MULTIDIMENSIONAL SCALING 3 WAY ANALYSIS l The CarrollChang

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MULTIDIMENSIONAL SCALING: 3 -WAY ANALYSIS l. The Carroll-Chang Individual Differences Scaling (INDSCAL) – SPSS

MULTIDIMENSIONAL SCALING: 3 -WAY ANALYSIS l. The Carroll-Chang Individual Differences Scaling (INDSCAL) – SPSS PROXSCAL/ ALSCAL version – Hierarchies of Distance Models MODULE 7 1 9/24/2020

Prologue INDSCAL is the most commonly-used 3 -way MDS program, brought in initially to

Prologue INDSCAL is the most commonly-used 3 -way MDS program, brought in initially to deal with Individual Differences in perception. l It is the program of choice if you have a SET (>5) of dis/similarity matrices you wish to analyse l MODULE 7 2 9/24/2020

Prologue … – Unlike PINDIS (qv)… l l which takes a set of CONFIGURATIONS

Prologue … – Unlike PINDIS (qv)… l l which takes a set of CONFIGURATIONS INDSCAL – takes individual pair-comparison similarity ratings or aggregate associations or correlations l l (no problem with neg. values). In SPSS, PROXSCAL – implements the INDSCAL model – as “Weighted Euclidean” Model MODULE 7 3 9/24/2020

Introduction to INDSCAL l 3 -way Scaling analyses a “cube” of DATA (rows, columns,

Introduction to INDSCAL l 3 -way Scaling analyses a “cube” of DATA (rows, columns, ways). – The “ 3 rd way” usually consists of a set of Individuals (or sub-groups), l but may equally be Times, Locations, Methods, Subgroups … – The most common form of 3 -way scaling is 3 W 2 M data – analysis of a set of 2 W 1 M dis/similarities data, implemented by the INDSCAL (INdividual Differences SCALing) Model of Carroll & Chang 1970) (INDSCAL-S in New. MDSX) MODULE 7 4 9/24/2020

INDSCAL & AGGREGATION l The “Problem of Aggregation” is: – How do you appropriately

INDSCAL & AGGREGATION l The “Problem of Aggregation” is: – How do you appropriately represent the variation in a set of individuals’ data? – Two extreme answers: l l l 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 5 9/24/2020

INDSCAL & AGGREGATION l A Middle view: – Individuals/groups may [or may not] have

INDSCAL & AGGREGATION l 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 l Original development was Horan (1969) l who proposes a DIMENSIONAL answer to the Aggregation problem: MODULE 7 6 9/24/2020

INDSCAL: Representation of Subject Weights MODULE 7 7 9/24/2020

INDSCAL: Representation of Subject Weights MODULE 7 7 9/24/2020

INDSCAL & AGGREGATION l Horan proposed a “Master [Reference]Space” formed by the union of

INDSCAL & AGGREGATION l Horan proposed a “Master [Reference]Space” formed by the union of all the dimensions which the subjects use. Each subject either: l l uses does not use (1) (0) or each of these (common) dimensions, creating their own “Private Space” l which is a “sub-space” – a subset of the dimensions of the Master Space – For example … l l Master Space = D 1, D 2, D 3, D 4, D 5 Private Spaces: – – MODULE 7 Laura = Thomas = Marie = Nicolas = [0 1 1 0 1] [1 0 0 1 0] nothing in common with Laura [1 1 1 0 1] shares D 1 with Thomas [0 0 1 0 0] a 1 -dimensional man! 8 9/24/2020

Carroll’s INDSCAL MODEL: IS DIMENSIONAL Carroll & Chang (1970) took up (and generalised) Horan’s

Carroll’s INDSCAL MODEL: IS DIMENSIONAL Carroll & Chang (1970) took up (and generalised) Horan’s model l 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: l INDSCAL dimensions are crucial : – INDSCAL produces unique orientation of Axes – rotation is therefore not permissible l (or if done, destroys optimal properties ) – Nothing constrains axes to be orthogonal Moreover … MODULE 7 9 9/24/2020

Carroll’s INDSCAL MODEL: Subjects’ DIMENSIONAL WEIGHTS l Each subject i DIFFERENTIALLY (+ly) WEIGHTS each

Carroll’s INDSCAL MODEL: Subjects’ DIMENSIONAL WEIGHTS l Each subject i DIFFERENTIALLY (+ly) WEIGHTS each of these fixed dimensions l [0 … wia … +1 ] – These weights ( wia ) are often interpreted to mean differential importance, salience, discrimination … – Each subject is characterised by a pattern of saliences (relative importance) l MODULE 7 for 2 D, often form ratio of : wi 1 / wi 2 10 9/24/2020

INDSCAL MODEL: Subject Space The pattern of subjects’ weights is represented in the INDSCAL

INDSCAL MODEL: Subject Space The pattern of subjects’ weights is represented in the INDSCAL Subject Space l Subject Space ONLY portrays subjects, NOT stimuli (so it’s NOT a joint mapping / biplot). l In the SUBJECT SPACE … l MODULE 7 11 9/24/2020

Subject Space, continued each individual subject is represented by a point (strictly a vector

Subject Space, continued each individual subject is represented by a point (strictly a vector from the origin) in the same (Group Space) dimensions l The similarity between two subjects is the angular separation of their vectors, l – n. b. NOT the distance between the points l 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 12 9/24/2020

INDSCAL MODEL: “Private” Spaces The individual’s set of dimensional weights, wia when applied to

INDSCAL MODEL: “Private” Spaces The individual’s set of dimensional weights, wia when applied to the Group space, X ij differentially shrinks/stretches each dimension l and hence “distorts” the configuration) to form subject i’s own idiosyncratic … “Private Space” (Yi ). l Each individual thus has a Private Space! l MODULE 7 13 9/24/2020

INDSCAL MODEL Simple Illustration MODULE 7 14 9/24/2020

INDSCAL MODEL Simple Illustration MODULE 7 14 9/24/2020

Political Imagery Study: Gp Sp MODULE 7 15 9/24/2020

Political Imagery Study: Gp Sp MODULE 7 15 9/24/2020

Young Plot l In ordinary INDSCAL Subject Space l l l N. b. “Line

Young Plot l In ordinary INDSCAL Subject Space l l l N. b. “Line of Equal weighting” is a useful tool For few dimensions, log (wi 1 / wi 2 ) is wellbehaved Young Plot (2 D) allows both l relative salience – (deflection from line of equal weighting ) and l % variation explained to be represented separately MODULE 7 16 9/24/2020

INDSCAL: Representation of Subject Weights MODULE 7 17 9/24/2020

INDSCAL: Representation of Subject Weights MODULE 7 17 9/24/2020

Young: ALSCAL & PROXSCAL l PROXSCAL (SPSS) version of INDSCAL is same model: Weighted

Young: ALSCAL & PROXSCAL l PROXSCAL (SPSS) version of INDSCAL is same model: Weighted Euclidean Distance – and allows Ord/Int/Spline transformations – BUT. . . Beware Young’s ALSCAL version! l l 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 (New. MDSX) and/or MULTISCALE and PROXSCAL in SPSS MODULE 7 18 9/24/2020

3 -way Scaling: Hierarchy of Models l The “Bell Labs” Hierarchy: l Carroll: IDIOSCAL

3 -way Scaling: Hierarchy of Models l The “Bell Labs” Hierarchy: l Carroll: IDIOSCAL – Idiosyncratic Rotation + Differential Weighting l Carroll INDSCAL (Weighted Distance) – Fixed Common Axes + Differential Weighting l Kruskal KYST=MINISSA/MRSCAL – Simple Distance MODULE 7 19 9/24/2020

3 -way Scaling: Hierarchy of Models l Ramsay: MLE MULTISCALE Hierarchy l l l

3 -way Scaling: Hierarchy of Models l Ramsay: MLE MULTISCALE Hierarchy l l l M 1 -M 2 -M 3/INDSCAL Multiple Functions (Splines); Error Theory; Confidence Ellipses Lingoes: PINDIS (Procrustean Individual Differences Scaling ) l l Take CONFIGURATIONS as data P 0 -P 1 - P 2 (Distance Models) – Parallel to Bell’s KYST – INDSCAL- IDIOSCAL l MODULE 7 Also Vector Models (P 3, P 4) & Mixed P 5 20 9/24/2020