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**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 ) What is MDS? Prof APM Coxon, U Cardiff Mod: 11/ 2009

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**ORIGINS / DEVELOPMENT OF MDS**

MDS (aka “Smallest Space Analysis”) Has origins in Psychometrics in 1920-’60s: 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) What is MDS? Prof APM Coxon, Cardiff U U Winchester 12/09

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**Yes! And it works!! CONSTRUCTING A MAP …**

Given a map, it’s 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 User’s 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 What is MDS? Prof APM Coxon, Cardiff U U. Winchester 12/09

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**WHAT IS MULTIDIMENSIONAL SCALING?**

A student’s 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) What is MDS? Prof APM Coxon, Cardiff Uni Uni Winchester 12/09

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**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 What is MDS? Prof APM Coxon, Cardiff U U Winchester, 12/2009

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**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 What is MDS? Prof APM Coxon, Cardiff Uni Uni Winchester 12/09

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**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” What is MDS? Prof APM Coxon, Cardiff U Uni Winchester 12/09

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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 What is MDS? Prof APM Coxon, Cardiff Uni Uni Winchester 12/09

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**What is MDS? Prof APM Coxon, Cardiff Uni**

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 (don’t ask: calculus! Derivatives!) How far to move? Step-size (call it ‘heuristic’ ; “parachute & mist”) (c) MOVE configuration/points BACK TO (a) What is MDS? Prof APM Coxon, Cardiff Uni Uni Winchester 12/09

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**What is MDS? Prof APM Coxon, Cardiff Uni**

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 What is MDS? Prof APM Coxon, Cardiff Uni Uni Winchester 12/2009

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**What is MDS? Prof APM Coxon, Cardiff Uni**

SITES & SOFTWARE: SITES NEWMDSX AND DOCUMENTATION: INTERACTIVE PERMAP (Heady) (presently obtained via NewMDSX) THREE-WAY SCALING (Kroonenberg) FORREST YOUNG’S VISTA (Visual Statistics) What is MDS? Prof APM Coxon, Cardiff Uni Uni Winchester, 12/09

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**What is MDS? Prof APM Coxon, Cardiff Uni**

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

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**Interpretation: Headlines MVA & MDS**

APPENDICES Interpretation: Headlines MVA & MDS Professor APM Coxon

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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 Waern’s 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 Coxon

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**MDS & other “Dimensional” Multivariate Analysis models**

What is MDS? Prof APM Coxon, Cardiff Uni Uni Winchester 12/09

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

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