Timing processes in motor imagery

Previous research shows inconsistencies in the timing of imagined and actual actions. Little is known about the timing in imagery, or how it relates to other forms of timing. Two studies examined whether imagery timing followed Weber's law, where variations in judgements grow linearly as the interval duration increases, or Vierordt's law, where short durations are overestimated and longer durations underestimated. In Study 1 participants (n=22) mentally walked and estimated journey times for flat paths and stairways, with and without a load. The timing patterns that emerged did not conform to Weber's law. In Study 2 participants (n=20) completed imagery, reproduction, production, and estimation timing tasks. Timing errors for imagery along a straight path, reproduction, estimation, and production all showed “Vierordt-like” effects. However, when imagining walking in a square participants consistently overestimated. It was concluded that imagery and interval timing processes are similar, but imagery timing is task dependent.     

Webers Idealtypus Als Methode Zur Bestimmung Des Begriffsinhaltes Theoretischer Begriffe in Den Kulturwissenschaften

Weber's Ideal Type as a Method of Forming the Content of Theoretical Concepts in Social Sciences}. Max Weber introduced the ideal type as the specific method of concept formation in social sciences. But the ideal type is not established in social research. Instead, authors in philosophy of science until today try to reconstruct and interpret what Weber said about ideal types as well as what might be their importance in Weber's social theory. The thesis of the following paper is that the difficulties in understanding Weber's ideal types are linked with Weber's intensional logic of concept formation. The thesis is defended in three steps. The first step deals with Weber's understanding of what is a scientific question in cultural sciences. Secondly Weber's critical arguments against positivism, hegelianism and historism are worked out. Thirdly, Weber's method of concept formation is reconstructed.     

Affine representations in psychophysics and the near-miss to Weber’s law

A theoretical account for the near-miss to Weber’s law in the form of a power function, with a special emphasis on the interpretation of the exponent, was proposed by Falmagne [Falmagne, J.-C. (1985). Elements of psychophysical theory. New York: Oxford University Press] within the framework of a subtractive representation, P(x,y)=F(u(x)−g(y)). In this paper, we examine a more general affine representation, P(x,y)=F(u(x)h(y)+g(y)). We first obtain a uniqueness theorem for the affine representation. We then study the conditions that force an affine representation to degenerate to a subtractive one. Part of that study involves the case for which two different affine representations co-exist for the same data. We also show that the balance condition P(x,y)+P(y,x)=1 constrains an affine representation to be a special kind of subtractive representation, a Fechnerian one. We further show that Falmagne’s power law takes on a special form for a so-called weakly balanced system of probabilities, in which case the affine representation is Fechnerian. Finally, following Iverson [Iverson, G.J. (2006a). Analytical methods in the theory of psychophysical discrimination I: Inequalities, convexity and integration of just noticeable differences. Journal of Mathematical Psychology, 50, 271-282], we generalize the Fechner method to construct the sensory scales in a weakly balanced affine representation by integrating (derivatives of) just noticeable differences.     

Shared timing variability in eye and finger movements increases with interval duration: Support for a distributed timing system below and above one second

The origins of the ability to produce action at will at the hundreds of millisecond to second range remain poorly understood. A central issue is whether such timing is governed by one mechanism or by several different mechanisms, possibly invoked by different effectors used to perform the timing task. If two effectors invoke similar timing mechanisms, then they should both produce similar variability increase with interval duration (interonset interval) and thus adhere to Weber's law (increasing linearly with the duration of the interval to be timed). Additionally, if both effectors invoke the same timing mechanism, the variability of the effectors should be highly correlated across participants. To test these possibilities, we assessed the behavioural characteristics across fingers and eyes as effectors and compared the timing variability between and within them as a function of the interval to be produced (interresponse interval). Sixty participants produced isochronous intervals from 524 to 1431?ms with their fingers and their eyes. High correlations within each effector indicated consistent performance within participants. Consistent with a single mechanism, temporal variability in both fingers and eyes followed Weber's law, and significant correlations between eye and finger variability were found for several intervals. These results can support neither the single clock nor the multiple clock hypotheses but instead suggest a partially overlapping distributed timing system.     

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.     

Measurement foundations for multiattribute psychophysical theories based on first order polynomials

The class of first order polynomial measurement representations is defined, and a method for proving the existence of such representations is described. The method is used to prove the existence of first order polynomial generalizations of expected utility theory, difference measurement, and additive conjoint measurement. It is then argued that first order polynomial representations provide a deep and far reaching characterization of psychological invariance for subjective magnitudes of multiattributed stimuli. To substantiate this point, two applications of first order polynomial representation theory to the foundations of psychophysics are described. First, Relation theory, a theory of subjective magnitude proposed by Shepard (Journal of Mathematical Psychology, 1981, 24, 21–57) and Krantz (Journal of Mathematical Psychology, 1972, 9, 168–199), is generalized to a theory of magnitude for multiattributed stimuli. The generalization is based on a postulate of context invariance for the constituent uniattribute magnitudes of a multiattribute magnitude. Second, the power law for subjective magnitude is generalized to a multiattribute version of the power law. Finally, it is argued that a common logical pattern underlies multiattribute generalizations of psychological theories to first order polynomial representations. This abstract pattern suggests a strategy for theory construction in multiattribute psychophysics.     

Analysis of multinomial models under inequality constraints: Applications to measurement theory

Multinomial random variables are used across many disciplines to model categorical outcomes. Under this framework, investigators often use a likelihood ratio test to determine goodness-of-fit. If the permissible parameter space of such models is defined by inequality constraints, then the maximum likelihood estimator may lie on the boundary of the parameter space. Under this condition, the asymptotic distribution of the likelihood ratio test is no longer a simple χ² distribution. This article summarizes recent developments in the constrained inference literature as they pertain to the testing of multinomial random variables, and extends existing results by considering the case of jointly independent mutinomial random variables of varying categorical size. This article provides an application of this methodology to axiomatic measurement theory as a means of evaluating properly operationalized measurement axioms. This article generalizes Iverson and Falmagne’s [Iverson, G. J. & Falmagne, J. C. (1985). Statistical issues in measurement. Mathematical Social Sciences, 10, 131-153] seminal work on the empirical evaluation of measurement axioms and provides a classical counterpart to Myung, Karabatsos, and Iverson’s [Myung, J. I., Karabatsos, G. & Iverson, G. J. (2005). A Bayesian approach to testing decision making axioms. Journal of Mathematical Psychology, 49, 205-225] Bayesian methodology on the same topic.     

The tune in-and-out model: A random walk and its application to a presidential election survey

The stochastic model for the evolution of preferences proposed by Falmagne, Regenwetter, and Grofman [1997. Journal of Mathematical Psychology, 41, 129-143] and tested by Regenwetter, Falmagne, and Grofman [1999. Psychological Review, 106, 362-384], as well as the alternative Thurstonian model of Böckenholt [Falmagne, J.-C., Regenwetter, M., & Grofman, B. (1997). A stochastic model for the evolution of preferences. In A. A. J. Marley (Ed.), Choice, decision and measurement: Essays in honor of R. Duncan Luce (pp. 113-131). Mahwah, NJ: Lawrence Erlbaum.], gave a good statistical account of attitudinal panel data from the 1992 US presidential election. We show, however, that both models have the defect of underestimating the number of respondents who did not change their order of preference for the candidates across different polls. We present a generalization of Falmagne et al.'s model based on the idea that some individuals may become momentarily impervious to all matters related to the campaign and ‘tune out.’ This behavior could be triggered by some personal reason or by some external event related to the campaign. Like the original model, the resulting model is a random walk, but on an augmented set of states. A respondent in a ‘live’ state behaves as in the previous model, except when receiving a ‘tune-out’ token, which effectively freezes the respondent's preference state until it is reversed by a ‘tune-in’ token. We describe and successfully test the new model on the same 1992 National Election Study panel data as those used by Böckenholt (2002) and Regenwetter et al. (1999).     

Early alternative derivations of Fechner's law

Historians of psychology, notably Boring, fostered Fechner's idea that Weber's law is the indispensable basis for the derivation of the logarithmic psychophysical law. However, it is shown here that Bernoulli in 1738 and Thurstone in 1931 derived the logarithmic law using principles other than Weber's law and that Fechner and Thurstone based their derivations on the principles originally employed by Bernoulli. It is concluded that awareness of researchers about Bernoulli's and Thurstone's derivations could expand the directions of research on the form of the psychophysical law. © 2009 Wiley Periodicals, Inc.     

Estimation of probabilities subject to linear inequality constraints

Based on Falmagne's representation theorem for finite random scale systems (J. C. Falmagne, Journal of Mathematical Psychology, 1978, 18, 52–72), simulated data are used to estimate choice probabilities such that a random representation exists. For the estimation a least-squares method is used constrained by linear inequalities. The minimization problem is solved by the method of quadratic programming.     

Black and white college women's maternal employment outcome expectations and their desired timing of maternal employment

In an attempt to better understand college women's maternal employment plans, this study compared the ability of different types of maternal employment outcome expectations to predict college women's desired timing of their maternal employment. A mailed questionnaire examined Black (n= 113) and White (n= 189) college women's maternal employment timing preference and their perceptions of the likelihood of 30 possible outcomes of maternal employment. Regressions showed that White women's expectations about the benefits for themselves and the costs for their children predicted their desired maternal employment timing. No variables predicted the timing preference of Black women. Additionally, the Black compared to White respondents expected maternal employment would bring fewer personal costs and more benefits for their children and they desired employment earlier in their child's life. Discussion focuses on Black and White women's conceptions of the integration of the employment and domestic roles.     

Finite and infinite state confusion models

A confusion model is defined as a model that decomposes response probabilities in stimulus identification experiments into perceptual parameters and response parameters. Historically, confusion models fall into two groups. Models in Group I, which includes Townsend's (Perception and Psychophysics, 1971, 9, 40–50) overlap model, were developed on the basis of the notion that stimulus identification is mediated by a finite number of internal states. We call the general class of models that have this processing interpretation finite state confusion models. Models in Group II, which includes Luce's (R. O. Luce et al., Eds., Handbook of Mathematical Psychology (Vol. I), New York: Wiley, 1963) biased choice model, were not developed on the basis of an explicit processing interpretation. It is shown here that models in Group II are not finite state confusion models. We prove in addition that except for Falmagne's (Journal of Mathematical Psychology, 1972, 9, 206–224) simply biased model models in Group II belong to a certain class of infinite state confusion models, namely, models asserting that stimulus identification is mediated by a continuous space of vectors representing detector activation levels.     

Distinct Visual Cues Mediate Aperture Shaping for Grasping and Pantomime-Grasping Tasks

The authors examined whether the top-down requirements of dissociating the spatial relations between stimulus and response in a goal-directed grasping task renders the mediation of aperture trajectories via relative visual information. To address that issue, participants grasped differently sized target objects (i.e., grasping condition) and also grasped to a location that was dissociated from the target object (i.e., pantomime-grasping condition). Just noticeable difference (JND) values associated with the early through late stages of aperture shaping were computed to examine the extent to which motor output adhered to, or violated, the psychophysical principles of Weber's law. As expected, JNDs during the late stages of the grasping condition violated Weber's law: a result evincing the use of absolute visual information. In contrast, JNDs for the pantomime-grasping condition produced a continuous adherence to Weber's law. Such a result indicates that dissociating a stimulus from a response is a perception-based task and results in aperture shaping via relative visual information.     

Mixture Models of Categorization

Many currently popular models of categorization are either strictly parametric (e.g., prototype models, decision bound models) or strictly nonparametric (e.g., exemplar models) (F. G. Ashby & L. A. Alfonso-Reese, 1995, Journal of Mathematical Psychology, 39, 216-233). In this article, a family of semiparametric classifiers is investigated where categories are represented by a finite mixture distribution. The advantage of these mixture models of categorization is that they contain several parametric models and nonparametric models as a special case. Specifically, it is shown that both decision bound models (F. G. Ashby & W. T. Maddox, 1992, Journal of Experimental Psychology: Human Perception and Performance, 16, 598-612; 1993, Journal of Mathematical Psychology, 37, 372-400) and the generalized context model (R. M. Nosofsky, 1986, Journal of Experimental Psychology: General, 115, 39-57) can be interpreted as two extreme cases of a common mixture model. Furthermore, many other (semiparametric) models of categorization can be derived from the same generic mixture framework. In this article, several examples are discussed and a parameter estimation procedure for fitting these models is outlined. To illustrate the approach, several specific models are fitted to a data set collected by S. C. McKinley and R. M. Nosofsky (1995, Journal of Experimental Psychology: Human Perception and Performance, 21, 128-148). The results suggest that semi-parametric models are a promising alternative for future model development.     

A characterization of the concept of independence in knowledge structures

In knowledge space theory a knowledge structure provides a deterministic representation of the implications among the items in a given set Q. Concrete procedures for the efficient assessment of knowledge by means of a knowledge structure have been proposed by Doignon and Falmagne [Falmagne, J.-C., & Doignon, J.-P. (1988a). A class of stochastic procedures for the assessment of knowledge. British Journal of Mathematical and Statistical Psychology, 41, 1-23; Falmagne, J.-C., & Doignon, J.-P. (1988b). A markovian procedure for assessing the state of a system. Journal of Mathematical Psychology, 32, 232-258]. The primitive idea at the core of such procedures is that the (correct or wrong) answers of a student to a subset A⊆Q of items could be inferred from the answers to a subset B⊆Q of items that were previously presented to that student. Since B provides information about A, from the viewpoint of the teacher these two subsets are not independent. This idea of dependence vs. independence is formalized in this paper in terms of an independence relation on the power set of Q. A nice characterization of this relation allows to express an arbitrary knowledge structure as the combination of a number of substructures each of which is independent of each other. An algorithm is then proposed which checks for independence in a knowledge structure and decomposes this last into a collection of independent substructures.     

Analytical methods in the theory of psychophysical discrimination I: Inequalities, convexity and integration of just noticeable differences

We re-examine the theoretical status of Fechner's Mathematical Auxiliary Principle [Fechner, G. T. (1889). Elemente der psychophysik. Leipzig: Breitkopf und Härtel] which underlies Fechner's method of constructing a sensory scale by integrating just noticeable differences (jnds). That “Principle” has been roundly criticized [Luce, R. D., & Edwards, W. (1958). The derivation of subjective scales from just noticeable differences. Psychological Review, 65, 227-237] as being inconsistent with the very basis of Fechner's psychophysical theory, and indeed this is the case. In important papers Pfanzagl [(1962). Über die stochastische Fundierung des psychophysischen Gesetzes (On the stochastic foundations of the psychophysical law). Biometrische Zeitschrift, 4, 1-14] and Krantz [(1971). Integration of just noticeable differences. Journal of Mathematical Psychology, 8, 591-599] resurrected Fechner's method; their analysis showed that the sensory scale could be written as the limit of a sequence of integrals, each of the form suggested by the auxiliary principle. In this work, we investigate the properties of a typical member of the Krantz-Pfanzagl sequence of integrals; we do so with the view to obtaining useful approximations to the true scale. Weber's inequality [Falmagne, J.-Cl. (1977). Note: Weber's inequality and Fechner's problem. Journal of Mathematical Psychology, 16, 267-271] plays an important role in our developments. That inequality, and other inequalities of a similar nature, allows us to place bounds on the error incurred by approximating a true scale u by an integral of jnds. Under appropriate conditions these bounds are sufficiently tight that the relative error is very small over the entire stimulus domain. We illustrate our theoretical results with a number of examples.     

On resolving conflicting views of cognitive conflict

F. B. Murray's (Developmental Review, 1983, 3, 54–61) conclusion that a theoretically ambiguous relationship exists between cognitive conflict and Piaget's notion of equilibration was considered. It was suggested that theoretical clarity can be achieved if a distinction is drawn between conflict induction (disequilibrium) and conflict resolution (equilibration). Murray's suggestion that in Piaget's theory internal (cognitive) conflict can be mentally created was disputed. G. N. Cantor's (Developmental Review, 1983, 3, 39–53) concerns about the adequacy of the measures of internal conflict in B. J. Zimmerman & D. E. Blom's (Developmental Review, 1983, 3, 18–38) study due to their independence from external conflict manipulations was discussed. We agreed with Cantor that such findings obviate direct disconfirmation of an internal conflict construct; however, we suggested that evidence of conservation learning in the absence of either external or internal conflict renders Piaget's account less tenable.     

Classes,collections, and distinctive features: Alternative strategies for solving inclusion problems

This study analyzed the strategies that children ages 5 through 8 years used on two modified versions of Inhelder and Piaget's (The early growth of logic in the child. New York: Norton, 1964) class inclusion task. In two experiments, children were tested on Wilkinson's (Cognitive Psychology, 1976, 8, 64–85) “percept” inclusion task in which distinctive features marked both supraordinate and subclasses. It was hypothesized that children who fail standard Piagetian inclusion tasks succeed on the “percept” task by counting and comparing mutually exclusive features rather than using features as markers for classes and subclasses. The hypothesis was supported by children's performances on “percept” tasks in which solutions based on feature counting conflicted with solutions based on consideration of class inclusion relations. In two other experiments, children answered part-whole and part-part comparison questions in which both terms were described as classes and/or subclasses, or in which one of the two terms was described as a collection (e.g., a bunch of grapes). These experiments contrasted Markman and Seibert's (Cognitive Psychology, 1976, 8, 561–577) “organization” hypothesis that the greater psychological integrity of collections facilitates reasoning on part-whole comparison problems with the hypothesis that the faciltative effect results from the “large number” connotation of collective nouns. Results on collection problems in which parts were described as collections supported the “large number” hypothesis. Results were discussed in terms of their implications for Piaget's theory.     

Sufficient conditions for an ordered-ring isomorphism onto a positive subinterval: A lemma for polynomial measurement

This paper discusses a sufficient set of conditions for a structure to be ordered-ring isomorphic to a positive subinterval of the real numbers. It extends our previous results concerning bounded versions of Hölder's Theorem (Falmagne, 1971). The main result can serve as a basic lemma for establishing representation theorems in polynomial conjoint measurement when the empirical sets contain neither large elements nor ones close to the natural zero.     

Time, trace, memory

Objections to a trace hypothesis for interval timing do not apply to the multiple-time-scale (MTS) theory, which incorporates a dynamic trace tuned by the system history and can easily accommodate interval timing over a 1,000:1 range. The MTS model can also account for Weber's law as well as systematic deviations from it. Contrary to our critics, we contend that patterns of variance in interval timing experiments are not fully described by scalar expectancy theory, and that attempting to understand timing by assigning variance to different elements of a flexible model that lacks inductive support is a flawed strategy, because the attempt may be successful even if the model is wrong. We further argue that biological plausibility is an unreliable guide to the development of behavioral theory, that prediction is not the same as test, that induction should precede deduction, and that a rat is not a clock.