Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 1 What is Multidimensional Scaling [MDS] ? Tony Macmillan Coxon –Honorary Professorial

Research Methods Festival 2006

What is MDS?Prof APM Coxon, U Edinburgh

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What is What is Multidimensional Multidimensional Scaling [MDS] ?Scaling [MDS] ? • Tony Macmillan Coxon

– Honorary Professorial Research Fellow, University of Edinburgh

– Emeritus Professor of Sociological Research Methods, University of Wales

– Co-founder &Co- Director of MDS software packages, • MDSX [OS] (freeware)and • NewMDSX for Windows (not-for-profit)• Website: www.newmdsx.com

• see my entry on multidimensional scaling in Lewis-Beck, M.S. et al, eds (2004) The Sage Encyclopaedia of Social Science Research Methods. London Sage Publications )



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What is What is Multidimensional Multidimensional Scaling [MDS] ?Scaling [MDS] ?

• A quick answer …– the most flexible, useful,

user-friendly – but grossly under-used,

under-taught – family of Research Methods

in Social Science!



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WHAT IS MULTIDIMENSIONAL SCALING?WHAT IS MULTIDIMENSIONAL SCALING?

• MDS (aka “Smallest Space Analysis”)– Has origins in Psychometrics in 1920-’60s:

• Analysis of similarity data as distances• Scale construction and dimensionality

reduction• Underwent major burst of development in

1960s due to “non-metric revolution”(Coombs) and computing developments



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WHAT IS MULTIDIMENSIONAL SCALING?

– Given a map, it’s easy to calculate the distances between the points……• MDS operates the other way round:

– Given the “distances” [data] find the [configuration] which generated them

• ….and can do so when all but the ordinal information has been jettisoned

– (the “non-metric revolution”)

• Even when there are missing data and in the presence of considerable “noise”/error

– (MDS is robust)



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WHAT IS MULTIDIMENSIONAL SCALING?WHAT IS MULTIDIMENSIONAL SCALING?

• MDS is a family of models differentiated by …

– (DATA) the empirical inter-relationships between a set of “objects”/variables which are given in a set of dis/similarity data

– (FUNCTION) are then re-scaled (according to permissible transformations for the data level of measurement), and in terms of

– (MODEL) the assumptions of the model chosen to represent the data



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• To produce a SOLUTION consisting of:– a CONFIGURATION, which is a

• pattern of points representing the “objects”

• located in a space of a small number of dimensions (hence “SSA”)

• where the distances (or angular separation) between the points represents the dis/similarities between the data-points

• as perfectly as possible (the imperfection/badness of fit is measured by Stress – “No stress is perfection”)



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– MDS thus provides….•a useful and easily-assimilable graphic visualisation

– Tukey: “A picture is worth a thousand words”

• and/or a graphical representation of the structure underlying a complex data set



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VARIANTS OF MDS MODELS

MDS can be used with a wide variety of DATA:

– direct data (pair comparisons, ratings, rankings)

– derived data (profiles, co-occurrence matrices, textual data, aggregated data)

– measures of association etc derived from simpler data, and

– tables of data.



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VARIANTS OF MDS MODELS

MDS can also be used with a wide variety of: Transformations (“levels of measurement”)

• monotonic (ordinal), • linear/metric (interval),

but also– Splines (SPSS PROXSCAL)– log-interval (MRSCAL), – Power (MULTISCALE)– “smoothness” (local preservation of distance)

Models– chiefly distance (Euclidean, but also City-block),– Factor/vector, Composition (additive)



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MDS PROGRAMS: • Usually either “General Purpose” (e.g. SPSS

PROXSCAL)• or specific to Data-shape, Trans & Model (e.g.

NewMDSX)– BASIC 2W1M SCALING:

• Non-metric (ordinal), Metric (linear, log-int)• Principal Components• Clustering (Hierarchical & Non-hierarchical)

– 2W2M (“Rectangular”) SCALING:• Multidimensional … Preference, Sorting, Unfolding• Property-fitting, Profile Analysis (continuity)• Triadic MD Scaling

– 3W2M (and higher)• Individual Differences (INDSCAL), (Tucker) Points-of-View • Procrustean IndDiffs (Lingoes’ PINDIS)• Canonical Decomposition ( 3-7 Mode)

– Utilities:• Comparison of configurations (by Procrustes rotation) • WOMBATS (Measures & Data-manipulation)



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SITES & SOFTWARE:

SITES

1. NEWMDSX AND DOCUMENTATION:

http://www.newmdsx.com

2. INTERACTIVE PERMAP (Heady)

http://www.ucs.louisiana.edu/~rbh8900/

3. THREE-WAY SCALING (Kroonenberg)

http://three-mode.leidenuniv.nl/encyclopedia/

encyclopedia.htm

FORREST YOUNG’S VISTA (Visual

Statistics)

http://forrest.psych.unc.edu/research/index.html



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WHAT IS MDS? … and now for an example

Perception of road accident causes by Ward Vanlaar (Traffic Injury Research Foundation, Ottawa) & APMC

An application of INDSCAL &c

Other examples available at: www.tonycoxon.com :

• Molinero: Predicting Bank Failure• Kendall: Maps from Marriages• Coxon: Subjective Categorization of Drugs



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WHAT IS MULTIDIMENSIONAL SCALING? WHAT IS MULTIDIMENSIONAL SCALING? APPENDIXAPPENDIX

• Multidimensional Scaling … • Is often subsumed with Cluster analysis &

Seriation/sequencing as Combinatorial Data Analysis

– (Arabie: http://www.ec-securehost.com/SIAM/DT06.html)

• Forms basis of Graphical Data Visualization

– http://lib.stat.cmu.edu/general/XGobi/

• and Data Mining– http://www.the-data-mine.com/

• As well as computer-assisted text-analysis– http://www.tlab.it/en/ (T-LAB)– http://www.apb.cwc.net/ (HAMLET)



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WHAT IS MULTIDIMENSIONAL SCALING?WHAT IS MULTIDIMENSIONAL SCALING?APPENDIXAPPENDIX : Related and Special-case

Models: – Metric Scalar Products SVDModels

*PRINCIPAL COMPONENTS ANALYSIS

FACTOR ANALYSIS (+ communalities)

– Discrete Clustering

*HIERARCHICAL CLUSTERING

*PARTITION & ADDITIVE CLUSTERING

– Tables

*SIMPLE (2W2M) AND MULTIPLE (3W) CORRESPONDENCE ANALYSIS

– Other *M/ANOVA & *SIMPLE COMPOSITION



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SOME POSSIBLE WEAKNESSES in MDS

There ARE any??!• Relative ignorance of the sampling/inferential

properties of stress• But, simulation (Spence), MLE estimation

• Prone-ness to local minima solutions• but less so, and multiple starts & interactive programs like

PERMAP allow thousands of runs to check

• A few forms of data/models are prone to degeneracies– especially MD Unfolding, but see new PREFSCAL in SPSS14)

• difficulty in representing the asymmetry of causal models – though external analysis is very akin to dependent-

independent modelling, – there are convergences with GLM in hybrid models such as

CLASCAL (INDSCAL with parameterization of latent classes)

Documents

Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 1 What is Multidimensional Scaling [MDS] ? Tony Macmillan Coxon –Honorary Professorial