Mathematical Psychology Research Articles

Unconditionally Selective Dependence of Random Variables on External Factors

What is the meaning of saying that random variables {X1, …, Xn} (such as aptitude scores or hypothetical response time components), not necessarily stochastically independent, are selectively influenced respectively by subsets {Γ1, …, Γn} of a factor set Φ upon which the joint distribution of {X1, …, Xn} is known to depend? One possible meaning of this statement, termed conditionally selective influence, is completely characterized in Dzhafarov (1999, Journal of Mathematical Psychology, 43, 123–157). The present paper focuses on another meaning, termed unconditionally selective influence. It occurs when two requirements are met. First, for i=1, …, n, the factor subset Γi is the set of all factors that effectively change the marginal distribution of Xi. Second, if {X1, …, Xn} are transformed so that all marginal distributions become the same (e.g., standard uniform or standard normal), the transformed variables are representable as well-behaved functions of the corresponding factor subsets {Γ1, …, Γn} and of some common set of sources of randomness whose distribution does not depend on any factors. Under the constraint that the factor subsets {Γ1, …, Γn} are disjoint, this paper establishes the necessary and sufficient structure of the joint distribution of {X1, …, Xn} under which they are unconditionally selectively influenced by {Γ1, …, Γn}. The unconditionally selective influence has two desirable properties, uniqueness and nestedness: {X1, …, Xn} cannot be influenced selectively by more than one partition {Γ1, …, Γn} of the factor set Φ, and the components of any subvector of {X1, …, Xn} are selectively influenced by the components of the corresponding subpartition of {Γ1, …, Γn}.

Ranked data analysis of a gamut-mapping experiment

We analyze data from a gamut-mapping experiment using several statistical procedures for ranked data. In this experiment six gamut-mapping algorithms were applied to six different images and the results were ranked by 31 judges according to how well the images matched an original. We fitted two distance-based statistical models to the data: both analyses showed that aggregate preference among the six algorithms depended on the image viewed. Based on the first model we classified the images into four classes or clusters. We applied unidimensional unfolding, a technique from mathematical psychology, to extract latent reference frames upon which judges plausibly ordered the algorithms. Four color experts gave interpretations of the derived reference frames. We used the second model to generate confidence sets for the consensus rankings, and another cluster analysis.

The fate of Herbart's mathematical psychology.

In the 30 years following its fullest presentation in 1824, the mathematical psychology of J. F. Herbart (1776-1841) was widely taught and often favorably discussed by scholars such as M. W. Drobisch. However, Herbart's theory received considerable criticism after the introduction of experimentation into psychology by Fechner and Wundt; Ebbinghaus was more friendly toward it. In the period from about 1885 to 1910, with the focus on unconscious processing in Freud's theory and the revival of interest in Herbart's educational theory, one might have expected interest in Herbart's mathematical psychology to rekindle, but this was not the case. The theory remained neglected throughout the 20th century, but it deserves reappraisal at the start of the 21st century.

The fate of Herbart's mathematical psychology.

The fate of Herbart's mathematical psychology.

Theory of reference frames and biological control

This article is aimed at introducing the theory of reference frames in mathematical psychology to the control theory community and to point out its relevance in understanding the differences between biological and engineering control systems. As an example, we discuss the problem of human posture control and point out how the reference frames theory lead to a natural solution which is in agreement with everyday observations on how a child learns to do posture control.

Using Conjoint Analysis To Value Ecosystem Change

Economists have developed a variety of methods to measure the values of environmental goods and services. Conjoint analysis (CJ) is a technique developed by mathematical psychologists to establish the structure of preferences across multi-attribute alternatives. It is a type of stated-preference method that has captured the attention of economists for purposes of analyzing preferences toward environmental goods and services. This paper outlines the applicability of CJ to environmental valuation, illustrates applications and valuation issues in the literature, and presents an application of CJ to valuing a Pennsylvania watershed quality improvement.

Accuracy, Scope, and Flexibility of Models

Traditionally, models are compared on the basis of their accuracy, their scope, and their simplicity. Simplicity is often represented by parameter counts; the fewer the parameters, the simpler the model. Arguments are presented here suggesting that simplicity has little place in discussions of modeling; instead, the concept of flexibility should be substituted. When comparing two models one should be wary of the possibility of their differential flexibility. Several methods for assessing relative flexibility are possible, as represented in this special issue of the Journal of Mathematical Psychology. Here, the method of cross-validation is applied in the comparison of two models, a linear integration model (LIM) and the fuzzy-logical model of perception (FLMP), in the fitting of 44 data sets concerning the perception of layout seen among three panels with the presence or absence of four sources of information for depth. Prior to cross-validation the two models performed about equally well; after cross-validation LIM was statistically superior to FLMP, but the overall pattern of fits remained nearly the same for both models.

Where is Mathematical Modeling in Psychology Headed?

The current state of mathematical psychology is characterized in terms of five paired contrasts: behavioral vs information-processing; linearity vs non-linearity; static vs dynamic; mathematical vs computational; and deterministic vs probabilistic. Each contrast is evaluated in terms of three basic facts of psychology: complexity, irreversibility of experience, and individual differences; and also in terms of three contemporary developments: computational power, brain imaging and recent mathematical developments. From these I draw some (low-confidence) conclusions about where I think the field is headed, and why.

Edward C. Carterette, 1921-1999

Edward C. Carterette passed away after a long battle with throat cancer on July 7, 1999. He was associate editor of Music Perception for 17 years, from 1982 until his death. Ed Carterette was born in 1921 in Mount Tabor, North Carolina. He entered the United States Army in 1937, leaving in 1946 at the rank of Lieutenant Colonel, the youngest to reach that rank at the time. Ed Carterette spent the next 10 years pursuing various degrees, including one in mathematics from the University of Chicago, one in psychology from Harvard, and a Ph.D. in experimental and mathematical psychology from Indiana University. During this period, from 1947 to 1956, Ed Carterette worked in the laboratories of M. A. B. Brazier (Brain Wave Laboratory, Massachusetts General Hospital, Boston), J. C. R. Licklider (Acoustics Laboratory, Massachusetts Institute of Technology) and J. P. Eagan (Hearing and Communication Laboratory, Indiana University). Ed arrived at the University of California, Los Angeles, in 1956. He remained at UCLA his entire academic career, becoming Professor Emeritus of Psychology in 1991. Ed was devoted to UCLA and its Department of Psychology, helping to establish it as one of the best-ranked departments of psychology in the United States. At various times he was a visiting scholar at such distinguished institutions as Stanford University, M.I.T., College of William and Mary, Cambridge University, and the Royal Institute of Technology, Stockholm. During his long career, Ed Carterette 's accomplishments led him to inclusion in more than 10 Who s Who publications, including Who's Who in the World. He was included in American Men and Women of Science. He was a fellow of the Acoustical Society of America, the American Association for the Advancement of Science, the American Psychological Association, the American Psychological Society, the Society of Experimental Psychologists, and the Society for Music Perception and Cognition.

Una lectura histórica de las relaciones conceptuales entre psicología teórica y psicología cuantitativa

Este artículo presenta una revisión de la historia de las relaciones conceptuales entre psicología teórica y psicología cuantitativa. Se revisan los conceptos de psicología matemática y psicometría, así como los vínculos epistemológicos que estas áreas tienen con los estructuralismos y funcionalismos psicológicos. El trabajo resalta la necesidad de dotar a los puntos de vista cuantitativos en psicología con herramientas conceptuales propiamente psicológicas. Se propone la teoría de la inteligencia de Spearman como un modelo de integración de psicología teórica y psicología cuantitativa; también se propone la integración de la psicometría con la psicología experimental en el terreno de las pruebas de hipótesis derivadas de marcos psicomatemáticos y psicoestadísticos.

Reference frames in learning and control

This article is aimed at introducing the theory of reference frames in mathematical psychology to the control theory community and to point out its relevance in reducing the complexity of control systems design. As an example we. discuss the balancing problem for a multi-segment pendulum.

A p* primer: logit models for social networks

Abstract A major criticism of the statistical models for analyzing social networks developed by Holland, Leinhardt, and others [Holland, P.W., Leinhardt, S., 1977. Notes on the statistical analysis of social network data; Holland, P.W., Leinhardt, S., 1981. An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association. 76, pp. 33–65 (with discussion); Fienberg, S.E., Wasserman, S., 1981. Categorical data analysis of single sociometric relations. In: Leinhardt, S. (Ed.), Sociological Methodology 1981, San Francisco: Jossey-Bass, pp. 156–192; Fienberg, S.E., Meyer, M.M., Wasserman, S., 1985. Statistical analysis of multiple sociometric relations. Journal of the American Statistical Association, 80, pp. 51–67; Wasserman, S., Weaver, S., 1985. Statistical analysis of binary relational data: Parameter estimation. Journal of Mathematical Psychology. 29, pp. 406–427; Wasserman, S., 1987. Conformity of two sociometric relations. Psychometrika. 52, pp. 3–18] is the very strong independence assumption made on interacting individuals or units within a network or group. This limiting assumption is no longer necessary given recent developments on models for random graphs made by Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association. 81, pp. 832–842] and Strauss and Ikeda [Strauss, D., Ikeda, M., 1990. Pseudolikelihood estimation for social networks. Journal of the American Statistical Association. 85, pp. 204–212]. The resulting models are extremely flexible and easy to fit to data. Although Wasserman and Pattison [Wasserman, S., Pattison, P., 1996. Logit models and logistic regressions for social networks: I. An introduction to Markov random graphs and p*. Psychometrika. 60, pp. 401–426] present a derivation and extension of these models, this paper is a primer on how to use these important breakthroughs to model the relationships between actors (individuals, units) within a single network and provides an extension of the models to multiple networks. The models for multiple networks permit researchers to study how groups are similar and/or how they are different. The models for single and multiple networks and the modeling process are illustrated using friendship data from elementary school children from a study by Parker and Asher [Parker, J.G., Asher, S.R., 1993. Friendship and friendship quality in middle childhood: Links with peer group acceptance and feelings of loneliness and social dissatisfaction. Developmental Psychology. 29, pp. 611–621].

The role of mathematical psychology in experimental psychology

(1998). The role of mathematical psychology in experimental psychology. Australian Journal of Psychology: Vol. 50, No. 3, pp. 129-130.

Selective serotonin reuptake inhibitors alone and in association with pindolol in the treatment of delusional depression

A positive n × n matrix, R = (rij), is said to be reciprocal if its entries verify rji = 1/rij > 0 for all 1 ≤ i,j ≤ n. In the context of the analytic hierarchy process, where such matrices arise from the pairwise comparison of n ≥2 decision alternatives on an arbitrary ratio scale, Saaty (in Journal of t Mathematical Psychology, 1977) proposed to use μ = (Λmax − n)/(n − 1) ≥ 0, a linear transform of the Perron eigenvalue Λmax of R, as a measure of the cardinal consistency in an agent's responses and posed the problem of determining how it might vary as a function of the rij's. He also suggested that an upper bound could be found for that consistency index when the entries of R are restricted to take their values in a bounded set. Both of these questions are answered here using classical results from linear algebra.

A test of the gravity lens theory.

Naito and Cole [1994, in Contributions to Mathematical Psychology: Psychometrics and Methodology Eds G H Fischer and D Laming (New York: Springer)] provide a configuration which they describe as the Gravity Lens illusion. In this configuration, four small dots are presented in proximity to four large disks, and one is asked to compare the slope of an imaginary line which connects one pair of dots with the slope of a line which connects the other pair. In fact the slopes are the same, i.e. their axes are parallel, but because of the positioning of the large disks they appear to be at different orientations. Naito and Cole propose that the perceptual bias is analogous to the effects of gravity on the metrics of physical space, such that mental projections in the vicinity of a disk (or an open circle) are distorted just as the path of light is bent as it passes a massive body such as a star. Here we provide a simple test of this concept by having subjects judge alignments of dots which lie near tangents to a circle. Subjects were asked to project straight lines through pairs of stimulus dots, selecting and marking points in open space which were collinear with each pair. As would be predicted by the Gravity Lens theory, the locations selected by subjects were displaced from straight lines. However, the error magnitudes were substantially larger for judgments of dot pairs which had an oblique alignment, as compared with dot pairs which were aligned with a cardinal axis. This differential of effect as a function of stimulus orientation is not predicted by the gravity concept.

Choosing subsets: a size-independent probabilistic model and the quest for a social welfare ordering

“Subset voting” denotes a choice situation where one fixed set of choice alternatives (candidates, products) is offered to a group of decision makers, each of whom is requested to pick a subset containing any number of alternatives. In the context of subset voting we merge three choice paradigms, “approval voting“ from political science, the “weak utility model” from mathematical psychology, and “social welfare orderings” from social choice theory. We use a probabilistic choice model proposed by Falmagne and Regenwetter (1996) built upon the notion that each voter has a personal ranking of the alternatives and chooses a subset at the top of the ranking. Using an extension of Sen's (1966) theorem about value restriction, we provide necessary and sufficient conditions for this empirically testable choice model to yield a social welfare ordering. Furthermore, we develop a method to compute Borda scores and Condorcet winners from subset choice probabilities. The technique is illustrated on an election of the Mathematical Association of America (Brams, 1988).

Introduction of contagious animal diseases into The Netherlands: elicitation of expert opinions

This paper describes an experiment aimed at the derivation of information on Foot-and-Mouth Disease, Classical and African Swine Fever, Swine Vesicular Disease, Newcastle Disease and Avian Influenza to be used in an economic model to simulate introduction of a virus into the Netherlands. Several elicitation techniques, including three-point estimation, Conjoint analysis and ELI were used in the experiment. Three-point estimation (asking for minimum, most likely, and maximum values) was used to derive information concerning the length of the High Risk Period (HRP), during which the virus may spread freely. Conjoint analysis is widely used in marketing and consumer science and was used in this experiment to elicit the relative importance of risk factors which may introduce disease into the Netherlands (such as import of livestock, import of animal products, etc.). The ELI-technique (ELIcitation of uncertain knowledge) originates from the area of mathematical psychology and was used to derive information on the expected number of outbreaks of the diseases under consideration in European countries. The experiment was conducted with 43 (out of 50) invited people, all involved in or related to disease control in one way or another. The participants expected, for all diseases, the longest HRP in countries of eastern Europe. Import of livestock was evaluated as the most important risk factor for all diseases under consideration. Most outbreaks within the next five years were expected to occur in countries of eastern Europe, the smallest number in the Netherlands and the Scandinavian countries. Results especially showed that Conjoint analysis and ELI are useful tools to help quantify subjective knowledge. These subjective elicitation methods may not reveal the `Golden Standard' but as long as `objective' predictive information is not available, use of these tools may be valuable to enhance quality of input and output of economic simulation models to support policy-making in the prevention and control of contagious animal diseases.

Why energy conservation fails

Introduction Fundamentals of Why Energy Conservation Fails Strange Terminology Psychology of Energy Conservation Mathematics of Energy Conservation Petroleum and Gasoline In the Comics Marxism and Energy Conservation Environment, Pollution and Recycling What Stanley Jevons Found Fast Forward to the Past How the Government Does Crime and Conservation Public Utilities The Question of Waste The Economic Aspect Selling Energy Conservation Resource Depletion Workers and Energy Conservation Examples They Don't Want You to Hear About The International Scene Morality, Politics, and Energy Conservation Food and Energy Conservation Where Do We Go from Here? Annotated Bibliography Index

The Reform Calculus Debate and the Psychology of Learning Mathematics

The Reform Calculus Debate and the Psychology of Learning Mathematics

Several Unresolved Conceptual Problems of Mathematical Psychology

This article is a personal commentary on the following, to my mind, unresolved issues of mathematical psychology: (1) the failure of the field to have become a fully accepted part of most departments of psychology; (2) the great difficulty we have in studying dynamic mechanisms, e.g., learning, because large samples are difficult to obtain: time samples wipe out the phenomena and subject samples are unrepresentative because of profound and ill understood individual differences; (3) the failure to unify successfully statistical and measurement theories which, I believe, are two facets of a common problem; (4) the proliferation of free parameters in many types of theories with little success in developing theories of such parameters; (5) the difficulties we have had in successfully formulating the mathematics of uncertainty and vagueness; and (6) the issues of modeling what are presumably discrete attributes and phenomena by continuous mathematics: when and how is this justified?