3-Way Analysis: INDSCAL in Hierarchies of Distance Models

1. MULTIDIMENSIONAL SCALING: 3-WAY ANALYSIS The Carroll-Chang Individual Differences Scaling (INDSCAL) SPSS PROXSCAL/ ALSCAL version Hierarchies of Distance Models MODULE 7

2. Prologue • INDSCAL is the most commonly-used 3-way MDS program, brought in initially to deal with Individual Differences in perception. • It is the program of choice if you have a SET (>5) of dis/similarity matrices you wish to analyse MODULE 7

3. Prologue … • Unlike PINDIS (qv)… • which takes a set of CONFIGURATIONS • INDSCAL • takes individual pair-comparison similarity ratings or aggregate associations or correlations • (no problem with neg. values). • In SPSS, PROXSCAL • implements the INDSCAL model • as “Weighted Euclidean” Model MODULE 7

4. Introduction to INDSCAL • 3-way Scaling analyses a “cube” of DATA (rows, columns, ways). • The “3rd way” usually consists of a set of Individuals (or sub-groups), • but may equally be Times, Locations, Methods, Sub-groups … • The most common form of 3-way scaling is 3W2M data – analysis of a set of 2W1M dis/similarities data, implemented by the INDSCAL (INdividual Differences SCALing) Model of Carroll & Chang 1970) (INDSCAL-S in NewMDSX) MODULE 7

5. INDSCAL & AGGREGATION • The “Problem of Aggregation” is: • How do you appropriately represent the variation in a set of individuals’ data? • Two extreme answers: • Individuals are so unique their data can’t be compared • There’s only one real structure; the rest is individual random variation (“error”) • Researchers often “wash out” systematic differences (variation) by averaging • That’s OK if variation is simply random error • But if these are significant or systematic, the average may represent nobody and be artefactual MODULE 7

6. INDSCAL & AGGREGATION • A Middle view: • Individuals/groups may [or may not] have distinctly different perspectives • … And still share some commonality with others • This is the assumption underlying INDSCAL and allied models • Original development was Horan (1969) who proposes a DIMENSIONAL answer to the Aggregation problem: MODULE 7

7. INDSCAL:Representation of Subject Weights MODULE 7

8. INDSCAL & AGGREGATION • Horan proposed a “Master [Reference]Space” formed by the union of all the dimensions which the subjects use.Each subject either: • uses (1) or • does not use (0) each of these (common) dimensions, creating their own “Private Space” • which is a “sub-space” – a subset of the dimensions of the Master Space • For example … • Master Space = D1, D2, D3, D4, D5 • Private Spaces: • Laura = [0 1 1 0 1] • Thomas = [1 0 0 1 0] nothing in common with Laura • Marie = [1 1 1 0 1] shares D1 with Thomas • Nicolas = [0 0 1 0 0] a 1-dimensional man! MODULE 7

9. Carroll’s INDSCAL MODEL: IS DIMENSIONAL Carroll & Chang (1970) took up (and generalised) Horan’s model • There is a common “GROUP SPACE” • ( X ij ) aka Stimulus Space), • spanned by a fixed set of shared common dimensions … BUT there are critical differences: • INDSCAL dimensions are crucial : • INDSCAL produces unique orientation of Axes • rotation is therefore not permissible • (or if done, destroys optimal properties ) • Nothing constrains axes to be orthogonal Moreover … MODULE 7

10. Carroll’s INDSCAL MODEL: Subjects’ DIMENSIONAL WEIGHTS • Each subject i DIFFERENTIALLY (+ly) WEIGHTS each of these fixed dimensions • [ 0 … wia … +1 ] • These weights ( wia ) are often interpreted to meandifferential importance, salience, discrimination … • Each subject is characterised by a pattern of saliences (relative importance) • for 2D, often form ratio of : wi1 / wi2 MODULE 7

11. INDSCAL MODEL: Subject Space • The pattern of subjects’ weights is represented in the INDSCAL Subject Space • Subject Space ONLY portrays subjects, NOT stimuli (so it’s NOT a joint mapping / biplot). • In the SUBJECT SPACE … MODULE 7

12. Subject Space, continued • each individual subject is represented by a point (strictly a vector from the origin) in the same (Group Space) dimensions • The similarity between two subjects is the angular separation of their vectors, • n.b. NOT the distance between the points • Length of a subject’s vector is proportional to the amount of his/her data variance explained (so size does matter!): the further a vector is from the origin, the better it is possible to account for his/her data • strictly, only holds for uncorrelated axes MODULE 7

13. INDSCAL MODEL: “Private” Spaces • The individual’s set of dimensional weights, wiawhen applied to the Group space, X ijdifferentially shrinks/stretches each dimension • and hence “distorts” the configuration) to form subject i’s own idiosyncratic … “Private Space” (Yi). • Each individual thus has a Private Space! MODULE 7

14. INDSCAL MODEL Simple Illustration MODULE 7

15. Political Imagery Study: Gp Sp MODULE 7

16. Young Plot • In ordinary INDSCAL Subject Space • N.b. “Line of Equal weighting” is a useful tool • For few dimensions, log (wi1 /wi2 ) is well-behaved • Young Plot (2D) allows both • relative salience • (deflection from line of equal weighting ) and • % variation explained to be represented separately MODULE 7

17. INDSCAL: Representation of Subject Weights MODULE 7

18. Young: ALSCAL & PROXSCAL • PROXSCAL (SPSS) version of INDSCAL is same model: Weighted Euclidean Distance • and allows Ord/Int/Spline transformations • BUT ... Beware Young’s ALSCAL version! • S-Stress and Large distances (to the fourth power!) produce distortion & exaggerate error (Ramsay) -- so great as to make solution dubious. Cavete! • S-INDSCAL outperforms ALSCAL • Weinberg, S. L., and Menil, V. C. (1993). • Therefore use instead S-INDSCAL (NewMDSX) and/or MULTISCALE and PROXSCAL in SPSS MODULE 7

19. 3-way Scaling: Hierarchy of Models • The “Bell Labs” Hierarchy: • Carroll: IDIOSCAL • Idiosyncratic Rotation + Differential Weighting • Carroll INDSCAL (Weighted Distance) • Fixed Common Axes + Differential Weighting • Kruskal KYST=MINISSA/MRSCAL • Simple Distance MODULE 7

20. 3-way Scaling: Hierarchy of Models • Ramsay: MLE MULTISCALE Hierarchy • M1-M2-M3/INDSCAL • Multiple Functions (Splines); Error Theory; Confidence Ellipses • Lingoes: PINDIS (Procrustean Individual Differences Scaling ) • Take CONFIGURATIONS as data • P0 -P1- P2 (Distance Models) • Parallel to Bell’s KYST – INDSCAL- IDIOSCAL • Also Vector Models (P3, P4) & Mixed P5 MODULE 7