Single-parameter power law psychophysics of auditory numerosity and the psychological moment hypothesis

In the present study, numerosity estimation was investigated. A two-parameter Stevens power law analysis was performed on a total of944 subjects in six experiments. Two pulse ranges (2–17 or 17–253 pulses) and six pulse rates (either constant or randomly varied within trial blocks) were used, variously, in an unsuccessful attempt to find evidence for a psychological moment, under the supposition that the exponent (or, possibly, the measure constant) would become smaller as increasing numbers of pulses fell within the interval determined by each psychological moment. A single-parameter reanalysis of these six experiments under the initial value condition that a (standard) stimulus of one pulse be assigned a theoretical response (modulus) of one yielded single-parameter equations whose exponents were reliably less varied than those for conventional two-parameter equations in Experiments 1–4 (with randomly varying pulse rates from trial to trial) but not less varied in Experiments 5 and 6 (in which pulse rates were constant within trial blocks). It was concluded that the variable pulse rate condition, with its reduced exponent variability and presumed reduced temporal confounding, provides a more valid estimate of the single-parameter power law exponent for numerosity, which was found to be 0.80.     

Perceived numerosity

In Experiments 1 and 2, dots appeared less numerous when bunched together on a sheet than when spread out over a larger area, and apparent numerosity proved to be a power function, with exponents of 0.72 (Experiment 1) and 0.78 (Experiment 2), of objective numerosity. In Experiment 3, where Xs were shown instead of dots, the exponent was 0.77. In Experiment 4, where Ss made magnitude productions rather than magnitude estimations of the Xs, the exponent was 0.94. The overall results indicate an exponent of about 0.85 for dots or Xs presented.     

Effects of instruction presentation mode in comparative judgments

In each of two experiments, the comparative instructions in a symbolic comparison task were either varied randomly from trial to trial (mixed blocks) or left constant (pure blocks) within blocks of trials. In the first experiment, every stimulus was compared with every other stimulus. The symbolic distance effect (DE) was enhanced, and the semantic congruity effect (SCE) was significantly larger, when the instructions were randomized than when they were blocked. In a second experiment, each stimulus was paired with only one other stimulus. The SCE was again larger when instructions were randomized than when they were blocked. The enhanced SCE and DE with randomized instructions follow naturally from evidence accrual views of comparative judgments.     

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 impact of stimulus and response variability on S-R correspondence effects

Six experiments investigated how variability on irrelevant stimulus dimensions and variability on response dimensions contribute to spatial and nonspatial stimulus-response (S-R) correspondence effects. Experiments 1-3 showed that, when stimuli varied in location and number, S-R correspondence effects for location or numerosity occurred when responses varied on these dimensions but not when responses were invariant on these dimensions. These results are consistent with the response-discrimination account, according to which S-R correspondence effects should only arise for a dimension that is used for discriminating between responses in working memory. Experiments 4-6 showed that, when responses varied in location and number, both invariant and variable stimulus number produced correspondence effects in S-R numerosity. In summary, the present results indicate that the usefulness of a particular dimension for response discrimination can be sufficient for producing S-R correspondence effects, whereas variability of a stimulus dimension is not sufficient for producing such effects.     

Power laws for the perception of rotation and the oculogyral illusion

Magnitude estimation was used to measure subjective motion for two indicators of vestibular function. Twelve as made estimates of 5-sec pulses of angular acceleration across the range of angular acceleration × time (at) =10-150 deg/sec. Results were: (1) the power law describes subjective motion for all individual as, (2) the power function exponent (1.41) for the perception of rotation is slightly greater than the exponent (1.25) for the oculogyral illusion, (3) a significant number of as gave higher exponents for the perception of rotation, and (4) the magnitude estimates of the oculogyral illusion and perception of rotation were highly correlated within and across as.     

The effects of constant versus varied reinforcers on preference and resistance to change

Previous research has demonstrated that factors such as reinforcer frequency, amount, and delay have similar effects on resistance to change and preference. In the present study, 4 boys with autism made choices between a constant reinforcer (one that was the same food item every trial) and a varied food reinforcer (one that varied randomly between three possible food items). For all 4 boys, varied reinforcers were preferred over constant reinforcers, and they maintained higher response rates than constant reinforcers. In addition, when a distraction (a video clip) was introduced, responding maintained by varied reinforcers was more resistant to distraction than responding maintained by constant reinforcers. Thus, the present experiment extended the generality of the relation between preference and resistance to change to variation in reinforcer quality.     

The effect of line configuration on perceived numerosity of dotted lines

Estimates of the number of objects in a line are made in many different situations. This paper demonstrates that besides the actual number of dots, aspects of line configuration affect the perceived numerosity of dotted lines. Experiment 1 provides evidence that the highly studied “clutter effect” in distance perception research replicates to the numerosity domain so that lines made up of more segments are perceived to contain more dots. Experiments 2–5 provide nomological validity for the recently proposed “direct distance” effect in distance perceptions by showing that numerosity perceptions are higher the greater the euclidean length between the line end points and by manipulating euclidean length in three orthogonal ways: the relative length of segments (Experiment 2), the angle between segments (Experiment 3), and the general direction of segments (Experiment 4). Experiment 5 conceptually replicates the results of Experiments 2–4 utilizing stimuli-based versus memory-based judgments and a discrimination task. Experiments 6 and 7 extend the research on spatial perception by demonstrating that the use of euclidean length as a source of information is inversely related to line width, with width varied through clutter (Experiment 6) and total line length (Experiment 7). Overall, the results demonstrate that the robustness of the euclidean length effect is contingent on the salience of alternative spatial heuristics-specifically, euclidean width. Theoretical implications are discussed.     

Laterality differences and shifts as a function of tactile temporal pattern complexity: Implications for differential processing strategies of the cerebral hemispheres

Summary The ability of subjects with a right-hand or left-hand preference to report accurately the number of tactile pulses in temporal sequences presented to preferred and nonpreferred hands was investigated as a function of temporal pattern complexity which varied in terms of intrinsic temporal relations among successive pulses and ranged from slow regular (periodic) sequences of small numbers of pulses to fast irregular (aperiodic) sequences of large numbers of pulses. As the complexity of the pattern increased in terms of pulse number, presentation rate, pulse density, and pulse aperiodicity, all subjects, for both preferred- and nonpreferred-hand stimulation, were increasingly less accurate in reporting the number of pulses in a temporal pattern. Most interesting was the finding that although hand-preference groups did not differ reliably in overall report accuracy, both groups showed a consistent shift in report accuracy from preferred- to nonpreferred-hand stimulation when patterns had more than seven pulses; furthermore, these shifts occurred for all presentation rates and pulse periodicities. A possible relationship between hand preference and cerebral psychological organization is suggested and the data are discussed in terms of the laterality differences reflecting basic underlying cerebral asymmetries in stimulus processing.These investigations were supported by Grant A8621 from the National Research Council of Canada. This paper was prepared while the author was a Visiting Researcher at the Department of Psychology, University of Stockholm, and supported, in part, by a Leave Fellowship awarded by the Social Sciences and Humanities Research Council of Canada. I would like to thank Gordon Tanne for his assistance in data collection and analysis. Requests for offprints should be sent to E.C. Lechelt, Department of Psychology, University of Alberta, Edmonton, Alberta, Canada T6G 2E9.     

To honor Fechner and obey Stevens: relationships between psychophysical and neural nonlinearities

G. T. Fechner (1860/1966) famously described two kinds of psychophysics: Outer psychophysics captures the black box relationship between sensory inputs and perceptual magnitudes, whereas inner psychophysics contains the neural transformations that Fechner's outer psychophysics elided. The relationship between the two has never been clear. Moreover, psychophysical power laws are found in almost every sensory system, yet the vast majority of neurons show sigmoid nonlinearities. Here, we selectively review the literatures on psychophysical and physiological nonlinearities and show how they can be placed within a framework for understanding the relationship between inner and outer psychophysics: a neural organization with a logical structure commensurate to outer psychophysical theory. In theoretical treatments of Stevens's law, the power law is a consequence of combining a Weber's law scaling of inputs with a Weber's law-like scaling of sensation magnitudes, yielding an exponent that is the ratio of the Weber constants. A neural derivation using physiological sigmoid nonlinearities should be commensurate to this internal logic. There is a class of models in which two nonlinear neural mechanisms (e.g., a sensory channel and the cortical numerosity mechanism tapped by magnitude estimation) are coupled through feedback, yielding power law behavior as an emergent property of the system, with an exponent that is a ratio of neural coupling strengths. Rather than a discrepancy between psychophysics and physiology, these models suggest complementarity between inner and outer psychophysics, because the Weber constants required for outer psychophysics modeling can be derived from the sigmoid nonlinearities of inner psychophysics.     

Compositional inductive biases in function learning

How do people recognize and learn about complex functional structure? Taking inspiration from other areas of cognitive science, we propose that this is achieved by harnessing compositionality: complex structure is decomposed into simpler building blocks. We formalize this idea within the framework of Bayesian regression using a grammar over Gaussian process kernels, and compare this approach with other structure learning approaches. Participants consistently chose compositional (over non-compositional) extrapolations and interpolations of functions. Experiments designed to elicit priors over functional patterns revealed an inductive bias for compositional structure. Compositional functions were perceived as subjectively more predictable than non-compositional functions, and exhibited other signatures of predictability, such as enhanced memorability and reduced numerosity. Taken together, these results support the view that the human intuitive theory of functions is inherently compositional.     

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     

The role of feedback and dot presentation format in younger and older adults’ number estimation

Numerosity estimation, the rapid assessment of the number of items in a visual scene, is historically inaccurate. We assessed whether providing feedback regarding the correct numerosity on either 0%, 50%, or 100% of the trials would affect younger and older adults’ estimation accuracy for randomized, clustered (i.e., groups of 3 or 7 dots), and stacked (i.e., column) dot formats. Participants provided estimates and confidence ratings in six blocks, each containing 48 trials (16 numerosities shown in each format). Feedback frequency was manipulated between participants during blocks 1–4; no feedback was provided during blocks 5 and 6, which contained old and new numerosities and previously estimated presentations rotated 90°. Estimation accuracy was age equivalent across blocks despite younger adults initially being more accurate than older adults. Feedback improved both age groups’ accuracy. Stacked presentations were most accurately estimated but were more likely to be over-estimated than clustered and randomized presentations. Older adults gave lower confidence ratings than younger adults despite both age groups showing increased confidence across blocks, for more structured presentation formats, and as feedback frequency increased. These results expand our understanding of the role of presentation format and feedback in producing age equivalence or age-related differences in numerosity estimation.     

Six does not just mean a lot: preschoolers see number words as specific

This paper examines what children believe about unmapped number words - those number words whose exact meanings children have not yet learned. In Study 1, 31 children (ages 2-10 to 4-2) judged that the application of five and six changes when numerosity changes, although they did not know that equal sets must have the same number word. In Study 2, 15 children (ages 2-5 to 3-6) judged that six plus more is no longer six, but that a lot plus more is still a lot. Findings support the hypothesis that children treat number words as referring to specific, unique numerosities even before they know exactly which numerosity each word refers to.     

The stroop effect: Failure to focus attention or failure to maintain focusing?

Subjects matched the position (high or low) of a first stimulus with either the position or the meaning of a word (HI or LO) presented after an interstimulus interval (ISI). The form of the first stimulus (X or O) varied randomly within trial blocks and specified the relevant feature (meaning or position) of the second stimulus to the subject. When the relevant feature was meaning, Stroop interference from the second stimulus position was observed. This interference decreased as ISI increased. At the longest ISI, it was eliminated completely. A number of explanations for these effects are discussed and then tested in two further matching experiments. In a fourth experiment a similar decrease in interference was observed. These reductions in interference with increase in ISI appear to represent the development of attentional focusing on the relevant feature of the second stimulus. Stroop interference may occur in experiments where the relevant attribute is constant because of a difficulty associated with repeated focusing on the same attribute.     

Magnitude estimates for electrical pulses: evidence for two neural mechanisms

The hypothesis that there are two neural mechanisms for electrocutaneous stimulation--one that is sensitive to low current and is adaptive to repeated stimulation and another that is responsive to high current and is less adaptive--was tested in a control and four main experiments. In the main experiments, magnitude estimates obtained for single electrical pulses (of 2-msec duration) were described by a simple power function for each combination of high- and low-current levels and 10 trial blocks. The results were: (1) The slope of the power function for low current was steeper than was that for high current; (2) for low current, the intercept of the power function decreased with increasing block, whereas for high current, it remained constant over blocks; (3) this decrease of the intercept for low current disappeared when judgmental blocks were separated by a rest period of 8 min; (4) the modulus did not affect the slope; (5) for a large modulus combined with low current, the intercept decreased rapidly over trial blocks, whereas for a small modulus combined with high current, the intercept increased over trial blocks. The first four findings support the two-mechanism hypothesis, but the last one may also be interpretable in terms of the regression to absolute scale values.     

基于数量感知的数量适应

数量适应是指观察者在知觉上适应刺激数量的过程, 这一过程可以通过考察适应后其数量感知的变化加以揭示。目前对这一现象的争论焦点在于它是对数量的单独适应还是对非数量表面信息的适应。本研究通过改变刺激点大小和刺激组块两种方式, 探讨非数量表面信息对数量感知和数量适应的影响。实验一证明刺激点大小不会对数量感知产生显著影响。实验二发现刺激组块会对数量感知产生显著影响。实验三在前两个实验基础上发现：刺激大小不对数量感知产生显著影响, 也不影响数量适应, 而刺激组块能显著改变数量感知, 并按照改变后的数量感进行适应。三个实验的结果说明：表面信息只有在影响到数量感知时才会影响适应, 数量适应是基于数量感知的加工而产生的适应现象。     

Context-specific control and the Stroop negative priming effect

The present study highlights the utility of context-specific learning for different probe types in accounting for the commonly observed dependence of negative priming on probe selection. Using a Stroop priming procedure, Experiments 1a and 1b offered a demonstration that Stroop priming effects can differ qualitatively for selection and no-selection probes when probe selection is manipulated between subjects, but not when it is manipulated randomly from trial to trial within subject (see also Moore, 1994). In Experiments 2 and 3, selection and no-selection probes served as two contexts that varied randomly from trial to trial, but for which proportion repeated was manipulated separately. A context-specific proportion repeated effect was observed in Experiment 2, characterized by modest quantitative shifts in the repetition effects as a function of the context-specific proportion repeated manipulation. However, with a longer intertrial interval in Experiment 3, a context-specific proportion repeated manipulation that focused on the no-selection probes changed the repetition effect qualitatively, from negative priming when the proportion repeated was .25 to positive priming when the proportion repeated was .75. The results are discussed with reference to the role of rapid, context-specific learning processes in the integration of prior experiences with current perception and action.     

Concurrent schedules: reinforcer magnitude effects

Five pigeons were trained on pairs of concurrent variable-interval schedules in a switching-key procedure. The arranged overall rate of reinforcement was constant in all conditions, and the reinforcer-magnitude ratios obtained from the two alternatives were varied over five levels. Each condition remained in effect for 65 sessions and the last 50 sessions of data from each condition were analyzed. At a molar level of analysis, preference was described well by a version of the generalized matching law, consistent with previous reports. More local analyses showed that recently obtained reinforcers had small measurable effects on current preference, with the most recently obtained reinforcer having a substantially larger effect. Larger reinforcers resulted in larger and longer preference pulses, and a small preference was maintained for the larger-magnitude alternative even after long inter-reinforcer intervals. These results are consistent with the notion that the variables controlling choice have both short- and long-term effects. Moreover, they suggest that control by reinforcer magnitude is exerted in a manner similar to control by reinforcer frequency. Lower sensitivities when reinforcer magnitude is varied are likely to be due to equal frequencies of different sized preference pulses, whereas higher sensitivities when reinforcer rates are varied might result from changes in the frequencies of different sized preference pulses.     

An Elementary Syllabus of Psychological Tests

The purpose of this study was to examine the universality of the “inverse square root law” insofar as it purports to show a constant relationship between emotional involvement and subjectively perceived geographical distance. It was argued that the square root exponent in the “law” was derived from experiments that used a limited range of stimuli (place names). With students as subjects, two experiments were conducted in order to discover the relationship between emotional involvement and subjective distance when both very close and very distant places were considered. It was shown that the inverse square root function was not constant but rather varied with the range of stimuli.