Maximum likelihood additivity analysis

A maximum likelihood estimation procedure was developed to fit unweighted and weighted additive models to conjoint data obtained by the categorical rating, the pair comparison or the directional ranking method. The scoring algorithm used to fit the models was found to be both reliable and efficient, and the program MAXADD is capable of handling up to 300 parameters to be estimated. Practical uses of the procedure are reported to demonstrate various advantages of the procedure as a statistical method.The research reported here was supported by Grant A6394 to the author from the Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Iowa City, Iowa, in May, 1980.Thanks are due to Jim Ramsay, Justine Sergent and anonymous reviewers for their helpful comments.Two MAXADD programs which perform the computations discussed in this paper may be obtained from the author.     

Rule following and rule use in the balance-scale task

Quinlan et al. [Quinlan, p., van der Mass, H., Jansen, B., Booij, O., & Rendell, M. (this issue). Re-thinking stages of cognitive development: An appraisal of connectionist models of the balance scale task. Cognition, doi:10.1016/j.cognition.2006.02.004] use Latent Class Analysis (LCA) to criticize a connectionist model of development on the balance-scale task, arguing that LCA shows that this model fails to capture a torque rule and exhibits rules that children do not. In this rejoinder we focus on the latter problem, noting the tendency of LCA to find small, unreliable, and difficult-to-interpret classes. This tendency is documented in network and synthetic simulations and in psychological research, and statistical reasons for finding such unreliable classes are discussed. We recommend that LCA should be used with care, and argue that its small and unreliable classes should be discounted. Further, we note that a preoccupation with diagnosing rules ignores important phenomena that rules do not account for. Finally, we conjecture that simple extensions of the network model should be able to achieve torque-rule performance.     

Regularized Partial and/or Constrained Redundancy Analysis

Methods of incorporating a ridge type of regularization into partial redundancy analysis (PRA), constrained redundancy analysis (CRA), and partial and constrained redundancy analysis (PCRA) were discussed. The usefulness of ridge estimation in reducing mean square error (MSE) has been recognized in multiple regression analysis for some time, especially when predictor variables are nearly collinear, and the ordinary least squares estimator is poorly determined. The ridge estimation method was extended to PRA, CRA, and PCRA, where the reduced rank ridge estimates of regression coefficients were obtained by minimizing the ridge least squares criterion. It was shown that in all cases they could be obtained in closed form for a fixed value of ridge parameter. An optimal value of the ridge parameter is found by G-fold cross validation. Illustrative examples were given to demonstrate the usefulness of the method in practical data analysis situations. We thank Jim Ramsay for his insightful comments on an earlier draft of this paper. The work reported in this paper is supported by Grants 10630 from the Natural Sciences and Engineering Research Council of Canada to the first author.     

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.     

Three notes on ALSCAL

It is reported that (1) a new coordinate estimation routine is superior to that originally proposed for ALSCAL; (2) an oversight in the interval measurement level case has been found and corrected; and (3) a new initial configuration routine is superior to the original.     

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.