Forrest W. Young

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.

An analysis and synthesis of multiple correspondence analysis, optimal scaling, dual scaling, homogeneity analysis and other methods for quantifying categorical multivariate data

We discuss a variety of methods for quantifying categorical multivariate data. These methods have been proposed in many different countries, by many different authors, under many different names. In the first major section of the paper we analyze the many different methods and show that they all lead to the same equations for analyzing the same data. In the second major section of the paper we introduce the notion of a duality diagram, and use this diagram to synthesize the many superficially different methods into a single method.

Multidimensional Scaling : History, Theory, and Applications

Contents: F.W. Young, Part I:History. Introduction. History. F.W. Young, Part II:Theory. Data Theory. Axiomatic Foundations. Unweighted Distance Models. Weighted Distance Models. Part III:Applications. J.A. Cameron, Nuclear Physics: Pattern Recognition in Nuclear Gamma-ray Spectra. L.M. Collins, Sociometry: Deriving Sociograms via Asymmetric Multidimensional Scaling. D.L. Hoffman, W.D. Perreault, Jr., Market Research: Consumer Preference and Perception. D.V. Easterling, Political Science: Using the Generalized Euclidian Model to Study Ideological Shifts in the U.S. Senate. M.R. Jones, R. MacCallum, Psychology: An Application of Principal Directions Scaling to Auditory Pattern Perception. B.H. Forsyth, Psychology: The Subjective Attributes of Natural Categories -- An Application of a Constrained Generalized Euclidian Model.

Perceptual dimensions of tactile surface texture: A multidimensional scaling analysis

The purpose of this study was to examine the subjective dimensionality of tactile surface texture perception. Seventeen tactile stimuli, such as wood, sandpaper, and velvet, were moved across the index finger of the subject, who sorted them into categories on the basis of perceived similarity. Multidimensional scaling (MDS) techniques were then used to position the stimuli in a perceptual space on the basis of combined data of 20 subjects. A three-dimensional space was judged to give a satisfactory representation of the data. Subjects’ ratings of each stimulus on five scales representing putative dimensions of perceived surface texture were then fitted by regression analysis into the MDS space. Roughness-smoothness and hardness-softness were found to be robust and orthogonal dimensions; the third dimension did not correspond closely with any of the rating scales used, but post hoc inspection of the data suggested that it may reflect the compressional elasticity (“springiness”) of the surface.

Quantitative analysis of qualitative data

This paper presents an overview of an approach to the quantitative analysis of qualitative data with theoretical and methodological explanations of the two cornerstones of the approach, Alternating Least Squares and Optimal Scaling. Using these two principles, my colleagues and I have extended a variety of analysis procedures originally proposed for quantitative (interval or ratio) data to qualitative (nominal or ordinal) data, including additivity analysis and analysis of variance; multiple and canonical regression; principal components; common factor and three mode factor analysis; and multidimensional scaling. The approach has two advantages: (a) If a least squares procedure is known for analyzing quantitative data, it can be extended to qualitative data; and (b) the resulting algorithm will be convergent. Three completely worked through examples of the additivity analysis procedure and the steps involved in the regression procedures are presented.

Individual differences in perceptual space for tactile textures: Evidence from multidimensional scaling

Ratio scaling was used to obtain from 5 subjects estimates of the subjective dissimilarity between the members of all possible pairs of 17 tactile surfaces. The stimuli were a diverse array of everyday surfaces, such as corduroy, sandpaper, and synthetic fur. The results were analyzed using the multidimensional scaling (MDS) program ALSCAL. There was substantial, but not complete, agreement across subjects in the spatial arrangement of perceived textures. Scree plots and multivariate analysis suggested that, for some subjects, a two-dimensional space was the optimal MDS solution, whereas for other subjects, a three-dimensional space was indicated. Subsequent to their dissimilarity scaling, subjects rated each stimulus on each of five adjective scales. Consistent with earlier research, two of these (rough/smooth andsoft/hard) were robustly related to the space for all subjects. A third scale,sticky/slippery, was more variably related to the dissimilarity data: regressed into three-dimensional MDS space, it was angled steeply into the third dimension only for subjects whose scree plots favored a nonplanar solution. We conclude that thesticky/slippery dimension is perceptually weighted less than therough/smooth andsoft/hard dimensions, materially contributing to the structure of perceptual space only in some individuals.

Regression with qualitative and quantitative variables: An alternating least squares method with optimal scaling features

A method is discussed which extends canonical regression analysis to the situation where the variables may be measured at a variety of levels (nominal, ordinal, or interval), and where they may be either continuous or discrete. There is no restriction on the mix of measurement characteristics (i.e., some variables may be discrete-ordinal, others continuous-nominal, and yet others discrete-interval). The method, which is purely descriptive, scales the observations on each variable, within the restriction imposed by the variables measurement characteristics, so that the canonical correlation is maximal. The alternating least squares algorithm is discussed. Several examples are presented. It is concluded that the method is very robust. Inferential aspects of the method are not discussed.

Additive structure in qualitative data: An alternating least squares method with optimal scaling features

A method is developed to investigate the additive structure of data that (a) may be measured at the nominal, ordinal or cardinal levels, (b) may be obtained from either a discrete or continuous source, (c) may have known degrees of imprecision, or (d) may be obtained in unbalanced designs. The method also permits experimental variables to be measured at the ordinal level. It is shown that the method is convergent, and includes several previously proposed methods as special cases. Both Monte Carlo and empirical evaluations indicate that the method is robust.

The principal components of mixed measurement level multivariate data: An alternating least squares method with optimal scaling features

A method is discussed which extends principal components analysis to the situation where the variables may be measured at a variety of scale levels (nominal, ordinal or interval), and where they may be either continuous or discrete. There are no restrictions on the mix of measurement characteristics and there may be any pattern of missing observations. The method scales the observations on each variable within the restrictions imposed by the variables measurement characteristics, so that the deviation from the principal components model for a specified number of components is minimized in the least squares sense. An alternating least squares algorithm is discussed. An illustrative example is given.

Nonmetric multidimensional scaling: Recovery of metric information

The degree of metric determinancy afforded by nonmetric multidimensional scaling was investigated as a function of the number of points being scaled, the true dimensionality of the data being scaled, and the amount of error contained in the data being scaled. It was found 1) that if the ratio of the degrees of freedom of the data to that of the coordinates is sufficiently large then metric information is recovered even when random error is present; and 2) when the number of points being scaled increases the stress of the solution increases even though the degree of metric determinacy increases.