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).     

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.

     

A counterexample to Fishburn's conjecture on finite linear qualitative probability

Kraft, Pratt and Seidenberg (Ann. Math. Statist. 30 (1959) 408) provided an infinite set of axioms which, when taken together with de Finetti's axiom, gives a necessary and sufficient set of “cancellation” conditions for representability of an ordering relation on subsets of a set by an order-preserving probability measure. Fishburn (1996) defined f(n) to be the smallest positive integer k such that every comparative probability ordering on an n-element set which satisfies the cancellation conditions C₄,…,C_k is representable. By the work of Kraft, Pratt, and Seidenberg (1959) and Fishburn (J. Math. Psychol. 40 (1996) 64; J. Combin. Design 5 (1997) 353), it is known that n-1?f(n)?n+1 for all n?5. Also Fishburn proved that f(5)=4, and conjectured that f(n)=n-1 for all n?5. In this paper we confirm that f(6)=5, but give counter-examples to Fishburn's conjecture for n=7, showing that f(7)?7. We summarise, correct and extend many of the known results on this topic, including the notion of “almost representability”, and offer an amended version of Fishburn's conjecture.     

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.     

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.     

Stereokinetic phenomena revisited: The oscillating tilted bar, the swinging gate and the vertical contracting bar

We present here a revised version of our mathematical modelling of the stereokinetic phenomena known as the “oscillating tilted bar”, the “swinging gate” and the stereokinetic phenomenon elicited by a vertical, periodically contracting line segment simultaneously undergoing a lateral displacement from left to right and conversely in the frontal plane of an observer. The criticisms of Liu, Z. [(2004). On the minimal relative motion principle—The oscillating tilted bar. Journal of Mathematical Psychology, 48, 196-198] and Rokers and Liu [(2004). On the minimal relative motion principle—Lateral displacement of a contracting bar. Journal of Mathematical Psychology, 48, 292-295.] helped us in reformulating our models, eliminating discrepancies and ambiguities. Characteristic of the present modelling is the clear definition of a multi-stage mathematical procedure matching different requirements posed by the Visual System, as we know them from our experimental observations.     

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.     

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.     

Fast and accurate calculations for cumulative first-passage time distributions in Wiener diffusion models

We propose an improved method for calculating the cumulative first-passage time distribution in Wiener diffusion models with two absorbing barriers. This distribution function is frequently used to describe responses and error probabilities in choice reaction time tasks. The present work extends related work on the density of first-passage times [Navarro, D.J., Fuss, I.G. (2009). Fast and accurate calculations for first-passage times in Wiener diffusion models. Journal of Mathematical Psychology, 53, 222–230]. Two representations exist for the distribution, both including infinite series. We derive upper bounds for the approximation error resulting from finite truncation of the series, and we determine the number of iterations required to limit the error below a pre-specified tolerance. For a given set of parameters, the representation can then be chosen which requires the least computational effort.     

The analysis of semigroups of multirelational systems

A finite family of binary relations on a finite set, termed here a relational system, generates a finite semigroup under the operation of relational composition. The relationship between simplifications of the semigroup of a relational system in the form of homomorphic images, and simplifications of the relational system itself is examined. First of all, the list of relational conditions establishing a relationship between a homomorphic image of the semigroup of a relational system and a simplified, or derived, version of that relational system, is reviewed and extended. Then a definition of empirical relationship is introduced (the Correspondence Definition) and it is shown how, in conjunction with a factorization procedure for finite semigroups (P. E. Pattison & W. K. Bartlett, Journal of Mathematical Psychology, 1982, in press), it leads to a systematic and efficient analysis for a relational system. Applications of the procedure to an empirical blockmodel and to a class of simple relational systems are presented.     

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.     

An experimental and theoretical investigation of the constant-ratio rule and other models of visual letter confusion

The constant-ratio rule (CRR) and four interpretations of R. D. Luce's (In R. D. Luce, R. R. Bush, & E. Galanter (Eds.), Handbook of mathematical psychology (Vol. 1). New York: Wiley, 1963) similarity choice model (SCM) were tested using an alphabetic confusion paradigm. Four stimulus conditions were employed that varied in set size (three, four or five stimulus elements) and set constituency (block letters: A, E, X; F, H, X; A, E, F, H; A, E, F, H, X), and were presented to each subject in independent blocks. The four interpretations of the SCM were generated by constraining one, both, or neither of its similarity and bias parameter sets to be invariant in across-stimulus set model predictions. The strictest interpretation of the SCM (both the similarity and bias parameters constrained), shown to be a special case of the CRR, and the CRR produced nearly equivalent across-set predictions that provided a reasonable first approximation to the data. However, they proved inferior to the least strict SCM (neither the similarity nor bias parameters were constrained; the common interpretation of the SCM in visual confusion). Additionally, the least strict SCM was compared to J. T. Townsend's (Perception and Psychophysics, 1971, 9, 40–50, 449–454) overlap model, the all-or-none model (J. T. Townsend, Journal of Mathematical Psychology, 1978, 18, 25–38), and a modified version of L. H. Nakatani's (Journal of Mathematical Psychology, 1972, 9, 104–127) confusion-choice model. Both the least strict SCM and confusion-choice models produced nearly equivalent within stimulus set predictions that were superior to the overlap and all-or-none within-set predictions. Measurement conditions related to model structure and equivalence relations among the models, many of them new, were examined and compared with the statistical fit results of the investigation.     

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.     

Finite automata and S−R models

Suppes (1969, Journal of Mathematical Psychology, 6) apparently proved that any finite automaton could be mimicked by the asymptotic behavior of a stimulussampling S?R model, broadly implying that cognitive theory could be reduced to S?R theory. Here it is shown that the finite automata used by Suppes are more restricted and less powerful than finite automata in general. Furthermore, the S?R models proposed by Suppes are limited to producing the behavior of only these restricted automata, and cannot mimic all finite automata. Hence, the formal claim that S?R models are adequate to produce all behaviors obtainable from finite automata, and the informal claim that cognitive theory reduces to S?R theory, do not follow from Suppes's (1969) result. Some alternative S?R models and their problems are also briefly discussed.     

The case of Piaget's group INRC

What is it that distinguishes Piaget's transformations N, R, and C from the rest of the 16! transformations of the 16 binary propositional operations? Here Piaget's INRC is considered as a subgroup of the group $\text{M}$₂ of all automorphisms and dual automorphisms of the free Boolean algebra with two generators. This group is isomorphic to S₄ × C₂. Its elements are given explicitly. Many other psychologically relevant subgroups of $\text{M}$₂ play an important role. They are discussed and their connections shown. Particular attention is given to involutions, even if the view that they constitute the sole representation of reversibility is abandoned. Piaget's transformation R turns out not to be the inverse operation of relations. The group of automorphisms, dual automorphisms, anti-automorphisms of the algebra of binary relations on a finite set is found. A crystallographic presentation of these groups is given and related work by Bart (Journal of Mathematical Psychology, 1971, 5, 539–553), Leresche (Revue Européenne des Sciences Sociales, 1976, 14, 219–241), and Pólya (The Journal of Symbolic Logic, 1940, 5, 98–103) is discussed.     

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 characterization theorem for random utility variables

In this note we investigate the condition that the distribution of the maximum of a set of random variables does not depend on which variable attains the maximum. This problem arises in random utility theory. When the random variables are independent, the property implies that all the marginal distributions must be Double Exponential (with distribution function exp(?e^?x) in standard form). When dependence is allowed the property characrerizes a much broader class consisting of arbitrary functions of arbitrary homogeneous functions of the variables e^?xi, a result stated without proof in D. J. Strauss (Journal of Mathematical Psychology, 1979, 20, 35–52). These are the distributions such that the maximum has the same distribution (apart from a location shift) as the marginals, provided the marginals are the same.     

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.     

Utility of extension of functional equations—when possible

Jean-Claude Falmagne observed in 1981 [On a recurrent misuse of classical functional equation result. Journal of Mathematical Psychology, 23, 190-193] that, even under regularity assumptions, not all solutions of the functional equation k(s+t)=k(s)+k(t), important in many fields, also in the theory of choice, are of the form k(s)=Cs. This is certainly so when the domain of the equation (the set of (s,t) for which the equation is satisfied) is finite. We mention an example showing that this can happen even on some infinite, open, connected sets (open regions). The more general equations k(s+t)=?(s)+n(t) and k(s+t)=m(s)n(t), called Pexider equations, have been completely solved on R². In case they are assumed valid only on an open region, they have been extended to R² and solved that way (the latter if k is not constant). In this paper their common generalization

k(s+t)=?(s)+m(s)n(t)     

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.