Individual loudness functions determined from direct comparisons of loudness intervals

Five subjects were required in each trial to directly compare two pairs of tones and indicate which pair of tones had the greater loudness difference. Ten 1,200-Hz tones differing only in intensity were employed. Subjects made binary comparisons among the 45 tone pairs that can be formed from these 10 tones. The loudness difference comparisons of each subject were found to satisfy two properties (transitivity and monotonicity) that are required for an interval scale representation of loudness. Therefore, individual loudness scales were constructed using a nonmetric scaling technique designed for comparisons of sensory intervals. These loudness scales differed significantly from subject to subject. Since a nonnumerical scaling procedure was employed, these individual differences could not be attributed to biases in the way in which observers use numbers or numerical concepts to describe the loudness of tones. Hence, they suggest strong individual differences in the coding of sound intensity.     

Eliciting Uncertainties: A Two Structure Approach

We recast subjective probabilities by rejecting behaviourist accounts of belief by explicitly distinguishing between judgements of uncertainty and expressions of those judgements. We argue that this entails rejecting that orderings of uncertainty be complete. This in turn leads naturally to several generalizations of the probability calculus. We define probability-like functions over incomplete algebras that reflect a subject’s incomplete judgements of uncertainty. These functions can be further generalized to (partial) inner and outer measures that reflect approximate elicitations.     

The effect of order of presentation of verbal probability expressions on numerical estimates in a medical context

Many medical decisions are made on a probability estimate. Models of risk benefit calculations in health behaviour rarely articulate how people perceive and appraise such probability. Cognitive processes may mediate the processing of probability expressions and may be important to understand the meaning or the range of meanings that probability statements portray to patients. Past studies have indicated that verbal expressions of probability are vague and subject to individual interpretation. On the theoretical level when subjects are asked to translate a set of verbal probability expressions, ranging from high to low, into their equivalent numerical expressions subjects usually produce a continuum of numerical equivalents also ranging from high to low. In practice clinicians frequently communicate information about uncertainty to the patients by verbal probability estimates. This study explored the effect of the order of presentation of the verbal expressions on the numerical probabilities produced by a group of medical students (n?=?87) in relation to medical probabilities. The results showed that the order of presentation (descending vs. random) of the verbal probability expressions was found to have a significant effect on three of the seven numerical probabilities produced by the subjects. The order effect is discussed to together with implications for clinical practice.     

Patterns of Preference for Numerical and Verbal Probabilities

We report results of an experiment designed to test a principle formulated by Budescu and Wallsten (1995), that, when communicating uncertainty information, mode choices are sensitive to sources and degrees of vagueness. In addition, we examined subjects’ efficacy in using such uncertainty information as a function of communication mode, source, and vagueness. In phase one of the experiment, subjects in a dyad used precise (numerical) or imprecise (verbal) expressions to communicate to a remote partner precise or vague uncertainty about the likelihoods of events. Spinner outcomes were used to generate precise uncertainty while answers to almanac questions were used to elicit vague uncertainty. In phase two, subjects saw the events paired with their partners’ estimates of similar events, and were asked to gamble on one event from each pair. Communication mode preferences were measured as the relative frequency that subjects chose the numerical mode to either express or receive uncertainty information regarding the events. Efficacy was measured as the relative frequency that subjects choose from the pair the event associated with the objectively more probable uncertainty expression. Underlying uncertainty interacted with direction of communication to affect preferences for modes of expression of the probabilities. Subjects preferred precise (numerical) information, especially for precise events (spinners). For vague events (questions), their preference for precise (numerical) information was stronger when receiving than when communicating information. Similar preferences were reflected in the efficiency of subsequent gamble decisions based on the probability estimates. Specifically, decisions were more efficacious (i.e. consistent with Expected Utility) when degrees of precision in events and estimates matched. © 1997 by John Wiley & Sons, Ltd.     

Exploiting illusory effects to disclose similarities in numerical and luminance processing

Recent studies have suggested that numerical and physical magnitudes are similarly processed by a generalized magnitude system. The present study investigates the number?Cluminance interaction, taking advantage of illusory effects in a cued line bisection task with numerical or nonnumerical flankers and varying levels of luminance. The results showed that both dimensions influenced bisection performance. Whereas numbers (Experiment 1) induced a systematic shift of the subjective midpoint toward the larger digit, luminance (Experiment 2) modulated the bisection performance toward the darker flanker. By combining these two illusions (Experiments 3 and 4), the two dimensions interfered with each other. This pattern of results suggests overlapping representations for physical and numerical magnitudes and highlights the value of illusory effects in cognitive research.     

用文字概率衡量不确定性：特征和问题

文字概率是衡量不确定性的方式之一, 即人们使用诸如“也许”、“未必”的词汇来描述特定事件发生的可能性。文字概率不同于数字概率, 主要体现在文字概率的模糊性、非概率运算性和语义特性上。这使得相对于数字概率, 用文字概率衡量不确定性既有优势也有问题, 进而对人们的不确定信息沟通和风险决策造成影响。虽然文字概率与数字概率存在特征上的差异并且人们在日常交流中偏爱文字概率, 但是大部分风险领域的研究却仅局限于数字概率, 今后有必要研究使用文字概率测量的风险决策。在已有文字概率特征研究的基础上, 还可以进一步探究其不同于数字概率的其它特征(文字/数字概率与双系统模型的联系、文字概率的跨文化差异等)及其对风险决策的影响。     

A logistic approach to knowledge structures

Models for quantitative (or numerical) testing like e.g. educational testing have a relatively long tradition in psychology, while the qualitative (or nonnumerical) approach to psychometrics is more recent. The approach presented in this paper can be regarded as an attempt to integrate, to some extent, the numerical and nonnumerical fields. In numerical testing a subject is characterized by some real-valued parameter representing her level or ability. In the nonnumerical approach the knowledge state of an individual is represented by the subset of problems that the individual is capable of solving. We propose a model in which the relationship between the ability levels and the knowledge states is worked out on a probabilistic basis. The central idea is that the ability parameters and the knowledge states are not independent. A logistic model is derived which specifies the probabilities of the knowledge states conditional on the ability levels. We show that the Rasch model arises as a special case of the proposed model.     

Motion direction identification in random cinematograms: a general model

The cinematograms of 12 two-state elements arranged in the clock positions in space and in a sequence of adjacent 100-ms frames in time were used as stimuli. Some positions in each frame (or all 12 of them) could be labeled as "domain" ones, and every element that was T frames and S positions (clockwise or counterclockwise) apart from a domain element could repeat the latter's state with probability P. The probability of the rotation direction identification was obtained as a function of T, S, P, number of frames, and the domain positions selection scheme. A generalized version of the reversed phi phenomenon was obtained: if P less than .5, then the psychometric value lies below .5 level. All the data can be accounted for by a simple model according to which the choice of direction is based on the counts of the different types of dipoles, each type being characterized by the probability and the weight of its count: In most situations all dipoles but the shortest ones (connecting the neighboring elements of successive frames) can be ignored.     

Decision Making under Strict Uncertainty: An Experimental Test of Competitive Criteria

In decisions under strict uncertainty individual decision makers have to choose one of a finite number of alternatives with complete information about their outcomes but in the absence of any information about the probabilities of the various states of nature. We report two experiments with multiple decision tasks in which subjects were instructed to rank the alternatives (Experiment 1), or choose the best one (Experiment 2). We test the major decision rules (criteria) which have been proposed for this case in the normative literature with special attention on Hurwicz′s "optimism-pessimism" criterion which is the only one that can accommodate individual differences. Although none of the criteria can account for the results of the majority of the subjects, choices made by a sizable minority of the subjects′ are consistent with Hurwicz′s model. There is no evidence for a systematic relationship between the model′s single "optimism-pessimism" parameter (inferred form choices) and, independently derived, individual measures of optimism (obtained from responses to personality scales). We conclude with a discussion of the difficulty of experimentally implementing strict uncertainty.     

Evidence with uncertain likelihoods

An agent often has a number of hypotheses, and must choose among them based on observations, or outcomes of experiments. Each of these observations can be viewed as providing evidence for or against various hypotheses. All the attempts to formalize this intuition up to now have assumed that associated with each hypothesis h there is a likelihood function  μ _h , which is a probability measure that intuitively describes how likely each observation is, conditional on h being the correct hypothesis. We consider an extension of this framework where there is uncertainty as to which of a number of likelihood functions is appropriate, and discuss how one formal approach to defining evidence, which views evidence as a function from priors to posteriors, can be generalized to accommodate this uncertainty.     

Patients' interpretations of verbal expressions of probability: implications for securing informed consent to medical interventions

Physicians often use verbal expressions of probability to characterize their uncertainty about outcomes and the risks or side effects of proposed therapies. However, there is an extensive literature that documents the inherent vagueness of such expressions. Because of the potential importance of probability terms to physician-patient communication and decision-making, we asked patients to tell us the odds they thought applicable to the term "rare," as used by their physician to discuss the likelihood of an adverse outcome from surgery. Patients were randomly assigned to one of three outcome groups: death, severe heart attack, or severe pneumonia. Demographic data were elicited from each subject, as were indicators of present health status, medical history for certain diseases and surgery, and life expectancy. Linear regression and ANOVA analyses of the responses indicate that patient age, education level, perceived health status, and recency of experience with disease and medical care influence patients' numeric interpretations. We discuss the implications of these results.     

The probability heuristics model of syllogistic reasoning

A probability heuristic model (PHM) for syllogistic reasoning is proposed. An informational ordering over quantified statements suggests simple probability based heuristics for syllogistic reasoning. The most important is the "min-heuristic": choose the type of the least informative premise as the type of the conclusion. The rationality of this heuristic is confirmed by an analysis of the probabilistic validity of syllogistic reasoning which treats logical inference as a limiting case of probabilistic inference. A meta-analysis of past experiments reveals close fits with PHM. PHM also compares favorably with alternative accounts, including mental logics, mental models, and deduction as verbal reasoning. Crucially, PHM extends naturally to generalized quantifiers, such as Most and Few, which have not been characterized logically and are, consequently, beyond the scope of current mental logic and mental model theories. Two experiments confirm the novel predictions of PHM when generalized quantifiers are used in syllogistic arguments. PHM suggests that syllogistic reasoning performance may be determined by simple but rational informational strategies justified by probability theory rather than by logic.     

Interpreting claims in offender profiles: the role of probability phrases,base‐rates and perceived dangerousness

Offender profilers use verbal and numerical probability expressions to convey uncertainty surrounding claims made about offender's characteristics. No previous research has examined how these expressions might affect the recipient's interpretation of the information. Seventy participants completed an online questionnaire and results showed a diverse range of interpretations of these uncertainty expressions. Moreover, characteristic base‐rates and dangerousness affected the perceived likelihood of the profiling claim, such that increased base‐rates and perceived dangerousness resulted in an increased perception of the claim being likely. Perceived likelihoods also depended on the framing of characteristics as well as the framing of the claim itself. Finally, where claims involved presenting a characteristic qualified by a low probability these claims were interpreted as more likely than not to be present. These findings have practical implications for profilers and more general theoretical implications for the study of risk perception. Copyright © 2008 John Wiley & Sons, Ltd.     

A disposition-behavior congruity model of perceived freedom

Three experiments were conducted to test a disposition-behavior congruity model of perceived freedom. The model was applied to attributions of freedom to actors who possess one of several dispositions and who can choose a behavior that is congruent or incongruent with this disposition. The model posits that attributions of freedom to an actor can be predicted from the conditional probabilities that the actor would choose each behavior available to him if he possessed the disposition congruent with it. This hypothesis was supported. However, contrary to expectations, the contribution to judgment of each conditional probability was not a function of the probability that the actor actually possessed the behavior-congruent disposition.     

A Strength Model of Probability Judgments for Tournaments

Fans of the National Basketball Association (NBA) assigned probability judgments to the outcomes of upcoming NBA games, and rated the strength of each team involved. The probability judgments obtained from these “expert” subjects exhibited high intersubject agreement and also corresponded closely to the eventual game outcomes. A simple model that associates a single strength value with each team accurately accounted for the probability judgments and their relationship to the ratings of team strength. The results show that, in this domain at least, probability judgments can be derived from direct assessments of strength which make no reference to chance or uncertainty.     

A local model of concurrent performance

Concurrent procedures may be conceptualized as consisting of two pairs of schedules with only one pair operating at a time. One schedule of each pair arranges reinforcers for staying in the current alternative, and the other schedule arranges reinforcers for switching to the other alternative. These pairs alternate operation as the animal switches between choices. This analysis of the contingencies suggests that variables operating within an alternative produce behavior that conforms to the generalized matching law. Rats were exposed to one pair of stay and switch schedules in each condition, and the probabilities of reinforcement varied across conditions. Both run length and visit duration were power functions of the ratio of the probabilities of reinforcement for staying and switching. The local model, a model of performance on concurrent procedures, was derived from this power function. Performance on concurrent schedules was synthesized from the performances on the separate pairs. Both the generalized matching law and the local model fitted the synthesized concurrent performances. These results are consistent with the view that the contingencies in the alternative, the probability of stay and switch reinforcement, are responsible for performance consistent with the generalized matching law. These results are compatible with momentary maximizing and molar maximizing accounts of concurrent performance. Models of concurrent performance that posit comparisons among the alternatives are not easily applied to these results.     

Comparing and unifying slope estimates across psychometric function models

The psychometric function relating stimulus intensity to response probability generally presents itself as a monotonically increasing sigmoid profile. Two summary parameters of the function are particularly important as measures of perceptual performance: the threshold parameter, which defines the location of the function over the stimulus axis (abscissa), and the slope parameter, which defines the (local) rate at which response probability increases with increasing stimulus intensity. In practice, the psychometric function may be modeled by a variety of mathematical structures, and the resulting algebraic expression describing the slope parameter may vary considerably between different functions fitted to the same experimental data. This variation often restricts comparisons between studies that select different functions and compromises the general interpretation of slope values. This article reviews the general characteristics of psychometric function models, discusses three strategies for resolving the issue of slope value differences, and presents mathematical expressions for implementing each strategy.     

Simple use of BIC to Assess Model Selection Uncertainty: An Illustration using Mediation and Moderation Models

Abstract

The Bayesian information criterion (BIC) has been used sometimes in SEM, even adopting a frequentist approach. Using simple mediation and moderation models as examples, we form posterior probability distribution via using BIC, which we call the BIC posterior, to assess model selection uncertainty of a finite number of models. This is simple but rarely used. The posterior probability distribution can be used to form a credibility set of models and to incorporate prior probabilities for model comparisons and selections. This was validated by a large scale simulation and results showed that the approximation via the BIC posterior is very good except when both the sample sizes and magnitude of parameters are small. We applied the BIC posterior to a real data set, and it has the advantages of flexibility in incorporating prior, addressing overfitting problems, and giving a full picture of posterior distribution to assess model selection uncertainty.     

Random effects and extended generalized partial credit models

In this paper it is shown that under the random effects generalized partial credit model for the measurement of a single latent variable by a set of polytomously scored items, the joint marginal probability distribution of the item scores has a closed-form expression in terms of item category location parameters, parameters that characterize the distribution of the latent variable in the subpopulation of examinees with a zero score on all items, and item-scaling parameters. Due to this closed-form expression, all parameters of the random effects generalized partial credit model can be estimated using marginal maximum likelihood estimation without assuming a particular distribution of the latent variable in the population of examinees and without using numerical integration. Also due to this closed-form expression, new special cases of the random effects generalized partial credit model can be identified. In addition to these new special cases, a slightly more general model than the random effects generalized partial credit model is presented. This slightly more general model is called the extended generalized partial credit model. Attention is paid to maximum likelihood estimation of the parameters of the extended generalized partial credit model and to assessing the goodness of fit of the model using generalized likelihood ratio tests. Attention is also paid to person parameter estimation under the random effects generalized partial credit model. It is shown that expected a posteriori estimates can be obtained for all possible score patterns. A simulation study is carried out to show the usefulness of the proposed models compared to the standard models that assume normality of the latent variable in the population of examinees. In an empirical example, some of the procedures proposed are demonstrated.     

Monkeys (Macaca mulatta and Cebus apella) track, enumerate, and compare multiple sets of moving items

Despite many demonstrations of numerical competence in nonhuman animals, little is known about how well animals enumerate moving stimuli. In this series of experiments, rhesus monkeys (Macaca mulatta) and capuchin monkeys (Cebus apella) performed computerized tasks in which they had to enumerate sets of stimuli. In Experiment 1, rhesus monkeys compared two sets of moving stimuli. Experiment 2 required comparisons of a moving set and a static set. Experiment 3 included human participants and capuchin monkeys to assess all 3 species' performance and to determine whether responding was to the numerical properties of the stimulus sets rather than to some other stimulus property such as cumulative area. Experiment 4 required both monkey species to enumerate subsets of each moving array. In all experiments, monkeys performed above chance levels, and their responses were controlled by the number of items in the arrays as opposed to nonnumerical stimulus dimensions. Rhesus monkeys performed comparably to adult humans when directly compared although capuchin performance was lower.