multi dimensional scaling -angrau - prashanth
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P.PRASHANTH
PhD SCHOLAR
Dept. of Agricultural ExtensionI.D. No. RAD/11-10
MULTI DIMENSIONAL
SCALING
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Out Line of Presentation
Some Historical Milestones
Statistics and Terms Associated with MDS
What is MDS? How to Conduct Multidimensional Scaling?
How to Decide Number of Dimensions?
MDS Applications
Assumptions and Limitations of MDS
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MDS is relatively more complicated scaling device, but withthis sort of scaling one can scale objects, individuals or both
with a minimum of information MDS allows a researcher to measure an item in more than
one dimension at a time
MDS is a class of procedures for representing perceptionsand preferences of respondents spatially by means of a
visual display Perceived or psychological relationships among stimuli are
represented as geometric relationships among points in amultidimensional space
These geometric representations are often called spatial
maps The axes of the spatial map are assumed to denote the
psychological bases or underlying dimensions respondentsuse to form perceptions and preferences for stimuli
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The MDS analysis will reveal the most salient attributeswhich happen to be the primary determinants formaking a specific decision
A/C to Beri (1999), MDS is a data reduction technique,the primary purpose of which is to uncover the 'hiddenstructure' of a set of data. It enables the researcher torepresent the proximities between objects spatially as
in a map The term 'proximities' means any set of numbers that
express the amount of similarity or difference betweenpairs of objects
The term 'objects' refers to things or events
T he proximity data can come from similarity judgments, identification confusion matrices,grouping data, same-different errors or any other measure of pair wise similarity
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Some Historical Milestones
1635 Van Langren provides a distance matrix and map 1958 Torgerson provides a solution for classical MDS based
on eigendecomosition
Torgerson proposed the first MDS method and coined theterm
1966 Gower provides independently the same solution forclassical MDS and gives connection to principal componentanalysis
Classical MDS For minimizing stress
Shepard provides heuristic for MDS in psychology
1954 Guttman facet theory, extra information (externalvariables)is available on the objects according to thefacete design by which the objects are generated
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1986-1998 Meulman: integration of (Non linear) multivariate
analysis and MDS
Much emphasis on the representation of objects less on thevariables
Findings by MDS through stress as a dimension reduction
technique
Including a wide variety of MVA techniques- (Non linear) PCA
- Multiple Correspondence Analysis
- Correspondence analysis
-Generalized Canonical Correlation Analysis
-Discriminant Analysis
1999 Heiser, Meulman, Busing: PRPXSCAL (i.e.SMACOF) in
SPSS (PASW)
2009: De Leeuw& Mair SMACOF in R
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Statistics and Terms Associated with MDS Similarity judgments. Similarity judgments are ratings on all possible pairs
of brands or other stimuli in terms of their similarity using a Likert type
scale.
Preference rankings. Preference rankings are rank orderings of the brands
or other stimuli from the most preferred to the least preferred. They are
normally obtained from the respondents.
Stress. This is a lack-of-fit measure; higher values of stress indicate poorer
fits.
R-square. R-square is a squared correlation index that indicates the
proportion of variance of the optimally scaled data that can be accounted
for by the MDS procedure. This is a goodness-of-fit measure.
Spatial map. Perceived relationships among brands or other stimuli are
represented as geometric relationships among points in amultidimensional space called a spatial map.
Coordinates. Coordinates indicate the positioning of a brand or a stimulus
in a spatial map.
Unfolding. The representation of both brands and respondents as points
in the same space, by using internal or external analysis, is referred to as
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What is MDS?
Table of travel times by train in french cities
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It contains the flying mileages between 10
American cities. The cities are the "objects." andthe mileages are the "similarities.
An MDS of these data gives the picture in Fig. 1,
a map of the relative locations of these 10 cities
in the United States.
This map has 10 points, one for each of the 10
cities. Cities that are similar (have short flying
mileages) are represented by points that areclose together, and cities that are dissimilar (have
large mileages) by points far apart.
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Multidimensional scaling (MDS) is a method that
represents measurements of similarity ordissimilarity among pairs of objects as distancebetween points of a low dimensional space
Who uses MDS?
-Psychology -Medicine
-Sociology -Chemistry
-Archaeology -Net work analysis
-Biology -Economics etc.
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Similarities and dissimilarities
Large similarity approximated by small distance inMDS
The similarity between stimuli is inverselyrelated to the distances of the correspondingpoints in the multidimensional space
Large dissimilarity approximated by large distancein MDS
General term-proximity
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T he Minkowski distance metric
A Euclidian distance metric The city-block distance metric
Euclidian distance metric is often used
because of mathematical convenience inMDS procedures
MDS is used when all the variables (whether
metric or non-metric) in a study are to beanalyzed simultaneously and all such variables
happen to be independent
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Euclidean distance to model dissimilarity. That
is, the distance d ij between points i and j is
defined as
Where xi Specifies the position (coordinate) of
point i on dimension
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Analysis of a face similarity
judgment task
Similarity ratings is shown for 10 faces is to revealsome of the perceptual dimensions thatsubjects might have used when generatingsimilarity judgments for these faces
The two dimensional scaling solution is shownfor the 10 faces.
After visual inspection,
the configuration can be
interpreted as the perceptual
dimensions of age and adiposity.
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A priori knowledge - Theory or past research may suggest aparticular number of dimensions.
Interpretability of the spatial map - Generally, it is difficult tointerpret configurations or maps derived in more than three
dimensions. Elbow criterion - A plot of stress versus dimensionality should
be examined.
Ease of use - It is generally easier to work with two-dimensional maps or configurations than with those involvingmore dimensions.
Statistical approaches - For the sophisticated user, statisticalapproaches are also available for determining thedimensionality.
C onducting Multidimensional Scaling
Decide on the Number of Dimensions
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Conducting Multidimensional ScalingFig. 21.1
Formulate the Problem
Obtain Input Data
Decide on the Number of Dimensions
Select an MDS Procedure
Label the Dimensions and Interpret
the Configuration
Assess Reliability and Validity
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C onducting Multidimensional Scaling
Formulate the Problem
Specify the purpose for which the MDS results would be used.
Select the brands or other stimuli to be included in the
analysis. The number of brands or stimuli selected normally
varies between 8 and 25. The choice of the number and specific brands or stimuli to be
included should be based on the statement of the marketing
research problem, theory, and the judgment of the
researcher.
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Obtain Input Data for MDS
i. Obtaining Input Data
a. Perception Data: Direct Approaches
b. Perception Data: Derived Approaches
c. Direct Vs. Derived Approaches
d. Preference Data
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Approaches to Create Perceptual
Maps
Attribute based approaches
Non attribute based approaches
Attribute Based Approaches:
If MDS used on attribute data, it is known as attributebased MDS
Assumption ± The attributes on which the individuals' perceptions of objects
are based, can be identified
Methods Used to Reduce the Attributes to a Small Number of Dimensions ± Factor Analysis
± Discriminant Analysis
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While attempting to construct a space containing mpoints such that m(m -·1)/2 interpoint distances reflect
the input data The metric (quantitative) approach to MDS treats the
input data as interval scale data and solves applyingstatistical methods for the additive constant whichminimizes the dimensionality of the solution space
The non-metric (qualitative) approach first gathers thenon-metric similarities by asking respondents to rankorder all possible pairs that can be obtained from a set of objects. Such non-metric data is then transformed intosome arbitrary metric space and then the solution is
obtained by reducing the dimensionality After this sort of mapping is performed, the dimensions
are usually interpreted and labelled by the researcher
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Input Data for Multidimensional Scaling
Direct (Similarity
Judgments)
Derived (Attribute
Ratings)
MDS Input Data
Perceptions Preferences
Fig. 21.2
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Perception Data: Direct Approaches. In direct approaches to gatheringperception data, the respondents are asked to judge how similar ordissimilar the various brands or stimuli are, using their own criteria. Thesedata are referred to as similarity judgments.
Very Very
Dissimilar SimilarCrest vs. Colgate 1 2 3 4 5 6 7
Aqua-Fresh vs. Crest 1 2 3 4 5 6 7
Crest vs. Ai 1 2 3 4 5 6 7
.
.
.
Colgate vs. Aqua-Fresh 1 2 3 4 5 6 7
The number of pairs to be evaluated is n (n -1)/2, where n is the numberof stimuli.
C onducting Multidimensional Scaling
Obtain Input Data
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Similarity Rating Of Toothpaste BrandsTable 21.1
Aqua-Fresh Crest Colgate Aim Gleem Macleans Ultra Bri te Close-Up Pepsodent Dentagard
Aqua-Fresh
Crest 5
Colgate 6 7
Aim 4 6 6Gleem 2 3 4 5
Macleans 3 3 4 4 5
Ultra Brite 2 2 2 3 5 5
Close-Up 2 2 2 2 6 5 6
Pepsodent 2 2 2 2 6 6 7 6
Dentagard 1 2 4 2 4 3 3 4 3
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Perception Data: Derived Approaches. Derived approachesto collecting perception data are attribute-based approaches requiring therespondents to rate the brands or stimuli on the identified attributes usingsemantic differential or Likert scales.
Whitens Does not
teeth ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ whiten teeth
Prevents tooth Does not prevent
decay ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ tooth decay
.
.
.
.
Pleasant Unpleasant
tasting ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ tasting
If attribute ratings are obtained, a similarity measure (such as Euclideandistance) is derived for each pair of brands.
C onducting Multi Dimensional Scaling
Obtain Input Data
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The direct approach has the following advantages and
disadvantages:
The researcher does not have to identify a set of salient
attributes. The disadvantages are that the criteria are influenced by the
brands or stimuli being evaluated.
Furthermore, it may be difficult to label the dimensions of the
spatial map.
C onducting Multi dimensional Scaling
Obtain Input Data Direct vs. Derived Approaches
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The attribute-based approach has the followingadvantages and disadvantages:
It is easy to identify respondents with homogeneous perceptions.
The respondents can be clustered based on the attribute ratings.
It is also easier to label the dimensions. A disadvantage is that the researcher must identify all the salient
attributes, a difficult task.
The spatial map obtained depends upon the attributes identified.It may be best to use both these approaches in a
complementary way. Direct similarity judgments may beused for obtaining the spatial map, and attribute ratings maybe used as an aid to interpreting the dimensions of theperceptual map.
C onducting Multidimensional Scaling
Obtain Input Data Direct vs. Derived Approaches
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Preference data order the brands or stimuli in terms of respondents' preference for some property.
A common way in which such data are obtained is throughpreference rankings.
Alternatively, respondents may be required to make pairedcomparisons and indicate which brand in a pair they prefer.
Another method is to obtain preference ratings for the variousbrands.
The configuration derived from preference data may differgreatly from that obtained from similarity data. Two brandsmay be perceived as different in a similarity map yet similar ina preference map, and vice versa..
C onducting Multidimensional Scaling
Preference Data
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Decide the no. of Dimensions Scree
test (Elbow test)
An important issue in MDS is choosing the number of dimensionsfor the scaling solution.
A configuration with a high number of dimensions achievesvery low stress values but cannot easily be comprehended bythe human eye, and is opt to be determined more by noisethan by the essential structure in the data.
On the other hand, a solution with too few dimensions might notreveal enough of the structure in the data
Stress (or other lack of fit measure) is plotted against thedimensionality.
stress decreases smoothly with increasing dimensionality makingthe choice of appropriate dimensionality very difficult with this
method In Figure 1b, the filled circles shows the scree plot for the face
similarity dataset
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Plot of Stress Versus Dimensionality
0.1
0.2
1
Number of Dimensions
432 500.0
0.3
S t r e s s
Fig. 21.3
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Even if direct similarity judgments are obtained, ratings of thebrands on researcher-supplied attributes may still becollected. Using statistical methods such as regression, theseattribute vectors may be fitted in the spatial map.
After providing direct similarity or preference data, therespondents may be asked to indicate the criteria they used inmaking their evaluations.
If possible, the respondents can be shown their spatial mapsand asked to label the dimensions by inspecting theconfigurations.
If objective characteristics of the brands are available (e.g.,horsepower or miles per gallon for automobiles), these couldbe used as an aid in interpreting the subjective dimensions of the spatial maps.
C onducting Multidimensional ScalingLabel the Dimensions and Interpret the Configuration
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A Spatial Map of Toothpaste Brands
0.5
-1.5
Dentagard
-1.0-2.0
0.0
2.0
0.0
Close Up
-0.5 1.0 1.50.5 2.0
-1.5
-1.0
-2.0
-0.5
1.5
1.0
Pepsodent
Ultrabrite
MacleansAim
Crest
Colgate
Aqua- Fresh
Gleem
Fig. 21.4
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Using Attribute Vectors to Label DimensionsFig. 21.5
0.5
-1.5
Dentagard
-1.0-2.0
0.0
2.0
0.0
Close Up
-0.5 1.0 1.50.5 2.0
-1.5
-1.0
-2.0
-0.5
1.5
1.0
Pepsodent
Ultrabrite
MacleansAim
Crest
Colgate
Aqua- Fresh
Gleem Fights
Cavities
Whitens Teeth
Cleans Stains
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The index of fit, or R-square is a squared correlation indexthat indicates the proportion of variance of the optimallyscaled data that can be accounted for by the MDS procedure.Values of 0.60 or better are considered acceptable.
Stress values are also indicative of the quality of MDSsolutions. While R-square is a measure of goodness-of-fit,stress measures badness-of-fit, or the proportion of varianceof the optimally scaled data that is not accounted for by theMDS model. Stress values of less than 10% are consideredacceptable.
If an aggregate-level analysis has been done, the original datashould be split into two or more parts. MDS analysis shouldbe conducted separately on each part and the resultscompared.
C onducting Multidimensional Scaling
Assess Reliability and Validity
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Stimuli can be selectively eliminated from the input data and
the solutions determined for the remaining stimuli.
A random error term could be added to the input data. The
resulting data are subjected to MDS analysis and the solutions
compared.
The input data could be collected at two different points in
time and the test-retest reliability determined.
C onducting Multidimensional Scaling
Assess Reliability and Validity
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Assessment of Stability by Deleting One Brand
0.5
-1.5 -1.0-2.0
0.0
2.0
0.0
Close Up
-0.5 1.0 1.50.5 2.0
-1.5
-1.0
-2.0
-0.5
1.5
1.0
PepsodentUltrabrite
Macleans
Aim
Crest
Colgate
Aqua- Fresh
Gleem
Fig. 21.6
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External Analysis of Preference Data
0.5
-1.5
Dentagard
-1.0-2.0
0.0
2.0
0.0
Close Up
-0.5 1.0 1.50.5 2.0
-1.5
-1.0
-2.0
-0.5
1.5
1.0
Pepsodent
Ultrabrite
Macleans
AimCrest
Colgate
Aqua- Fresh
Gleem Ideal Point
Fig. 21.7
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MDS Applications Exploratory data analysis; by placing objects as points
in a low dimensional space
The observed complexity in the original data matrix canoften be reduced while preserving the essentialinformation in the data
Consumers generally prefer a particular brand of a productnot on the basis of one attribute but on a number of attributes. The need for multidimensional scaling arises tounderstand such situations
It helps in the identification of attributes on the basis of which consumers perceive or evaluate products or brands.
It enables the positioning of different products or brands onthe basis of these attributes.
It helps to generate a perceptual map indicating location of the brands on the basis of attributes
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Assumptions and Limitations of MDS
It is assumed that the similarity of stimulus A to B is the same as thesimilarity of stimulus B to A.
MDS assumes that the distance (similarity) between two stimuli issome function of their partial similarities on each of severalperceptual dimensions.
When a spatial map is obtained, it is assumed that interpointdistances are ratio scaled and that the axes of the map are multidimensional interval scaled.
A limitation of MDS is that dimension interpretation relatingphysical changes in brands or stimuli to changes in the perceptualmap is difficult at best
MDS is not widely used because of the computation complicationsinvolved under it
Many of its methods are quite laborious in terms of both thecollection of data and the subsequent analyses
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Political party comparison website for Dutch parliamentelections 2010 asks to rate 30 political statements e.g.
1. the government needs to cut the budget by billions . Thebudget deficit should at the latest in 2015
Agree Dont Know Dis agree
2. those with high income should pay more taxes
Agree Dont know Dis agree
11 Political parties also rated these 30 items What is the political land cape in the Dutch elections of
2010?
Do IMDS on the distance between the 11 parties in 30dimensional space
Data on spatial map reducing the data about 11 parties intwo dimensions
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