The powers of noise-fitting: reply to Barth and Paladino

Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated by averaging over changing proportions of logarithmic and linear estimation patterns. To evaluate this possibility, we used conventional model fitting techniques with individual as well as group average data; simulations that varied the proportion of data generated by different functions; comparisons of alternative models' prediction of new data; and microgenetic analyses of rates of change in experiments on children's learning. Both new data and individual participants' data were predicted less accurately by power functions than by logarithmic and linear functions. In microgenetic studies, changes in the best fitting power function's exponent occurred abruptly, a finding inconsistent with Barth and Paladino's interpretation that development of numerical representations reflects a gradual shift in the shape of the power function. Overall, the data support the view that change in this area entails transitions from logarithmic to linear representations of numerical magnitude.     

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.     

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.     

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.     

Individual pitch functions and pitch-duration cross-dimensional matching

Eight young adult subjects scaled the auditory dimensions of duration and pitch using a modified method of numerical magnitude balance and adjustment of the stimuli for individual equal-loudness differences. Individual duration and pitch functions for magnitude estimation and for magnitude production fit the power law well. When compared with the revised reel scale and the pitch function obtained by S. S. Stevens and Galanter (Journal of Experimental psychology, 1957,54, 377–411), the group magnitude estimation and magnitude production pitch functions plotted in log-log coordinates showed high degrees of linearity. This was due mostly to the absence of a rollover in the high-frequency range of the continuum. It was hypothesized that pitch may be viewed as a linear function. This hypothesis was further supported when the exponents of the pitch and duration functions were used to predict closely the group exponent of cross-dimension matches.     

Scaling apparent distance in a large open field: some new data

Judged distance in a large open field, scaled by the method of magnitude estimation, is related to physical distance by a power function with an exponent smaller than unity. The exponents obtained with two ranges of distance were not affected by the availability of a standard. The mean exponent for all 80 individual power functions was 0.86, with a standard deviation of 0.11.     

SCALES OF UNPLEASANTNESS OF ELECTRICAL STIMULATION

The subjective unpleasantness elicited by an A.C. current of 50 c/sec. applied to two fingers was scaled by 14 subjects using the methods of magnitude and of category estimation. Nine current intensities, ranging between 2 and 10 times the individual sensation thresholds, were used as stimuli. A power function yielded a good fit to the magnitude-estimation data; the exponent was 1.81. The difference between this value and a considerably higher exponent reported by previous investigators was interpreted in terms of the wider stimulus range used in the present study. The relation between the category scale and the magnitude scale had the same general form as that found in several previous investigations.     

Deriving the loudness exponent from categorical judgments

The power function exponent for loudness is traditionally determined by means of a process of magnitude estimation. It is demonstrated in this paper that the exponent can also be obtained by using the procedure of absolute identification of sound intensity. It has been shown that subjects' responses to tones of a given intensity are distributed in a normal distribution whose variance depends on the range, R, over which the tones are distributed. By means of a standard statistical transformation, the normal density in log space is converted to the corresponding probability density in linear space. The power function exponent can then be obtained directly from the linear probability density. We also suggest that there is a direct relationship between the information calculated from experiments on absolute identification of sound intensity and the neurophysiological, poststimulus histogram measured in a nerve fiber in the auditory nerve.     

Stimulus range,number of categories,and the “virtual” exponent

Three different stimulus modalities (line length, number, and sound pressure) were judged by magnitude scaling techniques and by 7-, 15-, 31-, and 75-point category scales. All of the 40 subjects were given the same number stimuli, but two different sound-pressure ranges were presented (each to 20 subjects) and four different line-length ranges were presented (each to 10 subjects). Analyses of lack of fit for various simple functions were performed to determine bestfitting functions. The simple power function was often found to be an adequate fit to the data for all the response modalities used, although all of the response modalities were sensitive to changes in stimulus range. For simple power functions, the category-scale exponent was a function of both the range of stimuli and the number of categories provided. Category scales did not always produce exponents smaller than those obtained with magnitude estimation, which calls into question the concept of a virtual exponent for category scales.     

Pain combines additively across different sensory systems: a further support for the functional theory of pain

Subjects made magnitude estimations of noxious stimuli produced by a 6 X 6 factorial design of electric shocks (pulse trains) and loud tones. Group data and all individual results conformed to a linear additive model of pain. The estimates of pain approximated the linear sum of the pain estimates of the individual electrocutaneous and auditory components. Pain related differently to the two inducing stimuli. It grew as a mildly expansive power function of current intensity (with an exponent of about 1.2) but as a mildly compressive power function of sound-pressure level (with an exponent of about 0.8). These results replicate recent findings by the same authors in 1986 using a more aversive type of electric stimulation. They are interpreted as supportive of a new functional approach to understand pain and pain-related phenomena.     

Scaling occupational prestige by magnitude estimation and category rating methods: A comparison with the sensory domain

In a field study, models for magnitude estimation and for category ratings are applied to the scaling of occupational prestige. The two respective models provide sufficient conditions for magnitude estimates to yield logarithmic interval scales and for category ratings to lead to interval scales. Both models are found to hold reasonably well for the majority of respondents. As implied by a third model, the relation between magnitude estimation and category rating scales can well be described by a generalized power function. Although overall results do not favour one method over the other individual data analyses reveal substantial interindividual differences with respect to the capability of performing magnitude estimates and category ratings, respectively. The findings are compared to results recently found in psychophysical laboratory experiments, and it is concluded that the individual scale properties the two methods provide do not differ across the attitudinal and the sensory domains.     

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.     

Magnitude scales for electrocutaneous stimulation

The application of Stevens’s power law to the sensation resulting from electrocutaneous stimulation is reviewed. Its use for data from individual observers as well as pooled data from several observers is discussed. Magnitude estimates were obtained from 33 observers of the sensation resulting from electrocutaneous stimulation over the median nerve. Seven mathematical functions were applied to the data and tested for goodness of fit. The power function with or without threshold correction factor did not emerge as better than alternative functions. Difficulties in using the power function in studies of individual differences are reviewed. It is concluded that there is no adequate reason at present to discard the linear function in favor of more complex functions in psychophysical scaling of sensation induced by electric shock.     

Integration of noxious stimulation across separate somatosensory communications systems: a functional theory of pain

Functional measurement analyses and psychophysical techniques were used to assess how separate, cross-modal, aversive events are integrated in judgements of pain. Subjects made magnitude estimations of noxious stimuli produced by a 6 X 6 factorial design of electric shocks and loud tones. Group data and most of the individual results were consistent with a model of linear pain summation: The estimates of pain approximated the linear sum of the pain estimates of the individual electrocutaneous and auditory components. The relation between painful sensation and current intensity could be described by a mildly expansive power function with an exponent of about 1.1. Auditorily produced painful sensation related to sound pressure level by a mildly compressive power function with an exponent of about 0.90 as a representative figure. Results are interpreted in terms of a functional theory of pain. Noxious events are first transformed to psychological scale values via stimulus-specific psychophysical transfer functions. The outputs of these functions are then combined with other pain-related internal representations of either sensory or cognitive origin, according to simple algebraic models.     

Comparison of magnitude estimation of loudness in children and adults

This study compared children (mean age 10.9 years) and college students on the magnitude estimation of loudness. Both the 20 children and the 20 adults were unpracticed observers. In one condition, the standard tone was assigned the number 10, and in the other condition, the number 20. Under both conditions the power function was found to fit the data of the children quite well, and to give approximately the same exponent. Of particular interest was the similarity between the data of the children and adults.     

Individual differences in long-range time representation

On the basis of experimental data, long-range time representation has been proposed to follow a highly compressed power function, which has been hypothesized to explain the time inconsistency found in financial discount rate preferences. The aim of this study was to evaluate how well linear and power function models explain empirical data from individual participants tested in different procedural settings. The line paradigm was used in five different procedural variations with 35 adult participants. Data aggregated over the participants showed that fitted linear functions explained more than 98% of the variance in all procedures. A linear regression fit also outperformed a power model fit for the aggregated data. An individual-participant-based analysis showed better fits of a linear model to the data of 14 participants; better fits of a power function with an exponent β?>?1 to the data of 12 participants; and better fits of a power function with β?<?1 to the data of the remaining nine participants. Of the 35 volunteers, the null hypothesis β?=?1 was rejected for 20. The dispersion of the individual β values was approximated well by a normal distribution. These results suggest that, on average, humans perceive long-range time intervals not in a highly compressed, biased manner, but rather in a linear pattern. However, individuals differ considerably in their subjective time scales. This contribution sheds new light on the average and individual psychophysical functions of long-range time representation, and suggests that any attribution of deviation from exponential discount rates in intertemporal choice to the compressed nature of subjective time must entail the characterization of subjective time on an individual-participant basis.     

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.     

Psychophysical judgment of curvatures

In this study we explored psychophysical judgments (magnitude estimation) of the curvature of visual lines. Prior research has lacked a rigorous definition of curvature, but here it was defined as the reverse of the ray of the osculating circle tangent to the curve in the given point. Four different functions were evaluated: hyperbole, sinusoid, parabola, and a cubic function. Using the method of magnitude estimation to judge 20 exemplar curves from each of the families, three respondents made a total of 1440 estimations each. Responses to all four families of curves were fit well by Stevens' power function with an exponent less than 1. One application of this research is the use of such curves as variables in assessing illusions of curvature.     

METHODOLOGICAL NOTE ON SCALES OF GUSTATORY INTENSITY

E kman , G. Methodological note on scales of gustatory intensity. Scand. J. Psychol ., 1961, 2, 185–100.—The subjective salt intensity of seven concentrations of sodium chloride in water was measured by the method of ratio estimation and by three variants of the method of magnitude estimation. The four scales thus constructed were not in agreement. The magnitude scales varied systematically with the stimulus used as standard. The average magnitude scale was in good agreement with the scale constructed by the method of ratio estimation. A combined scale was constructed from all the data. This scale is a power function of stimulus concentration. The exponent of the function is 1.59. The function includes an additive constant, which may indicate either the lack of an absolute zero point of the scale, or the presence of a basic sensation without external stimulation.     

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.