Nested models of relative judgment: Applications to a similarity averaging model

A theoretical analysis of the Eisler and Ekman (1959) model of similarity judgments for unidimensional continua is presented, based on a general model of relative judgment. This general model assumes that judgments are mediated by perceived relations of pairs of stimuli, that there exists a transformation of the judgmental response that is a function of the sensory ratio of the two stimuli, and that response bias operates in a multiplicative manner. Three structural conditions are presented, each imposing constraints on the structure of observed judgments. The structural conditions define threenested models of relative judgment, with the second a weakened version of the first, and the third a weakened version of the second. The special virtue of the general model is that it is applicable to a variety of judgmental tasks (e.g., ratio estimation, similarity, pair comparison), the key being derivation of theresponse transformation conforming to the structural conditions. The structural conditions thus constitute necessary conditions for several different judgmental models. The theory was first applied with success to ratio estimation judgments (Fagot, 1978), and this paper applies the general model to the Eisler and Ekman similarity “averaging” model. Empirical tests were carried out on published data for pitch, darkness, visual area, and heaviness judgments. Although the strong form of the model presented by Eisler and Ekman was rejected, weakened versions were generally supported by the data. These results were similar to those obtained for ratio estimation (Fagot, 1978), and are interpreted to be very promising for the general model of relative judgment.     

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.     

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.     

The frequency principle for comparative judgments)

This experiment supplements a recetlf study of the effects of an interpolated weight upon memory for a standard (Parducci, Marshall, & Degner, 1966). The earlier data suggested that the stimulus-averaging approach as represented by the theory of adaptation level should be modified to incorporate the principle that Ss use alternative categories with equal frequency. The present data support this modification by demonstrating that even when the stimulus mean is held constant, judgments are affected by the relative frequencies of the comparison stimuli.     

Predicting similarity and categorization from identification.

In this article, the relation between the identification, similarity judgment, and categorization of multidimensional perceptual stimuli is studied. The theoretical analysis focused on general recognition theory (GRT), which is a multidimensional generalization of signal detection theory. In one application, 2 Ss first identified a set of confusable stimuli and then made judgments of their pairwise similarity. The second application was to Nosofsky's (1985b, 1986) identification-categorization experiment. In both applications, a GRT model accounted for the identification data better than Luce's (1963) biased-choice model. The identification results were then used to predict performance in the similarity judgment and categorization conditions. The GRT identification model accurately predicted the similarity judgments under the assumption that Ss allocated attention to the 2 stimulus dimensions differently in the 2 tasks. The categorization data were predicted successfully without appealing to the notion of selective attention. Instead, a simpler GRT model that emphasized the different decision rules used in identification and categorization was adequate.     

Mental size scaling examined

Observers made same-different shape judgments of stimuli that were identical in shape and size different in shape but not in size, or differed in relative size along a number of steps for both same- and different- shaped forms. "Same" judgment RTs increased monotonically with increases in the magnitude of the relative size difference. In contrast, "different" judgment RTs were unaffected by changes in relative size. A second experiment in which stimulus presentation was successive rather than simultaneous yielded essentially the same results. Consideration was given to a dual- process model in which a time-consuming analog process normalizes stimuli that are size discrepant prior to a comparison stage that is operative for those structures responsible for "same" decisions but not for "different" judgments. Some evidence that seems inconsistent with a normalization process which operates prior to contact with memory was discussed, and an alternative explanation in which the early detection of a size difference causes changes in decision criteria was suggested.     

Comparison requirements and attention in identical-nonidentical stimulus discriminations

Perceptual matching data show several puzzling effects. Particularly problematic are the disparities between the processing rates for same and different stimuli--the fast-same effect--and between the processing rates for two same-different judgment tasks that are related as mirror images--the task effect. Current models have difficulty accounting simultaneously for both effects. Central to these models is a stimulus comparison process that derives relative judgments of sameness and difference from tests of the congruence of stimulus representations. A contrasting view holds that same-different judgments can be modeled as absolute, rather than relative, judgments. This latter view is shown to be supported by experimental data. Reaction times for judgments of identical letter strings increase with string length at the same rate whether judgments are based on all the information in the strings or just the information in a single pair of component letters. The data show that stimulus comparisons of the sort described by previous models are not involved in these judgments. An attentional model accounts for the data and for the fast-same and task effects as well.     

Weight of evidence supports one operation for "ratios" and "differences" of heaviness

This paper investigates an apparent contradiction between recent studies of "ratios" and "differences" of heaviness. Birnbaum and Veit (1974) found a single rank order for judgments in the two tasks, whereas Rule, Curtis, and Mullin (1981), who used a different stimulus set, procedure, and experimental design, reported two orders. To investigate the cause of this discrepancy, the present study manipulated the experimental design using the same stimuli and procedure as Rule et al. (1981). In one experiment (within-subject designs), each subject judged all combinations of the standard and comparison stimulus; in the other experiment (between-subjects designs) each subject received only one standard, and different groups of subjects were given different standards. "Ratios" and "differences" of heaviness were monotonically related for the majority of subjects who judged all combinations of standards and comparisons. Variations in the modulus and response examples did not affect the rank order of "ratios" within subjects. These results suggest that the contradiction in results is due to the difference in experimental design rather than differences in stimuli or procedure. In the between-subjects designs, the rank order of the "ratio" judgments depended on the standards and examples. Both previous and present results are consistent with the theory that subjects use one operation, subtraction, for both tasks and that the judgment function varies with between-subjects manipulations of the standard, examples, and modulus.     

Relative judgment: A phenomenon and a theory

An experiment is described in which it was demonstrated that relative judgments of the more probable of two statements are quicker if the statements are both probable rather than improbable. For judgments of the less probable, the reverse result was obtained. This phenomenon is discussed in relation to various theories of judgment and choice. A theory is presented which assumes that judgment involves a relation between a stimulus and a word and follows Thurstone’s notion that stimuli differ in their discrirninal dispersions. This new theory is shown to be consistent with recent results in psycholinguistics.     

Modifying the gestalt factor of proximity: theories compared

A theoretical controversy between adaptation-level (AL) theory and response-frequency-equalization (RFE) theory was examined. The stimulus presented for judgment was a matrix of dots which varied in appearance as the relative horizontal and vertical separation between the dots was altered. The matrix appeared to be columns of dots (larger horizontal separation), rows of dots (larger vertical separation), or neither row-like nor column-like (when the two separations were equal). A stimulus with an extreme row organization was presented on alternate trials and resulted in more vertical judgments. Both AL theory and RFE theory are shown to be inadequate on two counts. They both predict that the response latencies to the vertical stimuli should be affected, whereas these were unaffected. They both predict that in terms of the categorical decision, 'horizontal' or 'vertical', the responses to both horizontal and vertical stimuli should be affected. It is shown that the context effect is almost entirely on the horizontal stimuli. These results are consistent with a spatial-adaptation model.     

重复启动对时序知觉和时距知觉的影响

采用时序判断和时距估计任务,探讨重复启动对时序知觉和时距知觉的影响。结果表明：在时序判断中,当未启动的靶刺激先出现时出现了显著的启动效应,启动的靶刺激被知觉为早出现;在启动的靶刺激先出现时产生了启动效应的反转,启动的靶刺激被知觉为后出现。在时距估计中,启动条件下的两个靶刺激之间的时距长度被知觉为显著长于未启动条件下的时距长度。因此,在同一时间历程中,重复启动对时序知觉和时距估计都存在显著影响,启动刺激所引发的表征激活和反应抑制双加工过程在时序知觉中是按照表征匹配机制、在时距知觉中是基于唤醒机制起作用,这为时间认知分段综合模型的理论构想提供了支持证据。     

Higher order sequential effects in psychophysical judgments

We found that the depth of sequential effects depends on the judgment task. An experiment with squares indicated that stimulus-response pairs up to two trials back were included in the judgment process when subjects were required to make category judgments of size, whereas only the immediately preceding event was incorporated when subjects were making magnitude estimations. In the case of category judgment, interactions between the current stimulus and prior stimuli as well as configural effects indicated that events one and two trials back meet an equivalent function in the judgment process and that these events may jointly operate in one trial. These findings can be explained by a class of models that assume that the position of preceding stimuli relative to the current stimulus is decisive in the judgment process. The multiple-standards model is a representative of this class according to which there are two types of standards: (1) the endpoints of the range as long-term standards and (2) traces of preceding stimuli as short-term standards.     

Higher order sequential effects in psychophysical judgments

We found that the depth of sequential effects depends on the judgment task. An experiment with squares indicated that stimulus-response pairs up to two trials back were included in the judgment process when subjects were required to make category judgments of size, whereas only the immediately preceding event was incorporated when subjects were making magnitude estimations. In the case of category judgment, interactions between the current stimulus and prior stimuli as well as configural effects indicated that events one and two trials back meet an equivalent function in the judgment process and that these events may jointly operate in one trial. These findings can be explained by a class of models that assume that the position of preceding stimuli relative to the current stimulus is decisive in the judgment process. The multiple-standards model is a representative of this class according to which there are two types of standards: (1) the endpoints of the range as long-term standards and (2) traces of preceding stimuli as short-term standards.     

Size-distance paradox with accommodative micropsia

Two experiments tested the hypothesis that the paradoxical relative distance judgment associated with the size-distance paradox is due to the visual system’s assuming equal linear size and perceiving a smaller angular size for the closer stimulus equal in visual angle. In Experiment I, two different sized coins were presented successively, and 16 Ss were asked to give ordinal judgments of apparent distance and apparent size. When the two coins depicted the same figures, the closer stimulus was judged to be farther and smaller, more frequently, than when two coins depicted different figures. In Experiment II, 48 Ss were asked to give ratio judgments of apparent distance, apparent linear size, and apparent angular size for two stimuli which were presented successively. When the stimuli were of equal shape, the mean ratios of the far stimulus to the near stimulus were smaller for the apparent distance but larger for the apparent linear size and angular size than when the stimuli were of different shape. The obtained distance judgments were consistent with the hypothesis but the obtained judgments of linear size and angular size were not.     

Quantitative functions for size and distance judgments

A psychophysical approach was used to obtain judgments of visual extent under three conditions. In tuvo conditions a comparison stimulus at each of two distances was matched in size to a standard which varied in distance. Stimuli were presented on a well-lighted table and were judged by two observers under Objective instructions. Both the standard and comparison were located in either a frontal or longitudinal plane. In a third condition relative distance estimates were given of two stimuli which varied in their relative positions along the table. The mean results for all conditions were described as a power function of physical stimulus measures. The exponent was greater than 1.0 for frontal size and usually less than 1.0 for flat size and distance. The position of the comparison affected the magnitude of the exponents to a lesser degree. These findings have relevance for interpretations of size and distance judgments.     

An empirical test of two psychophysical models

Two theoretical relationships between sensitivity measures (Weber fractions, Ekman fractions, and their logarithms) and the exponents of the psychophysical power function were tested empirically with the brightness attribute. One model was based on Weber and Ekman fractions, the other on the logarithms of these measures. The stimulus parameters were time interval between standard and comparison targets and position of the standard in the luminance series. Weber fractions were based on data obtained by the method of constant stimuli, whereas Ekman fractions and exponents were based on data obtained by magnitude estimation. The results were in closer agreement with the theoretical predictions generated by the logarithmic model when group data were analyzed. With individual subjects, a detailed correspondence between fact and theory was not found with either model.     

Relative judgment and knowledge of the category structure

For evenly spaced stimuli, a purely relative judgment account of unidimensional categorization performance is trivial: All that is required is knowledge of the size of stimulus difference corresponding to the width of a category. For unevenly spaced stimuli, long-term knowledge of the category structure is required. In the present article, we will argue that such knowledge does not necessitate a direct, absolute mapping between (representations of ) stimulus magnitudes and category labels. We will show that Stewart, Brown, and Chater’s (2005) relative judgment model can account for data from absolute identification experiments with uneven stimulus spacing.     

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.     

A critical comparison of two random-walk models for two-choice reaction time

Two random-walk models for two-choice reaction time, being the sequential probability ratio test model initially suggested by Stone and the relative judgment theory recently proposed by Link, are here compared. The comparison shows the precise mathematical relation between the models, the predictions they have in common, and two points at which they differ. Three experiments concerned with the effect of stimulus probability on two-choice reaction time are examined with respect to three predictions from the models. In this three-way comparison the two models are more like each other than either is to the data; a greater variety of phenomena is revealed than either model can at present accommodate.     

An eye fixation analysis of choice and judgment with multiattribute stimuli

The present paper examined the selection of processing heuristics for choice and judgment across a range of stimuli. Eye fixations were monitored while six subjects made choices or judgments of pairs of gambles, vacations, or gifts. Each stimulus was represented by three attributes that varied in the amount of dimensional interdependency. The two heuristics of interest were dimensional and holistic evaluation. Each manifested a characteristic pattern of eye fixations. Dimensional evaluation required alternating fixations from a single attribute of one stimulus to the same attribute of the other stimulus. Holistic evaluation was characterized by transitions from attribute to attribute within a single stimulus. The results demonstrated that when the stimulus attributes were either interdependent (gambles) or dissimilar (gifts), the processing heuristic was determined by stimulus characteristics. When the stimulus dimensions were neither interdependent nor dissimilar (vacations), the selection of a processing strategy was determined by the prescribed task. This study suggests that any global theory of choice or judgment must be validated over a wide range of stimuli.