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.     

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.     

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.     

Cognitive reference points in judgments of symbolic magnitude

Research on speeded symbolic magnitude comparisons indicates that decisions are made more quickly when the magnitudes of the stimuli being compared are relatively close to an explicit or implicit reference point. Alternative explanations of this phenomenon are tested by seeking similar effects in nonspeeded rating tasks. In accord with the predictions of discriminability models, rated magnitude differences between stimuli in the vicinity of a reference point are expanded relative to differences between stimuli far from it. The inferred locations of cities along a west-east axis varies systematically depending on which coast, Pacific or Atlantic, is specified as the reference point. Scales derived from the rating data are correlated with the pattern of reaction times obtained in a comparable speeded comparison task. In addition, the distance between the cities nearest the locale of our subjects is subjectively stretched. Reference point effects are also observed when the form of the comparative specifies an implicit reference point at either end of a continuum of subjective size; however, these effects are very small and do not clearly support a discriminability interpretation. Stronger evidence for discriminability effects is obtained when an explicit reference point is established at an arbitrary size value. An implicit scaling model, related to range-frequency theory, is proposed to account for the influence of reference points on relative discriminability of stimulus magnitudes. The implicit scaling model is used to develop an account of how symbolic magnitudes may be learned and of how habitual reference points can produce asymmetries in distance judgments.     

Subjective ratios and differences in perceived heaviness

In previous studies, judgments of ratios and differences in subjective magnitude have yielded similar orders, consistent with a hypothesis that a single perceived relation underlies both judgment tasks. In the present research, 15 subjects estimated heaviness differences between 28 pairs of eight weights and each of 8 groups of 10 subjects evaluated heaviness ratios of eight variable stimuli with respect to a different standard stimulus. Presenting stimuli that were equally spaced on a cube-root scale of weight enhanced expected ordinal discrepancies between ratio and difference estimates, and employing independent groups for each standard stimulus in ratio estimation eliminated a possible bias due to varying standards within the presentation sequence. Differences in orders of ratio and difference estimates together with differences in scales obtained from non-metric analyses in terms of a difference model indicated that the judgments were based on two perceived relations that are ordinally consistent with arithmetic operations of ratios and differences. A ratio scale of heaviness was derived from the combined orders of subjective ratios and differences.     

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.     

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.     

Intra- and interindividual variation in the power law exponent for area summation

Subjects were required to indicate the magnitudes of subjective sums of areas of regular 20-sided polygons by three methods—generation of a polygon of matching area, estimation of the ratio of the subjective sum to a fixed area, and magnitude estimation. Power law exponents were estimated for each subject by maximum likelihood estimation. A very large amount of interindividual exponent variation was observed. Exponents resulting from magnitude estimation were consistent on the average with previous results, but exponents from area matching and ratio estimation were much larger and close to unity. When subjects were given feedback on the discrepancy between their responses and physical area summation, they readily shifted their exponents toward unity.     

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.     

Reevaluation of two models for judgments of perceptual intervals

In Marks and Cain’s (1972) experiment, 12 Ss estimated magnitudes of stimuli presented one at a time and differences in magnitudes of stimuli presented in pairs for heaviness of weight, roughness of emery papers, and area of circles. Their results appeared inconsistent with the two-stage model of magnitude estimation. But the expressions fitted to the data were different from those previously used in tests of the model. An additive constant was omitted. A reanalysis of their data with the constant included yielded support for the model for all three continua. An alternative model proposed by Marks and Cain was reasonably consistent with the da~a for weight, while inconsistent with the data for roughness and area.     

The dimensionality of judgments of heaviness

Two groups of Ss judged differences in heaviness between all pairs of stimuli in three series of five weights each. One group judged differences in the 100-300 g range, the other group judged differences in the 700–900 g range. The three series of weights in the two ranges consisted of one series of constant size varying in both weight and density, one series varying only in weight, and one series varying only in density. Multidimensional scaling analysis revealed at least two dimensions, and indicated that where both weight and density are allowed to vary, both will affect judgment of the magnitude of differences in heaviness. With heavier weights, density appears to exert an increasingly important influence on judgments of heaviness. Predictions from Ross and Di Lollo’s vector theory were confirmed.     

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.     

Neutral vs ego-orienting instructions: Effects on judgments of magnitude estimation

Three experiments were performed to examine the relative constancy of the exponent in the psychophysical power law under varying motivating conditions. The method of magnitude estimation was used to obtain judgments of apparent tactual roughness or of apparent area size of squares. Patterns of the qualitative observations of the three Es and of the various exponents for the six groups of Ss indicated that neutral instructions and “ego-orienting” instructions, which were perceived as unbelievable coming from an equal fellow student, both yielded exponents identical to those reported in the literature. Believable ego-orienting instructions given by an E of clearly perceived higher social status produced a statistically significantly lower exponent than neutral. Intermediate conditions, wherein Ss apparently disbelieved both types of instructions, but assumed that the superior-status E was “analyzing” them, yielded exponents of intermediate size. Results and supplementary trend analyses are discussed as possible, highly sensitive indicators of motivational impacts on sensory judgments.     

On the "absoluteness" of category and magnitude scales of pain

The concept of "absolute scaling" (Zwislocki & Goodman, 1980) implies that direct judgments of sensory magnitude not only reflect the relative positions of the stimuli being judged, but also permit us to assess level differences in sensation. In order to explore this notion for different scaling methods, in the present investigation we compared magnitude estimation with category partitioning, a verbally anchored categorization procedure, in scaling painful pressure stimuli covering different intensity ranges. The results indicate that when the same stimulus range was presented after 1 week, both methods appeared to be highly reliable, with category partitioning faring somewhat better than magnitude estimation. When the stimulus range was unobtrusively changed between sessions, both methods reflected the within-subjects shift in absolute level. When two different sets of subjects judged the slightly different stimulus ranges, both methods resulted in scale values consistent with absolute scaling, though only category partitioning was sensitive enough to differentiate the two stimulus ranges. The results are discussed in the context of different possibilities of anchoring direct scaling methods in order to obtain "absolute" level information.     

Contextual effects in difference judgments

Manipulating stimulus spacing, stimulus frequency, or stimulus range usually affects intensity judgments. In six experiments, I investigated the locus of analogues of these contextual effects in a “difference” estimation task. When all stimuli elicited the same taste quality, stimulus distribution affected the scale values only when water was included in the stimulus set (Experiments 1–3). When the subjective ranges of two taste qualities were manipulated, different scale values were obtained for the separate qualities in the two conditions (Experiment 4). Manipulation of the expected response distribution did not affect the scale values or the responses (Experiments 5–6). It is concluded that shifts in stimulus distributions or stimulus ranges result in shifts in subjective scale values. The contextual effects can be interpreted as relative shifts of a number of gustatory continua, with water lying on a separate continuum. Proposed is a model for context-dependent judgments, consisting of four stages: stimulus classification, stimulus placement, continuum placement, and continuum projection.     

On the “absoluteness” of category and magnitude scales of pain

The concept of “absolute scaling” (Zwislocki & Goodman, 1980) implies that direct judgments of sensory magnitude not only reflect the relative positions of the stimuli being judged, but also permit us to assess level differences in sensation. In order to explore this notion for different scaling methods, in the present investigation we compared magnitude estimation with category partitioning, a verbally anchored categorization procedure, in scaling painful pressure stimuli covering different intensity ranges. The results indicate that when the same stimulus range was presented after 1 week, both methods appeared to be highly reliable, with category partitioning faring somewhat better than magnitude estimation. When the stimulus range was unobtrusively changed between sessions, both methods reflected the within-subjects shift in absolute level. When two different sets of subjects judged the slightly different stimulus ranges, both methods resulted in scale values consistent with absolute scaling, though only category partitioning was sensitive enough to differentiate the two stimulus ranges. The results are discussed in the context of different possibilities of anchoring direct scaling methods in order to obtain “absolute” level information.     

The effect of object shape and mode of presentation on judgments of apparent volume

Apparent volume for geometric solids and life-sized two dimensional representations of these solids was scaled by magnitude estimation. Data were adequately fit by power functions whose exponents were less than those previously reported for judgments of apparent length and area. Object shape and mode of presentation affected both magnitude estimations of apparent volume and the best fitting power function exponents. The influence of shape, both across object shape classes and within the cylinder shape class, appears to depend upon the relative elongation of the vertical dimension.     

Time errors and differential sensation weighting

Sixteen pairs of successive tones, with different amplitude combinations, were presented with 16 combinations of tone duration and interstimulus interval. A separate group of 12 subjects was assigned to each presentation condition and made comparative loudness judgments for each of the pairs. Perceived within-pair loudness differences were scaled by a Thurstonian method using the subjective width of the "equal" category as the unit. The scale differences were well described by weighted linear combinations of the sensation magnitudes of the tones in the pairs. The time error can be regarded as an effect of this differential weighting. For the longer interstimulus intervals, the weight of the second tone was the greater, causing the usual inverse relation between time error and stimulus intensity level. For the shorter interstimulus intervals, these effects were reversed. An analysis of the pattern of weights led to the development of two models, one of which is a generalization of Michels and Helson's time error model. The weights could be interpreted as reflecting the differential efficiency of the loudness information from the two compared stimuli.     

The effect of the induction of involvement on psychophysical judgments

The present investigation was based on the concept of invariance, which holds that identical principles govern the judgment of stimuli arrayed on both physically and socially defined scales. Two experiments were conducted in which involvement was manipulated through the use of instructions presented in conjunction with category judgments obtained in a training session. This was followed by an anchor session. Experiment 1 employed a series of weights as stimuli, while Experiment 2 used random pattern dot slides. The experiments were similar, except for the inclusion of two additional features in the weight study. These were: (1) positive feedback, introduced as an independent variable between the judgment sessions and tested for its effect as an enhancer of involvement; and (2) two stimuli in the anchor session not part of the original stimulus series. These weights provided a test for the generality of involvement set. Among various hypotheses tested in both studies was the expectancy that the introduction of involvement associated with the formation of a judgment scale would lessen the impact of an anchor on judgment. This expectancy was based on the observed effects of involvement in social judgment contexts. Results of both experiments supported the judgment maintenance hypothesis.     

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).