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.     

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.     

Numerosity judgments for tactile stimuli distributed over the body surface

A large body of research now supports the claim that two different and dissociable processes are involved in making numerosity judgments regarding visual stimuli: subitising (fast and nearly errorless) for up to 4 stimuli, and counting (slow and error-prone) when more than 4 stimuli are presented. We studied tactile numerosity judgments for combinations of 1-7 vibrotactile stimuli presented simultaneously over the body surface. In experiment 1, the stimuli were presented once, while in experiment 2 conditions of single presentation and repeated presentation of the stimulus were compared. Neither experiment provided any evidence for a discontinuity in the slope of either the RT or error data suggesting that subitisation does not occur for tactile stimuli. By systematically varying the intensity of the vibrotactile stimuli in experiment 3, we were able to demonstrate that participants were not simply using the 'global intensity' of the whole tactile display to make their tactile numerosity judgments, but were, instead, using information concerning the number of tactors activated. The results of the three experiments reported here are discussed in relation to current theories of counting and subitising, and potential implications for the design of tactile user interfaces are highlighted.     

Effect on numerosity judgment of grouping of tones by auditory channels

Previous research suggests that numerosity judgments for sequences of tones improve when the sequence is structured such that equal and small groups of tones alternate between the ears. The present. study systematically investigates the effects of the structure of tone sequences on number judgment. Tone frequency is chosen as the grouping principle instead of spatial location. In the first experiment, sequences with equal groups of 2, 3, 4, 5, 6, or 7 tones (groups alternating between 800 and 1, 250 Hz) were compared with monotonous sequences. At a slow repetition rate of the tones (280 msec onset to onset), grouping the sequence deteriorated the numerosity judgment. At II fast repetition rate of the tones (100 msec onset to onset) grouping improved numerosity judgment, but only if the group size did not exceed 4 to 5 tones. In the second experiment, the equality of group size in the sequence as a necessary condition for the improvement was investigated. It was found that tone sequences comprised of equal groups were judged more accurately in number than sequences comprised of unequal groups. These results seem to give support for the existence of an auditory subitizing process. The results of this study are also compared with the results of studies in which sequences of tones, alternating one by om! between locations or frequencies, had to be judged in number.     

Judged Likelihood of Different Second Crimes: A Function of Judged Similarity

Numerous studies have concluded that the judged probability of conviction for a crime is higher when information concerning a prior conviction is disclosed. This is even more true when the prior and present crimes are the same. This paper examines the argument that the influence of a hypothetical previous conviction on the judgment of predilection toward a hypothetical present crime is a continuous function of the degree of similarity between past and present crimes. The subjects were presented with all possible pairs of 10 crimes, the first being considered as a prior conviction and the second as a presently charged crime. The subjects judged the subjective likelihood that a person convicted of the first crime (C1) would, in fact, he of a mind and inclination to commit the second one (C2). Independent judgments of intercrime, undefined similarity also were obtained. The main experimental findings, as expected, were that the judged probability of C2. given C1, was: (a) greatest when C2 was a repeat of C1; (b) uniquely different for each C1; (c) predicted quite well by degree of intercrime similarity; (d) poorly predicted by crime seriousness values. Thus, the subject appears to make judgments of predilection on the basis of simple representativeness heuristics, which specify that certain crimes will be considered more likely if they are more semantically related to earlier ones. Such a mental device unfortunately would he systematically biased in courtroom settings because judgments of intercrime similarity are not influenced by real-world probabilities.     

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.     

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     

Subitizing: Magical numbers or mere superstition?

Summary It is widely believed that humans are endowed with a specialized numerical process, calledsubitizing, which enables them to apprehend rapidly and accurately the numerosity of small sets of objects. A major part of the evidence for this process is a purported discontinuity in the mean response time (RT) versus numerosity curves at about 4 elements, when subjects enumerate up to 7 or more elements in a visual display. In this article, RT data collected in a speeded enumeration experiment are subjected to a variety of statistical analyses, including several tests on the RT distributions. None of these tests reveals a significant discontinuity as numerosity increases. The data do suggest a strong stochastic dominance in RT by display numerosity, indicating that the mental effort required to enumerate does increase with each additional element in the display, both within and beyond the putative subitizing range.     

Connectedness affects dot numerosity judgment: Implications for configural processing

Participants judged the number of dots in visual displays with brief presentations (200 msec), such that the numerosity judgment was based on an instantaneous impression without counting. In some displays, pairs of adjacent dots were connected by line segments, whereas, in others, line segments were freely hanging without touching the dots. In Experiments 1, 2A, and 2B, connecting pairs of dots by line segments led to underestimation of dot numbers in those patterns. In Experiment 3, we controlled for the number of freely hanging line segments, whereas Experiment 4 showed that line segments that were merely attached to dots without actually connecting them did not produce a considerable underestimation effect. Experiment 5 showed that a connectedness effect existed when stimulus duration was reduced (50 msec) or extended (1,000 msec). We conclude that connectivity affects dot numerosity judgments, consistent with earlier findings of a configural effect in numerosity processing. Implications of the role of connectedness in object representation are discussed.     

Judgments of a Woman's Emotions: The Influence of Observer Attitudes1

The influence that observers' attitudes toward lesbians have on their judgments of a lesbian target's emotional display was examined. In Phase One, college undergraduates' attitudes toward lesbians were measured. In Phase Two, 185 subjects returned and viewed a videotape in which a female target described as a lesbian, a novel individual, or a heterosexual, responded to an interviewer's questions regarding her recent romantic breakup. The subjects then judged the appropriateness of the target's display of 11 emotions. Contrary to predictions, attitudinally negative subjects evaluated the Lesbian target as highly appropriate. The potential applications of this result for social psychologists interested in examining the contact hypothesis, as well as for outgroup members themselves, are discussed.     

Criterion adjustment and probability matching

Twenty subjects made all possible (136) pairwise-comparison judgments of pattern goodness between 17 five-dot patterns. Using multidimensional scaling and hierarchical clustering analyses, a two-dimensional representation was judged the most appropriate for the derived pattern space. The interpreted structure of the spatial-hierarchical configuration strongly supported previous findings that pattern redundancy is the most salient dimension underlying pattern goodness judgments and that patterns can be partitioned on this dimension according to the size of their equivalence sets, i.e., the number of patterns resulting from rigid rotations in 90-deg increments or reflection. A psychological mechanism that accounts for the effect of pattern redundancy on judged goodness was proposed.     

Development of children's ability to judge relative numerosity

Three- to seven-year-old chilren were trained through reinforcement to select either the more or less numerous of two rows of squares. One group was shown rows in which number covaried with row length, one group saw rows in which number covaried with row density, and a third group saw rows in which number did not covary with other dimensions. The children's operational stage was assessed by a number conservation test. All children successfully judged relative numerosity when number covaried with length or with density, but only concrete operational children were successful when number did not covary with other dimensions. Preoperational children are thus not able to base judgments of relative numerosity solely on number.     

Underestimating numerosity of items in visual search tasks

Previous research on numerosity judgments addressed attended items, while the present research addresses underestimation for unattended items in visual search tasks. One potential cause of underestimation for unattended items is that estimates of quantity may depend on viewing a large portion of the display within foveal vision. Another theory follows from the occupancy model: estimating quantity of items in greater proximity to one another increases the likelihood of an underestimation error. Three experimental manipulations addressed aspects of underestimation for unattended items: the size of the distracters, the distance of the target from fixation, and whether items were clustered together. Results suggested that the underestimation effect for unattended items was best explained within a Gestalt grouping framework.     

Memory for psychophysical scaling judgments

In three experiments, we examined memory for responses and stimuli experienced in a single direct psychophysical scaling session in which subjects made absolute magnitude estimations of the loudnesses of pure tones. Recall of scaling responses was found to be accurate for the softest and loudest stimuli, but systematically greater than actual judgments for the intermediate stimulus amplitudes, yielding distorted psychophysical functions for the recall data which nonetheless had the same power function exponent as that for the judged stimuli. Also, memory for the range of stimulus amplitudes was fairly accurate, but subjects could not distinguish between judged and nonjudged amplitudes within that range. The results are consistent with the role of extreme stimuli as anchors for judgment, and indicate that memories for these stimuli and responses made to them can be expected to influence future scaling judgments. These results also are consistent with the uncertainty hypothesis of mnemophysics. The present research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada to L.M.W. Thanks to the reviewers for insightful critiques.     

The dilution effect: Nondiagnostic information weakens the implications of diagnostic information

In a series of studies, subjects were asked to make predictions about target individuals. Some subjects were given information about the target which pretest subjects had judged to be “diagnostic”—that is, had judged to be usefully predictive of the outcome. Other subjects were given a mix of information judged to be diagnostic and information judged to be “nondiagnostic” by pretest subjects—that is, judged to be of little value for predicting the outcome. Subjects given mixed information made much less extreme predictions than did subjects given only diagnostic information. It was argued that this “dilution effect” occurs because people make predictions by making simple similarity judgments. That is, they compare the information they have about the target with their conception of outcome categories. The presence of individuating but nondiagnostic information about the target reduces the similarity between the target and those outcomes that are suggested by the diagnostic information. One of the major implications is that stereotypes and other “social knowledge structures” may be applied primarily to abstract, undifferentiated individuals and groups and may be largely set aside when judgments are made about concrete, individuated people.     

When Fluency Signals Truth: Prior Successful Reliance on Fluency Moderates the Impact of Fluency on Truth Judgments

Repeated statements are more frequently judged to be true. One position relates this so‐called “truth effect” to metacognitive experiences of fluency, suggesting that repeated statements are more frequently judged to be true because they are processed more fluently. Although most prior research focused on why repetition influences truth judgments, considerably less is known about when fluency is used as information. The present research addresses this question and investigates whether reliance on fluency is moderated by learning experiences. Specifically, we focus on changes in the reliance on fluency over the course of time. A series of experiments reveals that fluency is more likely to be used in truth judgments when previous reliance on fluency has resulted in valid judgments, compared with when previous reliance on fluency was misleading. These findings suggest that reliance on fluency in judgments is a finely tuned process that takes prior experiences with fluency‐based judgments into account. Copyright © 2013 John Wiley & Sons, Ltd.     

A pointing technique for rapidly collecting localization responses in auditory research

A technique is described for rapidly collecting responses in auditory-localization experiments. Subjects indicate the perceived direction of the sound by pointing at a 20-cm-diam spherical model. In Experiment 1, the subjects judged the direction of a broadband signal, which could originate from any of 239 directions ranging through 360° of azimuth and between ?45° and +90° of elevation. Using this technique, the subjects responded 2–8 times more rapidly than previous subjects who used either a verbal-reporting technique or a head-pointing technique. Localization judgments were as accurate as they had been with verbal reports, but were not as accurate as judgments collected using the head-pointing technique. In Experiment 2, the signal was turned off and the experimenter read the spherical coordinates of the signal location to the subjects. The subjects pointed to these coordinates more accurately than they had judged the direction of the sounds in Experiment 1, suggesting that the response technique had not been the limiting factor in that experiment. Circumstances relevant to the choice of response techniques for auditory-localization experiments are discussed.     

Cerebral asymmetry in time perception

Eighteen right-handed male subjects estimated the duration of dot patterns of varying numerosity which were briefly flashed to the left and right visual fields. The mean judged duration of patterns flashed to the left visual field was significantly less than the mean judged duration of patterns flashed to the right visual field. However, the duration of all patterns was underestimated, regardless of visual field. In addition, apparent duration increased linearly when plotted as a function of log duration and increased monotonically when plotted as a function of stimulus numerosity. Overall accuracy was nearly equivalent in both visual fields, but there were interactions between actual durations and visual field and between numerosity and visual field. Thomas and Weaver’s [1975) model for time perception is applied to the data, and it is suggested that the left hemisphere relies on a timer to estimate duration, while the right hemisphere relies on a visual information processor to estimate duration. nt]mis|The author would like to thank A. B. Kristofferson, S. F. Witelson, an anonymous referee, Mary Kay Fortman, and Allen Gouse for their helpful comments. Some of these data were presented at the meeting of the Midwestern Psychological     

Children's judgments of numerical quantity: A new view of early quantification

Three experiments investigated the role of length, density, and number dimensions in children's numerosity judgments of linear arrays of beads. The number dimension is physically defined by a length × density rule. Judgments based, even in part, on the number dimension would show signs of a length × density rule. Experiment 1 examined numerosity judgments of large arrays. Results showed that judgments by 3- and 4-year-olds obeyed a length + density rule, providing cogent evidence for a general adding strategy in young children's judgments of quantity. The physical length × density rule emerged gradually with increasing age. Experiment 2 showed that numerosity judgments of small arrays by 3- and 4-year-olds obeyed a length × density rule, indicating response to the number dimension. This result was expected and thereby validated the linearity of the response scale at the youngest ages. Experiment 3 verified the integration rules for individual 3- and 4-year-olds. The integration rules were interpreted in terms of Piaget's stages of the development of quantification. No evidence for Piaget's initial stage was found. This leads to a new view of early quantification which grants young children the ability to integrate stimulus dimensions.     

The effect of target group size on risk judgments and comparative optimism: the more, the riskier

In 5 experiments, college students exhibited a group size effect on risk judgments. As the number of individuals in a target group increased, so did participants' judgments of the risk of the average member of the group for a variety of negative life events. This happened regardless of whether the stimuli consisted of photographs of real peers or stick-figure representations of peers. As a result, the degree to which participants exhibited comparative optimism (i.e., judged themselves to be at lower risk than their peers) also increased as the size of the comparison group increased. These results suggest that the typical comparative optimism effect reported so often in the literature might be, at least in part, a group size effect. Additional results include a group size effect on judgments of the likelihood that the average group member will experience positive and neutral events and a group size effect on perceptual judgments of the heights of stick figures. These latter results, in particular, support the existence of a simple, general cognitive mechanism that integrates stimulus numerosity into quantitative judgments about that stimulus.