Single judgments of numerosity

In previous studies, subjects generally underestimated the number of elements present in a display. To eliminate the range and intertrial effects that arise when several displays are judged in succession and that might have produced the underestimation, subjects in the present study judged only a single display. The single judgments fitted a power function having an exponent of .83, which is consistent with previous data. Single judgments of loudness, area, and duration, by contrast, have produced abnormally low exponents apparently because the built-in scale unit, or modulus, available on numerosity is lacking for other modalities. The tendency to underestimate numerosity was much stronger for female than for male subjects.     

Perceived numerosity, item arrangement, and expectancy

Ten sets of dots ranging in number from 7 to 91 per set were presented to 54 subjects. Half were arranged regularly and half randomly. The regular sets were overestimated by an average of 22%, while the random were underestimated by 16% compared with the physical number. Evidence was presented showing that subjects expect results in the opposite direction. The findings are discussed in relation to the expectancy-contrast model of Birnbaum and Veit.     

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.     

Perceived numerosity of spatiotemporal events

Numerosity discrimination was examined when items were varied in space-time position rather than in space only. Observers were instructed to indicate which of two adjacent streamsof visual events contained more items. The precision of numerosity discrimination of dynamic events was not remarkably different from that of static patterns. Two basic numerosity biases previously found for static dot patterns—inhibitory overestimation and satellite underestimation—were demonstrated for items distributed randomly over a spatiotemporal interval. It was also demonstrated that two streams, equated in the number and luminous energy of items, are not judged equal in their visible number if items in one of these two streams have longer duration than items in the second stream. These findings can be accounted for by the occupancy model of perceived numerosity (Allik & Tuulmets, 1991a) if it is supposed that the impact that each element has on its neighborhood is spread along both spatial and temporal coordinates. Perceived numerosity decreases with both spatial and temporal proximity between the-visual-items. Space and time have interchangeable effects on perceived numerosity: the amount of numerosity bias caused by the spatial proximity of items can also be produced by the properly chosen temporal proximity of items     

A diffusion model for discrimination of temporal numerosity

This paper presents a model for sequential discrimination between the temporal frequencies of two discrete events. The model results from adjoining a theory for continuous estimation of temporal numerosity to a mathematical representation of the response criterion. A simple set of assumptions is used to describe the time evolution of numerosity estimates by a stochastic diffusion process. The predictions of the model are extensively compared to a body of results presented in a previous report, with an emphasis on individual performances. It is shown that a number of response strategies, observed in actual experiments, can be predicted by generalizing the classical notion of a single-valued response threshold to a stochastic, time-varying process.     

Multisensory numerosity judgments for visual and tactile stimuli

To date, numerosity judgments have been studied only under conditions of unimodal stimulus presentation. It is therefore unclear whether the same limitations on correctly reporting the number of unimodal visual or tactile stimuli presented in a display might be expected under conditions in which participants have to count stimuli presented simultaneously in two or more different sensory modalities. In Experiment 1, we investigated numerosity judgments using both unimodal and bimodal displays consisting of one to six vibrotactile stimuli (presented over the body surface) and one to six visual stimuli (seen on the body via mirror reflection). Participants had to count the number of stimuli regardless of their modality of presentation. Bimodal numerosity judgments were significantly less accurate than predicted on the basis of an independent modality-specific resources account, thus showing that numerosity judgments might rely on a unitary amodal system instead. The results of a second experiment demonstrated that divided attention costs could not account for the poor performance in the bimodal conditions of Experiment 1. We discuss these results in relation to current theories of cross-modal integration and to the cognitive resources and/or common higher order spatial representations possibly accessed by both visual and tactile stimuli.     

Comparative judgment of numerosity and numerical magnitude: attention preempts automaticity

It is commonly believed that humans are unable to ignore the meanings of numerical symbols, even when these meanings are irrelevant to the task at hand. In 5 experiments, the authors tested the notion of automatic activation of numerical magnitude by asking participants to compare, while timed, pairs of numerical arrays on either numerosity or numerical value. Garner and Stroop effects were used to gauge the degree of interactive processing. The results showed that both effects were sensitive to the discriminability of values along the constituent dimensions, to the number of stimulus values used, and to practice and motivation. Notably, Stroop and Garner effects were eliminated under several conditions. These findings are incompatible with claims of obligatory activation of meaning in numerical processing, and they cast doubt on theories positing automatic processing of semantic information for alphanumerical symbols.     

Magnitude production and magnitude estimation of taste intensity

Magnitude productions of sodium chloride (salty), quinine hydrochloride (bitter), and sucrose (sweet) yielded steeper psychophysical functions than those obtained with magnitude estimation. Hydrochloric acid (sour) produced the opposite effect. The results are discussed with respect to previous findings in taste intensity scaling and to general psychophysical considerations.     

Sensation seeking and estimation of numerosity

In a sample of 81 subjects, significant negative correlations were obtained between scores on Zuckerman's Sensation Seeking Scale and estimates of numerosity. A cognitive view of sensation seeking was suggested, according to which high scores are a function of an underestimation of the amount of stimulation experienced and low scores a function of overestimation. This approach was discussed in relation to motivational theories and those involving strength of the nervous system.     

Perceived numerosity as a function of item size

In a sample of 48 subjects, there was a significant inverse relation between item size and estimation of numerosity. This result was shown to be consistent with Gestalt theory. According to this approach, the components of good figures appear more numerous, and smaller items produce better figures.     

The effect of different methods to construct non-symbolic stimuli in numerosity estimation and comparison

Numerosity estimation and comparison tasks are often used to measure the acuity of the approximate number system (ANS), a mechanism which allows extracting numerosity from an array of dots independently from several visual cues (e.g. area extended by the dots). This idea is supported by studies showing that numerosity can be processed while these visual cues are controlled for. Different methods to construct dot arrays while controlling their visual cues have been proposed in the past. In this paper, these methods were contrasted in an estimation and a comparison task. The way of constructing the dot arrays had little impact on estimation. In contrast, in the comparison task, participants' performance was significantly influenced by the method that was used to construct the arrays of dots, indicating better performance when the visual cues of the dot arrays (partly) co-varied with numerosity. The present study therefore shows that estimates of ANS acuity derived from comparison tasks are inconsistent and dependent on how the stimuli are constructed. This makes it difficult to compare studies which utilised different methods to construct the dot arrays in numerosity comparison tasks. In addition, these results question the currently held view of the ANS as capable of robustly extracting numerosity independently from visual cues.     

Effects of exposure to psychophysical scaling on lingual vibrotactile magnitude estimation and magnitude production

The purpose of the present study was to determine if the results obtained by the scaling methods of magnitude estimation and magnitude production could be influenced by providing subjects with prior exposure to psychophysical scaling in the form of magnitude estimation or magnitude production. Group 1 (n = 10, Mage = 21.1 yr.) performed lingual vibrotactile-magnitude estimation followed by lingual vibrotactile magnitude production. Group 2 (n = 10, Mage = 19.7 yr.) performed lingual vibrotactile-magnitude production (using the magnitude-estimation responses provided by Group 1), followed by lingual vibrotactile-magnitude estimation. For the magnitude estimations there was no over-all statistically significant difference between the two groups, but there was for the magnitude-production values. Magnitude-estimation scaling was apparently not influenced by prior exposure to magnitude production, while magnitude-production scaling was influenced by prior exposure to magnitude estimation. The results are discussed in terms of how subjective scaling behavior in psychophysical experimentation may be influenced by the interaction between an absolute internal scaling mechanism and parameters set by the experimenter, such as scaling method and range of stimulus intensity.     

Temporal information affects the performance of numerosity discrimination: Behavioral evidence for a shared system for numerosity and temporal processing

In this study, we tested whether and how temporal information would affect the performance of numerosity discrimination in sequential events, in an attempt to make the relationship clear between temporal and numerosity processing. We manipulated the duration of event presentation (i.e., the stimulus duration) and the duration of the sequence (i.e., the total interval). We also employed three levels of standard event numbers (i.e., 5, 10, and 20) to test whether and how the effect would differ among event numbers. The results showed that temporal information affected the performance of numerosity discrimination; precision deteriorated when stimulus duration and the total interval were manipulated, and the number of events in the longer total interval were judged as less numerous than those in the shorter total interval across standard numbers. Accordingly, our results suggested the existence of a shared system between time and numerosity processing.     

Scaling clinical judgments of symptom pathology by means of the psychophysiological method of magnitude estimation

Summing scores across heterogeneous symptom items without consideration of their differing psychopathological significance has been criticized as producing an inadequate picture of an individual's clinical status. The purpose of this study was to derive symptom item weights representing clinically judged seriousness of each symptom through the application of Stevens's psychophysical method of magnitude estimation. A nationwide sample of 129 clinicians rated the pathological significance of 221 symptom items in a design such that every rater rated 121 items, 21 of which were rated by all raters and 100 of which were rated only by the A or B subgroup to which each rater was randomly assigned. Each item was rated as to the seriousness of the pathology it would represent if manifested by either a boy child, girl child, boy adolescent, or girl adolescent, with one-fourth of the raters assigned to each condition. The results of 211 two-way analyses of variance revealed that age and age and sex in interaction, but not sex alone, significantly influenced the clinical ratings. The resulting magnitude estimation ratings of symptom pathology ranged from 1.0 to 9.9. They were demonstrated to have satisfactory reliability and convergent validity and to have the psychophysical characteristics of a prothetic continuum.A preliminary version of this paper was presented at the 85th Annual Convention of the American Psychological Association, San Francisco, August 1977. The research was supported by HEW grant MH24886. The authors wish to express appreciation to Walter W. Wengel for his work in developing the median quotient ratings used in the study, to Terrance Auger for making available unpublished raw data, and to Merton S. Krause for helpful suggestions regarding the interpretation of results.     

Temporal discrimination increases in precision over development and parallels the development of numerosity discrimination

Time perception is important for many aspects of human behavior, and a large literature documents that adults represent intervals and that their ability to discriminate temporal intervals is ratio dependent. Here we replicate a recent study by vanMarle and Wynn (2006) that used the visual habituation paradigm and demonstrated that temporal discrimination in 6-month-old infants is also ratio dependent. We further demonstrate that between 6 and 10 months of age temporal discrimination increases in precision such that by 10 months of age infants succeed at discriminating a 2:3 ratio, a ratio that 6-month-old infants were unable to discriminate. We discuss the potential implications of the fact that temporal discrimination follows the same developmental progression that has been previously observed for number discrimination in infancy (Lipton & Spelke, 2003).     

Stability of individual loudness functions obtained by magnitude estimation and production

A correlational analysis of individual magnitude estimation and production exponents at the same frequency was perfor.med, as well as an analysis of individual exponents produced in different sessions by the same procedure across frequency(250, 1, 000, and 3, 000 Hz). Taken together, results show, first, that individual exponent differences do not decrease by counterbalancing magnitude estimation with magnitude production, and, second, that individual exponent differences remain stable over time despite changes in stimulus frequency. Further results disclose that although individual magnitude estimation and production exponents do not necessarily obey the .6 power law, it is possible to predict the slope (exponent) of an equal-sensation function averaged for a group of listeners from individual magnitude estimation and production data. Assuming that individual listeners with sensorineural hearing loss also produce stable and reliable magnitude functions, it is also shown that the slope of the loudness-recruitment function measured by magnitude estimation and production can be predicted for individuals with bilateral losses of long duration. Thus, results obtained in normal and in pathological ears suggest that individual listeners can produce loudness judgments that reveal, albeit indirectly, the input-output characteristic of the auditory system.