Perceptual Mapping by Multidimensional
Scaling: A Step by Step Primer
By
Brian F. Blake, Ph.D.
Stephanie Schulze, M.A.
Jillian M. Hughes, M.A. Candidate
Methodology Series
September 2003
Cleveland State University
Brian F. Blake, Ph.D.
Senior Editor
Jillian M. Hughes
Co-Editor
Entire Series available: http://academic.csuohio.edu:8080/cbrsch/home.html
RESEARCH REPORTS IN CONSUMER BEHAVIOR
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AVAILABLE PUBLICATIONS:
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Lyttle, B. & Weizenecker, M. Focus groups: A basic introduction, February, 2005.
Arab, F., Blake, B.F., & Neuendorf, K.A. Attracting Internet shoppers in the Iranian
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Liu, C., Blake, B.F., & Neuendorf , K.A. Internet shopping in Taiwan and U.S.,
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Jurik, R., Blake, B.F., & Neuendorf, K.A. Attracting Internet shoppers in the
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Saaka, A., Sidon , C., & Blake, B.F. Laddering: A “How to do it” manual – with a
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Blake, B.F., Schulze, S., & Hughes, J.M. Perceptual mapping by multidimensional
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RESEARCH REPORTS IN CONSUMER BEHAVIOR
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
Cleveland’s 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.
Forward
Perceptual Maps are widely used by market researchers, e.g., to portray a brand’s
image or consumer’s 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 appeal or attractiveness 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.
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.
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) “It’s a people thing” 3) “When it’s your
health, experience counts” and 4) “Your partner in good health”. For this report, let us
abbreviate them as 1) “doctor’s 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 respondent’s 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”).
Classic Multidimensional Scaling CMDS
Hospitals by Advertisements
2.0
metro
1.5
swgener
1.0
fairview
hrtsurg
.5
Dimension 2
0.0
-.5
dietexe
parmacom
-1.0
magbull
hrtcent
-1.5
-2.0
-1.5
-1.0
-.5
0.0
.5
1.0
1.5
Dimension 1
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
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.
WMDS Hospitals by Ads (Males)
Magic Bullet
Heart Surgery
Diet and Exercise
SW General
Parma Community
Metrohealth
Fairview
Heart Center
WMDS Hospitals by Ads (Females)
SW General
Fairview
Metrohealth
Magic Bullet
Diet and Exercise
Heart Surgery
Heart Center
Parma Community
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 Blake’s 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.
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 person’s 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 respondent’s evaluations are desired, the
researcher should input the respondent’s judgments into a matrix where the units used for
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.
A
B
C
D
E
F
G
H
TABLE 1
A
0
6
8
5
3
5
4
6
B
C
D
E
F
G
H
0
10
6
8
6
7
6
0
9
10
3
6
10
0
4
7
9
7
0
5
2
4
0
3
8
0
2
0
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 doesn’t 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 X plus 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
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 C’s 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
Euclidean Distance
3.5
3
2.5
Y
2
A
C
1.5
1
0.5
0
0
1
2
3
4
5
X
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.
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 respondent’s options were: “A)
I have never seen it before, and know nothing about it, B) I have seen it before, but don’t
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 it’s a tagline for _______ hospital.” This questioning was
repeated for the other three taglines, which again were ‘It’s a people thing’, ‘When it’s
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 ‘It’s 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 respondent’s options
were: “A) I have never seen it before, and know nothing about it, B) I have seen it
before, but don’t 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 it’s 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 respondent’s options were: “A) I know
practically nothing about it, B) I have heard the name, but don’t 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.
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 “doctor’s 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
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
After creating the desired matrix, the next step in conducting a CMDS analysis is
under the Analyze option at the top of the SPSS screen. After selecting Analyze, the
researcher should select the Scale option. Under the scale option, the analyst will find
Multidimensional Scaling. Next, a box will appear illustrating the objects on the lefthand side. Each object should be highlighted and moved over to the empty area on the
right-hand side. In the lower right-hand corner of the square, one will find the choices
model and options. Under the model selection, the researcher can specify the desired
level of measurement. Ratings or numerical scales, such as here, are typically treated as
interval. There are several measures of distance; the most popular one in consumer
research is Euclidean. For conditionality, select matrix. Finally, the researcher should
enter the desired number of dimensions. The number of dimensions that will be
employed in the final solution will depend upon the interpretability of the solution and
upon the stress measure. However, it is recommended to begin with two dimensions and
then to increase the dimensions to three. You will be looking for the dimensionality that
is better in regard to interpretability and to stress, a measure of goodness of fit of the
mapping solution to the data. Dimensionalities higher than three are rarely employed in
perceptual mapping. In professional applications, researchers strongly prefer the simpler
two dimensional solution.
Finally, the options selection allows the researcher to specify group plots,
individual subject plots, the data matrix, and the model and options summary. In the
initial analyses of the data, all of these choices should be selected and included in the
SPSS output. The criteria box simply shows the options that are already set by default in
the SPSS program. The researcher does not need to manually select any of these options
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
4
5
1
2
3
4
5
.000
Magic Bullet
Heart Surgery
Diet & Exercise
Heart Center
Fairview General
.000
4.690
4.440
4.740
.000
.000
4.020
3.710
5.120
.000
5.920
4.230
.000
4.040
6 MetroHealth
5.120
.000
5.430
4.970
7 Parma Comm.
4.230
5.430
.000
4.140
8 SouthWest Gen.
4.040
4.970
4.140
.000
4.060
4.960
4.180
6
6 MetroHealth
7 Parma Comm.
8 SouthWest Gen.
SPSS OUTPUT 1
.000
4.960
4.750
7
.000
4.500
8
.000
CMDS (as well as the other variants of MDS) proceeds in a series of steps. The
SPSS output provides the iteration history of the solution. Iterations as the original input
data are transformed step-by-step to produce the final solution (map). Next, stress tests
and the RSQ value (i.e., r-squared correlation) are also shown in SPSS Output 2 on the
next page. Young’s S-Stress formula is a measure of statistical fit that ranges from 1
indicating the worst possible fit to 0 indicating a perfect fit. It can be seen that there is an
improvement (decrease) in Young’s S-Stress as the iterations proceed. The iterations will
continue until there is no more improvement in S-Stress or until the specified number of
iterations is made. SPSS by default sets a maximum of 30 iterations. A value of .10 or
less is considered a good fit for two dimensions; a lower stress (.07 or so) is considered
good for a three dimensional solution. The RSQ value is the squared correlation
coefficient between the distances and the data, and it is the variance accounted for in the
solution. The RSQ and Kruskal’s stress index are used as the measures of goodness of fit
of the solution. Here, the stress is too high and the RSQ is too low for comfort. In
practice, we would want a better fit.
Iteration history for the 2 dimensional solution (in
squared distances)
Young's S-stress formula 1 is used.
Iteration
S-stress
Improvement
1
2
3
4
.37295
.34050
.33447
.33378
.03246
.00603
.00069
Iterations stopped because
S-stress improvement is less than
.001000
Stress and squared correlation (RSQ) in
distances
RSQ values are the proportion of variance of the scaled
data (disparities)
in the partition (row, matrix, or entire data)
which
is accounted for by their corresponding
distances.
Stress values are Kruskal's stress formula 1.
Stress =
SPSS OUTPUT 2
For matrix
.20196
RSQ =
.75785
The 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 dimensions
Stimulus Coordinates
Dimension
Stimulus
Number
1
2
Stimulus
Name
MAGIC BULLET
HEART SURGERY
1
2
1.1437
-1.3325
.7485
-.7954
3
DIET & EXERCISE
4
HEART CENTER
5
FAIRVIEW
6
METROHEALTH
7
PARMA COMMUNITY
8
SW GENERAL
SPSS OUTPUT 3
.9450
-.8620
.8389
-1.3411
.9433
-.3353
-1.2516
1.1193
.8995
-.7282
-1.0970
1.1048
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) for
subject
1
1
2
3
4
.000
2.137
1.033
3.370
2.843
.000
1.033
3.010
2.651
.000
3.280
2.330
.000
2.715
6 MetroHealth
1.842
1.996
1.791
1.393
7 Parma Comm.
2.997
2.997
2.959
4.230
8 SW General
1.996
2.728
2.407
2.805
6
7
8
.000
3.601
3.010
.000
1.945
.000
5
1
2
3
4
5
Magic Bullet
Heart Surgery
Diet & Exercise
Heart Center
Fairview General
.000
3.203
2.060
1.816
6
7
8
MetroHealth
Parma Comm.
SW General
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
advertisement “diet & exercise” to fit Southwest General Hospital. On the contrary,
respondent’s 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.
CMDS Hospitals by Ads
Heart Center
SW General
Fairview
Magic Bullet
Metrohealth
Heart Surgery
Parma Community
Diet and Exercise
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.
Scatterplot of Linear Fit
Euclidean distance model
3.5
3.0
2.5
2.0
1.5
Distances
1.0
.5
0.0
.5
1.0
Disparities
SPSS OUTPUT 6
1.5
2.0
2.5
3.0
Besides looking at statistical indicators of fit, one should “eyeball” the matrix of
raw (unscaled) input data against the perceptual map’s 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.
Magic
Bullet
MB
HS
DE
HC
Fair
Met
SW
G
Par
Heart
Surgery
Diet &
Exe.
Heart
Center
Fairvie Metr SW
w
o
General
Parm
a
0
4.69
4.44
4.74
0
5.12
4.23
0
4.02
3.71
5.12
0
5.43
0
5.92
4.23
5.43
0
0
4.04
4.97
4.14
0
4.06
0
4.96 4.96
0
4.04
4.97
4.14
0
4.18 4.75
4.50
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.
0
♦
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 doesn’t 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 brand’s image because CMDS does
not incorporate respondent’s 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 brand’s image.
III.
A.
Weighted Multidimensional Scaling (WMDS)
General Rationale
♦
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.
♦
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,
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
0
5
6
3
10
9
8
7
0
2
5
6
3
2
5
7
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
B
C
D
E
F
G
H
0
3
2
1
4
5
8
0
6
4
5
10
8
0
2
4
3
6
0
2
8
5
0
7
4
0
5
0
0
9
8
3
3
6
6
0
7
8
6
4
9
0
4
6
8
2
0
4
6
5
0
6
2
0
6
0
The first step in conducting a WMDS analysis is under the Analyze option. The
steps are exactly the same as above if you were conducting a CMDS solution; however,
the only difference is under the Model tab, 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 subject’s data transformation for ordered categorical (ordinal)
data only.
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. Young’s S-Stress formula, Kruskal’s stress formula, and the R squared
correlation are shown below in SPSS Output 8 for the WMDS solution.
Iteration history for the 2 dimensional solution (in
squared distances)
Young's S-stress formula 1 is used.
Iteration
0
1
2
3
4
S-stress
.27308
.27251
.25375
.25254
.25236
Improvement
.01876
.00122
.00017
Iterations stopped because
S-stress improvement is less than
.001000
Stress and squared correlation (RSQ) in
distances
RSQ values are the proportion of variance of the scaled
data (disparities)
in the partition (row, matrix, or entire data)
which
is accounted for by their corresponding
distances.
Stress values are Kruskal's stress formula 1.
Matrix
Stress
RSQ
Matrix
.243
.644
2
Stress
RSQ
1
.197
.777
Averaged (rms) over matrices
Stress =
.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 dimensions
Stimulus Coordinates
Dimension
Stimulus
1
Name
1
2
3
4
5
6
7
8
MAGIC BULLET
HEART SURGERY
DIET & EXERCISE
HEART CENTER
FAIRVIEW
PARMA COMMUNITY
SW GENERAL
METROHEALTH
2
Number
-.3858
-.6418
-.3909
-1.0139
1.0387
-1.1670
1.7995
.7612
1.1261
1.0086
.9122
-1.8631
-.7798
-.3413
.5537
-.6164
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.
WMDS Euclidean Distance
Males and Females Group Space
1.5
magbull
hrtsurg
dietexe
1.0
swgener
.5
0.0
parmacom
metro
fairvw
Dimension 2
-.5
-1.0
-1.5
hrtcent
-2.0
-1.5
-1.0
-.5
0.0
.5
1.0
1.5
2.0
Dimension 1
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
Subject Weights
.52
1
.51
Dimension 2
.50
.49
2
.48
.60
.62
.64
.66
.68
.70
.72
.74
Dimension 1
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/ segment’s 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 Weights
Dimension
Subject Weirdness
1
.0759
.6155
1
.5146
2
2
.0714
.7347
.4871
Overall 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 .6155½ and on Dimension 2 was .5146½. These weights
were multiplied by the stimulus coordinate location for Magic Bullet on both dimensions,
which were -.3858 & 1.1261. Therefore, (.6155*-.3858) ½ = -.3027 was the male
segment space for Magic Bullet on dimension 1 and (.5146*1.1261) ½ = .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 .7347½ and .4871½. The
following table on the next page was found for each of the segments.
Males
Magic Bullet
Heart
Surgery
Diet/Exercis
e
Heart
Center
Fairview
Parma
Comm.
SW General
MetroHealth
Dimension Dimension
1
2
-.3027
-.5035
.8078
.7235
-.3067
.6544
-.7954
-1.3365
.8149
-.9156
-.5594
-.2448
1.4118
.5972
.3972
-.4422
Females
Dimension Dimension
1
2
Magic Bullet
Heart
Surgery
Diet/Exercis
e
Heart Center
-.3307
-.5501
.7859
.7039
-.3351
.6366
-.8691
-1.3003
Fairview
Parma
Comm.
SW General
MetroHealth
.8903
-1.0003
-.5442
-.2382
1.5424
.6525
.3864
-.4302
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.
WMDS Hospitals by Ads (Males)
Magic Bullet
Heart Surgery
Diet and Exercise
SW General
Parma Community
Metrohealth
Fairview
Heart Center
WMDS Hospitals by Ads (Females)
SW General
Fairview
Metrohealth
Magic Bullet
Diet and Exercise
Heart Surgery
Heart Center
Parma Community
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)½ , which is .2178. This was repeated for all 28 pairs of the
8 stimuli and the following table was generated.
PAIR
Magic Bullet and Heart Surgery
Magic Bullet and Diet and Exercise
Magic Bullet and Heart Center
Magic Bullet and Fairview
Magic Bullet and Parma
Community
Magic Bullet and Southwest
General
Magic Bullet and Metrohealth
Heart Surgery and Diet and
Exercise
Heart Surgery and Heart Center
Heart Surgery and Fairview
Heart Surgery and Parma
Community
Heart Surgery and SW General
Heart Surgery and Metrohealth
Diet and Exercise and Heart
Center
Diet and Exercise and Fairview
Diet and Exercise and Parma
Community
Diet and Exercise and SW General
Diet and Exercise and Metrohealth
Heart Center and Fairview
Heart Center and Parma
Community
Heart Center and SW General
Heart Center and Metrohealth
Fairview and Parma Community
MALE
DISTANCES
0.2178
1.535
2.002
1.7659
FEMALE
DISTANCES
0.2342
0.1494
2.1546
1.8055
1.218
1.2236
1.763
1.5402
1.9152
1.5638
0.2086
2.0806
1.8396
0.2253
2.0294
1.9059
1.0523
1.9429
1.6032
1.0441
2.1165
1.653
2.05
1.6527
2.0092
1.7017
1.086
1.7376
1.4211
1.788
1.099
1.8941
1.4538
1.915
1.0983
2.8067
1.655
1.7589
1.0702
2.9428
1.7528
1.9152
Fairview and SW General
Fairview and Metrohealth
Parma Community and SW
General
Parma Community and
Metrohealth
SW General and Metrohealth
1.1276
0.2472
1.1363
0.2637
2.4143
2.6183
1.5256
1.1697
1.6639
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.
WMDS
Scatterplot of Fit
3.0
2.5
2.0
1.5
Distances
1.0
.5
0.0
0.0
.5
1.0
1.5
2.0
2.5
3.0
Disparities
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.
♦ 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
doesn’t 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.
♦ Respondent fatigue may occur during the questionnaire process
because of the repeated paired comparisons. This problem holds for
CMDS also.
♦ Finally, WMDS does not indicate the degree of preference for the
stimuli; it only indicates similarity among the objects. It suggests,
then, what people see, but not what they want.
MDS Conclusion
CMDS and WMDS are useful when an investigator is interested in perceived
similarity or perceived fit between one set of items and another set. In a professional
setting, though, additional information to label the perceptual dimensions and to assess
the preferences of the market would typically be necessary to make the CMDS/ WMDS
maps the basis for action.
APPENDIX A:
SPSS OUTPUT
Classic Multidimensional Scaling: Two Dimensions
Alscal Procedure Options
Data OptionsNumber of Rows (Observations/Matrix).
Number of Columns (Variables) . . .
Number of Matrices
. . . . . .
Measurement Level . . . . . . .
Data Matrix Shape . . . . . . .
Type . . . . . . . . . . .
Approach to Ties . . . . . . .
Conditionality . . . . . . . .
Data Cutoff at . . . . . . . .
8
8
1
Interval
Symmetric
Dissimilarity
Leave Tied
Matrix
.000000
Model OptionsModel . . . . . .
Maximum Dimensionality
Minimum Dimensionality
Negative Weights . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Euclid
2
2
Not Permitted
Job Option Header . . . . . .
Data Matrices . . . . . . .
Configurations and Transformations
Output Dataset . . . . . . .
Initial Stimulus Coordinates . .
.
.
.
.
.
Printed
Printed
Plotted
Not Created
Computed
.
.
.
.
30
.00100
.00500
Ulbounds
Output Options-
Algorithmic OptionsMaximum Iterations
. .
Convergence Criterion
.
Minimum S-stress . . .
Missing Data Estimated by
.
.
.
.
.
.
.
.
.
.
.
.
Raw (unscaled) Data for Subject 1
1
2
3
4
5
1
2
3
4
5
.000
4.690
4.440
4.740
.000
.000
4.020
3.710
5.120
.000
5.920
4.230
.000
4.040
6
5.120
.000
5.430
4.970
7
4.230
5.430
.000
4.140
8
4.040
4.970
4.140
.000
.000
4.060
4.960
4.180
6
6
7
8
.000
4.960
4.750
7
.000
4.500
8
.000
_
Iteration history for the 2 dimensional solution (in
squared distances)
Young's S-stress formula 1 is used.
Iteration
1
2
3
4
S-stress
.37295
.34050
.33447
.33378
Improvement
.03246
.00603
.00069
Iterations stopped because
S-stress improvement is less than
.001000
Stress and squared correlation (RSQ) in
distances
RSQ values are the proportion of variance of the scaled
data (disparities)
in the partition (row, matrix, or entire data)
which
is accounted for by their corresponding
distances.
Stress values are Kruskal's stress formula 1.
Stress
=
For matrix
.20196
RSQ =
.75785
_
Configuration derived in 2 dimensions
Stimulus Coordinates
Dimension
Stimulus
Number
Stimulus
Name
1
2
1
2
3
4
5
6
7
8
MAGBULL
HRTSURG
DIETEXE
HRTCENT
FAIRVW
METRO
PARMACOM
SWGENER
1.1437
-1.3325
.9450
-.8620
.8389
-1.3411
.9433
-.3353
.7485
-.7954
-1.2516
1.1193
.8995
-.7282
-1.0970
1.1048
_
Optimally scaled data (disparities) for
subject
1
1
2
3
4
5
1
2
3
4
5
.000
2.317
2.229
2.335
.662
.000
2.081
1.971
2.469
.000
2.751
2.155
.000
2.088
6
2.469
.662
2.578
2.416
7
2.155
2.578
.662
2.123
8
2.088
2.416
2.123
.662
.000
2.095
2.412
2.137
6
6
7
8
7
.000
2.412
2.338
8
.000
2.250
Classical Multidimensional Scaling CMDS
Hospitals by Advertisements
1.5
hrtcent
swgener
fairvw
1.0
magbull
.5
Dimension 2
0.0
-.5
metro
hrtsurg
parmacom
dietexe
-1.0
-1.5
-1.5
-1.0
Dimension 1
-.5
0.0
.5
1.0
1.5
.000
Scatterplot of Linear Fit
Euclidean distance model
3.5
3.0
2.5
2.0
1.5
Distances
1.0
.5
0.0
.5
1.0
1.5
2.0
2.5
3.0
Disparities
Classic Multidimensional Scaling: Three Dimensions
Alscal Procedure Options
Data OptionsNumber of Rows (Observations/Matrix).
Number of Columns (Variables) . . .
Number of Matrices
. . . . . .
Measurement Level . . . . . . .
Data Matrix Shape . . . . . . .
Type . . . . . . . . . . .
Approach to Ties . . . . . . .
Conditionality . . . . . . . .
Data Cutoff at . . . . . . . .
8
8
1
Interval
Symmetric
Dissimilarity
Leave Tied
Matrix
.000000
Model OptionsModel . . . . . .
Maximum Dimensionality
Minimum Dimensionality
Negative Weights . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Euclid
3
3
Not Permitted
Output OptionsJob Option Header . . . . . .
Data Matrices . . . . . . .
Configurations and Transformations
Output Dataset . . . . . . .
Initial Stimulus Coordinates . .
.
.
.
.
.
Printed
Printed
Plotted
Not Created
Computed
.
.
.
.
30
.00100
.00500
Ulbounds
Algorithmic OptionsMaximum Iterations
. .
Convergence Criterion
.
Minimum S-stress . . .
Missing Data Estimated by
_
.
.
.
.
.
.
.
.
.
.
.
.
Raw (unscaled) Data for Subject 1
1
2
3
4
5
1
2
3
4
5
.000
4.690
4.440
4.740
.000
.000
4.020
3.710
5.120
.000
5.920
4.230
.000
4.040
6
5.120
.000
5.430
4.970
7
4.230
5.430
.000
4.140
8
4.040
4.970
4.140
.000
.000
4.060
4.960
4.180
6
6
7
8
_
.000
4.960
4.750
7
.000
4.500
8
.000
Iteration history for the 3 dimensional solution (in
squared distances)
Young's S-stress formula 1 is used.
Iteration
1
2
3
S-stress
.12073
.11583
.11551
Improvement
.00490
.00032
Iterations stopped because
S-stress improvement is less than
.001000
Stress and squared correlation (RSQ) in
distances
RSQ values are the proportion of variance of the scaled
data (disparities)
in the partition (row, matrix, or entire data)
which
is accounted for by their corresponding
distances.
Stress values are Kruskal's stress formula 1.
Stress
_
=
For matrix
.10963
RSQ =
.95017
Configuration derived in 3 dimensions
Stimulus Coordinates
Dimension
Stimulus
Number
Stimulus
Name
1
2
3
1
2
3
4
5
6
7
8
MAGBULL
HRTSURG
DIETEXE
HRTCENT
FAIRVW
METRO
PARMACOM
SWGENER
-.6025
1.5085
-1.2598
.4510
-.3146
1.5989
-1.3418
-.0398
-.6484
.9025
1.2315
-1.2801
-.7253
.8474
.8997
-1.2272
1.4347
-.0435
-.3078
-1.0987
1.4425
.3309
-.6977
-1.0604
_
Optimally scaled data (disparities) for
subject
1
1
2
3
4
5
1
2
3
4
5
.000
2.824
2.732
2.843
1.089
.000
2.577
2.462
2.983
.000
3.279
2.654
.000
2.584
6
2.983
1.089
3.098
2.928
7
2.654
3.098
1.089
2.621
8
2.584
2.928
2.621
1.089
.000
2.591
2.924
2.636
6
6
7
8
.000
2.924
2.847
7
.000
2.754
8
.000
Derived Stimulus Configuration
Euclidean distance model
1.5
metro
hrtsurg
dietexe
1.0
parmacom
.5
Dimension 2
0.0
magbull
fairvw
-.5
-1.0
hrtcent
swgener
2.0 1.5
1.0 .5
0.0 -.5
-1.0
Dimension 1
0.0
-1.0-.5
1.5
.5 1.0
2.0
Dimension 3
Scatterplot of Linear Fit
Euclidean distance model
3.5
3.0
2.5
2.0
1.5
Distances
1.0
.5
0.0
1.0
1.5
2.0
2.5
3.0
3.5
Disparities
Weighted Multidimensional Scaling: Two Dimensions
Alscal Procedure Options
Data OptionsNumber of Rows (Observations/Matrix).
Number of Columns (Variables) . . .
Number of Matrices
. . . . . .
8
8
2
Measurement Level
Data Matrix Shape
Type . . . .
Approach to Ties
Conditionality .
Data Cutoff at .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Interval
Symmetric
Dissimilarity
Leave Tied
Matrix
.000000
Model . . . . . .
Maximum Dimensionality
Minimum Dimensionality
Negative Weights . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Indscal
2
2
Not Permitted
Job Option Header . . . . . .
Data Matrices . . . . . . .
Configurations and Transformations
Output Dataset . . . . . . .
Initial Stimulus Coordinates . .
Initial Subject Weights . . . .
.
.
.
.
.
.
Printed
Not Printed
Plotted
Not Created
Computed
Computed
.
.
.
.
30
.00100
.00500
Ulbounds
Model Options-
Output Options-
Algorithmic OptionsMaximum Iterations
. .
Convergence Criterion
.
Minimum S-stress . . .
Missing Data Estimated by
_
.
.
.
.
.
.
.
.
.
.
.
.
Iteration history for the 2 dimensional solution (in
squared distances)
Young's S-stress formula 1 is used.
Iteration
S-stress
Improvement
0
.27308
1
.27251
2
.25375
.01876
3
.25254
.00122
4
.25236
.00017
Iterations stopped because S-stress improvement is less
than
.001000
Stress and squared correlation (RSQ) in
distances
RSQ values are the proportion of variance of the scaled
data (disparities)
in the partition (row, matrix, or entire data)
which
is accounted for by their corresponding
distances.
Stress values are Kruskal's stress formula 1.
Matrix
Stress
RSQ
Matrix
.243
.644
2
Stress
RSQ
1
.197
.777
Averaged (rms) over matrices
Stress =
.22105
RSQ = .71037
_
Configuration derived in 2 dimensions
Stimulus Coordinates
Dimension
Stimulus
Number
Stimulus
Name
1
2
1
2
3
4
5
6
7
8
MAGBULL
HRTSURG
DIETEXE
HRTCENT
FAIRVW
PARMACOM
SWGENER
METRO
-.3858
-.6418
-.3909
-1.0139
1.0387
-1.1670
1.7995
.7612
1.1261
1.0086
.9122
-1.8631
-.7798
-.3413
.5537
-.6164
_
Subject weights measure the importance of each dimension to
each subject.
Squared weights sum to RSQ.
A subject with weights proportional to the average weights
has a weirdness of
zero, 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 weirdness
of one,
the maximum value for nonnegative weights.
Subject Weights
Subject
Number
Weirdness
1
1
.0759
.6155
2
.0714
.7347
Overall importance of
each dimension:
.4594
_
Dimension
2
.5146
.4871
.2510
Flattened Subject Weights
Subject
Number
1
2
Variable
Plot
1
Symbol
1
-1.0000
2
1.0000
WMDS Euclidean Distance
Males and Females Group Space
1.5
magbull
hrtsurg
dietexe
1.0
swgener
.5
0.0
parmacom
metro
fairvw
Dimension 2
-.5
-1.0
-1.5
hrtcent
-2.0
-1.5
-1.0
-.5
0.0
.5
1.0
1.5
2.0
Dimension 1
WMDS
Subject Weights
.52
1
.51
Dimension 2
.50
.49
2
.48
.60
.62
.64
.66
.68
.70
.72
.74
Dimension 1
WMDS
Scatterplot of Fit
3.0
2.5
2.0
1.5
Distances
1.0
.5
0.0
0.0
.5
Disparities
1.0
1.5
2.0
2.5
3.0
Flattened Subject Weights
Individual differences (weighted) Euclidean distance model
1.5
2
1.0
.5
0.0
Variable 1
-.5
1
-1.0
-1.5
-.6
-.4
-.2
-.0
.2
.4
.6
One Dimensional Plot
Weighted Multidimensional Scaling: Three
Dimensions
Alscal Procedure Options
Data OptionsNumber of Rows (Observations/Matrix).
Number of Columns (Variables) . . .
Number of Matrices
. . . . . .
Measurement Level . . . . . . .
Data Matrix Shape . . . . . . .
Type . . . . . . . . . . .
Approach to Ties . . . . . . .
Conditionality . . . . . . . .
Data Cutoff at . . . . . . . .
8
8
2
Interval
Symmetric
Dissimilarity
Leave Tied
Matrix
.000000
Model OptionsModel . . . . . .
Maximum Dimensionality
Minimum Dimensionality
Negative Weights . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Indscal
3
3
Not Permitted
Output OptionsJob Option Header . . . . . .
Data Matrices . . . . . . .
Configurations and Transformations
Output Dataset . . . . . . .
Initial Stimulus Coordinates . .
Initial Subject Weights . . . .
.
.
.
.
.
.
Printed
Not Printed
Plotted
Not Created
Computed
Computed
.
.
.
.
30
.00100
.00500
Ulbounds
Algorithmic OptionsMaximum Iterations
. .
Convergence Criterion
.
Minimum S-stress . . .
Missing Data Estimated by
_
.
.
.
.
.
.
.
.
.
.
.
.
Iteration history for the 3 dimensional solution (in
squared distances)
Young's S-stress formula 1 is used.
Iteration
0
1
2
3
4
5
6
7
S-stress
.18562
.18467
.17557
.17387
.17235
.17094
.16971
.16876
Improvement
.00911
.00170
.00152
.00142
.00122
.00095
Iterations stopped because
S-stress improvement is less than
.001000
Stress and squared correlation (RSQ) in
distances
RSQ values are the proportion of variance of the scaled
data (disparities)
in the partition (row, matrix, or entire data)
which
is accounted for by their corresponding
distances.
Stress values are Kruskal's stress formula 1.
Matrix
Stress
RSQ
Matrix
.104
.853
2
Stress
RSQ
1
.104
.884
Averaged (rms) over matrices
Stress =
.10424
RSQ = .86887
_
Configuration derived in 3 dimensions
Stimulus Coordinates
Dimension
Stimulus
Number
Stimulus
Name
1
2
3
1
2
3
4
5
6
7
8
MAGBULL
HRTSURG
DIETEXE
HRTCENT
FAIRVW
PARMACOM
SWGENER
METRO
.0752
-.6683
-.2696
-1.2907
.7337
-1.2219
1.7006
.9412
1.1438
1.0738
1.1741
-1.6781
-1.0397
-.1611
-.0265
-.4863
1.1247
-.9630
-.0447
.3795
-1.1813
.7880
-1.4006
1.2974
_
Subject weights measure the importance of each dimension to
each subject.
Squared weights sum to RSQ.
A subject with weights proportional to the average weights
has a weirdness of
zero, 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 weirdness
of one,
the maximum value for nonnegative weights.
Subject Weights
Subject
Number
Weirdness
Dimension
2
3
1
1
.1742
.5804
2
.2027
.7548
Overall importance of
each dimension:
.4533
_
.5666
.5105
.4420
.2326
.2908
.1247
Flattened Subject Weights
Subject
Number
1
2
Variable
Plot
1
2
Symbol
1
-1.0000
1.0000
2
1.0000 -1.0000
Derived Stimulus Configuration
Individual differences (weighted) Euclidean distance model
magbull
dietexe
hrtsurg
1.5
1.0
.5
Dimension 2
swgener
metro
0.0
parmacom
-.5
fairvw
-1.0
-1.5
hrtcent
2.0 1.5
1.0 .5
0.0 -.5
-1.0
Dimension 1
-.5 0.0
-1.5-1.0
1.5
.5 1.0
Dimension 3
Derived Subject Weights
Individual differences (weighted) Euclidean distance model
1
.57
.56
.55
Dimension 2
.54
.53
.52
2
.51
.8
.7
Dimension 1
.6
.3
.4
.5
Dimension 3
Scatterplot of Linear Fit
Individual differences (weighted) Euclidean distance model
3.5
3.0
2.5
2.0
Distances
1.5
1.0
.5
.5
Disparities
1.0
1.5
2.0
2.5
3.0
Flattened Subject Weights
Individual differences (weighted) Euclidean distance model
1.5
1
1.0
.5
0.0
Variable 2
-.5
2
-1.0
-1.5
-1.5
-1.0
Variable 1
-.5
0.0
.5
1.0
1.5