Title: What is MDS? Prof APM Coxon, U Edinburgh
1What 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 )
2What 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!
3WHAT 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
4WHAT 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)
5WHAT 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
6WHAT 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)
7WHAT 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
8VARIANTS 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.
9VARIANTS 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)
10MDS 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)
11SITES 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
12WHAT 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
13WHAT 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)
14WHAT 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
15SOME 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)