Systematic covariation of the parameters in the near-miss to Weber's law, pointing to a new law

The exponent in the power function modeling of the near-miss to Weber's law is known to depend on several experimental factors, such as tone frequency and the presence or absence of noise. This paper presents data that confirm a theoretical argument that the near-miss exponent also depends on the definition of ‘just-noticeably different’ used empirically. Intensity discrimination data were collected for pure, 1000-Hz tones presented in quiet. Referent sound pressure levels ranged from 40 to 80 dB SPL in 10-dB steps. For each referent, the tone judged louder than the referent with probability equal to ν was determined, with ν ranging from 0.16 to 0.84. It was found that the power function exponent decreases monotonically with ν. Moreover, it was found that the two power law parameters co-vary systematically with ν, implying a submodel that has an interesting fixed-point property.     

The parameters in the near-miss-to-Weber's law

Many empirical data support the hypothesis that the sensitivity function grows as a power function of the stimulus intensity. This is usually referred to as the near-miss-to-Weber's law. The aim of the paper is to examine the near-miss-to-Weber's law in the context of psychometric models of discrimination. We study two types of psychometric functions, characterized by the representations P_a(x)=F(ρ(a)x^γ(a)) (type A), and P_a(x)=F(γ(a)+ρ(a)x) (type B). A central result shows that both types of psychometric functions are compatible with the near-miss-to-Weber's law. If a representation of type B exists, then the exponent in the near-miss is necessarily a constant function, that is, does not depend on the criterion value used to define “just noticeably different”. If, on the other hand, a representation of type A exists, then the exponent in the near-miss-to-Weber's law can vary with the criterion value. In that case, the parameters in the near-miss co-vary systematically.     

Counting and timing models in psychophysics and the conjoint Weber's law

J. C. Falmagne, G. Iverson, and S. Marcovici ((1979). Psychological Review, 86, 25–43) proposed a generalization of Weber's law, which they called the conjoint Weber's law. Empirically, the law sometimes holds. When it fails, the data satisfy a relation that Falmagne et al. identify as the conjoint Weber's inequality. This paper investigates the ability of counting and timing models of psychophysics to predict Weber's law and the conjoint Weber's law. It is shown that although the timing model naturally predicts both laws to hold, all reasonable counting models predict them to fail. Instead, counting models naturally predict Weber's inequality and the conjoint Weber's inequality.     

Weber's Law influences numerical representations in rhesus macaques (Macaca mulatta)

We present the results of two experiments that probe the ability of rhesus macaques to match visual arrays based on number. Three monkeys were first trained on a delayed match-to-sample paradigm (DMTS) to match stimuli on the basis of number and ignore continuous dimensions such as element size, cumulative surface area, and density. Monkeys were then tested in a numerical bisection experiment that required them to indicate whether a sample numerosity was closer to a small or large anchor value. Results indicated that, for two sets of anchor values with the same ratio, the probability of choosing the larger anchor value systematically increased with the sample number and the psychometric functions superimposed. A second experiment employed a numerical DMTS task in which the choice values contained an exact numerical match to the sample and a distracter that varied in number. Both accuracy and reaction time were modulated by the ratio between the correct numerical match and the distracter, as predicted by Weber's Law.     

A generalization of Piéron's law to include background intensity and latency distribution

Piéron's law, which formulates the effect of the signal intensity i on the mean or the median reaction time (MRT) as a power function, MRT=R+i^-βC, in which R, β and C are positive parameters, is of much interest in the simple reaction time literature. In particular, consistent estimates of β are obtained. However, this law offers no information regarding the effect of the signal intensity on other aspects of the reaction time distribution. In addition, the background intensity, which also has some impact on the detectability of the signal, is not considered in Piéron's law. In this paper, we generalize Piéron's law based on the above comments. That is, we assume that both β and C are functions of the background intensity and of the percentile rank. We then investigate from a theoretical viewpoint some possible functional forms of β and C, with a special emphasis on the dependency of β on the background intensity. It is shown that the so-called “Fechner-Thurstone-type representation” forces β and C to be “separable.” If we further assume a near-miss to Weber's law-type representation, then the functional form of β can be determined.     

Remarks on the identifiability of thurstonian paired comparison models under multiple judgment

It is well-known that the representations of the Thurstonian models for difference judgment data are not unique. It has been shown that equivalence classes can be formed to provide a more meaningful partition of the covariance structures of the Thurstonian ranking models. In this paper, we examine the equivalence relations between Thurstonian covariance structure models for paired comparison data obtained under multiple judgment and discuss their implications on the general identification constraints and methods to check for parameter identifiability in restricted models.The author is indebted to Ulf Böckenholt and Albert Maydeu-Olivares for their significant comments and suggestions which led to considerable improvement in this article.     

Timescale invariance and Weber's law in choice

Pigeons were exposed to concurrent schedules for which reinforcement was alternately available at different times for each of two choices. In Experiment 1 (in which reinforcement times progressed arithmetically), overall, but not relative, response rate was timescale invariant. In Experiment 2 (in which reinforcement times progressed geometrically and were more spaced out), there was temporal control at all reinforcement times, but the amplitude of left-right response alternation decreased as time in the trial increased. These results indicate that the temporal regulation of both overall and relative response rates conforms to Weber's law although relative rate is heavily influenced by processes other than timing. It also adds support to the idea that overall and relative response rate reflects the operation of two independent processes.     

Analytical methods in the theory of psychophysical discrimination II: The Near-miss to Weber's Law, Falmagne's Law, the Psychophysical Power Law and the Law of Similarity

We examine the possibility of accounting for an empirical phenomenon in auditory psychophysics called the “Near-Miss to Weber's Law” [Jesteadt, W., Weir, C., & Green, D. (1977). Intensity discrimination as a function of frequency and sensation level. Journal of the Acoustical Society of America, A, 61, 169-177], within the framework of Fechner's Psychophysics. To this end, we introduce a pair of Psychophysical Laws governing the behavior of Weber sensitivities. One of these, the Power Law, embodies ideas that Jesteadt et al. introduced in their original discussion of the near-miss phenomenon. Falmagne's Law captures ideas of Falmagne and colleagues [Falmagne, J.-Cl. (1985). Elements of psychophysical theory. New York: Oxford University Press; Doble, C., Falmagne, J.-Cl., & Berg, B.G. (2003). Theoretical note: Recasting (the near-miss to) Weber's Law. Psychological Review, 110, 365-375]. These two laws, which account for the near-miss phenomenon in quite different ways, are shown to be special cases of a more general Psychophysical Law of Similarity. We find all (smooth) solutions of this Law of Similarity that are compatible with Fechnerian psychophysics. Of the 11 solutions, four are relevant to the near-miss phenomenon; two of these solutions reproduce the models of Jesteadt et al. and Doble et al. and the other two solutions each provide a new accounting of the near-miss phenomenon. We discuss the possibility of distinguishing among these four models in empirical data.     

Ambiguous comparative judgment: Fuzzy set model and data analysis

Abstract:  This paper proposed two types of fuzzy set models for ambiguous comparative judgments, which did not always hold transitivity and comparability properties. The first type of model was a fuzzy theoretical extension of the additive difference model for preference that was used to explain ambiguous preference strength. The second was a fuzzy logic model for explaining ambiguous preference in which preference strength was bounded, such as a probability measure. In both models, multi-attribute weighting parameters and all attribute values were assumed to be asymmetric fuzzy L-R numbers. For each model, a method of parameter estimation using fuzzy regression analysis was proposed. Numerical examples were also provided for comparison. Finally, the theoretical and practical implications of the proposed models were discussed.     

Inconsistency and uncertainty in multi-attribute judgment of human performance

This paper examines the effect of uncertainty and inconsistency on the judgment of human performance. The results indicate that the effect of inconsistency on judgment is not mediated by subjective uncertainty. We find that both the level and the extremity of judgment decrease with uncertainty. These effects are explained, respectively, by uncertainty aversion and by regressiveness. We also find that both the level and the extremity of judgment of human performance increase with inconsistency. These effects are explained by reliance on integration rules in which judgment is based primarily on some aspects of the information, while other aspects are, to some extent, ignored.     

Moral judgment and self-reported moral events in adulthood

As part of a 13-year study of the development of reasoning about the Good Life and moral judgment, 29 middle-class, well-educated adults, ranging in age from 18 to 80 years, were interviewed twice, 4 years apart on Standard Form Moral Judgment interviews and were asked to describe their own moral events that occurred recently. Both times, the average moral judgment score on the hypothetical dilemmas was about stage 4. The reasoning in the spontaneously reported moral events was significantly lower, although the two scores were highly correlated. Difference scores between the two were not related to scores on the hypothetical dilemmas. While there were no significant gender differences in scores on the hypothetical dilemmas, there were in scores on spontaneously reported events favoring males. The philosophic quality of the reported events could be easily categorized by traditional moral-philosophic categories; about half of the events contained traditional deontic moral content (moral right); the other half contained material defined as the moral good. The context of most events was interpersonal as opposed to societal. The most frequent issue discussed concerned honesty about financial issues, particularly income tax evasion. Ways to expand the research on adult moral experience are discussed.     

A common representational system governed by Weber's law: nonverbal numerical similarity judgments in 6-year-olds and rhesus macaques

This study compared nonverbal numerical processing in 6-year-olds with that in nonhuman animals using a numerical bisection task. In the study, 16 children were trained on a delayed match-to-sample paradigm to match exemplars of two anchor numerosities. Children were then required to indicate whether a sample intermediate to the anchor values was closer to the small anchor value or the large anchor value. For two sets of anchor values with the same ratio, the probability of choosing the larger anchor value increased systematically with sample number, and the psychometric functions superimposed when plotted on a logarithmic scale. The psychometric functions produced by the children also superimposed with the psychometric functions produced by rhesus monkeys in an analogous previous experiment. These examples of superimposition demonstrate that nonverbal number representations, even in children who have acquired the verbal counting system, are modulated by Weber's law.     

A clean self can render harsh moral judgment

Physical cleanliness has many medical benefits, such as protection from the dangers of contagion. We explore a potential unintended consequence of cleanliness. Given the metaphorical association between physical cleanliness and moral purity (Zhong & Liljenquist, 2006), we contend that a clean self may also be linked to a virtuous self. This enhanced moral self-perception can in turn license harsher moral judgment. Three experiments found that cleanliness, whether induced via physical cleansing or through a visualization task, licensed severe judgment on morally contested issues such as abortion and pornography. Further, we found that an inflated moral self mediated the relationship between cleanliness and moral judgment. These results provide unique insight to the social significance of cleanliness and may have important implications for discrimination and prejudice.     

Knowledge activation after information encoding: Implications of trait priming on person judgment

It is widely assumed that traits primed after the encoding of person information do not lead to assimilation effects on the judgment of that person. The authors challenge this view by providing evidence that post-encoding trait primes can result in assimilative person judgments under certain conditions. In Experiments 1 and 2, we identify the conditions under which these assimilation effects occur. Experiment 1 shows the importance of participants’ goals during person information encoding: assimilation is observed when person information is encoded as part of a memorization goal (as opposed to an impression formation goal). The findings of Experiment 2 further reveal that the encoded person information should imply trait concepts rather than being merely vague with respect to the primed trait category. Finally, the results of Experiment 3 suggest that the obtained assimilation effect is driven by differential accessibility for prime-congruent person information.     

相似性判断和差异性判断的不对称性及其机制探索

本文阐述了相似性判断和差异性判断不对称性的相关研究。首先介绍了不对称性的发展及理论解释。其次,根据事件相关电位的研究结果,阐述了图-词信息的双编码模型对相似性判断的影响。接着分析和介绍了相似性判断的双编码过程及其时间进程。最后,作者提出有待深入研究的几个问题：（1）差异性策略在分类和推理中的应用;（2）相似性双编码过程的实证研究及其应用;（3）从发展认知神经科学的角度探讨不对称的认知神经机制。     

The power law and Weber's law in fixed-interval postreinforcement pausing: A scalar timing model

Some quantitative properties of the postreinforcement pause under fixed-interval schedules were simulated by a computer model embodying two processes, either of which could initiate responding in an interval. The first was a scalar timing system similar to that hypothesized to underlie behaviour on other tasks. The second was a process that initiated responding without regard to elapsed time in the interval. The model produced simulated pauses with a mean that varied as a power function of the interval value, and a standard deviation that appeared to grow as a linear function of the mean. Both these features were found in real data. The model also predicted several other features of pausing and responding under fixed-interval schedules and was also consistent with the results produced under some temporal differentiation contingencies. The model thus illustrated that behaviour that conformed to the power law could nevertheless be reconciled with scalar timing theory, if an additional non-timing process could also initiate responding.     

Effects of circular motion on judgment of rotation direction and depth order in visual motion

Abstract:  The rotation direction and depth order of a rotating sphere consisting of random dots often reverses while it is viewed under orthographic projection. However, if a short viewing distance is simulated under perspective projection, the correct rotation direction can be perceived. There are two motion cues for the rotation direction and depth order. One is the speed cue; points with higher velocities are closer to the observer. The other is the vertical motion cue; vertical motion is induced when the dots recede from or approach the observer. It was examined whether circular motion, which does not have any depth information but induces vertical velocities, masks the vertical motion cue. In Experiment 1, the effects of circular motion on the judgment of the rotation direction of a rotating sphere were examined. The magnitude of the two cues (the speed cue and the vertical velocity cue) as well as the angular speed of circular motion was varied. It was found that the performance improved as the vertical velocity increased and that the speed cue had slight effects on the judgment of the rotation direction. It was also found that the performance worsened as the angular speed of the circular motion was increased. In Experiment 2, the effects of circular motion on depth judgment of a rotating half sphere were investigated. The performance worsened as the angular speed of the circular motion increased, as in Experiment 1. These results suggest that the visual system cannot compensate perfectly for circular motion for the judgment of the rotation direction and depth order.     

Speed discrimination in 6- and 10-month-old infants follows Weber's law

The speed of a moving object is a critical variable that factors into actions such as crossing a street and catching a ball. However, it is not clear when the ability to discriminate between different speeds develops. Here, we investigated speed discrimination in 6- and 10-month-old infants using a habituation paradigm showing infants events of a ball rolling at different speeds. The 6-month-olds looked longer at novel speeds that differed by a 1:2 ratio than at the familiar ones but showed no difference in looking time to speeds that differed by a 2:3 ratio. In contrast, the 10-month-olds succeeded at discriminating a 2:3 ratio. For both age groups, discrimination was modulated by the ratio between novel and familiar speeds, suggesting that speed discrimination is subject to Weber's law. These findings show striking parallels to previous results in infants' discrimination of duration, size, and number and suggest a shared system for processing different magnitudes.