
MULTIDIMENSIONAL SCALING
Overview
Multidimensional scaling (MDS) uncovers underlying dimensions based on a series of similarity or distance judgments by subjects. MDS is popular in marketing research for brand comparisons and in psychology, where it has been used to study the dimensionality of personality traits. Other uses include analysis of particular academic disciplines using citation data (Small, 1999) and any application involving ratings, rankings, differences in perceptions, or voting. In spite of being designed for judgment data, MDS can be used to analyze any correlation matrix, treating correlation as a type of similarity measure. That is, the higher the correlation of two variables, the closer they will be located in the map created by MDS.
MDS may be thought of as a way of representing subjective attributes in objective scales. A type of perceptual mapping, the central MDS output takes the form of a set of scatterplots ("perceptual maps") in which the axes are the underlying dimensions and the points are the products, candidates, opinions, or other objects of comparison. The objective of MDS is to array points in multidimensional space such that the distances separating points physically on the scatterplot(s) reflect as closely as possible the subjective distances obtained by surveying subjects. That is, MDS shows graphically how different objects of comparison do or do not cluster. Goodness of fit of an MDS model is shown by the stress statistic, phi.
MDS is mainly used to compare objects when the bases (dimensions) of comparison are not known and may differ from objective dimensions (ex., color, size, shape, weight, etc.) which are observable beforehand by the researcher. Though it is possible to use MDS with quantitative variables, it is more common to use factor analysis to group variables whose dimensions are objective and measurable, or to use Qmode factor analysis. This is because factor analysis uses partial coefficients, controlling similarity for other variables in the model, whereas MDS uses whole coefficients to assess similarity and does not take account of control relationships as factor analysis does. Nonetheless, because MDS does not require assumptions of linearity, metricity, or multivariate normality, sometimes it is preferred over factor analysis for these reasons even for objective data. Pros and cons of MDS vs. factor analysis are discussed below.
SPSS offers two MDS models, ALSCAL and PROXSCAL. For reasons discussed below, PROXSCAL is generally preferred. ALSCAL is provided in SPSS Base but PROXSCAL is an addon module in SPSS Categories.
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Below is the unformatted table of contents.
MULTIDIMENSIONAL SCALING Table of Contents Multidimensional Scaling 6 Overview 6 Key Terms and Concepts 7 Objects and subjects 7 Objects 7 Subjects 7 Data collection methods 7 Compositional and decompositional approaches 8 Decompositional MDS 8 Compositional MDS 9 Distance 9 Similarity vs. dissimilarity matrices 9 Default distance matrices 9 Creating distance matrices from metric variables 10 Example 12 Subject, object, and objective matrices 13 Subject matrices 14 Object matrices 14 Objective matrices 15 Matrix shape in SPSS 16 Square symmetric 16 Square asymmetric 16 Rectangular 16 SPSS matrix conditionality 17 Matrix 17 Row 17 Unconditional 17 Level of measurement 17 MDS as a test of nearmetricity of ordinal data 18 Dimensions 18 Optimal number of dimensions 18 Rotation of axes 19 Labeling of dimensions 19 Models in SPSS ALSCAL 20 Models 20 Classical MDS (CMDS) 20 Classical MDS (CMDS) is also known as Principal Coordinate Analysis or metric CMDS. In SPSS press the Model button in the MDS dialog, then in the Model dialog select"Euclidean distance" in the Scaling Model area. If data are a single matrix, CMDS is performed. 20 Nonmetric CMDS 20 Replicated MDS (RMDS) 20 Multiplematrix principal coordinates analysis 21 Individual differences Euclidean distance (INDSCAL) 21 Asymmetric Euclidean distance model (ASCAL) 22 Asymmetric individual differences Euclidean distance model (AINDS) 22 Generalized Euclidean metric individual differences model (GEMSCAL) 22 ALSCAL Output Options in SPSS 22 SPSS menu 22 Example 23 SStress and Interation History 24 Scree plots 25 Local minima 25 Interpretability 26 Goodness of fit measures 26 Stimulus coordinates and MDS plots 27 Fit plots 29 Other output options 32 PROXSCAL Input and Output Options in SPSS 34 SPSS 34 Scaling models 37 Example 42 Iteration history 43 Stress and Fit Measures 44 MDS coordinates 46 MDS maps 47 Assumptions 49 Proper specification of the model 49 Proper level of measurement 49 Objects and dimensions 49 Similar scales 49 Comparability 49 History 50 Sample size 50 Missing values 50 Few ties 50 Data distribution 50 SPSS limits 50 Frequently Asked Questions 51 What other procedures are related to MDS? 51 How does MDS work? 52 If one has multiple data matrices, why do RMDS or INDSCAL? Why not just do a series of CMDS models, one on each matrix? 52 What computer programs handle MDS? 53 What is Torgerson Scaling? 53 How does MDS relate to "smallest space analysis"? 53 Bibliography 54 Pagecount: 55