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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 1 What is Multidimensional 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: 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 2 What is Multidimensional 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 3 WHAT IS MULTIDIMENSIONAL SCALING? MDS (aka Smallest Space Analysis) –Has origins in Psychometrics in s: 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 4 WHAT IS MULTIDIMENSIONAL SCALING? – Given a map, its 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 5 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 6 WHAT IS MULTIDIMENSIONAL SCALING? 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 7 WHAT IS MULTIDIMENSIONAL SCALING? – 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 8 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 9 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 10 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 11 SITES & SOFTWARE: SITES 1.NEWMDSX AND DOCUMENTATION: 2.INTERACTIVE PERMAP (Heady) 3.THREE-WAY SCALING (Kroonenberg) FORREST YOUNGS VISTA (Visual Statistics)

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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 12 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: : Molinero: Predicting Bank Failure Kendall: Maps from Marriages Coxon: Subjective Categorization of Drugs

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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 13 WHAT IS MULTIDIMENSIONAL SCALING? APPENDIX Multidimensional Scaling … Is often subsumed with Cluster analysis & Seriation/sequencing as Combinatorial Data Analysis –(Arabie: Forms basis of Graphical Data Visualization –http://lib.stat.cmu.edu/general/XGobi/http://lib.stat.cmu.edu/general/XGobi/ and Data Mining –http://www.the-data-mine.com/http://www.the-data-mine.com/ As well as computer-assisted text-analysis –http://www.tlab.it/en/ (T-LAB)http://www.tlab.it/en/ –http://www.apb.cwc.net/ (HAMLET)http://www.apb.cwc.net/

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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 14 WHAT IS MULTIDIMENSIONAL SCALING? APPENDIX WHAT IS MULTIDIMENSIONAL SCALING? APPENDIX : 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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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 15 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)

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