On theoretical and realizable ideal conditions in psychophysics: Magnitude and category scales and their relation

By spacing 10 stimuli (white noise) between 40 and 110 dB according to two criteria [equal response ambiguity (ERA) and equal discriminability (ED)], an attempt was made to construct an “ideal” case for magnitude estimation and category rating. The “ideal” case is defined by linear and constant Weber functions (SDs as a function of scale values) for the two scales, respectively. Altogether, three group and two individual magnitude and category rating experiments were run with these two spacings. It was found that the ERA spacing approximated the ideal case well for both Weber functions and the ED spacing only for the Weber function of the category scale. The general psychophysical differential equation that relates scale values and Weber functions for the two scales allowed good prediction of the category scales from the magnitude scales and the Weber functions. The data suggested a distinction between phenotypic (empirical) and genotypic Weber functions, analogous to “real” and “ideal” cases in physics.     

Averaging: An empirical validity criterion for magnitude estimation

Ss judged grayness of neutral value Munsell chips under two response conditions: magnitude estimation and graphic rating. In addition, they judged average grayness of pairs of chips under the same two response modes. The averaging data were evaluated in terms of a simple model for subjective averaging. The graphic rating data fitted the model, but the magnitude estimates showed consistent discrepancy. It was concluded that the Ss were averaging, but that magnitude estimation distorted their judgments.     

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.     

Two subjective scales of number

The way in which the apparent magnitude of numbers grows with their absolute magnitude was measured with a modified version of the direct technique Marks and Slawson (1966) used to determine the psychophysical exponent for loudness. This modified technique required subjects to estimate how evenly and randomly a sequence of integers appeared to sample the numerical continuum. The results indicate that the apparent magnitude of numbers increases with a decelerated power function of their arithmetic magnitude when a series samples from an open-ended range. However, when an upper boundary of the range is specified, the subjective scale seems to be linear. Random productions of numbers parallel the results found with judgments of presented sequences. The two scales of number provide the basis for an interpretation of the difference between magnitude and category scales: that subjects use numbers differently when the response scale is open-ended Imagnitude estimation than when it has a fixed upper limit tcategory scale. Given the assumption that subjects use numbers in this way in the two tasks, the qualitative relation between magnitude and category scales is predicted.     

STUDIES IN VISUAL PERCEPTION OF ARCHITECTURAL SPACES AND ROOMS: II. Judgments of open and closed space by category rating and magnitude estimation

G ärling , T. Studies in visual perception of architectural spaces and rooms. II. Judgments of open and closed space by category rating and magnitude estimation. Scand. J. Psychol ., 1969, 10 , 257–268.—In one experiment, colour photographs of architectural spaces were employed as stimuli and judged with respect to open and closed space. Category scales of open and closed were complementary, magnitude scales were reciprocal. Judged open space correlated positively and judged closed space negatively with physical size of actual spaces. In another experiment, judged size was found to be directly related to judged open space and inversely related to judged closed space. Furthermore, open and closed space judged in situ correlated to a significant degree with corresponding judgments from photographs.     

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.     

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.     

EMPIRICAL TEST OF A MODEL RELATING MAGNITUDE AND CATEGORY SCALES

E isler , H. Empirical test of a model relating magnitude and category scales. Scand. J. Psychol ., 1962, 3 , 88–96.—The function K =α log( φ+ q/k) +β seems to describe the relation between category scale values K and subjective magnitudes φ. The additive constant q/k is obtained from the S ds of the magnitude estimates.
The model was empirically confirmed for the loudness and softness of white noise scaled by the methods of magnitude estimation and category rating.     

Stimulus discriminability in the magnitude estimation and category rating of loudness

Thirty-three Ss made category ratings and magnitude estimations of 10 auditory stimuli differing in loudness. The results from each task were examined in terms of the response uncertainty conditional upon each stimulus. The results did not support the suggestion that category judgments are influenced by the relative discriminability of stimuli in a way which is not characteristic of magnitude estimates, but were found to be consistent with the subjective Standard hypothesis. It is argued that the observed quasi-logarithmic relationship between category scale and ratio scale values reflects the constraints placed upon responses in category rating tasks.     

On scales of sensation: Prolegomena to any future psychophysics that will be able to come forth as science

Sensory scales fall into two classes. Type I scales of sensory intensity can be approximated by metric scaling procedures (magnitude estimation, magnitude production) and nonmetric procedures (conjoint measurement); Type I scales are supported by theoretical consideration of sensory processes. Type II scales of sensory dissimilarity can be approximated by metric scaling procedures (category rating, interval estimation, equisection) and nonmetric procedures (analysis of proximities). The psychophysical functions that relate Type I and Type II scales to their corresponding physical scales are in both cases power functions, but the exponents that govern Type I functions are typically about twice as large. Both Type I scales of sensory intensity and Type II scales of sensory dissimilarity are meaningful measures of perceptual experience, but they are measures of different aspects of perception. The duality of sensory scales helps to explain some apparent contradictions among divergent attempts to validate scales of sensation.     

Cross-modal additive conjoint structures and psychophysical scale convergence

I investigated a variety of issues related to the measurement of the magnitude of psychological experience, especially the magnitude of sensations. Different groups of subjects made pair comparisons, magnitude estimations, and category judgments of the "total sensory magnitude" of light and sound stimuli presented conjointly. Another group judged the dissimilarity of pairs of conjoint stimuli. Various axioms, especially double cancellation, were tested on the resulting rank orders of conjoint stimuli. Judgements of the total magnitude of conjoint combinations of sound and light stimuli formed an additive conjoint structure. Dissimilarity judgments gave rise to a closely related lattice structure. Moreover, various scales of the individual attributes (loudness and brightness) calculated from the two types of judgments of the conjoint stimuli displayed substantial convergence, each scale for a given modality being linear with all other scales for that modality.     

THE PROBLEM OF MAGNITUDE AND CATEGORY SCALES: INTERPRETATION OF RESULTS

J unge , K. The problem of magnitude and category scales: interpretation of results. Scand. J. Psychol ., 1962, 3 , 215–218.—It is proposed that three main factors are responsible for the relation between magnitude and category scales: (1) ratio judgments on a transformed category scale, (2) interval judgments, and (3) hybrid judgments. The last term refers to the tentative explanation that magnitude judgments are a cross between pure ratio and interval judgments.     

Remembrance of sounds past: memory and psychophysical scaling

What is the role of long-term memories of previous stimulus-response mappings, and of previous sensory and perceptual experiences generally, in psychophysical scaling judgments? I conducted four experiments in an attempt to provide some preliminary answers to this question. In each experiment, subjects made judgments of the loudness of sounds on 3 successive days. Stimulus intensities were drawn randomly from the same set on Days 1 and 3 but from a different set, either all 12 dB higher or all 12 dB lower, on Day 2. Four different types of psychophysical scaling judgments were studied: category judgment without an experimenter-induced identification function, ratio magnitude estimation with a variable standard, absolute magnitude estimation, and cross-modality matching. The first two methods required completely relative judgment, the last two completely absolute judgment. Data from all methods reveal profound effects of stimulus-response mappings experienced on previous days (long-term memory) and immediately previous stimuli and responses (short-term memory) on responses to current stimuli. Responses were typically a compromise between absolute and relative judgment. Individual differences were dramatic.     

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.     

Judgments of average magnitude: Analyses in terms of the functional measurement and two-stage models

Subjects made magnitude estimates of the average loudness of pairs of 1,000-Hz tones varying in sound pressure. A test of fit of an averaging model employing an analysis of variance suggested that the judgments were internally consistent. However, estimates of the parameters of a two-stage model based on the assumption that power transformations were imposed in both input and output implied a nonlinear output function, inconsistent with the averaging model. Additional analyses employing a nonmetric scaling solution also suggested that output was nonlinear, indicating that this implication was not an artifact of the strong assumptions of the two-stage model. Large differences were found among the output functions of individual subjects, and it was suggested that these may have inflated the error term in the analysis of variance, reducing its power to detect violations of the additive model. Similar analyses were performed on data from judgments of average grayness collected by Weiss (1972).     

The relation of lightness and stereoscopic depth in a simple viewing situation

The effect of stereoscopic depth on perceived lightness was studied using a simple, achromatic stimulus arrangement. In Experiment 1, depth/lightness interactions were sought between a single test field and a single induction field. In Experiment 2, depth/lightness interactions were looked for between a single test field and two induction fields. Stimuli were presented on a computer screen and viewed with a stereoscope. The subjects reported perceived lightness of the achromatic test field by rating its apparent blackness along a dimension of 0%–100%. In Experiment 1, they reported lightness judgments of the test field across 13 perceived depth levels and 8 contrast levels. In Experiment 2, they gave lightness judgments of the test field across 7 perceived depth levels and 16 contrast levels. We were particularly interested in observing the generality of Gilchrist’s coplanar ratio hypothesis. The results showed that when stereopsis and contrast levels are the available cues, depth and lightness percepts are independent, and it is retinal ratios, not coplanar ratios, that dictate lightness perception. We conclude that before the relative depth location of an object is determined, its lightness value is known through sensory-level processes.     

JUDGED CONSTITUTIONALITY OF FUNCTIONAL PSYCHOSIS DISORDER CLASSIFICATIONS

S tone , L. A. Judged constitutionality of functional psychosis disorder classifications. Scand. J. Psychol ., 1968, 9, 252–256.—Psychiatrists (N = 37) made magnitude estimations and category scale evaluations of 15 psychiatric classification stimuli with respect to their own opinions regarding biogenic predisposition associated with the stimuli. The scale from category judgments was a logarithmic transformation of the magnitude estimation scale. Judgmental variability, in subjective unit measurement, was linearly and positively related to subjective magnitude. The underlying judgmental continuum was tentatively Classified as prothetic. The results were in line with those obtained in several previous studies.     

Cross-task validation of functional measurement

Ss judged average heaviness of two unseen lifted weights on a 20-step rating scale. They also rated heaviness of a single visible object of variable size and weight. The data for each task obeyed the parallelism prediction of the linear integration model. This within-task consistency validates jointly the model and the response scale. Functional measurement procedure yielded independent scales of subjective heaviness from each task. These were linearly related, a criterion of external consistency. The plot of subjective heaviness against objective weight was slightly negatively accelerated. These results disagree with those that have been obtained with magnitude estimation. Since the rating data satisfy both internal and external consistency, they constitute the true measure of sensation. Magnitude estimation, in contrast, must be biased and invalid.     

Effects of stimulus context on magnitude estimations and category ratings of perceived loudness--the spacing of intensity intervals

Effects of stimulus context on magnitude estimations and on category ratings were examined for a range of stimulus intensities of a 1-kHz tone. The stimuli were distributed in equal-interval steps of energy so they formed a perceptual cluster of high-intensity tones with a perceptual outlier at the lowest intensity. According to the Invariance Principle, the shape of the response function should not be affected by the distribution of stimulus intensities. However, neither magnitude estimations nor category ratings yielded the linear functions predicted from the Invariance Principle when plotted on log-log axes. Instead, both procedures yielded concave-upward response functions for the group data as well as for the individual data sets of the six subjects. Moreover, unlike previous reports of a nonlinear relationship, we found a linear relationship between magnitude estimations and category ratings. Rather than implying an equivalence of the underlying sensory scales, however, our results may imply subjects used a similar attention strategy for both procedures. We consider some theoretical suggestions, including an attention-band concept, for modification of a multistage stimulus-response (S-R) transformation model.     

Categories and particulars: prototype effects in estimating spatial location

A model of category effects on reports from memory is presented. The model holds that stimuli are represented at 2 levels of detail: a fine-grain value and a category. When memory is inexact but people must report an exact value, they use estimation processes that combine the remembered stimulus value with category information. The proposed estimation processes include truncation at category boundaries and weighting with a central (prototypic) category value. These processes introduce bias in reporting even when memory is unbiased, but nevertheless may improve overall accuracy (by decreasing the variability of reports). Four experiments are presented in which people report the location of a dot in a circle. Subjects spontaneously impose horizontal and vertical boundaries that divide the circle into quadrants. They misplace dots toward a central (prototypic) location in each quadrant, as predicted by the model. The proposed model has broad implications; notably, it has the potential to explain biases of the sort described in psychophysics (contraction bias and the bias captured by Weber's law) as well as symmetries in similarity judgments, without positing distorted representations of physical scales.