Perceptual mapping by multidimensional scaling MDS

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These analyses address issues of concern to marketing and advertising professionals and to

academic researchers investigating consumer behavior. The reports present original research

and cutting edge analyses conducted by faculty and graduate students in the Consumer-

Industrial Research Program at Cleveland State University.

Subscribers to the series include those in advertising agencies, market research organizations,

product manufacturing firms, health care institutions, financial institutions and other

professional settings, as well as in university marketing and consumer psychology programs.

To ensure quality and focus of the reports, only a handful of studies will be published each

year.

ProfessionalSeries - Brief, bottom line oriented reports for those in marketing and

advertising positions. Included are both B2B and B2C issues.

How To Series - For marketers who deal with research vendors, as well as for

professionals in research positions. Data collection and analysis procedures.

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oriented.


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AVAILABLE PUBLICATIONS:

Professional Series

Lyttle, B. & Weizenecker, M. Focus groups: A basic introduction, February, 2005.

Arab, F., Blake, B.F., & Neuendorf, K.A. Attracting Internet shoppers in the Iranianmarket, February, 2003. Liu, C., Blake, B.F., & Neuendorf , K.A. Internet shopping in Taiwan and U.S.,

February, 2003. Jurik, R., Blake, B.F., & Neuendorf, K.A. Attracting Internet shoppers in the

Austrian market, January, 2003. Blake, B.F., & Smith, L. Marketers, Get More Actionable Results for Your Research

Dollar!, October, 2002.

How To Series

Blake, B.F., Valdiserri, J., Neuendorf, K.A., & Nemeth, J. Validity of the SDS-17measure of social desirability in the American context, November, 2005.

Blake, B.F., Dostal, J., & Neuendorf, K.A. Identifying constellations of websitefeatures: Documentation of a proposed methodology, February, 2005.

Saaka, A., Sidon , C., & Blake, B.F. Laddering: A How to do it manual with anote of caution, February, 2004.

Blake, B.F., Schulze, S., & Hughes, J.M. Perceptual mapping by multidimensionalscaling: A step by step primer, July, 2003.

Behavioral Science Series

Shamatta, C., Blake, B.F., Neuendorf, K.A, Dostal, J., &Guo, F. Comparing websiteattribute preferences across nationalities: The case of China, Poland, and the USA,

October, 2005. Blake, B.F., Dostal, J., & Neuendorf, K.A. Website feature preference constellations:

Conceptualization and measurement, February, 2005. Blake, B.F., Dostal, J., Neuendorf, K.A., Salamon, C., & Cambria, N.A. Attribute

preference nets: An approach to specifying desired characteristics of an innovation,

February, 2005. Blake, B.F., Neuendorf, K.A., Valdiserri, C.M., & Valdiserri, J. The Online

Shopping Profile in the cross-national context: The roles of innovativeness and

perceived newness, February, 2005. Blake, B.F., & Neuendorf , K.A. Cross-national differences in website appeal: A

framework for assessment, July, 2003.

Blake, B.F., Neuendorf , K.A., & Valdiserri , C.M. Appealing to those most likely toshop new websites, June, 2003.

Blake, B.F., Neuendorf , K.A., & Valdiserri , C.M. Innovativeness and variety ofinformation shopping, April, 2003.


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EDITORIAL DIRECTOR: DR.BRIAN BLAKE

Dr. Brian Blake has a wide variety of academic and professional experiences.

His early career... academically, rising from Assistant Professor to tenured Professor at Purdue

University, his extensive published research spanned the realms of psychology (especially

consumer, social, and cross-cultural), marketing, regional science, sociology, community

development, applied economics, and even forestry. Professionally, he was a consultant to the

U.S. State Department and to the USDA, as well as to private firms.

Later on...on the professional front, he co-founded a marketing research firm, Tactical Decisions

Group, and turned it into a million dollar organization. After merging it with another firm to

form Triad Research Group, it was one of the largest market research organizations based in

Ohio. His clients ranged from large national firms (e.g., Merck and Co., Dupont, Land o Lakes)

to locally based organizations (e.g., MetroHealth System, American Greetings, Progressive

Insurance, Liggett Stashower Advertising). On the academic side, he moved to Cleveland State

University and co-founded the Consumer-Industrial Research Program (CIRP). Some of

Clevelands best and brightest young marketing research professionals are CIRP graduates.

In the last few years...academically, he is actively focusing upon establishing CIRP as a center

for cutting edge consumer research. Professionally, he resigned his position as Chairman of

Triad and is now Senior Consultant to Action Based Research and consultants with a variety of

clients.

EDITOR(2003):JILLIAN HUGHES

Currently a CIRP graduate student, she graduated Magna Cum Laude from Mount Union

College, where she majored in Psychology, with a focus on Consumer Behavior, and minored in

Sociology. Among her many research interests; she focuses on Internet buying behavior, and the

effects of Social Desirability Bias on Innovativeness Scales. She had the honor of presenting

research concerning age differences in brand labeling at the Ohio Undergraduate Psychology

Conference in April of 2001 at Kenyon College. She also presented another piece of original

research on Internet buying behavior of college students at the Interdisciplinary Conference for

the Behavioral Sciences hosted by Mount Union College in April, 2001.


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Forward

Perceptual Maps are widely used by market researchers, e.g., to portray a brands

image or consumers reactions to product features. Although the major statistical

techniques have been available for several decades, these are still many questions about

those techniques among practicing professionals.

This report is intended for on the job professionals who are fairly unfamiliar

with the concrete procedures used to generate and to interpret such maps

In overview, three types of maps are especially popular among professional

researchers:

perceptual maps that identify the images of brands, products, services, etc.

preference maps that estimate differences among segments or individuals in

the appealorattractiveness of brands, products, services, features.

hybrid maps which portray both images and appeal.

A variety of statistical techniques can be used to generate each type of map.

Perceptual maps are usually constructed via multidimensional scaling - multiple

discriminant function correspondence analysis. Preference maps are typically

developed by a form of multidimensional unfolding. Hybrid maps are composed

by first devising a perceptual map and then introjecting preferences as ideal points

or as vectors.

This paper focuses upon one mapping technique, multidimensional scaling

(MDS), and executes it via a program package that is widely used by market

researchers, SPSS.


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Abstract

The overall objective of this report was to document the applications of two

widely applied forms of perceptual mapping, Classical and Weighted Multidimensional

Scaling. A step-by-step guide is provided for the use of these mapping techniques. It is

anticipated that this report will be valuable to the professional market researcher who is

new to perceptual mapping and to others looking for a detailed reference source for

performing the basics of these techniques.

The illustrative data pertain to the images of particular hospitals in the Northeast

Ohio area. The data were gathered from a convenience sample of family, friends, and

acquaintances of the researchers. A total of 107 took part in the study.

The techniques were Classic Multidimensional Scaling (CMDS) and Weighted

Multidimensional Scaling (WMDS). The statistical software program SPSS was used,

but the ideas can be generalized to other statistical packages and programs.

I. Overview of the Three Mapping Procedures

Before describing each technique in detail, let us present them in overview.

1) Classic Multidimensional Scaling (CMDS)

To begin, the data for CMDS and WMDS are indicators of the degree of

similarity among objects, brand names, etc. Here, the data are ratings of the degree of

perceived similarity among the twelve stimuli- four hospitals identified by name, four

unnamed hospitals described by taglines, and four unnamed hospitals described by

advertisements.


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The four named hospitals in our illustration were Fairview Hospital, Parma

Community Hospital, Southwest General Hospital, and MetroHealth System. The four

taglines were 1) In the hands of doctors 2) Its a people thing 3) When its your

health, experience counts and 4) Your partner in good health. For this report, let us

abbreviate them as 1) doctors hands 2) people thing 3) experience counts and 4)

partner. The four advertisements were labeled 1) magic bullet 2) heart surgery 3)

diet and exercise and 4) heart center. The advertisements are shown on pages 3-6.

Respondents rated the similarity of pairs of these items on a 0-10 point scale, higher

numbers meaning more similarity. For the CMDS example on the next page, we consider

just the four named hospitals and the four hospitals described by advertisements.

CMDS analyzed ratings to produce the positioning map on page 7. The goal of

mapping is to portray the respondents perceptions of similarity among the items along a

given number of dimensions or yardsticks. The closer together on the map are the items,

the

more similar they are perceived to be. In the figure on the next page, the following

abbreviations are used for the named hospitals: fairview (Fairview Hospital), parmacom

(Parma Community Hospital), swgener (Southwest General Hospital), and metro

(MetroHealth System). The following abbreviations were used for the hospitals

described by advertisements: magbull (magic bullet), hrtsurg (heart surgery),

dietexe (diet and exercise), and hrtcent (heart center).


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Here, the hospital described in the advertisement Heart Center is perceived to be

similar to Parma Community Hospital. However, the advertisement Heart Center is

seen as quite distinct from MetroHealth System. Thus, respondents as a group perceived

that the advertisement Heart Center fits Parma Community Hospital better than

MetroHealth System.

Among other uses, this kind of mapping can identify how well an advertisement

or tagline can fit a hospital, or even be incompatible with a particular brand name.

2) Weighted Multidimensional Scaling (WMDS)

In Weighted Multidimensional Scaling (WMDS), we are again considering the

perceived similarity among stimuli, but here we identify differences in perception among

specific segments of individuals. WMDS calculates the differences among the groups of

respondents on a given number of dimensions. Each dimension is essentially the same

Classic Multidimensional Scaling CMDS

Hospitals by Advertisements

Dimension 1

1.51.0.50.0-.5-1.0-1.5-2.0

Dim

ension2

2.0

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

swgener

parmacom

metro

fairview

hrtcent

dietexe

hrtsurg

magbull


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for all segments (e.g. world-wide reputation). In contrast, if we were to do a CMDS

separately for each segment, we could conceivably develop maps in which a dimension in

one group would have to be interpreted as different from the dimension found in other

groups (e.g. personal attention, location close by).

Respondents rated the same 12 objects (4 hospitals/ 4 advertisements/ 4 taglines)

by similarity for selected pairs of these items. Also, WMDS shows the importance of a

dimension to a particular segment when that segment perceives the stimuli in question.

One map is produced for each segment. The closer together the stimuli are on the map,

the more similar is the weight assigned to an object on a particular dimension. WMDS

analyzed these ratings to produce the positioning map illustrated on the next page. The

data were separated into male and female segments, and the maps can be seen on the next

two pages.


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Magic BulletHeart Surgery

Diet and Exercise

Heart Center

Fairview

Parma Community

SW General

Metrohealth

WMDS Hospitals by Ads (Males)


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Magic Bullet

Heart Surgery

Heart Center Parma Community

Diet and Exercise

SW General

Fairview

Metrohealth

WMDS Hospitals by Ads (Females)


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In the male WMDS map, the hospital described in the advertisement Heart

Center is perceived to be similar to Parma Community Hospital in the eyes of males.

However, the advertisement Heart Center is perceived to be much different from

Southwest General Hospital. Therefore, male respondents as a group perceive that the

advertisement Heart Center fits Parma Community Hospital better than Southwest

General Hospital.

This kind of perceptual mapping can aid in determining how appropriate or

inappropriate an advertisement or tagline can be for a hospital depending on the target

market segment of interest. Furthermore, this mapping technique can identify the most

appropriate advertisement or tagline for a particular brand name in general.

The information used for more detailed explanation in subsequent sections of this

paper was drawn from:

Young and Harris (19XX), Chapter 7 Multidimensional Scaling.

Hair, Anderson, Tatham, and Black (1998), Multivariate Data Analysis,

5th

Ed., Chapter 10.

Meyers (1996), Segmentation and Positioning for Strategic Marketing

Decisions. Perceptual Positioning Maps, Chapter 8.

Lecture notes from Professor Brian Blakes Advanced Consumer Research

course PSY 620.

George and Mallery (2001), SPSS for Windows Step by Step: A Simple

Guide and Reference 10.0 Update 3rd

Ed., Multidimensional Scaling

Chapter 19.


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II. CLASSIC MULTIDIMENSIONAL SCALING (CMDS)

A. Goals/ Objectives of CMDS

A researcher can use CMDS to reach two general goals or objectives. First,

CMDS can estimate the relative importance of the dimensions that respondents use to

judge the degree of similarity or dissimilarity among the stimuli. Second, the degree of

similarity among all of the stimuli on those dimensions can be assessed. To empirically

understand or to label the nature of a given dimension requires analyses above and

beyond CMDS itself.

B. CMDS General Rationale

Classic Multidimensional Scaling (CMDS) is a statistical technique created to

transform data indicating the degree of rated similarity or dissimilarity of objects to

scores indicating distances among the objects. Then, a map is constructed showing the

distances among the objects. Objects closer together on the map are perceived as more

similar and objects further apart are perceived as more dissimilar. The same unit of

measurement is used for all of the distances among the objects. One matrix of data is

used, displaying the perceptions of one person or the average persons answers in the

group of respondents in question.

C. Alternatives/ Options

CMDS can produce perceptual maps that portray disaggregated results (which

show evaluations of a single individual) or aggregated results (which show the combined

assessments of many individuals). If a single respondents evaluations are desired, the

researcher should input the respondents judgments into a matrix where the units used for


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comparison are listed across the columns and rows as shown in Table 1 on the next page.

On the contrary, if cumulative respondent answers are desired, the average or mean

evaluation can be computed in SPSS under the descriptives option. The researcher then

inputs the resulting mean scores into a matrix.

In this matrix, the hypothetical data below are the similarity ratings between the

two items in each pair. The higher the number, the more similar respondents perceived

the two items to be.


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A B C D E F G H

A 0

B 6 0

C 8 10 0D 5 6 9 0

E 3 8 10 4 0

F 5 6 3 7 5 0

G 4 7 6 9 2 3 0

H 6 6 10 7 4 8 2 0

TABLE 1

In the resulting positioning map, CMDS presents the distances between objects on

the dimensions along which the distances are calculated. The researcher must specify

the number of dimensions, which is usually two or three for ease of interpretation.

CMDS doesnt directly label these dimensions and the researcher must indirectly

estimate names for the dimensions based, for example, on the rank order of the objects on

a given dimension.

Specifically, each unit or object is illustrated by a plotted location in

multidimensional space on a positioning map. These plotted locations illustrate the

similarities that are perceived by respondents. CMDS determines the distances among all

combinations of pairs of the objects and plots the objects accordingly. Briefly, the

perceptual space is usually generated by a 2 or 3 dimensional Euclidean model. The

Euclidean geometrical representation is a multidimensional generalization of the

Pythagorean theorem we all learned in high school geometry class. The formula is: the

squared distance between the two points on Dimension Xplus the squared distance

between the two points on Dimension Y equals the squared straight line distance between

the two points. For example, we can figure the Euclidean straight-line distance on two


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dimensions between A and C if we know their coordinates on the X and Y axes

(dimensions). As shown below, if the plotted coordinates (locations) for A are 1 on X

and 3 on Y, and Cs coordinates are 4 on X and 2 on Y, then the distance between A & C

can be calculated using the formula. The difference between A and C on Dimension 1

for X is 4-1=3 and on Dimension 2 for Y is 3-2=1. Therefore, the Euclidean distance

formula would be (1)2 + (3)2 = 10 or 3.33. The Euclidean distance between A and C is

3.33.

D. Illustrative Case

Twelve presentation boards were presented to respondents representing twelve

different hospitals. The respondents were informed that the hospitals represented may or

may not be located in the Northeast Ohio area, and that they may or may not be familiar

with the hospitals portrayed. Four of the hospitals were described by taglines, four of the

hospitals were described by advertisements, and four of the hospitals were described by

their respective hospital names.

Euclidean Distance

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5

X

YA

C


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First, respondents were asked about their familiarity with a tagline. The

respondents were instructed to assume that the tagline was an accurate description of the

hospital represented. An illustrative question stated, Now, focus on tagline In the hands

of doctors. What do you know about this tagline? The respondents options were: A)

I have never seen it before, and know nothing about it, B) I have seen it before, but dont

know much about it, C) I have seen it before, and know quite a bit about it, or D) I have

seen it before, and know that its a tagline for _______ hospital. This questioning was

repeated for the other three taglines, which again were Its a people thing, When its

your health, experience counts, and Your partner in good health.

Respondents rated a total of six pairs of the four taglines. Ratings were made on a

scale of 0-10, where 0 indicated very different and 10 indicated very similar. The

interviewer presented the taglines and a question example stated, Rate how similar/

dissimilar are the two hospitals described by the two taglines. Use a scale of 0 to 10, with

0 meaning very different and 10 meaning very similar. The first two hospitals are the

ones described by In the hands of doctors and the one described by Its a people

thing.

Next, respondents were asked about their familiarity with an advertisement. The

respondents were instructed to assume that the advertisement was an accurate description

of a hospital. An illustrative question stated, Now, focus on advertisement that was

labeled magic bullet. What do you know about this ad? The respondents options

were: A) I have never seen it before, and know nothing about it, B) I have seen it

before, but dont know much about it, C) I have seen it before, and know quite a bit about


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it, or D) I have seen it before, and know that its a advertisement for _______ hospital.

This questioning was repeated for the other three advertisements.

Then, respondents rated a total of six pairs of the four advertisements. Ratings

were made using the same scale of 0-10, where 0 indicated very different and 10

indicated very similar. The interviewer then presented the advertisements and a question

example stated, Rate how similar/ dissimilar are the two hospitals described by the two

advertisements. Use a scale of 0 to 10, with 0 meaning very different and 10 meaning

very similar. The first two hospitals are the ones described by the advertisements labeled

magic bullet and people thing.

Finally, hospital names were presented to respondents to assess what the

respondents knew about the hospital names. An illustrative question stated, What do

you know about Parma Community Hospital? A respondents options were: A) I know

practically nothing about it, B) I have heard the name, but dont know much about it, C) I

have heard the name and know a little about it, or D) I have heard the name and know

quite a bit about it.

Next, respondents rated a total of six pairs of the four hospital names. Ratings

were made on the same scale of 0-10, again where 0 indicated very different and 10

indicated very similar. The interviewer presented the hospital names and a question

example stated, Rate how similar/ dissimilar are the two hospitals named. Use a scale of

0 to 10, with 0 meaning very different and 10 meaning very similar. The respondent

was then presented with each of the six pairs of hospitals: Fairview & MetroHealth

System, Fairview & Parma Community, etc. This questioning was repeated for the

other three hospital names.


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Respondents were then asked to indicate the degree of similarity or dissimilarity

of hospitals when taglines were paired with hospital names. The same 0-10 similarity

scale was employed. There were 16 pairs of hospitals and taglines. An illustrative

question stated, Rate how similar/ dissimilar are the two hospitals using a scale of 0 to

10, with 0 meaning very different and 10 meaning very similar. The respondent was

then presented with the pairs doctors hands and Fairview, etc.

Respondents were then asked to indicate the degree of similarity or dissimilarity

of advertisements paired with hospital names using the same 0-10 scale. There were 16

pairs of hospitals and advertisements. An illustrative question stated, Rate how similar/

dissimilar are the two hospitals. Use a scale of 0 to 10 with 0 meaning very different and

10 meaning very similar. The respondents were then given the pairs magic bullet &

Fairview, etc.

E. Data Needed

The data analyzed in CMDS are displayed in a single matrix showing the degree

of rated dissimilarity between hospitals in a pair. The matrix consists of rows and

columns listing each object paired with all other objects. The measurement level,

shape, and conditionality of the data can vary among various CMDS analyses and

must be specified for any MDS to be conducted.

A) Measurement level-- The data can be ordinal, interval, or ratio.

1) Ordinal data arranges objects in a rank order from high to low on a

dimension.

2) Interval data pertains to numbering in which one number is a fixed

amount more or less than another number. For example, the amount of


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the dimension (here, similarity) between the number 3 and 4 on a scale

is assumed to be the same as the amount between 4 and 5, and that

amount is the same as between 5 and 6 on the scale. In other words,

an interval scale is one in which the intervals between consecutive

numbers are equal.

3) Ratio data has a true zero point (absence of the factor) and allows one

to calculate ratios or proportions. One cannot do this with an interval

scale because there is no zero starting point for the numbers.

However, a ratio scale has a true zero point, and so one can calculate

ratios, such as one item is twice as much as another.

B) Shape-- The shape of the data for CMDS is square, where rows and

columns in the matrix represent the same set of units or objects.

C) Conditionality-- Data used in CMDS are typically matrix conditional

indicating that in the analysis all numbers in the data matrix can be compared

to each other, no matter what the row or column involved. The data matrix is

devised by averaging (mean) similarity ratings for all respondents for a given

pair of stimuli. In the illustrative case, this yields an 8 X 8 matrix.

By default, SPSS assesses higher numbers as more dissimilar, so it was necessary

to recode our values into higher numbers representing more dissimilarity and lower

numbers representing more similarity. Keep in mind that if the questionnaire uses higher

numbers to indicate dissimilarity and lower numbers to indicate similarity, then it is not

necessary to recode the values.

F. SPSS Specific Steps


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and are listed here only for explanatory purposes. These options include the S-Stress

convergence, which represents the lowest possible level of improvement that the

algorithm will configure. The criteria box also specifies the minimum S-Stress value,

which by default is .001 and the maximum number of iterations that the program will

attempt in finding the best possible solution for the data. The default for the latter is 30.

G. SPSS Output

After analyzing the ratings and specifying separate 2 and 3 dimensional solutions

for both taglines by hospitals and advertisements by hospitals, a 3 dimensional solution

was found to have the best fit. However, a 2 dimensional solution is shown for

illustrative purposes because a two dimensional solution is used far more frequently in

professional practice. It is important to note that the fit of the 2 dimensional solution

obtained in this data set would be too low to be considered useful in practice.

The output generated in SPSS for a CMDS solution provides an abundance of

information. First, the SPSS output provides the unscaled means of the similarity ratings

of the objects among the aggregated respondents. The ratings are shown in SPSS Output

1.Each unit or object used for the paired comparisons is shown on the left-hand side as

rows numbered 1-8 and across the top of the matrix as columns numbered 1-8.

Raw (unscaled) Data

1 2 3 45

1 Magic Bullet .0002 Heart Surgery 4.690 .0003 Diet & Exercise 4.440 4.020 .0004 Heart Center 4.740 3.710 5.920 .0005 Fairview General .000 5.120 4.230 4.040

.000


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Iteration history for the 2 dimensional solution (insquared distances)

Young's S-stress formula 1 is used.

Iteration S-stress Improvement

1 .372952 .34050 .032463 .33447 .006034 .33378 .00069

Iterations stopped becauseS-stress improvement is less than

.001000

Stress and squared correlation (RSQ) indistances

RSQ values are the proportion of variance of the scaleddata (disparities)

in the partition (row, matrix, or entire data)which

is accounted for by their correspondingdistances.

Stress values are Kruskal's stress formula 1.

For matrixStress = .20196 RSQ = .75785

SPSS OUTPUT 2The display of the stimulus coordinates on each dimension is provided next. The

coordinates of each object are the coordinates used to create the plots in the map.

Configuration derived in 2 dimensionsStimulus Coordinates

Dimension

Stimulus Stimulus 1 2Number Name

1 MAGIC BULLET 1.1437 .74852 HEART SURGERY -1.3325 -.7954


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3 DIET & EXERCISE .9450 -1.25164 HEART CENTER -.8620 1.11935 FAIRVIEW .8389 .89956 METROHEALTH -1.3411 -.72827 PARMA COMMUNITY .9433 -1.0970

8 SW GENERAL -.3353 1.1048SPSS OUTPUT 3

In SPSS Output 4 on the next page, the CMDS procedure presents a matrix of the

optimally scaled data for subject 1 (the aggregated respondents) in the aggregated

matrix. The data reflect the original ratings of respondents considered as a group. These

are the distances among the hospitals and advertisements in two-dimensional space.

Optimally scaled data (disparities) forsubject 1

1 2 3 45

1 Magic Bullet .0002 Heart Surgery 2.137 .0003 Diet & Exercise 1.033 1.033 .0004 Heart Center 3.370 3.010 3.280 .0005 Fairview General 2.843 2.651 2.330 2.715

.000 6 MetroHealth 1.842 1.996 1.791 1.3933.203

7 Parma Comm. 2.997 2.997 2.959 4.2302.060

8 SW General 1.996 2.728 2.407 2.8051.816

6 7 8

6 MetroHealth .0007 Parma Comm. 3.601 .0008 SW General 3.010 1.945 .000

SPSS OUTPUT 4

The perceptual map is then presented and shown on the next page in SPSS Output

5. The interpretation of this perceptual map indicates that respondents perceive the


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advertisement diet & exercise to fit Southwest General Hospital. On the contrary,

respondents perceive the advertisement diet & exercise to be quite distinct from

Fairview.

The researcher should estimate the nature of the two dimensions. The

dimensions can be interpreted as yardsticks or criteria people use to judge the similarity

of the items. Respondents may differentiate the hospitals/ advertisements in regard to

where the hospitals are located, the quality of the care, the prestige of the hospital, etc.

We can develop a feel for the nature of a dimension by looking at where the hospital is

located on a dimension. Other and better ways of labeling dimensions involve more

complex statistical procedures beyond the goals of this report.


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Magic Bullet

Fairview

Parma Community

Diet and Exercise

Metrohealth

Heart Surgery

SW General

Heart Center

CMDS Hospitals by Ads


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Finally, the CMDS Output 6 provides the scatterplot of fit between the scaled

input data (horizontal axis) against the distances (vertical axis). That is, this diagram

represents the fit of the distances with the data. It is important to examine the scatter of

the objects along a perfect diagonal line running from the lower left to the upper right to

assess the fit of the data to the distances. Ideally, when there is a perfect fit, the

disparities and the distances will show a straight line of points. As the points diverge

from the straight line, the fit or accuracy of the map decreases. When stress levels are

very low, the points are close to the straight line. The worse the fit (and the higher the

stress), the more the points diverge from the straight line. In SPSS Output 6, the

scatter of the objects shows that the objects are not a very good fit.

SPSS OUTPUT 6

Scatterplot of Linear Fit

Euclidean distance model

Disparities

3.02.52.01.51.0.5

Distances

3.5

3.0

2.5

2.0

1.5

1.0

.5

0.0


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Besides looking at statistical indicators of fit, one should eyeball the matrix of

raw (unscaled) input data against the perceptual maps distances. It is important to

ensure that the input data match the resulting perceptual map, especially for critically

important objects (e.g. an advertisement under evaluation, the client hospital, etc.). For

example, looking again at the CMDS map on page 24 and the raw data matrix below, if

the client hospital is Southwest General Hospital, the closest advertisement is diet &

exercise. On the contrary, the advertisement magic bullet is farther away from

Southwest General Hospital. The input data should convey the same message through

the numbers. Therefore, the two matrices should be juxtaposed together to verify that

both show the same pattern.

MagicBullet

HeartSurgery

Diet &Exe.

HeartCenter

Fairview

Metro

SWGeneral

Parma

MB 0

HS 4.69 0

DE 4.44 4.02 0

HC 4.74 3.71 5.92 0

Fair 0 5.12 4.23 4.04 0

Met 5.12 0 5.43 4.97 4.06 0SWG

4.23 5.43 0 4.14 4.96 4.96 0

Par 4.04 4.97 4.14 0 4.18 4.75 4.50 0

CMDS Summary

In summary, the use of CMDS has both advantages and disadvantages to the

researcher.

First, it is especially useful in finding unique brand images and distinctive

product concepts.


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Second, it is easy to determine the fit, or lack of fit, of advertisements to

brands.

Third, CMDS can also identify the competitors of a brand (if by

competitor, we mean a brand perceived to be comparable).

Fourth, it is relatively simple to understand the output.

Overall, CMDS shows the uniqueness of an object based on specific dimensions, which

represent distinguishing attributes.

However, it can also present obstacles for the researcher.

First, the researcher doesnt know the nature of the dimensions unless

additional analyses are conducted to label the dimensions.

Second, CMDS does not directly show any differences in individual

respondents or segments because it aggregates everyone.

Third, the program you are using may not show the goodness of fit for a

single stimulus object, although it estimates for the objects as a group.

Fourth, it does not inform the researcher whether differing from another

brand in the set is good or bad for the brands image because CMDS does

not incorporate respondents preferences into the map.

Fifth, there is a problem of actionability. In many applications, it cannot

be the sole guide to strategy because it does not provide information on

how to change a brands image.

III. Weighted Multidimensional Scaling (WMDS)

A. General Rationale


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WMDS is based upon CMDS, but extends the simpler CMDS to allow for

individual segment differences.

WMDS generates a group space, a mapping that pertains in general to

all individuals/ segments. The group space (or common space) does not

show the uniqueness of a specific individual/ segment.

Separate spaces (maps) are produced for each individual or segment.

The group space mapping is adjusted (through stretching or shrinking of

the dimensions) in an attempt to capture the uniqueness of the judgments

of each individual/ segment.

The more an individual/ segment is estimated to differentiate among

objects on a given dimension, the more important is that dimension

assumed to be to that individual/ segment.

The spaces (maps) for the various individuals/ segments must have the

same dimensions. That is, the rank order of objects on a given dimension

(e.g. Dimension 1) is the same for each individual/ segment. So, for

example, Fairview is the highest of all the hospitals/ ads/ taglines on

Dimension 1. In the map of each and every individual/ segment, it will be

the highest on Dimension 1. The maps of the various individuals/

segments differ, though, in how much the objects are spread out on a

dimension. For example, Fairview may be higher than Parma Community

on Dimension 1 in all maps. But the distance between the two hospitals

on Dimension 1 may be very small for one individual/ segment and be

very great for another individual/ segment.


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WMDS can be run for individuals, in which case a separate data matrix is

required for each individual. Or, more popularly, WMDS can consider

differences among preselected segments. Individuals can be grouped

together based on a wide variety of factors.

For simplicity in our example, respondents are grouped together based on their

gender. Persons can also be grouped together based on comparability of their individual

level maps. The latter would be a three phase analysis: (a) do a WMDS, in which each

individual is treated separately; (b) in the resulting solution, group together those persons

who have similar maps into a reasonable number of segments; (c) do a second WMDS

assessing differences among the segments. In keeping with the primer goals of this

report, we only note this application in passing.

B. Data Needed

Demographics and general background questions were asked in the questionnaire.

These questions pertained to respondent age, income, level of education, adults in

household, gender, and ethnicity. One of these options can be used to separate into

segments. WMDS can place more or less weight on the variable (for example, income)

depending on the goals/ objectives of the analysis. We used gender as the criterion to

divide the respondents into segments for the WMDS solution. The data needed for

WMDS (measurement level, shape, and conditionality) is the same as the data needed for

CMDS.

C. SPSS Specific Steps

The analysis proceeded in the following steps. First, averaging across all males,

the mean for each paired comparison was entered into each cell of the first matrix. Again,


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the data below are hypothetical and represent the dissimilarity ratings between each pair

combination. The higher the number, the more dissimilar respondents perceived the two

items to be. The data in SPSS for the second matrix of the next segment (females) should

begin immediately following the conclusion of the preceding matrix as shown below.

A B C D E F G H

A 0

B 5 0

C 6 3 0

D 3 2 6 0

E 10 1 4 2 0

F 9 4 5 4 2 0G 8 5 10 3 8 7 0

H 7 8 8 6 5 4 5 0

A 0

B 2 0

C 5 9 0

D 6 8 7 0

E 3 3 8 4 0

F 2 3 6 6 4 0

G 5 6 4 8 6 6 0

H 7 6 9 2 5 2 6 0

The first step in conducting a WMDS analysis is under theAnalyze option. The

steps are exactly the same as above if you were conducting a CMDS solution; however,

the only difference is under theModeltab, in which Individual Differences Euclidean

Distance should be selected. Also, under the options tab, the researcher should specify

group plots, the data matrix, and the model and options summary. Individual subject

plots need not be selected, as it would be in a CMDS solution. Individual subject plots

show separate plots of each subjects data transformation for ordered categorical (ordinal)

data only.


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D. SPSS Output

Again, after analyzing the ratings and specifying separate 2 and 3 dimensional

solutions for advertisements by hospitals, the data in the two matrices was found to have

the best fit with a 3 dimensional solution. However, again we present a 2 dimensional

solution for illustrative purposes. The fit of the 2 dimensional solution would not be used

in practice because of the high stress and the low Pearson R correlation.

The output created by SPSS for a WMDS solution first shows the iteration history

of the solution. Youngs S-Stress formula, Kruskals stress formula, and the R squared

correlation are shown below in SPSS Output 8 for the WMDS solution.




0 .273081 .272512 .25375 .018763 .25254 .001224 .25236 .00017


.001000


RSQ values are the proportion of variance of the scaled

data (disparities)in the partition (row, matrix, or entire data)

whichis accounted for by their corresponding

distances.Stress values are Kruskal's stress formula 1.


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Matrix Stress RSQ Matrix StressRSQ

1 .243 .644 2 .197.777

Averaged (rms) over matricesStress = .22105 RSQ = .71037

SPSS OUTPUT 8

In SPSS Output 8, the program provides the separate S-Stress and R Squared

values for both male and female matrices as well as the combined S-Stress and R Squared

value, which represents the average subject. In our output, it is shown that the Stress

value indicates a bad solution. The R Squared value is also low. Again, we use this

example only for illustration. In practice, we would want a lower stress value and a

higher R Squared value.

Next, the display of the stimulus coordinates on each dimension is provided in

SPSS Output 9 below. The coordinates of each object are the coordinates used to create

plots in the combined map, or group space.

Configuration derived in 2 dimensionsStimulus Coordinates

Dimension

Stimulus 1 2Name Number

1 MAGIC BULLET -.3858 1.12612 HEART SURGERY -.6418 1.00863 DIET & EXERCISE -.3909 .91224 HEART CENTER -1.0139 -1.86315 FAIRVIEW 1.0387 -.77986 PARMA COMMUNITY -1.1670 -.34137 SW GENERAL 1.7995 .55378 METROHEALTH .7612 -.6164


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SPSS OUTPUT 9

The combined WMDS solution (group space) for both segments, male and

female, is provided next. The computer algorithm uses the group space coordinates to

plot the stimuli accordingly. Whatever dimensions both male and females use to

differentiate among the stimuli are shown in SPSS Output 10 on the next page.

This group space can be interpreted in the same manner as the CMDS, (i.e. in

terms of what points are close (similar) to an ad or to a hospital).

Keep in mind that this group space is derived by giving equal emphasis to each

segment. Thus, both male and female segments have equal say in devising the group

space. Even though there may be more males or more females in the sample, by devising

just two matrices (produced by averaging across everyone with the same gender), the

WMDS gives equal weight to the two matrices. If we would want to give more weight

(emphasis) to one gender rather than to another, we would have to do the additional step

of weighting the two matrices.

Hence, the group space in WMDS is not the same solution (map) as one would

generate by combining all respondents, male and female, into one matrix and then doing

a CMDS. The CMDS will give more emphasis to whatever gender has the larger sample.


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SPSS OUTPUT 10

The derived subject weights are plotted in SPSS Output 11. The program plots

the weights according to their location on the two dimensions. The space or map is

computed for each segment or individual. This is done by combining the subject (i.e.

individual/ segment) weights and the coordinates of the items in group space.

WMDS Euclidean Distance

Males and Females Group Space

Dimension 1

2.01.51.0.50.0-.5-1.0-1.5D

imension2

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

-2.0

metro

swgener

parmacom

fairvw

hrtcent

dietexehrtsurgmagbull


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SPSS OUTPUT 11

Next, the subject weights and weirdness index are provided by SPSS. Subject

weights measure the importance of each dimension to each individual/ segment. The

higher the weight, the more stretched out is a dimension; the smaller the weight

conversely, the more is that dimension shrunk for that individual/ segment. The

weirdness index reflects the atypicality of an individual/ segments space. It has values

between 0 and 1, where subjects with weights that are proportional to the average weights

has a weirdness of 0. A subject with one large weight and many low weights has a

weirdness near 1. A subject with exactly one positive weight also has a weirdness of 1.

The subject weights and weirdness index can be seen in SPSS Output 12.

Subject WeightsDimension

Subject Weirdness 1 2

1 .0759 .6155 .5146

WMDS

Subject Weights

Dimension 1

.74.72.70.68.66.64.62.60D

imension2

.52

.51

.50

.49

.48

2

1


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2 .0714 .7347 .4871Overall importance of

each dimension: .4594 .2510

SPSS OUTPUT 12

Using the stimulus coordinates in group space (Output 9) and the subject weights

(Output 12), Euclidean distance was figured by multiplying the square root of the weight

for each segment in weight space by the stimulus coordinate location on each

dimension from their shared common space. For example, the square root of the weight

for males on Dimension 1 was .6155

and on Dimension 2 was .5146. These weights

were multiplied by the stimulus coordinate location for Magic Bullet on both dimensions,

which were -.3858 & 1.1261. Therefore, (.6155*-.3858)

= -.3027 was the male

segment space for Magic Bullet on dimension 1 and (.5146*1.1261)

= .8078 was the

male segment space for Magic Bullet on dimension 2. This procedure was repeated for

all stimuli for males. Next, the female segment was figured in the same manner.

However, the female weights for dimensions 1 and 2 were .7347

and .4871. The

following table on the next page was found for each of the segments.

Males Dimension1

Dimension2

Females Dimension1

Dimension2

Magic Bullet -.3027 .8078 Magic Bullet -.3307 .7859HeartSurgery

-.5035 .7235 HeartSurgery

-.5501 .7039

Diet/Exercise

-.3067 .6544 Diet/Exercise

-.3351 .6366

HeartCenter

-.7954 -1.3365 Heart Center -.8691 -1.3003

Fairview .8149 -.5594 Fairview .8903 -.5442ParmaComm.

-.9156 -.2448 ParmaComm.

-1.0003 -.2382

SW General 1.4118 .3972 SW General 1.5424 .3864MetroHealth .5972 -.4422 MetroHealth .6525 -.4302


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Next, the male and female maps were plotted and are shown on the next two

pages. Each of the two maps is interpreted in the same way as the CMDS maps.

The two maps appear to be quite comparable, but not exactly the same. How

comparable are the two maps? To determine their comparability, one can correlate the

interpoint distances in one map with the interpoint distances in the other maps. That is,

we would calculate the distance between each of the possible pairs of points in one map

and then correlate that with the corresponding distance on the other map.

We first calculate the Euclidean distance separating all points on a map. There

are 28 pairs of the 8 items, so there are 28 interpoint distances on each map.

Magic BulletHeart Surgery

Diet and Exercise

Heart Center

Fairview

Parma Community

SW General

Metrohealth

WMDS Hospitals by Ads (Males)


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Magic Bullet

Heart Surgery

Heart Center Parma Community

Diet and Exercise

SW General

Fairview

Metrohealth

WMDS Hospitals by Ads (Females)


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For example, pair 1 was Magic Bullet and Heart Surgery. As shown in the

previous table, the subtracted distance between Magic Bullet (-.3027) and Heart Surgery

(-.5035) was .2008 for dimension 1. The subtracted distance for the same items (.8078

and .7235) was .0843 for dimension 2. The Euclidean formula was then used to

determine the straight line distance between the pairs. According to the example, this

would be ((.2008)2 + (.0843)2), which is .2178. This was repeated for all 28 pairs of the

8 stimuli and the following table was generated.

PAIRMALE

DISTANCESFEMALE

DISTANCES

Magic Bullet and Heart Surgery 0.2178 0.2342Magic Bullet and Diet and Exercise 1.535 0.1494Magic Bullet and Heart Center 2.002 2.1546Magic Bullet and Fairview 1.7659 1.8055Magic Bullet and ParmaCommunity 1.218 1.2236Magic Bullet and SouthwestGeneral 1.763 1.9152Magic Bullet and Metrohealth 1.5402 1.5638Heart Surgery and Diet andExercise 0.2086 0.2253

Heart Surgery and Heart Center 2.0806 2.0294Heart Surgery and Fairview 1.8396 1.9059Heart Surgery and ParmaCommunity 1.0523 1.0441Heart Surgery and SW General 1.9429 2.1165Heart Surgery and Metrohealth 1.6032 1.653Diet and Exercise and HeartCenter 2.05 2.0092Diet and Exercise and Fairview 1.6527 1.7017Diet and Exercise and ParmaCommunity 1.086 1.099Diet and Exercise and SW General 1.7376 1.8941Diet and Exercise and Metrohealth 1.4211 1.4538Heart Center and Fairview 1.788 1.915Heart Center and ParmaCommunity 1.0983 1.0702Heart Center and SW General 2.8067 2.9428Heart Center and Metrohealth 1.655 1.7528Fairview and Parma Community 1.7589 1.9152


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Fairview and SW General 1.1276 1.1363Fairview and Metrohealth 0.2472 0.2637Parma Community and SWGeneral 2.4143 2.6183Parma Community and

Metrohealth 1.5256 1.6639SW General and Metrohealth 1.1697 1.2078

Next, a simple Pearson R correlation was calculated between the male and female

groups. If a Pearson R correlation is high, it can be concluded that the two spaces (male

and female maps) are comparable. If a Pearson R correlation is low, it can be concluded

that there is a huge difference between the two matrices. In our data, it was found that

the two matrices were highly correlated at .10 and significant at the .942 level.

As in the CMDS solution, the WMDS Output 13 on the next page provides the

scatterplot of fit between the scaled input data (horizontal axis) against the distances

(vertical axis). This diagram represents the fit of the distances with the data. Again,

when the stress levels are low, the points are close to the straight line running from the

lower left-hand corner to the upper right-hand corner. The worse the fit, the more the

points diverge from the straight line.


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SPSS OUTPUT 13

The scatter of the objects shows that the objects are running along the straight

line, but not necessarily a good fit. In a professional or academic setting, we would want

to use a map only if it had less scatter than is displayed here.

The researcher can now interpret the separate maps of each segment and draw

action implications specific to each segment.

WMDS Summary

In summary, WMDS has both advantages and disadvantages to the researcher.

Let us first consider the advantages of using WMDS:

It is especially useful for comparing sectors of the population or

market in terms of the way they see particular objects.

WMDS

Scatterplot of Fit

Disparities

3.02.52.01.51.0.50.0Distances

3.0

2.5

2.0

1.5

1.0

.5

0.0


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In WMDS, the dimensions on the maps are exactly the same for all

segments. If a CMDS were to be calculated independently for each

segment, the dimensions may have completely different meanings for

each segment. This is because WMDS calculates the separate

segment solutions using the same dimensions whereas CMDS does

not.

The ease of interpretability is evident through the use of WMDS due to

the dimensions meaning the same thing for all segments.

Actionability is easier because it clarifies the orientations of different

segments of the population.

Finally, interpretation is the same as CMDS because all interpoint

distances between the objects are on the same scale of distance

between each other.

However, WMDS has its disadvantages as well.

First, WMDS cannot be used as a scaling technique if there are

dramatic differences between the matrices. It may be difficult for

WMDS to find common dimensions that work for the groups.

Next, WMDS indicates the perceived similarity of the stimuli, but

doesnt necessarily explain the basis of the perceived similarity

(dimensions/ attributes). The researcher will need additional

information in the survey to determine labels for the dimensions. One

can guess at the dimensions, but it is not advisable.


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APPENDIX A:

SPSS OUTPUT

Classic Multidimensional Scaling: Two Dimensions

Alscal Procedure Options

Data Options-

Number of Rows (Observations/Matrix). 8Number of Columns (Variables) . . . 8Number of Matrices . . . . . . 1Measurement Level . . . . . . . Interval

Data Matrix Shape . . . . . . . SymmetricType . . . . . . . . . . . DissimilarityApproach to Ties . . . . . . . Leave TiedConditionality . . . . . . . . MatrixData Cutoff at . . . . . . . . .000000

Model Options-

Model . . . . . . . . . . . EuclidMaximum Dimensionality . . . . . 2

Minimum Dimensionality . . . . . 2Negative Weights . . . . . . . Not Permitted

Output Options-

Job Option Header . . . . . . . PrintedData Matrices . . . . . . . . PrintedConfigurations and Transformations . PlottedOutput Dataset . . . . . . . . Not CreatedInitial Stimulus Coordinates . . . Computed

Algorithmic Options-

Maximum Iterations . . . . . . 30Convergence Criterion . . . . . .00100Minimum S-stress . . . . . . . .00500Missing Data Estimated by . . . . Ulbounds


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Raw (unscaled) Data for Subject 1

1 2 3 45

1 .0002 4.690 .0003 4.440 4.020 .0004 4.740 3.710 5.920 .0005 .000 5.120 4.230 4.040

.0006 5.120 .000 5.430 4.970

4.0607 4.230 5.430 .000 4.140

4.9608 4.040 4.970 4.140 .000

4.180

6 7 8

6 .0007 4.960 .0008 4.750 4.500 .000

_

Iteration history for the 2 dimensional solution (in

squared distances)



1 .372952 .34050 .032463 .33447 .006034 .33378 .00069


.001000



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is accounted for by their corresponding

distances. Stress values are Kruskal's stress formula 1.


_

Configuration derived in 2 dimensions

Stimulus Coordinates

Dimension


1 MAGBULL 1.1437 .7485

2 HRTSURG -1.3325 -.79543 DIETEXE .9450 -1.25164 HRTCENT -.8620 1.11935 FAIRVW .8389 .89956 METRO -1.3411 -.72827 PARMACOM .9433 -1.09708 SWGENER -.3353 1.1048

_


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1 2 3 45

1 .0002 2.317 .0003 2.229 2.081 .0004 2.335 1.971 2.751 .0005 .662 2.469 2.155 2.088

.0006 2.469 .662 2.578 2.416

2.0957 2.155 2.578 .662 2.123

2.4128 2.088 2.416 2.123 .662

2.137

6 7 8

6 .0007 2.412 .0008 2.338 2.250 .000

Classical Multidimensional Scaling CMDS

Hospitals by Advertisements

Dimension 1

1.51.0.50.0-.5-1.0-1.5

Dimension2

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

swgener

parmacom

metro

fairvwhrtcent

dietexe

hrtsurg

magbull


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Disparities

3.02.52.01.51.0.5

Distances

3.5

3.0

2.5

2.0

1.5

1.0

.5

0.0

Classic Multidimensional Scaling: Three Dimensions


Data Options-

Number of Rows (Observations/Matrix). 8Number of Columns (Variables) . . . 8Number of Matrices . . . . . . 1Measurement Level . . . . . . . IntervalData Matrix Shape . . . . . . . SymmetricType . . . . . . . . . . . DissimilarityApproach to Ties . . . . . . . Leave TiedConditionality . . . . . . . . MatrixData Cutoff at . . . . . . . . .000000

Model Options-

Model . . . . . . . . . . . EuclidMaximum Dimensionality . . . . . 3Minimum Dimensionality . . . . . 3Negative Weights . . . . . . . Not Permitted


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Output Options-

Job Option Header . . . . . . . PrintedData Matrices . . . . . . . . PrintedConfigurations and Transformations . Plotted

Output Dataset . . . . . . . . Not CreatedInitial Stimulus Coordinates . . . Computed


Maximum Iterations . . . . . . 30Convergence Criterion . . . . . .00100Minimum S-stress . . . . . . . .00500Missing Data Estimated by . . . . Ulbounds_

Raw (unscaled) Data for Subject 1

1 2 3 45

1 .0002 4.690 .000

3 4.440 4.020 .0004 4.740 3.710 5.920 .0005 .000 5.120 4.230 4.040

.0006 5.120 .000 5.430 4.970

4.0607 4.230 5.430 .000 4.140

4.9608 4.040 4.970 4.140 .000

4.180

6 7 8

6 .0007 4.960 .0008 4.750 4.500 .000

_


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1 .120732 .11583 .004903 .11551 .00032


.001000




is accounted for by their correspondingdistances.

Stress values are Kruskal's stress formula 1.


_


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Dimension

Stimulus Stimulus 1 2 3Number Name

1 MAGBULL -.6025 -.6484 1.43472 HRTSURG 1.5085 .9025 -.04353 DIETEXE -1.2598 1.2315 -.30784 HRTCENT .4510 -1.2801 -1.09875 FAIRVW -.3146 -.7253 1.44256 METRO 1.5989 .8474 .33097 PARMACOM -1.3418 .8997 -.69778 SWGENER -.0398 -1.2272 -1.0604

_


1 2 3 45

1 .0002 2.824 .0003 2.732 2.577 .0004 2.843 2.462 3.279 .0005 1.089 2.983 2.654 2.584

.0006 2.983 1.089 3.098 2.928

2.5917 2.654 3.098 1.089 2.621

2.924

8 2.584 2.928 2.621 1.0892.636

6 7 8

6 .0007 2.924 .0008 2.847 2.754 .000


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Derived Stimulus Configuration


metro

Dimension 2

fairvw

2.0 2.0

magbull

hrtsurg

-1.0

-.5

1.51.5

0.0

.5

1.01.0

1.0

1.5

hrtcent

.5.5

Dimension 3Dimension 1

swgener

0.00.0 -.5 -.5-1.0-1.0

dietexe

parmacom

Scatterplot of Linear FitEuclidean distance model

Disparities

3.53.02.52.01.51.0

Dis

tances

3.5

3.0

2.5

2.0

1.5

1.0

.5

0.0

Weighted Multidimensional Scaling: Two Dimensions


Data Options-

Number of Rows (Observations/Matrix). 8Number of Columns (Variables) . . . 8Number of Matrices . . . . . . 2


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Measurement Level . . . . . . . IntervalData Matrix Shape . . . . . . . SymmetricType . . . . . . . . . . . DissimilarityApproach to Ties . . . . . . . Leave TiedConditionality . . . . . . . . Matrix

Data Cutoff at . . . . . . . . .000000

Model Options-

Model . . . . . . . . . . . IndscalMaximum Dimensionality . . . . . 2Minimum Dimensionality . . . . . 2Negative Weights . . . . . . . Not Permitted

Output Options-

Job Option Header . . . . . . . PrintedData Matrices . . . . . . . . Not PrintedConfigurations and Transformations . PlottedOutput Dataset . . . . . . . . Not CreatedInitial Stimulus Coordinates . . . ComputedInitial Subject Weights . . . . . Computed


Maximum Iterations . . . . . . 30Convergence Criterion . . . . . .00100

Minimum S-stress . . . . . . . .00500Missing Data Estimated by . . . . Ulbounds_




0 .273081 .272512 .25375 .018763 .25254 .001224 .25236 .00017

Iterations stopped because S-stress improvement is lessthan .001000


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in the partition (row, matrix, or entire data)whichis accounted for by their corresponding

distances.Stress values are Kruskal's stress formula 1.


1 .243 .644 2 .197.777


_



Dimension


1 MAGBULL -.3858 1.12612 HRTSURG -.6418 1.00863 DIETEXE -.3909 .91224 HRTCENT -1.0139 -1.86315 FAIRVW 1.0387 -.77986 PARMACOM -1.1670 -.34137 SWGENER 1.7995 .55378 METRO .7612 -.6164

_

Subject weights measure the importance of each dimension toeach subject.Squared weights sum to RSQ.


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A subject with weights proportional to the average weightshas a weirdness ofzero, the minimum value.A subject with one large weight and many low weights has a

weirdness near one.A subject with exactly one positive weight has a weirdnessof one,the maximum value for nonnegative weights.

Subject Weights

DimensionSubject Weird- 1 2Number ness

1 .0759 .6155 .51462 .0714 .7347 .4871

Overall importance ofeach dimension: .4594 .2510_

Flattened Subject Weights

Variable

Subject Plot 1Number Symbol1 1 -1.00002 2 1.0000


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WMDS Euclidean Distance

Males and Females Group Space

Dimension 1

2.01.51.0.50.0-.5-1.0-1.5

Dimension2

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

-2.0

metro

swgener

parmacom

fairvw

hrtcent

dietexehrtsurg

magbull

WMDSSubject Weights

Dimension 1

.74.72.70.68.66.64.62.60

Dim

ension2

.52

.51

.50

.49

.48

2

1

WMDS

Scatterplot of Fit

Disparities

3.02.52.01.51.0.50.0

Distances

3.0

2.5

2.0

1.5

1.0

.5

0.0


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Individual differences (weighted) Euclidean di

One Dimensional Plot

.6.4.2-.0-.2-.4-.6

Variable1

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

2

1

Weighted Multidimensional Scaling: ThreeDimensions


Data Options-

Number of Rows (Observations/Matrix). 8Number of Columns (Variables) . . . 8Number of Matrices . . . . . . 2Measurement Level . . . . . . . IntervalData Matrix Shape . . . . . . . SymmetricType . . . . . . . . . . . DissimilarityApproach to Ties . . . . . . . Leave TiedConditionality . . . . . . . . MatrixData Cutoff at . . . . . . . . .000000

Model Options-

Model . . . . . . . . . . . IndscalMaximum Dimensionality . . . . . 3Minimum Dimensionality . . . . . 3Negative Weights . . . . . . . Not Permitted


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Output Options-

Job Option Header . . . . . . . Printed

Data Matrices . . . . . . . . Not PrintedConfigurations and Transformations . PlottedOutput Dataset . . . . . . . . Not CreatedInitial Stimulus Coordinates . . . ComputedInitial Subject Weights . . . . . Computed


Maximum Iterations . . . . . . 30Convergence Criterion . . . . . .00100Minimum S-stress . . . . . . . .00500Missing Data Estimated by . . . . Ulbounds_




0 .185621 .184672 .17557 .009113 .17387 .001704 .17235 .001525 .17094 .001426 .16971 .001227 .16876 .00095

Iterations stopped because

S-stress improvement is less than.001000



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is accounted for by their corresponding

distances. Stress values are Kruskal's stress formula 1.


1 .104 .853 2 .104.884


_



Dimension

Stimulus Stimulus 1 2 3

Number Name

1 MAGBULL .0752 1.1438 1.12472 HRTSURG -.6683 1.0738 -.96303 DIETEXE -.2696 1.1741 -.04474 HRTCENT -1.2907 -1.6781 .37955 FAIRVW .7337 -1.0397 -1.18136 PARMACOM -1.2219 -.1611 .78807 SWGENER 1.7006 -.0265 -1.40068 METRO .9412 -.4863 1.2974

_


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Subject weights measure the importance of each dimension toeach subject.Squared weights sum to RSQ.

A subject with weights proportional to the average weights

has a weirdness ofzero, the minimum value.A subject with one large weight and many low weights has aweirdness near one.A subject with exactly one positive weight has a weirdnessof one,the maximum value for nonnegative weights.

Subject Weights

DimensionSubject Weird- 1 2 3Number ness

1 .1742 .5804 .5666 .44202 .2027 .7548 .5105 .2326

Overall importance ofeach dimension: .4533 .2908 .1247_


VariableSubject Plot 1 2Number Symbol1 1 -1.0000 1.00002 2 1.0000 -1.0000


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Derived Stimulus Configuration

Individual differences (weighted) Euclidean distance

metro

Dimension 2

2.0 1.5

magbull

-1.5

-1.0

1.01.5

swgener

-.5

0.0

fairvw

.5

.51.0

hrtcent

1.0

parmacom

1.5

0.0.5

dietexe

Dimension 3Dimension 1

-.50.0 -.5 -1.0

hrtsurg

-1.5-1.0

Derived Subject WeightsIndividual differences (weighted) Euclidean distance

Dimension 2

.8 .5

.51

.522

.53

.54

.55

1

.56

.7 .4

.57

Dimension 3Dimension 1.6 .3


Individual differences (weighted) Euclidean dis

Disparities

3.02.52.01.51.0.5

Distances

3.5

3.0

2.5

2.0

1.5

1.0

.5


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Individual differences (weighted) Euclidean di

Variable 1

1.51.0.50.0-.5-1.0-1.5

Variable2

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

2

1

Documents

Perceptual mapping by multidimensional scaling MDS