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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 Q-mode 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 add-on module in SPSS Categories.

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```MULTIDIMENSIONAL SCALING
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 near-metricity 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
Multiple-matrix 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
Example	23
S-Stress 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