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



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



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



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



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



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



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



What is MDS?

Table of travel times by train in french cities



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.



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.



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



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



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



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.



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



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




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.



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



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



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



Input Data for Multidimensional Scaling

Direct (Similarity

Judgments)

Derived (Attribute

Ratings)

MDS Input Data

Perceptions Preferences

Fig. 21.2



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.


Obtain Input Data



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



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



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



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.


Obtain Input Data Direct vs. Derived Approaches



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..


Preference Data



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



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



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



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



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



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.





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.





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



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



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



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



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

multi dimensional scaling -angrau - prashanth

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