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 )

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!

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

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)

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

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)

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

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.

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)

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)

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)

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

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 – and Data Mining – As well as computer-assisted text-analysis – (T-LAB) – (HAMLET)

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

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)