Trichotomous preferences for gambles

Let G_X be the set of all gambles on a set X of decision outcomes. Trichotomous preferences divide G_X into three disjoint subsets, viz., the set P⁺ of preferable gambles, the set P⁰ of indifferent gambles, and the set P^- of unpreferable gambles. Those preference comparisons are made against the status quo, i.e., the do-nothing alternative. This paper presents and discusses dyadic representations of such trichotomous preferences and identifies necessary and sufficient axioms for the existence of those representations.     

Additive Utilities on Densely Ordered Sets

This paper shows sufficient conditions for the existence of additive utilities without a restricted solvability axiom. Our conditions require that each essential component of the underlying Cartesian product be densely ordered.     

Influence of arm positions on EMG-reaction time of the biceps brachii for elbow flexion and forearm supination

The influence of starting positions of the arm on EMG-RTs of the biceps brachii muscle for elbow flexion and forearm supination was examined using 16 normal subjects. Two angles of the elbow joint, 45 degrees and 110 degrees flexion, and two positions of the forearm, 45 degrees supination and 90 degrees pronation, were used as the factorial combinations of all four. The EMG-RT for elbow flexion decreased in the order of 110 degrees Pronation greater than 45 degrees Pronation = 110 degrees Supination greater than 45 degrees Supination, and that for forearm supination decreased in the order of 45 degrees Supination greater than 45 degrees Pronation = 110 degrees Supination greater than 110 degrees Pronation. These results were kinesiologically interpreted that variations of EMG-RTs were based on the change in the number of synergic muscles participating in an intended movement and the muscle length of the prime mover at the start of the movement.     

Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities

A maximum likelihood estimation procedure is developed for multidimensional scaling when (dis)similarity measures are taken by ranking procedures such as the method of conditional rank orders or the method of triadic combinations. The central feature of these procedures may be termed directionality of ranking processes. That is, rank orderings are performed in a prescribed order by successive first choices. Those data have conventionally been analyzed by Shepard-Kruskal type of nonmetric multidimensional scaling procedures. We propose, as a more appropriate alternative, a maximum likelihood method specifically designed for this type of data. A broader perspective on the present approach is given, which encompasses a wide variety of experimental methods for collecting dissimilarity data including pair comparison methods (such as the method of tetrads) and the pick-M method of similarities. An example is given to illustrate various advantages of nonmetric maximum likelihood multidimensional scaling as a statistical method. At the moment the approach is limited to the case of one-mode two-way proximity data, but could be extended in a relatively straightforward way to two-mode two-way, two-mode three-way or even three-mode three-way data, under the assumption of such models as INDSCAL or the two or three-way unfolding models.The first author's work was supported partly by the Natural Sciences and Engineering Research Council of Canada, grant number A6394. Portions of this research were done while the first author was at Bell Laboratories. MAXSCAL-4.1, a program to perform the computations described in this paper can be obtained by writing to: Computing Information Service, Attention: Ms. Carole Scheiderman, Bell Laboratories, 600 Mountain Ave., Murray Hill, N.J. 07974. Thanks are due to Yukio Inukai, who generously let us use his stimuli in our experiment, and to Jim Ramsay for his helpful comments on an earlier draft of this paper. Confidence regions in Figures 2 and 3 were drawn by the program written by Jim Ramsay. We are also indebted to anonymous reviewers for their suggestions.     

Principal component analysis with external information on both subjects and variables

A method for structural analysis of multivariate data is proposed that combines features of regression analysis and principal component analysis. In this method, the original data are first decomposed into several components according to external information. The components are then subjected to principal component analysis to explore structures within the components. It is shown that this requires the generalized singular value decomposition of a matrix with certain metric matrices. The numerical method based on the QR decomposition is described, which simplifies the computation considerably. The proposed method includes a number of interesting special cases, whose relations to existing methods are discussed. Examples are given to demonstrate practical uses of the method.The work reported in this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first author. Thanks are due to Jim Ramsay, Haruo Yanai, Henk Kiers, and Shizuhiko Nishisato for their insightful comments on earlier versions of this paper. Jim Ramsay, in particular, suggested the use of the QR decomposition, which simplified the presentation of the paper considerably.     

Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features

A new procedure is discussed which fits either the weighted or simple Euclidian model to data that may (a) be defined at either the nominal, ordinal, interval or ratio levels of measurement; (b) have missing observations; (c) be symmetric or asymmetric; (d) be conditional or unconditional; (e) be replicated or unreplicated; and (f) be continuous or discrete. Various special cases of the procedure include the most commonly used individual differences multidimensional scaling models, the familiar nonmetric multidimensional scaling model, and several other previously undiscussed variants.The procedure optimizes the fit of the model directly to the data (not to scalar products determined from the data) by an alternating least squares procedure which is convergent, very quick, and relatively free from local minimum problems.The procedure is evaluated via both Monte Carlo and empirical data. It is found to be robust in the face of measurement error, capable of recovering the true underlying configuration in the Monte Carlo situation, and capable of obtaining structures equivalent to those obtained by other less general procedures in the empirical situation.This project was supported in part by Research Grant No. MH10006 and Research Grant No. MH26504, awarded by the National Institute of Mental Health, DHEW. We wish to thank Robert F. Baker, J. Douglas Carroll, Joseph Kruskal, and Amnon Rapoport for comments on an earlier draft of this paper. Portions of the research reported here were presented to the spring meeting of the Psychometric Society, 1975. ALSCAL, a program to perform the computations discussed in this paper, may be obtained from any of the authors.Jan de Leeuw is currently at Datatheorie, Central Rekeninstituut, Wassenaarseweg 80, Leiden, The Netherlands. Yoshio Takane can be reached at the Department of Psychology, University of Tokyo, Tokyo, Japan.     

Sequence of preparatory set for response movement

Electromyographic reaction times (EMG-RTs) of the right biceps brachii muscle were examined for two movement patterns, elbow flexion and forearm supination, in 8 healthy male subjects under simple and complex RT conditions with varied preparatory intervals (PIs): 0, 200, 400, 600, and 800 msec. In the simple RT condition, the subject was informed of the movement patterns to be performed prior to beginning the trials. In the complex RT condition the subject had to choose one of the two movement patterns at the time of the presentation of a warning signal. The results indicated that: (1) compared with the simple RT condition a delay of about 100 msec. in over-all mean EMG-RT was observed at PI = 0 msec. in the complex RT condition; (2) the difference of over-all mean EMG-RT between the two RT conditions disappeared when PI = 400 msec.; and (3) the difference in EMG-RTs between flexion and supination in the complex RT condition became the same as that in the simple RT condition when PI = 700 msec. It is assumed that the preparatory set for response movements is organized in an order, resulting in the differentiation of RT.     

Ideal point discriminant analysis

A new method of multiple discriminant analysis was developed that allows a mixture of continuous and discrete predictors. The method can be justified under a wide class of distributional assumptions on the predictor variables. The method can also handle three different sampling situations, conditional, joint and separate. In this method both subjects (cases or any other sampling units) and criterion groups are represented as points in a multidimensional euclidean space. The probability of a particular subject belonging to a particular criterion group is stated as a decreasing function of the distance between the corresponding points. A maximum likelihood estimation procedure was developed and implemented in the form of a FORTRAN program. Detailed analyses of two real data sets were reported to demonstrate various advantages of the proposed method. These advantages mostly derive from model evaluation capabilities based on the Akaike Information Criterion (AIC).The work reported in this paper has been supported by Grant A6394 from the Natural Sciences and Engineering Research Council of Canada and by a leave grant from the Social Sciences and Humanities Research Council of Canada to the first author. Portions of this study were conducted while the first author was at the Institute of Statistical Mathematics in Tokyo on leave from McGill University. He would like to express his gratitude to members of the Institute for their hospitality. Thanks are also due to T. Komazawa at the Institute for letting us use his data, to W. J. Krzanowski at the University of Reading for providing us with Armitage, McPherson, and Copas' data, and to Don Ramirez, Jim Ramsay and Stan Sclove for their helpful comments on an earlier draft of this paper.     

Analysis of contingency tables by ideal point discriminant analysis

Cross-classified data are frequently encountered in behavioral and social science research. The loglinear model and dual scaling (correspondence analysis) are two representative methods of analyzing such data. An alternative method, based on ideal point discriminant analysis (DA), is proposed for analysis of contingency tables, which in a certain sense encompasses the two existing methods. A variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and/or as direct constraints on model parameters. This, along with maximum likelihood estimation of the model parameters, allows interesting model comparisons. This is illustrated by the analysis of several data sets.Presented as the Presidential Address to the Psychometric Society's Annual and European Meetings, June, 1987. Preparation of this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada. Thanks are due to Chikio Hayashi of University of the Air in Japan for providing the ISM data, and to Jim Ramsay and Ivo Molenaar for their helpful comments on an earlier draft of this paper.