Sources and influence of perceptual variance: Comment on Dzhafarov's Regular Minimality Principle

Dzhafarov [(2002). Multidimensional Fechnerian scaling: Pairwise comparisons, regular minimality, and nonconstant self-similarity. Journal of Mathematical Psychology, 46, 583-608] claims that Regular Minimality (RM) is a fundamental property of “same-different” discrimination probabilities and supports his claim with some empirical evidence. The key feature of RM is that the mapping, h, between two observation areas based on minimum discrimination probability is invertible. Dzhafarov [(2003a). Thurstonian-type representations for “same-different” discriminations: Deterministic decisions and independent images. Journal of Mathematical Psychology, 47, 184-204; (2003b). Thurstonian-type representations for “same-different” discriminations: Probabilistic decisions and interdependent images. Journal of Mathematical Psychology, 47, 229-243] also demonstrates that well-behaved Thurstonian models of “same-different” judgments are incompatible with RM and Nonconstant Self-Similarity (NCSS). There is extensive empirical support for the latter. Stimulus and neural sources of perceptual noise are discussed and two points are made:

Point 1: Models that require discrimination probabilities for noisy stimuli to possess the property that h is invertible would be too restrictive.

Point 2: In the absence of stimulus noise, violations of RM may be so subtle that their detection would be unlikely.

     

On the existence of extremal cones and comparative probability orderings

We study the recently discovered phenomenon [Conder, M. D. E., & Slinko, A. M. (2004). A counterexample to Fishburn's conjecture. Journal of Mathematical Psychology, 48(6), 425-431] of existence of comparative probability orderings on finite sets that violate the Fishburn hypothesis [Fishburn, P. C. (1996). Finite linear qualitative probability. Journal of Mathematical Psychology, 40, 64-77; Fishburn, P. C. (1997). Failure of cancellation conditions for additive linear orders. Journal of Combinatorial Designs, 5, 353-365]—we call such orderings and the discrete cones associated with them extremal. Conder and Slinko constructed an extremal discrete cone on a set of n=7 elements and showed that no extremal cones exist on a set of n?6 elements. In this paper we construct an extremal cone on a finite set of prime cardinality p if p satisfies a certain number theoretical condition. This condition has been computationally checked to hold for 1725 of the 1842 primes between 132 and 16,000, hence for all these primes extremal cones exist.     

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.     

Identity development during the first two university years: Exploring intersections between micro-, ontogenetic processes

Identity development processes are explored through analyses of longitudinal interview data collected with ten students during the first two university years. Micro-level analyses, based upon the work of Josephs, I. E. and Valsiner, J. [(1998). How does autodialogue work? Miracles of meaning maintenance and circumvention strategies. Social Psychology Quarterly, 61, 68-83], entailed the identification of oppositional meaning complexes—statements of conflicting meaning about self—as well as circumvention strategies. Ontogenetic analyses of developmental trajectories were based on Freeman, M. and Robinson, R.'s [(1990). The development within: An alternative approach to the study of lives. New Ideas in Psychology, 8, 53-72] conceptualization of the developmental process. From the ontogenetic analyses, two primary developmental patterns, and several subpatterns, were identified. Linkages between micro and ontogenetic levels were indicated by differences in rates of oppositional meaning complexes for individuals engaged in “developmental work” compared to those not so engaged, and differences in across-time patterning of these complexes for developmental subgroups. Findings support the potential of these methods for identifying semiotic mechanisms that are associated with longer-term developmental change.     

Modeling an experiment on construing self and others

In this paper we analyze the results of two repertory grid experiments in which subjects categorized themselves and others on bipolar constructs (Adams-Webber and Rodney, Canadian Journal of Behavioral Sciences, 1983, 15, No 1, 52–59). We attempt to explain these results on the basis of the assumption that the subjects relied upon a special “algebraic processor” for modeling the self and others, previously described by Lefebvre (Journal of Mathematical Psychology, 1980, 22, 83–120; Algebra of conscience: A comparative analysis of western and soviet ethical systems, 1982, Reidel: Dordrecht; Journal of Mathematical Psychology, 1985, 29, 289–310).     

A simple derivation of Prelec's probability weighting function

Since Kahneman and Tversky [(1979). Prospect theory: An analysis of decision under risk. Econometrica, 47, 263-291], it has been generally recognized that decision makers overweight low probabilities. Of the several weighting functions that have been proposed, that of Prelec [(1998). The probability weighting function. Econometrica, 60, 497-528] has the attractions that it is parsimonious, consistent with much of the available empirical evidence and has an axiomatic foundation. Luce [(2001). Reduction invariance and Prelec's weighting functions. Journal of Mathematical Psychology, 45, 167-179] provided a simpler derivation based on reduction invariance, rather than compound invariance of Prelec [(1998). The probability weighting function. Econometrica, 60, 497-528]. This note introduces a behavioral assumption that we call power invariance and provides a simple derivation of Prelec's function. Thus, we have three a priori different behavioral assumptions all leading to Prelec's function.     

The executive demands of strategic reasoning are modified by the way in which children are prompted to think about the task: Evidence from 3- to 4-year-olds

We investigated a test of strategic reasoning (the Windows task) that in different studies has yielded contrasting pictures of young children's executive abilities [Russell, J., Mauthner, N., Sharpe, S., & Tidswell, T. (1991). The “windows task” as a measure of strategic deception in preschoolers and autistic subjects. British Journal of Developmental Psychology, 9, 331–349; Samuels, M. C., Brooks, P. J., & Frye, D. (1996). Strategic game playing through the windows task. British Journal of Developmental Psychology, 14, 159–172]. An experiment with 52 three- to four-year-olds showed robust effects of different wordings for the prompts used to ask children to respond, and found that a single exposure to the facilitating wording led to improved performance on subsequent trials where the standard wording was used. This suggests that the effect of the wording was to help children infer an appropriate basis for responding, and not to reduce the trial-by-trial working memory or inhibitory demands of the task.     

A new definition of well-behaved discrimination functions

A discrimination function shows the probability or degree with which stimuli are discriminated from each other when presented in pairs. In a previous publication [Kujala, J.V., & Dzhafarov, E.N. (2008). On minima of discrimination functions. Journal of Mathematical Psychology, 52, 116–127] we introduced a condition under which the conformity of a discrimination function with the law of Regular Minimality (which says, essentially, that “being least discriminable from” is a symmetric relation) implies the constancy of the function’s minima (i.e., the same level of discriminability of every stimulus from the stimulus least discriminable from it). This condition, referred to as “well-behavedness,” turns out to be unnecessarily restrictive. In this note we give a significantly more general definition of well-behavedness, applicable to all Hausdorff arc-connected stimulus spaces. The definition employs the notion of the smallest transitively and topologically closed extension of a relation. We provide a transfinite-recursive construction for this notion and illustrate it by examples.     

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.     

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.     

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.     

Notes on selective influence, probabilistic causality, and probabilistic dimensionality

The paper provides conceptual clarifications for the issues related to the dependence of jointly distributed systems of random entities on external factors. This includes the theory of selective influence as proposed in Dzhafarov [(2003a). Selective influence through conditional independence. Psychometrika, 68, 7-26] and generalized versions of the notions of probabilistic causality [Suppes, P., & Zanotti, M. (1981). When are probabilistic explanations possible? Synthese, 48, 191-199] and dimensionality in the latent variable models [Levine, M. V. (2003). Dimension in latent variable models. Journal of Mathematical Psychology, 47, 450-466]. One of the basic observations is that any system of random entities whose joint distribution depends on a factor set can be represented by functions of two arguments: a single factor-independent source of randomness and the factor set itself. In the case of random variables (i.e., real-valued random entities endowed with Borel sigma-algebras) the single source of randomness can be chosen to be any random variable with a continuous distribution (e.g., uniformly distributed between 0 and 1).     

A generalization of the Yule-Simon model,with special reference to word association tests and neural cell assembly formation

A generalization of the Yule-Simon model is suggested and such related questions as inversion problems and nonequilibrium behavior are solved. The generalization follows the work of Haight and Jones (Journal of Mathematical Psychology, 1974, 11, 237–244) and thus special reference is made to word association tests. Some new possible applications of the presented model are offered, namely, in the field of neural cell assemblies.     

Evaluating a model of global psychophysical judgments—I: Behavioral properties of summations and productions

The research presented is a partial empirical evaluation of the second author's proposed psychophysical theory [Luce (2002). Psychological Review, 109, 520-532; Luce (2004). Psychological Review, 111, 446-454]. The theory deals with the global percept of subjective intensity, in which there is a psychophysical function Ψ that maps pairs of physical intensities onto the positive real numbers and represents, in an explicit mathematical way, subjective summation and a form of ratio production. A number of behavioral properties have been shown to follow from these specific representations, and in the presence of certain plausible background assumptions these properties are also sufficient for the representations. In four auditory experiments, key behavioral properties of summation over the two ears and a form of generalized ratio production are evaluated empirically. Considerable support is reported for particular forms of Ψ for summations and ratio productions separately. A second article, Steingrimsson and Luce (Journal of Mathematical Psychology, in press), explores the behavioral properties that link summations and productions.     

Nearest neighbor analysis of point processes: Simulations and evaluations

Tversky, Rinott, and Newman (Journal of Mathematical Psychology, 1983, 27, 000) examine the asymptotic behavior of a measure of the centrality of the nearest neighbor relation. The applicability of their conclusions when the number of dimensions (d) and the number of points (n) take on the small-to-moderate values commonly encountered in the analysis of proximity data is investigated. The results suggest that convergence is fast when n is large relative to d and slow when d is large relative to n.     

The mediating influence of unconditional self-acceptance and labile self-esteem on the relationship between multidimensional perfectionism and exercise dependence

ObjectivesPrevious research has indicated that perfectionism may be an important antecedent of exercise dependence (Hagan, A. L., & Hausenblas, H. A. (2003). The relationship between exercise dependence and perfectionism. American Journal of Health Studies, 18, 133–137; Hausenblas, H. A., & Symons Downs, D. (2002a). Exercise dependence: a systematic review. Psychology of Sport and Exercise, 3, 89–123, Hausenblas, H.A., & Symons Downs, D. (2002b). How much is too much? The development and validation of the exsrcise dependence scale. Psychology and Health, 17, 387–404). To date, however, few studies have sought to examine the psychological processes that underpin this relationship. The purpose of the present investigation was to determine the degree to which self-oriented and socially prescribed perfectionism were associated with exercise dependence, and to ascertain whether the relationships were mediated by unconditional self-acceptance and labile self-esteem.MethodThree hundred and seven middle-distance runners completed a multi-section inventory that included 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] Multidimensional Perfectionism Scale, Chamberlain, J. M., & Haaga, D. A. F. [(2001). Unconditional self-acceptance and psychological health. Journal of Rational Emotive and Cognitive Behavior Therapy, 19, 163–176] Unconditional Self-acceptance Scale, Dykman, B. M. [(1998). Integrating cognitive and motivational factors in depression: initial tests of a goal orientation approach. Journal of Personality and Social Psychology, 74, 139–158] Labile Self-Esteem Scale, and Ogden, J., Veale, D., & Summers, Z. [(1997). The development and validation of the Exercise Dependence Questionnaire. Addiction Research, 5, 343–356] Exercise Dependence Questionnaire.ResultsStructural Equation Modeling provided support for three hypotheses. First, that self-oriented perfectionism had a direct positive effect on exercise dependence. Second, that unconditional self-acceptance fully mediated the relationship between socially prescribed perfectionism and exercise dependence. Third, that labile self-esteem mediated the relationship between unconditional self-acceptance and exercise dependence. Multi-group invariance analysis further indicated that there were subtle variations in the nature of these relationships for male and female samples.ConclusionsThe findings from the present study indicate that both self-oriented and socially prescribed perfectionism may be critical antecedents of exercise dependence, but that the psychological mechanisms underpinning their association with exercise dependence may differ.     

A statistical approach to Saaty's scaling method for priorities

This paper investigates the logarithmic least squares (LLSM) approach to Saaty's (Journal of Mathematical Psychology, 1977, 5, 234–281) scaling method for priorities in hierarchical structures. It is argued that statistical criteria are important in deciding the scaling method controversy. It is shown that LLSM is statistically optimal under a number of realistic and practical models. Variances and covariances of parameter estimates are derived. The covariance matrix associated with overall priority differences is also developed. These results allow for a significance analysis of apparent priority differences.     

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.     

The role of cognitive resources in determining our moral intuitions: Are we all liberals at heart?

Recent research provides evidence that one important difference between liberals and conservatives is their basic moral intuitions. These studies suggest that while liberals and conservatives respond similarly to considerations of harm/care and fairness (what Graham and Haidt call the “individualizing” foundations), conservatives also respond strongly to considerations of in-group, authority, and purity (the “binding” foundations) while liberals do not. Our study examined two alternative hypotheses for this difference—the first being that liberals cognitively override, and the alternative being that conservatives cognitively enhance, their binding foundation intuitions. Using self-regulation depletion and cognitive load tasks to compromise people's ability to monitor and regulate their automatic moral responses, we found support for the latter hypothesis—when cognitive resources were depleted/distracted, conservatives became more like liberals (de-prioritizing the binding foundations), rather than the other way around. This provides support for the view that conservatism is a form of motivated social cognition.     

On the relationship of data to theory: A reply to Ronis and Greenwald

Ronis and Greenwald (Dissonance theory revised again: Comment on the paper by Fazio, Zanna, and Cooper. Journal of Experimental Social Psychology, 15, 1979, 62–69) question our integrative formulation of dissonance and self-perception processes. We argue that our formulation provides an explanation of both the full range of results in the literature and our own experimental data, while neither dissonance nor self-perception theory can singly explain the existing data.