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.     

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).     

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.     

INFORMATION TRANSMISSION, PARTITIONING AND WEBER'S LAW

M ac R ae A. W. Information transmission, partitioning and Weber's law:. Some comments on Baird's cognitive theory of psychophysics. Scand. J. Psychol ., 1972, 13 , 73–80.—Baird's derivation of Weber's Law from information measures has three principal defects: ( a ) he miscalculates the transmission achieved by successive partitioning judgments, ( b ) in his theoretical arguments he fails to distinguish between empirical fact and his own definitions or postulates, ( c ) he misinterprets the experimental evidence chosen to illustrate his conclusions.     

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.     

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.     

On the assessment of procedural knowledge: From problem spaces to knowledge spaces

By generalizing and completing the work initiated by Stefanutti and Albert (2003, Journal of Universal Computer Science, 9, 1455), this article provides the mathematical foundations of a theoretical approach whose primary goal is to construct a bridge between problem solving, as initially conceived by Newell and Simon (1972, Human problem solving. Englewood Cliffs, NJ: Prentice-Hall.), and knowledge assessment (Doignon and Falmagne, 1985, International Journal of Man-Machine Studies, 23, 175; Doignon and Falmagne, 1999, Knowledge spaces. Berlin, Germany: Springer-Verlag.; Falmagne et al., 2013, Knowledge spaces: Applications in education. New York, NY: Springer-Verlag; Falmagne and Doignon, 2011, Learning spaces: Interdisciplinary applied mathematics. Berlin, Germany: Springer-Verlag.). It is shown that the collection of all possible knowledge states for a given problem space is a learning space. An algorithm for deriving a learning space from a problem space is illustrated. As an example, the algorithm is used to derive the learning space of a neuropsychological test whose problem space is well known: the Tower of London (TOL; Shallice, 1982, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 298, 199). The derived learning space could then be used for adaptively assessing individual planning skills with the TOL.     

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.     

Illumination-invariance of Plateau's midgray

In a classical experiment Plateau [(1872). Sur la mesure des sensations physiques, et sur la loi qui lie l’intensité de la cause excitante. Bulletins de L’Academie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, 2me Série, 33, 376-388] found that the midgray operation is illumination-invariant. Falmagne [(1985). Elements of psychophysical theory. New York: Oxford University Press] provided a formalization of this finding that gives rise to two functional equations, the solutions of which determine the possible forms of the psychophysical function. This approach is generalized so that it explicitly refers to the illuminance, and can be applied to the experimentally well-studied class of center-surround stimuli. It is shown that the illumination-invariance of both cross-context matches and Plateau's midgray specify the psychophysical function up to an increasing function of the relative luminance of stimulus/illumination and center/surround, respectively. Relating this result to that implied by Wallach's ratio principle clarifies the role of Plateau's midgray in experimental research on achromatic color perception. Further generalizing the framework extends its scope to more complex experimental settings and the therein observed invariances.     

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 role of experience in location estimation: Target distributions shift location memory biases

Research based on the Category Adjustment model concluded that the spatial distribution of target locations does not influence location estimation responses [Huttenlocher, J., Hedges, L., Corrigan, B., & Crawford, L. E. (2004). Spatial categories and the estimation of location. Cognition, 93, 75–97]. This conflicts with earlier results showing that location estimation is biased relative to the spatial distribution of targets [Spencer, J. P., & Hund, A. M. (2002). Prototypes and particulars: Geometric and experience-dependent spatial categories. Journal of Experimental Psychology: General, 131, 16–37]. Here, we resolve this controversy by using a task based on Huttenlocher et al. (Experiment 4) with minor modifications to enhance our ability to detect experience-dependent effects. Results after the first block of trials replicate the pattern reported in Huttenlocher et al. After additional experience, however, participants showed biases that significantly shifted according to the target distributions. These results are consistent with the Dynamic Field Theory, an alternative theory of spatial cognition that integrates long-term memory traces across trials relative to the perceived structure of the task space.     

Assessing the local identifiability of probabilistic knowledge structures

Given a collection Q of problems, in knowledge space theory Doignon & Falmagne, (International Journal of Man–Machine Studies 23:175–196, 1985) the knowledge state of a student is the collection K ? Q of all problems that this student is capable of solving. A knowledge structure is a pair (Q, ), where is a collection of knowledge states that contains at least the empty set and Q. A probabilistic knowledge structure (PKS) is a knowledge structure (Q, , π), where π is a probability distribution on the knowledge states. The PKS that has received the most attention is the basic local independence model BLIM; Falmagne & Doignon, (British Journal of Mathematical and Statistical Psychology 41:1–23, 1988a, Journal of Mathematical Psychology 32:232–258, 1988b). To the best of our knowledge, systematic investigations in the literature concerning the identifiability of the BLIM are totally missing. Based on the theoretical work of Bamber and van Santen (Journal of Mathematical Psychology 29:443–473, 1985), the present article is aimed to present a method and a corresponding computerized procedure for assessing the local identifiability of the BLIM, which is applicable to any finite knowledge structure of moderate size.     

Task performance and magnitude of goal valence

J. W. Brehm and his associates (J. W. Brehm, R. A. Wright, S. Solomon, L. Silka, & J. Greenberg, 1983, Journal of Experimental Social Psychology, 19, 21–48) recently argued that the magnitude of goal valence (the attractiveness or unattractiveness of a potential outcome) varies directly with motivational arousal level. Motivational arousal, in turn, is thought to be a function of the perceived difficulty of goal attainment. This formulation was tested in the present study by examining the relationship between goal attractiveness ratings and performance on an anagram task. According to the Yerkes-Dodson law (R. M. Yerkes & J. D. Dodson, 1908, Journal of Comparative Neurological Psychology, 18, 459–482), the relationship between motivational arousal and performance should be curvilinear; optimal performance is usually observed for moderate levels of motivation relative to either low or very high motivation levels. Consistent with the Brehm et al. hypothesis, optimum performance in the present study was observed for subjects who reported moderate levels of goal attractiveness relative to subjects who reported either low or high levels of goal attractiveness. Anticipatory ratings of the difficulty of the anagrams were also congruent with the Brehm et al. model. These findings converge with data from other studies supporting the utility of goal attractiveness as an index of motivational arousal and provide an additional dimension of support for the model proposed by Brehm et al.     

Relations between multidimensional perfectionism and burnout in junior-elite male athletes

ObjectivesThe purpose of the present investigation was to examine the moderating influence of perceptions of goal progress and achievement goal orientations on the relationship between multidimensional perfectionism and athlete burnout.Methods201 junior-elite male athletes, ranging from 11 to 21 years of age (M = 15.64, SD = 1.92), were recruited from professional sport clubs in the UK and completed a multi-section inventory assessing self-oriented and socially prescribed perfectionism [Hewitt, P. L., & Flett, G. L. (1991). Perfectionism in the self and social contexts: conceptualization, assessment, and association with psychopathology. Journal of Personality and Social Psychology, 60, 456–470], achievement goal orientations [Roberts, G. C., Treasure, D. C., & Balague, G. (1998). Achievement goals in sport: the development and validation of the perception of success questionnaire. Journal of Sport Sciences, 16, 337–347], perceived goal progress [Hill, A. P., Hall, H. K., Appleton, P. R., & Kozub, S. A. (2008). Perfectionism and burnout in junior-elite soccer players: the mediating influence of unconditional self-acceptance. Psychology of Sport and Exercise, 9, 630–644] and multidimensional athlete burnout [Raedeke, T. D., & Smith, A. L. (2001). Development and preliminary validation of an athlete burnout measure. Journal of Sport and Exercise Psychology, 23, 281–306].ResultsRegression analyses revealed that socially prescribed perfectionism demonstrated a significant positive association, and self-oriented perfectionism a significant negative association with burnout dimensions. However, the hypotheses for moderation of the perfectionism–burnout relationship were not supported.ConclusionsOverall, while there was no evidence to support the hypothesised moderation of the perfectionism–burnout relationship, the results provide support for a growing body of literature which indicates that maladaptive forms of perfectionism may contribute to burnout in elite junior athletes [Chen, L. H., Kee, Y. H., Chen, M., & Tsaim, Y. (2008). Relation of perfectionism with athletes' burnout: further examination. Perceptual and Motor Skills, 106, 811–820; Gould, D., Tuffey, S., Udrey, E., & Loehr, J. (1996). Burnout in competitive junior tennis players: II. Qualitative analysis. The Sport Psychologist, 10, 341–366; Gould, D., Udry, E., Tuffey, S., & Loehr, J. (1996). Burnout in competitive junior tennis players: I. A quantitative psychological assessment. The Sport Psychologist, 10, 332–340; Hall, H. K. (2006). Perfectionism: a hallmark quality of world class performers, or a psychological impediment to athletic development? In D. Hackfort, & G. Tenenbaum (Eds.), Perspectives in sport and exercise psychology: Essential processes for attaining peak performance (Vol. 1, pp. 178–211). Oxford, UK: Meyer & Meyer Publishers; Hill et al., 2008; Lemyre, P. N., Hall, H. K., & Roberts, G. C. (2008). A social cognitive approach to burnout in elite athletes. Scandinavian Journal of Medicine & Science in Sports, 18, 221–224].     

Subjective responses to alcoholic and non-alcoholic beverages by alcoholic respondents

Changes in the subjective response to alcohol following electrical aversive conditioning have been studied through the use of the semantic differential (Costello, 1974; Costello et al., 1974a, 1974b; Hallam et al., 1972; Miller et al., 1974; Miller et al. 1973). As pointed out by the Costello group and by Kaplan (1972) there are several problems with this technique. The midpoint of the 7-point scale may be psychologically impossible to define if the subject does not perceive the adjectives pairs as mutually antagonistic. It is frequently assumed that the adjective pairs are bipolar, based on the original work of Osgood et al. (1957). Recent evidence (Costello, 1974; Costello et al., 1974a, 1974b) has demonstrated that the scales used by Hallam et al. (1972) were in fact, not bipolar.The stimuli used by Costello and coworkers were an alcoholic and a non-alcoholic slide, following the work of Hallam et al. (1972). Recent studies (Miller et al., 1974; Miller et al., 1973; Wilson, 1973) have utilized rating the taste of real alcohol, as opposed to slides, in order to measure the effectiveness of behavioral therapies in alcoholism. The subject is asked to taste several beverages, some of which have alcohol, and rate the taste of each drink on a set of semantic differential adjective pairs. Miller et al. (1974), found that while the actual amount of pre-therapy alcoholic beverage consumed predicted which subjects did better at 6 months outcome, the semantic differential ratings did not. One reason might have been that the scales were not bipolar and thus the semantic measurement was highly unreliable.The purpose of the present study was to replicate the Costello studies, using real beverages, alcoholic and non-alcoholic as the stimuli.     

RIASEC interest and confidence cutoff scores: Implications for career counseling

One strategy commonly used to simplify the joint interpretation of interest and confidence inventories is the use of cutoff scores to classify individuals dichotomously as having high or low levels of confidence and interest, respectively. The present study examined the adequacy of cutoff scores currently recommended for the joint interpretation of the Strong Interest Inventory (SII: Donnay, Morris, Schaubhut, & Thompson, 2005) and the Skills Confidence Inventory (SCI: [Betz, Borgen et al., 1996] and [Betz et al., 2005]). Cutoff score criteria were evaluated based on a sample of college students and employed adults (n = 1000), and cross-validated with a second sample of college students (n = 1, 149) who completed an experimental measure of RIASEC interests and confidence. Results were interpreted in the context of social cognitive career theory (Lent, Brown, & Hackett, 1994) by testing predictions regarding the relative frequency of individuals with either matched or mismatched levels of interest and confidence.     

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.     

The relationship between peer servant leadership,social identity,and team cohesion in intercollegiate athletics

Servant leadership is characterized by a leader’s emphasis on serving their followers first. Within the sport context, servant leadership has been studied primarily in coaches and has been associated with positive athlete outcomes, such as increased athlete satisfaction, motivation, and performance. However, the impact of servant leadership from an athlete perspective (e.g., formal team captains) has been underexplored. As such, the purpose of the present study was to examine the relationships between peer servant leadership, cohesion, and social identity within intercollegiate athletes. Two hundred and eighty-eight Division I and Division III NCAA intercollegiate athletes participated in the present study (female n = 165; male n = 123; M_age = 19.41, SD_age = 1.09) and completed the Revised Servant Leadership Profile for Sport (RSLP-S; Hammermeister et al., 2008), Group Environment Questionnaire (GEQ; Carron et al., 1985; Eys et al., 2007), and Social Identity Questionnaire for Sport (SIQS; Bruner & Benson, 2018). Structural equation modeling was used to assess the relationships between peer servant leadership, cohesion, and social identity. Results revealed that peer servant leadership positively predicted cohesion, and this relationship was fully mediated by social identity. The current study supports the effectiveness of peer servant leadership in sport contexts and provides theoretical support for the social identity approach to leadership.