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.     

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.     

Is an equal interval scale an equal discriminability scale?

Stevens and Galanter’s (1957) iterative procedure for minimizing bias in category scaling was used for the scaling of loudness of white noise. The spacing obtained deviated systematically from a spacing constructed in accordance with an equal discriminability scale from a previous experiment (Eisler & Montgomery, 1972). For the stimulus spacing yielding a “pure” category scale, a magnitude scale was constructed too. Since the category scale could be predicted accurately by Fechnerian integration of this magnitude scale, it was concluded that the “pure” category scale is a pure discrimination scale. The discrepancy between the equal discriminability scale and the “pure” category scale was interpreted as a bias in the former scale due to greater recognizability of stimuli located at the extremes of the stimulus range.     

Ratio scale measurement of the perceived lengths of lines.

In Experiments 1 and 2, magnitude and category estimates of perceived line length difference were used to rank order 36 pairs of lines with respect to the psychological length difference between the lines forming a pair. In Experiment 3 magnitude estimates of line length ratios were used to order the same 36 pairs with respect to the psychological length ratio between the lines comprising a pair. A nonmetric analysis of the rank-order of the difference and ratio judgments was used to construct a ratio scale of perceived line length. Perceived line length was found to be a power function of physical line length with an exponent of .46. In Experiments 4 and 5, magnitude and category estimates of line length similarity were obtained. A nonmetric analysis of these data indicated that similarity judgments were monotonic inverses of judgments of psychological ratios.     

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.     

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.     

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.     

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.     

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.     

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.     

HOW PROTHETIC IS THE CONTINUUM OF SMELL?: A further comment on the relation between magnitude and category scales

The relation K =αlog( ψ + q / k ) +β between category scale K and magnitude scale ψ found in previous experiments on loudness and softness of white noise was confirmed for smell data, too. The small increase in variability (SDs) of judgments with magnitude for smell as compared with loudness and softness suggests the substitution of the prothetic-metathetic dichotomy with a quantifiable feature, protheticness, with metathetic continua as the limiting case.     

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.     

Psychophysics of Kinesthesis

A horizontal straight-arm movement was used to determine the relationship between the physical stimulus continuum and the psychological continuum of kinesthetic extent of arm movement. A power function with an exponent of 1.075 best described this relationship. In addition, to classify extent of movement as belonging to either a prothetic or metathetic continuum, the category scale was derived. According to the functional criteria described by Stevens (1957), extent of movement was subserved by a metathetic process which would indicate that, for the present task, judgments were made through the use of positional cues rather than amplitude cues.     

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.     

Perceived numerosity: A comparison of magnitude production,magnitude estimation,and discrimination judgments

In previous studies, modalities with a higher Weber fraction have tended to have a lower power-function exponent. Within a modality, however, the Weber fraction and power-function exponent for individual subjects were unrelated, and the present study largely confirms this finding for the numerosity dimension. More important than discriminability in the judgment of numerosity were cognitive factors. A single feedback trial considerably reduced intersubject variability on the magnitude-estimation exponent, although it failed to eliminate individual differences completely (precue and postcue exponents correlated signigicantly, r=+.50). Intrasubject variability, by contrast, seemingly did not involve the underlying exponent. As in previous studies, numerosity generally was underestimated and the power-function exponent was 1.08 for magnitude production and .80 for precue magnitude estimation. Contrary to previous results, however, males and females did not differ in exponent, perhaps because the present procedure allowed self-selection of individuals more interested in numerosity tasks.     

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 PROBLEM OF CATEGORY SCALES IN PSYCHOPHYSICS

E isler , H. On the problem of category scales in psychophysics. Scand. J. Psychol ., 1962, 3 , 81–87.—In the first Part of the paper it is shown that, if three assumptions are granted, the category scale must be logarithmically related to the magnitude scale: K =α log φ +β . In the second part of the paper, the relation K =α log ( kφ+ q) +β is derived, starting from the assumptions that the category scale is a pure function of discrimination and that discrimination is appropriately described by the linear generalization of Weber's law for prothetic continua. We can reconcile these two formulas by defining the zero-point of the magnitude scale as the point where variability vanishes.     

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.     

PSYCHOPHYSICAL SCALES OF THE ODOR INTENSITY OF AMYL ACETATE

The paper presents psychophysical scales of amyl acetate in benzyl benzoate sniffed from cotton. Four scales obtained by direct scaling procedures, ratio estimation and magnitude estimation, yield functions of the form R = kSⁿ , with n ranging from 0.39 to 0.57. These data support earlier findings that the intensity of smell is a negatively accelerated function of stimulus intensity. In addition, comparison of magnitude and category scales indicate that the subjective intensity of smell is a prothetic continuum. Finally, further analysis of subjective ratios as a function of stimulus ratios again shows that ( a ) the magnitude scale is a ratio scale and ( b ) the function obtained conforms to the power law.