An “open-transformed scale” for correcting ceiling effects and enhancing retest reliability: The example of pain

The use of closed scales (with anchors at each end) to measure pain was found to produce ceiling effects characterized by a deceleration of ratings toward the upper end of the scale. This was consistent with previous research. Apart from producing nonlinear functions, the closed scale also limited test-retest reliability because of subjects’ tendencies to correct their distorted ratings in subsequent trials. However, an open-ended scale coupled with transformation of reported ratings into a decile scale virtually eliminated the ceiling effect, thus producing consistently linear functions and maximizing test-retest reliability. This finding may have implications for the measurement of other sensory and psychological phenomena, especially those in which the property evaluated varies in a continuous fashion.     

Categories of Large Numbers in Line Estimation

How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words—thousand, million, billion—to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position and response time are expected near the boundaries between categories. In contrast to previous work (Landy, Silbert, & Goldin, 2013) that suggested only that a minority of college undergraduates employed categorical boundaries, we find that discontinuities near category boundaries occur in most or all participants, but that accurate and inaccurate participants respond in opposite ways to category boundaries. Accurate participants highlight contrasts within a category, whereas inaccurate participants adjust their responses toward category centers.     

Evidence against “absolute” scaling

Zwislocki and Goodman (1980) argue that there exists an “absolute coupling” of numbers to sensation magnitudes and conclude that, when subjects are left unconstrained by a designated stimulus-number pair, they may use this “absolute” scale. The purpose of the studies reported here was to test whether Zwislocki and Goodman’s (1980) absolute scaling procedure reduces contextual effects due to variations in the stimulus spacing. It was found that magnitude estimations vary as a function of the stimulus spacing, regardless of whether subjects are instructed to use a standard and modulus, and, furthermore, that category ratings yield effects of the stimulus spacing comparable to those obtained with magnitude estimations. It is argued that removing the so-called constraints of a standard and modulus does not yield an “absolute” scale of sensation. The absolute scaling procedure increases response variability and thereby lowers the power of a test for contextual effects.     

Relative olfactory intensity perception as mediated by ratio-range category scale responses

Scales whose categories are labeled with ranges of ratio values are compared with verbal category scales and magnitude estimation. Relative perceived intensities of Eugenol odor were scaled by power law methods, using 102 Ss, five scaling methods-one verbal, two numerical, and two magnitude estimation-and making comparisons against two alternate odor reference standards. Variations in the psychophysical exponent values derived under each condition were examined. Comparisons between scale types were made: numerical ratio-range category scales may behave as magnitude estimations or as category scales depending on the way responses are scored by the experimenter.     

Direct estimation: effect of methodological factors on scale tupe

Abstract.— An experiment was performed to investigate the importance of methodological differences between magnitude estimation and category rating. It was found that the form of a scale varied with (a) the range of responses used by the subject and (b) with his freedom of choosing a highest number. Other factors investigated played a minor role. A narrow range and fixed upper scale value yielded the typical category scale, a wide range with no restriction on the highest value the typical magnitude scale. The same factors that determined the form of the scale also affected the Weber functions.     

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.     

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.     

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.     

Judgments of average lightness and darkness: A further consideration of inverse attributes

Twenty subjects judged the average lightness and darkness of paired Munsell gray papers according to magnitude estimation and category rating instructions. The data from category rating were approximately consistent with an averaging model but those from magnitude estimation were not. An analysis in terms of a two-stage model suggested that category ratings and magnitude estimates of each attribute were produced on the basis of the same composition rule, but implicated different output transformations. In judging darkness, subjects appeared to have reversed the scales employed in judging lightness, by a linear transformation in the case of category rating and by a reciprocal transformation in the case of magnitude estimation. However, differences between the input parameters obtained from judgments of lightness and darkness suggested that the scales of these attributes constructed from judgments of average magnitude may be biased.     

An "open-transformed scale" for correcting ceiling effects and enhancing retest reliability: the example of pain

The use of closed scales (with anchors at each end) to measure pain was found to produce ceiling effects characterized by a deceleration of ratings toward the upper end of the scale. This was consistent with previous research. Apart from producing nonlinear functions, the closed scale also limited test-retest reliability because of subjects' tendencies to correct their distorted ratings in subsequent trials. However, an open-ended scale coupled with transformation of reported ratings into a decile scale virtually eliminated the ceiling effect, thus producing consistently linear functions and maximizing test-retest reliability. This finding may have implications for the measurement of other sensory and psychological phenomena, especially those in which the property evaluated varies in a continuous fashion.     

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.     

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.     

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.     

Sensory scales of taste intensity

The subjective intensity of taste was scaled by the method of magnitude estimation in which Os assigned numbers to designate the apparent strength ofstimulus concentrations. Substances used were sucrose, dextrose, maltose, fructose, saccharin, Sucaryl, sodium chloride, and quinine sulfate. For aqueous solutions of each substance, taste intensity was found to increase as a power function of concentration by weight. Some approximate exponents were: sucrose, 1.3; sodium chloride, 1.4; quinine sulfate. 1.0. The magnitude scale for sucrose was compared with the category scale obtained by a commonly used rating procedure. The category scale turned out to be highly nonlinear.     

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.     

Individual loudness functions determined from direct comparisons of loudness intervals

Five subjects were required in each trial to directly compare two pairs of tones and indicate which pair of tones had the greater loudness difference. Ten 1,200-Hz tones differing only in intensity were employed. Subjects made binary comparisons among the 45 tone pairs that can be formed from these 10 tones. The loudness difference comparisons of each subject were found to satisfy two properties (transitivity and monotonicity) that are required for an interval scale representation of loudness. Therefore, individual loudness scales were constructed using a nonmetric scaling technique designed for comparisons of sensory intervals. These loudness scales differed significantly from subject to subject. Since a nonnumerical scaling procedure was employed, these individual differences could not be attributed to biases in the way in which observers use numbers or numerical concepts to describe the loudness of tones. Hence, they suggest strong individual differences in the coding of sound intensity.     

刺激范畴的激活与抑制对预搜索的影响

考察范畴混合与预知靶子范畴对预搜索的影响。以2到9的汉字与阿拉伯数字为范畴材料,以搜索一个靶子数字（三、七、3或7）为任务,以预搜索为实验范式。在一些条件下,被试预知靶子的范畴,其它条件下则无法预知靶子的范畴。结果表明：（1）预知靶子范畴可以改进搜索;（2）当靶子与旧客体的范畴相同时,上述预期效应就会减弱;（3）当靶子与旧客体的范畴不同时,范畴预期效应又增加了。最后讨论了实验结果的理论意义     

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.     

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.     

Individual differences in loudness processing and loudness scales

Parameters of the psychophysical function for loudness (a 1000-Hz tone) were assessed for individual subjects in three experiments: (a) binaural loudness summation, (b) temporal loudness summation, and (c) judgments of loudness intervals. The loudness scales that underlay the additive binaural summation closely approximated S. S. Stevens's (1956) sone scale but were nonlinearly related to the scales that underlay the subtractive interval judgments, the latter approximating Garner's (1954) lambda scale. Interindividual differences in temporal summation were unrelated to differences in scaling performance or in binaural summation. Although the exponents of magnitude-estimation functions and the exponents underlying interval judgments varied considerably from subject to subject, exponents computed on the basis of underlying binaural summation varied less. The results suggest that interindividual variation in the exponent of magnitude-estimation functions largely reflects differences in the ways that subjects use numbers to describe loudnesses and that the sensory representations of loudness are fairly uniform, though probably not wholly uniform, among people with normal hearing. The magnitude of individual variation in at least one measure of auditory intensity processing, namely, temporal summation, seems at least as great as the magnitude of the variation in the underlying loudness scale.