Brightness as a function of retinal locus

Brightness functions were determined for the dark-adapted fovea and periphery. In one series of experiments, observers matched numbers to the brightness of a 1° white target at various intensities, presented half the time to the fovea, the other half to one of five peripheral loci: 5°, 12°, 20°, 35°, and 60°. In a second series, observers matched the brightness of a 1° white target in the fovea of one eye to the brightness of an identical target in the periphery of the other eye at various intensities. Thresholds were also determined for the fovea and for the five peripheral loci by a staircase procedure. The magnitude estimations and the interocular matches concur in showing that a stimulus of fixed luminance appears brighter in the periphery than in the fovea. The brightness was found to be maximal at 20°. Brightness grows as a similar power function of luminance at all six retinal positions.     

Brightness as a function of retinal locus

Brightness functions were determined for the dark-adapted fovea and periphery. In one series of experiments, observers matched numbers to the brightness of a 1° white target at various intensities, presented half the time to the fovea, the other half to one of five peripheral loci: 5°, 12°, 20°, 35°, and 60°. In a second series, observers matched the brightness of a 1° white target in the fovea of one eye to the brightness of an identical target in the periphery of the other eye at various intensities. Thresholds were also determined for the fovea and for the five peripheral loci by a staircase procedure. The magnitude estimations and the interocular matches concur in showing that a stimulus of fixed luminance appears brighter in the periphery than in the fovea. The brightness was found to be maximal at 20°. Brightness grows as a similar power function of luminance at all six retinal positions.     

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.     

A note on cross-modality matching

”Stevens' law”, that the psychophysical law is a power function, is often taken to be confirmed by results of cross-modality matching experiments, for it predicts both the fact that cross-modality matching experiments yield power functions, and the exponents of these power functions. Both these predictions, however, follow from a more general form of the psychophysical law, of which Fechner's law is a special case. In view of this, an interpretation of cross-modality matching based on stimulus discriminability rather than sensation magnitude is proposed.     

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.     

An analysis of U-shaped metacontrast

Using a brightness-discrimination task similar to that employed by Bernstein, Proctor, Proctor, and Schurman (1973), masking functions were obtained in two experiments. In Experiment I, test stimulus (TS) and mask stimulus (MS) energies were held constant but luminance and duration were varied reciprocally. The obtained masking functions, plotted as a function of stimulus onset asynchrony (SOA), were of an essentially identical U shape. This suggests that (a) SOA is a more suitable measure of delay than interstimulus interval, and (b) Bloch’s law holds for the requisite discrimination. In Experiment II, TS luminance and MS luminance were varied independently. This was to see whether the MS served as a frame of reference at short SOA, as suggested previously (Bernstein et al, 1973). The results were that this was, in fact, the case and that the transition from comparative to absolute judgment strategies as SOA increases is a major contributor to U-shaped masking functions.     

Loudness and reaction time: I

It is widely assumed, based on Chocholle’s (1940) research, that stimuli that appear equal in loudness will generate the same reaction times. In Experiment 1, we first obtained equal-loudness functions for five stimulus frequencies at four different intensity levels. It was found that equal loudness produced equal RT at 80 phons and 60 phons, but not at 40 phons and 20 phons. It is likely that Chocholle obtained equivalence between loudness and RT at all intensity levels because of relay-click transients in his RT signals. One main conclusion drawn from Experiment 1 is that signal detection (in reaction time) and stimulus discrimination (in loudness estimation) require different perceptual processes. In the second phase of this investigation, the RT-intensity functions from six different experiments were used to generate scales of auditory intensity. Our analyses indicate that when the nonsensory or “residual” component is removed from auditory RT measures, the remaining sensory-detection component is inversely related to sound pressure according to a power function whose exponent is about — 3. The absolute value of this exponent is the same as the .3 exponent for loudness when interval-scaling procedures are used, and is one-half the size of the .6 exponent which is commonly assumed for loudness scaling.     

Tests of the psychological meaning of the power law.

The purpose of this study was to establish a theoretical framework for Stevens' empirically derived power law. Three models were proposed to explain the power law. They respectively outline how sensory, stimulus, and response variables determine the judgmental behavior in a psychophysical task. A correlational study on individual differences in exponents was carried out to test the predictions derived from each model. The use of four different sensory continua and four scaling procedures provided the experimental means of manipulating the sensory, stimulus, and response variables in the scaling situation. The results showed that response variables are important determinants of judgmental behavior in psychophysical scaling. These findings suggest that subjects' responses to stimulus intensities in a scaling task are largely cognitive.     

BRIGHTNESS SCALES FOR MONOCHROMATIC LIGHT

By means of amethod of ratio estimation, scale values were obtained for the subjective brightness of various physical intensities of monochromatic light of various wave lengths. In a second experiment the scale was constructed by a method of magnitude estimation. The brightness functions were studied by plotting the scale values against stimulus intensity for each wave length. The two experiments gave essentially the same results. It was shown: (1) Brightness of monochromatic light is a power function of stimulus intensity. The exponent of the function is approximately one-third for all wave lengths. (2) Properties of the brightness functions can explain certain empirical relations between brightness, hue and saturation.     

Reaction time as a function of stimulus intensity for the monkey

Monkeys were trained to release a telegraph key in response to a visual or auditory stimulus. The latency of the key release response was measured for different stimulus intensities. In general, the relation between latency and intensity is inverse and exponential with greater variability of latency at the lower intensities. Some preliminary data involving differential reinforcement of short latencies are presented.     

The haptic perception of curvature

Ninety-two subjects, schoolchildren and undergraduate and postgraduate students, took part in a series of experiments on the haptic perception of curvature. A graded series of surfaces was produced using piano-convex lenses masked off to produce curved strips that could be explored without using arm movements. Thresholds were measured using the constant method and a staircase procedure. Experiments 1 and 2 yielded data on the absolute and difference thresholds for curvature. Experiment 3 demonstrated that the effective stimulus for curvature is represented by the overall gradient of a curved surface. Using this measure, it was shown that the present absolute thresholds for curvature are lower than those previously reported. In Experiment 4, absolute thresholds were compared using spherical and cylindrical curves: the results showed that, at least with the narrow strips used, the type of curvature does not exert a significant influence on performance. In Experiment 5, the subjective response to curvature was assessed using a rating procedure. Power functions are reported, although the relationship between stimuli and responses had a strong linear component. This suggests that haptically perceived curvature may be a metathetic rather than a prothetic continuum.     

Can the relationship between tangential velocity and radius of curvature explain motor constancy?

To address contributions of speed, efficiency, radius of curvature and joint complexity to the strength of the lawful relationship between tangential velocity and radius of curvature (power law), an experiment considered the strength of the power law when participants were instructed to perform circling movements using the elbow, finger, shoulder, or wrist. Five participants performed circling motions in a vertical plane upon a Smartboard that sampled finger tip position at 200Hz. Page's L tested whether the strength of the power law could be predicted by: (1) speed; (2) submovements; (3) joint complexity; (4) radius of curvature. A second experiment considered the strength of the power law when six participants were instructed to perform circling movements of different sizes (large, medium, and small) using their shoulders. Movement speed or efficiency could not explain the strength of the power law, instead the power law was stronger for movements with a smaller radius of curvature or fewer joints. The strength of the power law varied with effector, questioning the role of the power law in motor constancy.     

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.     

Conditioned suppression of behavior maintained by intracranial stimulation as a function of stimulation intensity.

Conditioned suppression was demonstrated in two experiments with rats lever pressing on a fixed-ration 1 schedule for lateral hypothalamic intracranaial stimulation (ICS)'n Experiment I, conditioned suppression of responding for low-intensity ICS was obtained with a moderate intensity of foot shock, In Experiment II, low and high intensities of ICS were alternated within the same session and the same animal The suppression that was exhibited with low intensity ICS was minimal or absent with high-intensity stimulation, despite the pairing of foot shock with each warning stimulus. Conditioned suppression was a function of ICS intensity, and was independent of response rates. The inverse relationship between ICS intensity and degree os suppression is consistent with a motivational analysis of conditioned suppression. Previous reports of resistance to suppression of behaviors maintained by ICS may now be attributed to the use of high-intensity stimulation.     

Magnitude estimation of short electrocutaneous pulses

Summary A comparison was made of electrocutaneous magnitude estimation data across two experiments with contextual differences not involving stimulus parameters, such as number and range of stimuli and relative position of the standard in the stimulus range. The data were fitted by 2-parameter linear, log-linear and power functions. When the data are fitted by either linear or log-linear equations, both intercept and slope parameters are significantly affected by the different contextual factors. When the data are fitted by a power function, however, only the intercept is altered; the slope remains invariant despite contextual changes introduced in the second experiment.Although the empirically derived psychophysical power law has been applied to magnitude estimation data for all other sensory modalities, its application to electrocutaneous stimuli has been less successful.     

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.