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

2. 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

3. 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!!  

4. 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

5. 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

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

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

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

9. 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

10. 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

11. 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

12. 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

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

14. Interpretation: Headlines MVA & MDS
APPENDICES
Interpretation: Headlines
MVA & MDS
Professor APM Coxon

15. 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

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

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