1. Quick & Simple Introduction to Multidimensional Scaling Professor Tony Coxon Hon. Professorial Research Fellow, University of Edinburgh ( apm.coxon@ed.ac.uk )apm.coxon@ed.ac.uk see www.tonycoxon.com for information on mewww.tonycoxon.com see www.newmdsx.com for information resource on MDS and NewMDSX programs/doc.www.newmdsx.com See: “The User’s Guide to MDS” and “Key Texts in MDS” (readings), Heineman 1982 Available as pdf at £15 from newmdsx

2. 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 is a family of models structured by D-T-M: (DATA) the empirical information on inter-relationships between a set of “objects”/variables are given in a set of dis/similarity data (TRANSFORMATION)which are then re-scaled ( according to permissible transformations for the data / level of measurement), in terms of (MODEL) the assumptions of the model chosen to represent the data

3. MDS Solution … to produce a SOLUTION, consisting of : 1. a CONFIGURATION, which is a i. pattern of points representing the “objects” ii. located in a space of a small number of dimensions (hence SSA – “Smallest-Space Analysis”) iii. where the distances between the points represent the dis/similarities between the data-points iv. as perfectly as possible (the imperfection/badness of fit is measured by Stress) “Low stress is desirable; No stress is perfection”

4. Distances & Maps Given a map, it’s easy to calculate the (Euclidean) distances between the points : MDS operates the other way round: Given the “distances” [data] find the map [configuration] which generated them … and MDS can do so when all but ordinal information has been jettisoned (fruit of the “non-metric revolution”) even when there are missing data and in the presence of considerable “noise”/error (MDS is robust). MDS thus provides at least [exploratory] a useful and easily-assimilable graphic visualization of a complex data set (Tukey: “A picture is worth a thousand words”)

5. What is like MDS? Related and Special-case Models: Metric Scalar Products Models: *PRINCIPAL COMPONENTS ANALYSIS FACTOR ANALYSIS (+ communalities) Metric and Non-Metric Ultrametric Distance, Discrete models  *Hierarchical Clustering  * Partition Clustering (CONPAR)  Additive Clustering ( 2 and 3-way) Metric Chi-squared Distance Model for 2W2M and 3W data / Tables  *Simple (2W2M) and Multiple (3W) Correspondence Analysis BECAUSE OF NON-METRIC (MONOTONE) REGRESSION, MDS ALSO OFFERS ORDINAL EQUIVALENTS OF:  *ANOVA  other simple composition models …* UNICON (All models with asterisk * exist as programs within NewMDSX)

6. How does MDS differ from other Multivariate Methods? Compared to other multivariate methods, MDS models are usually: distribution-free (though MLE models do exist – Ramsay’s MULTISCALE) make conservative (non-metric) demands on the structure of the data, are relatively unaffected by non-systematic missing data, can be used with a very wide variety of types of data: direct data (pair comparisons, ratings, rankings, triads, sortings) derived data (profiles, co-occurrence matrices, textual data, aggregated data) measures of association/correlation etc derived from simpler data, and tables of data. range of transformations monotonic (ordinal), linear/metric (interval), but also log-interval, power, “smoothness” – even “maximum variance non-dimensional scaling” (Shepard)

7. How does MDS differ from other Multivariate Methods (2)? Compared to other multivariate methods, MDS models are also offer: range of models (chiefly distance (Euclidean, but also City-block), factor/vector (scalar-products), simple composition (additive). Also there are hierarchies of models: Similarity models: 2W1M METRIC – 3W2M INDSCAL – IDIOSCAL (honest!) Preference models : Vector-distance-weighted distance-rotated, weighted (PREFMAP) Procrustes rotation for putting configurations into maximum conformity, and then increasingly complex transformations: PINDIS the solutions are visually assimilable & readily interpretable the structure is not limited to dimensional information – also other simple structures (“horseshoes”, radex/circumplex, clusters, directions).

8. Weaknesses in MDS There ARE any??! Relative ignorance of the sampling properties of stress prone-ness to local minima solutions (but less so, and 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 SPSS) 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)

9. CHARACTERIZATION OF BASIC MDS & TERMINOLOGY Structure of MDS specifiable in terms of D-T-M DATA (specifies input data shape and content) DATA MATRIX INPUT: WAY: ‘dimensionality’ of array (2,3,4...) MODALITY: No of distinct sets (to be represented) (1,2,3 …) NB: Modality < or = Way Common examples: 2W1Mbasic models (LTM,UTM,FSM) 2W2Mrectangular, joint (conditional )mapping 3W2M(“stack” of 2W1M) Individual differences Scaling

10. CHARACTERIZATION OF BASIC MDS (2) TRANSFORMATION (form or type of rescaling performed on data) o Non-Metric /Ordinal:  = M(d)  Monotonic Increasing (sims) or Decreasing (dissims)  Order/inequality o Strong / Guttman:  (j,k) >  (l,m) -> d(j,k) > d(l,m) o weak/Kruskal:  (j,k) >  (l,m) -> d(j,k)  d(l,m)  Equality / ties o Primary  (j,k) =  (l,m) -> d(j,k) = OR  d(l,m) o 2ndary  (j,k) =  (l,m) -> d(j,k) = d(l,m) o Metric / Linear  Linear:  = L(d)   = a + b(d)

11. CHARACTERIZATION OF BASIC MDS (3) MODEL: Euclidean Distance where x(i,a) is the co-ordinate of point i on dimension a in the solution configuration X of low dimension The basic model is Euclidean distance, but other Minkowski metrics are available, including: City Block Model

12. (Badness of) FIT: Stress

13. Types of Analysis INTERNAL: If the analysis depends solely on the input data, it is termed “internal”, but EXTERNAL: If the analysis uses additionally to the input data / solution information relating to the same points (but from another source), it is termed “external”.