J. Douglas Carroll

Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition

An individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common “psychological space”. A corresponding method of analyzing similarities data is proposed, involving a generalization of “Eckart-Young analysis” to decomposition of three-way (or higher-way) tables. In the present case this decomposition is applied to a derived three-way table of scalar products between stimuli for individuals. This analysis yields a stimulus by dimensions coordinate matrix and a subjects by dimensions matrix of weights. This method is illustrated with data on auditory stimuli and on perception of nations.

Candelinc: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters

Very general multilinear models, called CANDELINC, and a practical least-squares fitting procedure, also called CANDELINC, are described for data consisting of a many-way array. The models incorporate the possibility of general linear constraints, which turn out to have substantial practical value in some applications, by permitting better prediction and understanding. Description of the model, and proof of a theorem which greatly simplifies the least-squares fitting process, is given first for the case involving two-way data and a bilinear model. Model and proof are then extended to the case ofN-way data and anN-linear model for generalN. The caseN = 3 covers many significant applications. Two applications are described: one of two-way CANDELINC, and the other of CANDELINC used as a constrained version of INDSCAL. Possible additional applications are discussed.

Spatial, non-spatial and hybrid models for scaling

In this paper, hierarchical and non-hierarchical tree structures are proposed as models of similarity data. Trees are viewed as intermediate between multidimensional scaling and simple clustering. Procedures are discussed for fitting both types of trees to data. The concept of multiple tree structures shows great promise for analyzing more complex data. Hybrid models in which multiple trees and other discrete structures are combined with continuous dimensions are discussed. Examples of the use of multiple tree structures and hybrid models are given. Extensions to the analysis of individual differences are suggested.

Synthesized Clustering a Method for Amalgamating Alternative Clustering Bases with Differential Weighting of Variables

In the application of clustering methods to real world data sets, two problems frequently arise: (a) how can the various contributory variables in a specific battery be weighted so as to enhance some cluster structure that may be present, and (b) how can various alternative batteries be combined to produce a single clustering that “best” incorporates each contributory set. A new method is proposed (SYNCLUS, SYNthesizedCLUStering) for dealing with these two problems.

Three-way metric unfolding via alternating weighted least squares

Three-way unfolding was developed by DeSarbo (1978) and reported in DeSarbo and Carroll (1980, 1981) as a new model to accommodate the analysis of two-mode three-way data (e.g., nonsymmetric proximities for stimulus objects collected over time) and three-mode, three-way data (e.g., subjects rendering preference judgments for various stimuli in different usage occasions or situations). This paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis. We begin with an introduction of the problem and a review of three-way unfolding. The three-way unfolding model, weighted objective function, and new algorithm are presented. Monte Carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance. Finally, three applications of the methodology are reported illustrating the flexibility and robustness of the procedure.

The Estimation of Ultrametric and Path Length Trees from Rectangular Proximity Data

A least-squares algorithm for fitting ultrametric and path length or additive trees to two-way, two-mode proximity data is presented. The algorithm utilizes a penalty function to enforce the ultrametric inequality generalized for asymmetric, and generally rectangular (rather than square) proximity matrices in estimating an ultrametric tree. This stage is used in an alternating least-squares fashion with closed-form formulas for estimating path length constants for deriving path length trees. The algorithm is evaluated via two Monte Carlo studies. Examples of fitting ultrametric and path length trees are presented.

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.

A quasi-nonmetric method for multidimensional scaling VIA an extended euclidean model

An Extended Two-Way Euclidean Multidimensional Scaling (MDS) model which assumes both common and specific dimensions is described and contrasted with the “standard” (Two-Way) MDS model. In this Extended Two-Way Euclidean model then stimuli (or other objects) are assumed to be characterized by coordinates onR common dimensions. In addition each stimulus is assumed to have a dimension (or dimensions) specific to it alone. The overall distance between objecti and objectj then is defined as the square root of the ordinary squared Euclidean distance plus terms denoting the specificity of each object. The specificity,sj, can be thought of as the sum of squares of coordinates on those dimensions specific to objecti, all of which have nonzero coordinatesonly for objecti. (In practice, we may think of there being just one such specific dimension for each object, as this situation is mathematically indistinguishable from the case in which there are more than one.)We further assume that δij =F(dij) +eij where δij is the proximity value (e.g., similarity or dissimilarity) of objectsi andj,dij is the extended Euclidean distance defined above, whileeij is an error term assumed i.i.d.N(0, σ2).F is assumed either a linear function (in the metric case) or a monotone spline of specified form (in the quasi-nonmetric case). A numerical procedure alternating a modified Newton-Raphson algorithm with an algorithm for fitting an optimal monotone spline (or linear function) is used to secure maximum likelihood estimates of the paramstatistics) can be used to test hypotheses about the number of common dimensions, and/or the existence of specific (in addition toR common) dimensions.This approach is illustrated with applications to both artificial data and real data on judged similarity of nations.

Preference Mapping of Conjoint-Based Profiles: An INDSCAL Approach

This article describes an approach in which conjoint methodology and multidimensional scaling can be fruitfully applied, in tandem, to the measurement and representation of preference data for factorially designed profiles. The approach is described and applied to an illustrative data set. We conclude the article with a brief discussion of possible extensions of the approach and additional research needs.

Some Multidimensional Scaling and Related Procedures Devised at Bell Laboratories, With Ecological Applications

A large number of multidimensional scaling (MDS) and related models, methods, and computer programs (for all of which we use the generic term “MDS procedures”) have been developed over the years at Bell Laboratories. This paper focuses on probably the most widely known and used subset of Bell Labs MDS procedures involving spatial (as opposed to tree structure, overlapping or non-overlapping clustering, or other “discrete and hybrid”) models. These are: the MDSCAL and KYST family, for two-way (metric or nonmetric) MDS of proximities (e.g., similarities or dissimilarities); INDSCAL, SINDSCAL and IDIOSCAL, for three-way MDS, primarily of proximities (but also applicable to more general multiway data, in a manner to be described); MDPREF, for “internal analysis” of preference (or other “dominance”) data for different individual “subjects” (or other data sources) in terms of a vector model; and the PREFMAP family for “external analysis” of such data (where the “stimulus” or other “object” dimensions are externally provided by prior analysis or theory, only “subject” vectors, ideal points and/or other parameters being determined from preference/dominance data). A number of these Bell Labs MDS procedures areapplied to some ecological data on seaworm species due to E. Fresi and collaborators.