Fitting one matrix to another under choice of a central dilation and a rigid motion

A least squares method is presented for fitting a given matrixA to another given matrixB under choice of an unknown rotation, an unknown translation, and an unknown central dilation. The procedure may be useful to investigators who wish to compare results obtained with nonmetric scaling techniques across samples or who wish to compare such results with those obtained by conventional factor analytic techniques on the same sample.Part of this work was done while the senior author held a visiting research fellowship at the Educational Testing Service, Princeton, New Jersey.     

Gender differences in preference to offer social support to assertive men and women

Although gender bias is clearly evident in the provision of institutional social support, it is unclear whether such bias exists on a more subtle and interpersonal level. When women act appropriately assertive at work, they may actually alienate support. This paper reports an experimental study that addressed this general issue of social support among employed men and women. A sample of 61 males and 55 females (predominantly white; N=116), employed by four northeast Ohio employers participated. The extent of preference to offer support to assertive males and females was explored. An unexpected double bias was found, with each sex preferring to offer support to their own sex.This project was funded by a grant received by the Applied Psychology Center of Kent State University, from the Ohio Department of Development through the Northeast Ohio Center for the Advancement of Labor Management Cooperation. We gratefully acknowledge this financial assistance.     

A generalized solution of the orthogonal procrustes problem

A solutionT of the least-squares problemAT=B +E, givenA andB so that trace (EE)= minimum andTT=I is presented. It is compared with a less general solution of the same problem which was given by Green [5]. The present solution, in contrast to Green's, is applicable to matricesA andB which are of less than full column rank. Some technical suggestions for the numerical computation ofT and an illustrative example are given.This paper is based on parts of a thesis submitted to the Graduate College of the University of Illinois in partial fulfillment of the requirements for a Ph.D. degree in Psychology.The work reported here was carried out while the author was employed by the Statistical Service Unit Research, U. of Illinois. It is a pleasure to express my appreciation to Prof. K. W. Dickman, director of this unit, for his continuous support and encouragement in this and other work. I also gratefully acknowledge my debt to Prof. L. Humphreys for suggesting the problem and to Prof. L. R. Tucker, who derived (1.7) and (1.8) in summation notation, suggested an iterative solution (not reported here) and who provided generous help and direction at all stages of the project.     

Is everyday causation deterministic or probabilistic?

One view of causation is deterministic: A causes B means that whenever A occurs, B occurs. An alternative view is that causation is probabilistic: the assertion means that given A, the probability of B is greater than some criterion, such as the probability of B given not-A. Evidence about the induction of causal relations cannot readily decide between these alternative accounts, and so we examined how people refute causal assertions. In four experiments most participants judged that a single counterexample of A and not-B refuted assertions of the form, A causes B. And, as a deterministic theory based on mental models predicted, participants were more likely to request multiple refutations for assertions of the form, A enables B. Similarly, refutations of the form not-A and B were more frequent for enabling than causal assertions. Causation in daily life seems to be a deterministic concept.     

The evaluation of linear forms

This presentation deals with the evaluation and transformation of linear forms. Especial emphasis is given to implicit methods in which it is not necessary to find the explicit values,x _i . The relation of the Aitken triple product matrixCA ^–1 B to the result of a linear transformation of linear forms is noted, and the numerical computation of this triple product matrix is indicated with the use of the simple Abbreviated Doolittle solution. Application is also made to the evaluation ofA ^–1 and ofA ^–1 C.     

Conformity of two sociometric relations

The problem of comparing two sociometric relations or measurements (A andB) recorded in distinct sociomatrices was originally discussed by Katz and Powell in the early 1950's and Hubert and Baker in the late 1970's. The problem is considered again using a probabilistic model designed specifically for discrete-valued network measurements. The model allows for the presence of various structural tendencies, such as reciprocity and differential popularity. A parameter that isolates the tendency for actors to choose other actors on both relations simultaneously is introduced, and estimated conditional on the presence of other parameters that reflect additional important network properties. The parameter is presented as a symmetric index but is also generalized to the predictive (A onB orB onA) situation. This approach to the problem is illustrated with the same data used by the earlier solutions, and the unique nature of the two relations in the data set (A = received choices,B = perceived choices), as it affects the modeling, is discussed. Significance tests for the parameter and related parameters are described, as well as an extension to more than two relations.Research support provided by National Science Grant #SES84-08626 to the University of Illinois at Urbana-Champaign. I am grateful to Dawn Iacobucci and Sheila Weaver for assistance with the research reported here, and to Carolyn Anderson, James Green, David Holtgrave, and four anonymous referees for comments on the paper.     

Admissible values of γ in direct oblimin rotation

It is shown that for nonpositive values of the parameter in the oblimin criterion, the criterion achieves a minimum on the manifold of all possible oblique rotations of any given full rank initial loading matrixA. For every positive value of , on the other hand, it is shown that there exists a full rank initial loading matrixA for which the oblimin criterion does not achieve a minimum over the manifold of all oblique rotations ofA. These results help explain the sometimes divergent behavior that results from using direct oblimin algorithms with set to a positive value.This research was supported in part by National Institutes of Health Grant RR-3.     

An alternating least squares method for the weighted approximation of a symmetric matrix

Bailey and Gower examined the least squares approximationC to a symmetric matrixB, when the squared discrepancies for diagonal elements receive specific nonunit weights. They focussed on mathematical properties of the optimalC, in constrained and unconstrained cases, rather than on how to obtainC for any givenB. In the present paper a computational solution is given for the case whereC is constrained to be positive semidefinite and of a fixed rankr or less. The solution is based on weakly constrained linear regression analysis.The authors are obliged to John C. Gower for stimulating this research.     

A short-term longitudinal study of the early development of self-regulation

A short-term longitudinal study was conducted to determine whether self-regulation at 4 years could be predicted from child and maternal measures obtained when the children were age 24 months. The subjects were 69 children and their mothers drawn from the general community. Criterion behaviors assessed at 4 years were those suggestive of attention-deficit hyperactivity disorder. Of the 24-month child measures, maternal ratings of the child's impulsivity and attention span, plus an objective measure of delay ability, were the most effective predictors. Maternal negativity, as assessed in mother-child interaction, predicted the criterion behaviors, even after the child's behavior as an elicitor of maternal behavior was controlled. Maternal child-rearing attitudes on the dimensions of warmth and aggravation were also effective predictors, even after controlling for the child's emotionality as a possible determinant of maternal attitudes. It is argued that the results with the maternal attitude measures provide support for the hypothesis that maternal behavior is a contributor to the development of self-regulation.This research was supported by a State of Ohio Academic Challenge Grant to the Bowling Green State University Clinical Psychology Program, and by a Basic Grant awarded to the first author by the Bowling Green State University Faculty Research Committee. The authors are grateful to Eric Dubow, Douglas Ullman, and two anonymous reviewers for their helpful comments and criticisms on previous versions of this article.     

Naive Probability: Model‐Based Estimates of Unique Events

We describe a dual‐process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non‐numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non‐numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P(A), P(B), P(A and B), disjunctions: P(A), P(B), P(A or B or both), and conditional probabilities P(A), P(B), P(B|A). They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning.     

A Sequent Formulation of Conditional Logic Based on Belief Change Operations

Peter Gärdenfors has developed a semantics for conditional logic, based on the operations of expansion and revision applied to states of information. The account amounts to a formalisation of the Ramsey test for conditionals. A conditional A > B is declared accepted in a state of information K if B is accepted in the state of information which is the result of revising K with respect to A. While Gärdenfors's account takes the truth-functional part of the logic as given, the present paper proposes a semantics entirely based on epistemic states and operations on these states. The semantics is accompanied by a syntactic treatment of conditional logic which is formally similar to Gentzen's sequent formulation of natural deduction rules. Three of David Lewis's systems of conditional logic are represented. The formulations are attractive by virtue of their transparency and simplicity.     

An Infinitary Extension of Jankov’s Theorem

It is known that for any subdirectly irreducible finite Heyting algebra A and any Heyting algebra B, A is embeddable into a quotient algebra of B, if and only if Jankov’s formula χ _A for A is refuted in B. In this paper, we present an infinitary extension of the above theorem given by Jankov. More precisely, for any cardinal number κ, we present Jankov’s theorem for homomorphisms preserving infinite meets and joins, a class of subdirectly irreducible complete κ-Heyting algebras and κ-infinitary logic, where a κ-Heyting algebra is a Heyting algebra A with # ≥  κ and κ-infinitary logic is the infinitary logic such that for any set Θ of formulas with # Θ ≥  κ, ∨Θ and ∧Θ are well defined formulas.     

Matrix correlation

A correlational measure for ann byp matrixX and ann byq matrixY assesses their relation without specifying either as a fixed target. This paper discusses a number of useful measures of correlation, with emphasis on measures which are invariant with respect to rotations or changes in singular values of either matrix. The maximization of matrix correlation with respect to transformationsXL andYM is discussed where one or both transformations are constrained to be orthogonal. Special attention is focussed on transformations which causeXL andYM to ben bys, wheres may be any number between 1 and min (p, q). An efficient algorithm is described for maximizing the correlation betweenXL andYM where analytic solutions do not exist. A factor analytic example is presented illustrating the advantages of various coefficients and of varying the number of columns of the transformed matrices.This research was supported by grant APA 0320 from the Natural Sciences and Engineering Research Council of Canada. The authors wish to acknowledge valuable discussion of this problem with Jan de Leeuw, University of Leiden.     

Demonstrative induction,old and new evidence and the accuracy of the electrostatic inverse square law

Maxwell claimed that the electrostatic inverse square law could be deduced from Cavendish's spherical condenser experiment. This is true only if the accuracy claims made by Cavendish and Maxwell are ignored, for both used the inverse square law as a premise in their analyses of experimental accuracy. By so doing, they assumed the very law the accuracy of which the Cavendish experiment was supposed to test. This paper attempts to make rational sense of this apparently circular procedure and to relate it to some variants of traditional problems concerning old and new evidence.My interest in demonstrative induction developed as the result of being given the opportunity by Joseph Pitt to comment on an earlier version of John Norton's Science and Certainty (1994) at a conference held at the Virginia Polytechnic Institute and State University. I want to thank Joe and John for much stimulating discussion and John specifically for sharing his work on demonstrative induction with me. My gratitude also goes to the National Science Foundation (DIR-8920699) and Ohio State University for providing funding for a larger project of which this paper forms a part.     

Inference,explanation, and asymmetry

Explanation is asymmetric: if A explains B, then B does not explain A. Traditionally, the asymmetry of explanation was thought to favor causal accounts of explanation over their rivals, such as those that take explanations to be inferences. In this paper, we develop a new inferential approach to explanation that outperforms causal approaches in accounting for the asymmetry of explanation.

     

Differences in irrational beliefs of pregnant and never pregnant adolescents

Twenty-five pregnant and/or parenting adolescents were compared with sixteen nonpregnant adolescents on two irrationality measures: The Child and Adolescent Scale of Irrationality and the Adolescent Pregnancy Beliefs Questionnaire. Findings suggest that pregnant adolescents subscribe to a greater number of general irrational beliefs; to a greater number of irrational beliefs specific to sex, dating and birth control; to a greater number of beliefs consistent with promoting pregnancy; and to fewer beliefs that might deter pregnancy when compared to never pregnant adolescents.Karen Westphal holds a Ph.D. in School Psychology from the University of South Carolina. She is an Assistant Professor of Psychology, Department of Psychology, Cleveland State University, Cleveland, Ohio.Susan Wagner worked with Dr. Westphal on this research while she was a graduate student at Cleveland State University. She is presently employed in the Mood Disorders Research Project at University Hospitals, Cleveland, Ohio.     

The larger the cause,the larger the effect: evidence of speed judgment biases in causal scenarios

ABSTRACT

When two motions appear to be causally related, the spatiotemporal features of motions are sometimes distorted in order to increase the consistency with causal impressions. Here, in four experiments, we tested if varying the speed of an object A could affect the judged speed of an object B that appeared to be causally related to A. Participants were presented with classic launching stimuli (Experiment 1), a variant of launching stimuli in which A could move with uniformly accelerated or decelerated motion (Experiment 2), non-launching stimuli that elicited a causal impression (Experiment 3), and stimuli showing a three-object launching event (Experiment 4). Main results showed that the judged speed of B was systematically biased towards the speed of A, and moreover that the judged speed of B depended on the average speed of A, rather than on the speed of A at the moment of collision as it would be predicted by Newtonian mechanics. The results are consistent with the hypothesis that internal representations of causal events based on property transmission (for instance, impetus) can affect judgments of the low-level properties of causal scenarios.     

Answer Sets and Qualitative Decision Making

Logic programs under answer set semantics have become popular as a knowledge representation formalism in Artificial Intelligence. In this paper we investigate the possibility of using answer sets for qualitative decision making. Our approach is based on an extension of the formalism, called logic programs with ordered disjunction (LPODs). These programs contain a new connective called ordered disjunction. The new connective allows us to represent alternative, ranked options for problem solutions in the heads of rules: A × B intuitively means: if possible A, but if A is not possible then at least B. The semantics of logic programs with ordered disjunction is based on a preference relation on answer sets. We show that LPODs can serve as a basis for qualitative decision making.     

Book reviews

J. Roger Saydah. The Ethical Theory of Clarence Irving Lewis. Athens, Ohio: Ohio University Press, 1969. Pp. 171. Indexed. $7.50.

Nicholas Rescher. Many‐Valued Logic. New York: McGraw‐Hill Book Co., 1969. Pp. ix, 359.

EVA Schaper. Prelude to Aesthetics. New York: Humanities Press, 1968. Pp. 179. Indexed. $6.50.

N. Fotion. Moral Situations. Yellow Springs, Ohio: The Antioch Press, 1968. Pp. 124. Indexed. $4.75; paperback $1.05.

John and Shirley Lachs, eds. Physical Order and Moral Liberty. Previously Unpublished Essays by George Santavana. Nashville, Tennessee: Vanderbilt University Press. Pp. xiv, 322. $7.95.

D. C. Armstrong. A Materialist Theory of Mind. New York: Humanities Press, 1968. Pp. xi, 372. Indexed. $8.50.

Annali della Facolta di Lettere et Filosofia, Pubblicazioni dell’ Universita di Barni, Volume XII, Adriatica Editrice, Bari, 1967. Pp. 251.

Bernard Zylstra. From Pluralism to Collectivism: The Development of Harold Laski's Political Thought. New York: Humanities Press, 1968. Pp. 236. $8.00.

Andrew J. Reck. The New American Philosophers: An Exploration of Thought Since “World War II. Baton Rouge, La.: Louisiana State University Press, 1968. Pp. xxii, 362. $8.95.

Elizabeth Ramsden Eames. Bertrand Russell's Theory of Knowledge. New York: George Braziller, 1969. Pp. 240. Indexed. $6.00.