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Mod: 11/ 2009 What is MDS? Prof APM Coxon, U Cardiff 1 What is Multidimensional Scaling [MDS] ? Anthony P.M. Coxon –Emeritus Professor of Sociological Research Methods, University of Wales –Honorary Professor, Cardiff University –Honorary Professorial Research Fellow, University of Edinburgh –Co-founder & Co- Director of MDS software packages, MDSX [OS] (freeware)and NewMDSX for Windows (not-for-profit) Website: Course materials: 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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U Winchester 12/09 What is MDS? Prof APM Coxon, Cardiff U 2 ORIGINS / DEVELOPMENT OF MDS MDS (aka Smallest Space Analysis) –Has origins in Psychometrics in s: Scale construction and dimensionality reduction Underwent major burst of development in 1960s due to non-metric revolution(Coombs) and computing developments allowing iterative estimation –Originally designed for analysis of LTM of dis/similarities data, taking a range of measures (not just PM correlations): anything which, by an act of faith, can be considered a similarity (Shepard) –Extended rapidly to deal with wide range of other types of data Rectangular matrices ; triads, pair-comparisons, free-sorting stacks of matrices (3-way scaling – INDSCAL)

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3 CONSTRUCTING A MAP … –Given a map, its easy to calculate the distances between the points … –MDS operates the other way round: Given the data [ interpreted as quasi distances ] it attempts to find the configuration [location of points] which generated the distances »This is Classic MDS: developed in 1930s – but imperfect, not robust, & works only if data are ratio. Whereas more recent MDS can work when only the ordinal information exists: Non-metric = ordinal MDS (Coombs / Kruskal non-metric revolution ) What?? You can create an accurate map from only the rank –order of the distances??? Yes! And it works!!

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The RANK of distances can recover the Map… though not the coastline NEWMDSX (RUNSCRIPT + SYNTAX) RUN NAME Rank of Scottish distances, COMMENT 1 = smallest; 120 = max; dissimilarity data F3.3, p48 The Users Guide to MDS N OF STIMULI 16 PARAMETERS DATA TYPE(1) LABELS BERWICK EDINBURGH GLASGOW STRANRAER AYR PERTH DUNDEE ABERDEEN STIRLING OBAN FORT_WM INVERNESS KYLE_LOCHALSH BRAEMAR ULLAPOOL THURSO READ MATRIX = Perth-Dundee COMPUTE = Stranraer - Thurso U. Winchester 12/09 What is MDS? Prof APM Coxon, Cardiff U 4

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Uni Winchester 12/09 What is MDS? Prof APM Coxon, Cardiff Uni 5 WHAT IS MULTIDIMENSIONAL SCALING? A students definition: –If you are interested in how certain objects relate to each other … and if you would like to present these relationships in the form of a map then MDS is the technique you need (Mr Gawels, KUB) A good start! – MDS provides … a useful and easily-assimilable graphic visualisation of all sorts of data –Tukey: A picture is worth a thousand words In a user-chosen (small) # of dimensions providing a graphical representation of the structure underlying a complex data set And measure how well / badly the solution distances match the data dissimilarities (Stress)

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U Winchester, 12/2009 What is MDS? Prof APM Coxon, Cardiff U 6 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 »Basically, type of input data, defined by their Way and Mode [e.g. 2W1M]. (Cf observations vs data) –(FUNCTION) data are then optimally re- scaled (according to permissible trans- formations for the data) in terms of … »Choice of level of measurement [e.g. ordinal ] –(MODEL) the assumptions of the model chosen to represent the data »Usually (Euclidean) Distance model

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Uni Winchester 12/09 What is MDS? Prof APM Coxon, Cardiff Uni 7 VARIANTS OF MDS due to type of data MDS can be used with a wide variety of DATA e.g.: SORTS OF DATA –direct data (pair comparisons, ratings, rankings, triads, counts) –derived data (profiles, co-occurrence matrices, textual data, aggregated data) –measures of association etc derived from simpler data, and –tables of data. TYPES of DATA Described by WAY (2W=matrix; 3W=stack of matrices …) And MODE (# sets of distinct objects – eg variables, subjects) –E.G. 2W1M; 2W2M; 3W2M … 7W4M

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Uni Winchester 12/09 What is MDS? Prof APM Coxon, Cardiff U 8 VARIANTS OF MDS MODELS due to TRANSFORMATIONS MDS can also be used with a wide variety of: Transformations (levels of measurement) monotonic (ordinal), linear/metric (interval), … but also –Splines (SPSS PROXSCAL) local preservation of distance –log-interval (MRSCAL), –Power (MULTISCALE) –smoothness

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VARIANTS OF MDS due to type of MODEL DISTANCE Minkowski-r –Usually Euclidean (r=2) Less often City Block, r=1 –Sometimes Dominance, r= 32 SCALAR PRODUCTS/Factor scalar product : a b = |a| |b| cos θ –E.g. Covariance, PM Correlation –As used in PCA, FA, MDPREF COMPOSITION –Most usually, Additive (cf ANOVA), as in Impression Formation: –X(i.j) = a(i) + b(j) + … nb Ordinal.non-metric ANOVA –But also, difference, product, mixed Uni Winchester 12/09 What is MDS? Prof APM Coxon, Cardiff Uni 9

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HOW DOES MDS WORK? Iteratively! START: Produce Init. Guestimate Configuration (a) FIT –Calculate distances (d) –Compare with data (δ) [via Ordinal regression] –Calculate overall badness -of-fit measure » Stress (d- δ) … well, almost! Actually more complex »Perfect/Acceptable? EXIT (b) IMPROVE: For each point, –find direction of improvement (dont ask: calculus! Derivatives!) –How far to move? Step-size (call it heuristic ; parachute & mist) (c) MOVE configuration/points BACK TO (a) Uni Winchester 12/09 What is MDS? Prof APM Coxon, Cardiff Uni 10

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Uni Winchester 12/2009 What is MDS? Prof APM Coxon, Cardiff Uni 11 MDS PROGRAMS: 1.Usually either General Purpose Package ( SPSS ) –Basic Model for 2W1M data: PROXSCAL and 3W2M INDSCAL –Also contains CORRESP, HICLUS and (in >SPSS13 ) PREFSCAL (2W2M) 2.or Library : set of programs, each specific to Data- shape, Trans & Model (e.g. NewMDSX for Windows ); includes –BASIC 2W1M SCALING: Non-metric (ordinal) MINISSA, Metric (MRSCAL) linear, Clustering (Hierarchical & Non-hierarchical) –2W2M (Rectangular) SCALING: Multidimensional … Preference, Triads, Unfolding, Sorting –3W2M (and higher) SCALING: Individual Differences (INDSCAL), (Tucker) Points-of-View Procrustean IndDiffs (Lingoes PINDIS) Or Interactive Package (PERMAP via NewMDSX) primarily for basic model Visually animated Superb diagnostic procedures

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Uni Winchester, 12/09 What is MDS? Prof APM Coxon, Cardiff Uni 12 SITES & SOFTWARE: SITES –NEWMDSX AND DOCUMENTATION: –INTERACTIVE PERMAP (Heady) »(presently obtained via NewMDSX) –THREE-WAY SCALING (Kroonenberg) –http://www.leidenuniv.nl/fsw/three-mode/content.htm – FORREST YOUNGS VISTA (Visual Statistics)

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UniWinchester 12/09 What is MDS? Prof APM Coxon, Cardiff Uni 13 WHAT IS MDS? … and now for an example!

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APPENDICES 1.Interpretation: Headlines 2.MVA & MDS Professor APM Coxon14

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MDS: Interpretation: Headlines For Euclidean Distance MDS: "What information is stable/significant? Beware Local minima [PERMAP] Remember: You may translate, reflect, (rigidly) rotate the configuration: do so! [e.g. NewMDSX Graphics; PERMAP] CLEARING UP Configuration: [PERMAP] Map Evaluation & Diagnostics; Points and Links; selective removal and hints of structure via Waerns Graphic links. BASIC STRUCTURES: Regional: what points are close to each other and distant from others? CLUSTERING [(HI)CLUS, SPSS] Linear: directions in space where some property is increasing: External properties [PRO-FIT NewMDSX], If you must... dimensions -- remember changing the origin or dimensional orientation has no effect on relative distance. Most MDS rotated at end to PCA... Unlike FA, dimensions may/ may not have importance. SIMPLE STRUCTURES dimensions, yes -- but also other simple structures (horseshoes, radex/circumplex). Professor APM Coxon15

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MDS & other Dimensional Multivariate Analysis models Uni Winchester 12/09 What is MDS? Prof APM Coxon, Cardiff Uni 16

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Research Methods Festival 2006 What is MDS? Prof APM Coxon, U Edinburgh 17 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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