What is MDS? Prof APM Coxon, U Edinburgh

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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 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 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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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, 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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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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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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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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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
• NEWMDSX AND DOCUMENTATION
• http//www.newmdsx.com
• INTERACTIVE PERMAP (Heady) http//www.ucs.louisian
  a.edu/rbh8900/
• THREE-WAY SCALING (Kroonenberg)
• http//three-mode.leidenuniv.nl/encyclopedia/encyc
  lopedia.htm
• FORREST YOUNGS 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? APPENDIX

• Multidimensional Scaling
• Is often subsumed with Cluster analysis
  Seriation/sequencing as Combinatorial Data
  Analysis
• (Arabie http//www.ec-securehost.com/SIAM/DT06.ht
  ml)
• 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?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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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)