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 Entire Series available: http://academic.csuohio.edu:8080/cbrsch/home.html Brian F. Blake, Ph.D. Jillian M. Hughes Senior Editor Co-Editor
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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
Entire Series available: http://academic.csuohio.edu:8080/cbrsch/home.html
Brian F. Blake, Ph.D. Jillian M. Hughes Senior Editor Co-Editor
RRRRESEARCH ESEARCH ESEARCH ESEARCH RRRREPORTS IN EPORTS IN EPORTS IN EPORTS IN CCCCONSUMER ONSUMER ONSUMER ONSUMER BBBBEHAVIOREHAVIOREHAVIOREHAVIOR
These analyses address issues of concern to marketing and advertising professionals and to
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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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� Blake, B.F., Valdiserri, J., Neuendorf, K.A., & Nemeth, J. Validity of the SDS-17
measure of social desirability in the American context, November, 2005.
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 .37295 2 .34050 .03246 3 .33447 .00603 4 .33378 .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. For matrix Stress = .20196 RSQ = .75785
SPSS OUTPUT 2
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 Stimulus 1 2 Number Name 1 MAGIC BULLET 1.1437 .7485 2 HEART SURGERY -1.3325 -.7954
3 DIET & EXERCISE .9450 -1.2516 4 HEART CENTER -.8620 1.1193 5 FAIRVIEW .8389 .8995 6 METROHEALTH -1.3411 -.7282 7 PARMA COMMUNITY .9433 -1.0970 8 SW GENERAL -.3353 1.1048
SPSS 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.
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.
Magic Bullet
Fairview
Parma Community
Diet and Exercise
Metrohealth
Heart Surgery
SW General
Heart Center
CMDS Hospitals by Ads
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
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
Heart Surgery
Diet & Exe.
Heart Center
Fairview
Metro
SW General
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 0
SWG
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.
♦ 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. Weighted Multidimensional Scaling (WMDS)
A. 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 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 0
G 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 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 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.
SW General 1.4118 .3972 SW General 1.5424 .3864 MetroHealth .5972 -.4422 MetroHealth .6525 -.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.
Magic Bullet Heart Surgery
Diet and Exercise
Heart Center
Fairview
Parma Community
SW General
Metrohealth
WMDS Hospitals by Ads (Males)
Magic Bullet
Heart Surgery
Heart Center Parma Community
Diet and Exercise
SW General
Fairview
Metrohealth
WMDS Hospitals by Ads (Females)
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 MALE
DISTANCES FEMALE
DISTANCES
Magic Bullet and Heart Surgery 0.2178 0.2342 Magic Bullet and Diet and Exercise 1.535 0.1494 Magic Bullet and Heart Center 2.002 2.1546 Magic Bullet and Fairview 1.7659 1.8055 Magic Bullet and Parma Community 1.218 1.2236 Magic Bullet and Southwest General 1.763 1.9152 Magic Bullet and Metrohealth 1.5402 1.5638 Heart Surgery and Diet and Exercise 0.2086 0.2253 Heart Surgery and Heart Center 2.0806 2.0294 Heart Surgery and Fairview 1.8396 1.9059 Heart Surgery and Parma Community 1.0523 1.0441 Heart Surgery and SW General 1.9429 2.1165 Heart Surgery and Metrohealth 1.6032 1.653 Diet and Exercise and Heart Center 2.05 2.0092 Diet and Exercise and Fairview 1.6527 1.7017 Diet and Exercise and Parma Community 1.086 1.099 Diet and Exercise and SW General 1.7376 1.8941 Diet and Exercise and Metrohealth 1.4211 1.4538 Heart Center and Fairview 1.788 1.915 Heart Center and Parma Community 1.0983 1.0702 Heart Center and SW General 2.8067 2.9428 Heart Center and Metrohealth 1.655 1.7528 Fairview and Parma Community 1.7589 1.9152
Fairview and SW General 1.1276 1.1363 Fairview and Metrohealth 0.2472 0.2637 Parma Community and SW General 2.4143 2.6183 Parma Community and Metrohealth 1.5256 1.6639 SW 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.
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
3.02.52.01.51.0.50.0
Distances
3.0
2.5
2.0
1.5
1.0
.5
0.0
♦ 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 Options- Number of Rows (Observations/Matrix). 8 Number of Columns (Variables) . . . 8 Number of Matrices . . . . . . 1 Measurement Level . . . . . . . Interval Data Matrix Shape . . . . . . . Symmetric Type . . . . . . . . . . . Dissimilarity Approach to Ties . . . . . . . Leave Tied Conditionality . . . . . . . . Matrix Data Cutoff at . . . . . . . . .000000 Model Options- Model . . . . . . . . . . . Euclid Maximum Dimensionality . . . . . 2 Minimum Dimensionality . . . . . 2 Negative Weights . . . . . . . Not Permitted Output Options- Job Option Header . . . . . . . Printed Data Matrices . . . . . . . . Printed Configurations and Transformations . Plotted Output Dataset . . . . . . . . Not Created Initial Stimulus Coordinates . . . Computed Algorithmic Options- Maximum Iterations . . . . . . 30 Convergence Criterion . . . . . .00100 Minimum S-stress . . . . . . . .00500 Missing Data Estimated by . . . . Ulbounds
Raw (unscaled) Data for Subject 1 1 2 3 4 5 1 .000 2 4.690 .000 3 4.440 4.020 .000 4 4.740 3.710 5.920 .000 5 .000 5.120 4.230 4.040 .000 6 5.120 .000 5.430 4.970 4.060 7 4.230 5.430 .000 4.140 4.960 8 4.040 4.970 4.140 .000 4.180 6 7 8 6 .000 7 4.960 .000 8 4.750 4.500 .000 _ Iteration history for the 2 dimensional solution (in squared distances) Young's S-stress formula 1 is used. Iteration S-stress Improvement 1 .37295 2 .34050 .03246 3 .33447 .00603 4 .33378 .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. For matrix Stress = .20196 RSQ = .75785 _ Configuration derived in 2 dimensions Stimulus Coordinates Dimension Stimulus Stimulus 1 2 Number Name 1 MAGBULL 1.1437 .7485 2 HRTSURG -1.3325 -.7954 3 DIETEXE .9450 -1.2516 4 HRTCENT -.8620 1.1193 5 FAIRVW .8389 .8995 6 METRO -1.3411 -.7282 7 PARMACOM .9433 -1.0970 8 SWGENER -.3353 1.1048 _
Iteration history for the 3 dimensional solution (in squared distances) Young's S-stress formula 1 is used. Iteration S-stress Improvement 1 .12073 2 .11583 .00490 3 .11551 .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. For matrix Stress = .10963 RSQ = .95017 _
Weighted Multidimensional Scaling: Two Dimensions Alscal Procedure Options Data Options- Number of Rows (Observations/Matrix). 8 Number of Columns (Variables) . . . 8 Number of Matrices . . . . . . 2
Measurement Level . . . . . . . Interval Data Matrix Shape . . . . . . . Symmetric Type . . . . . . . . . . . Dissimilarity Approach to Ties . . . . . . . Leave Tied Conditionality . . . . . . . . Matrix Data Cutoff at . . . . . . . . .000000 Model Options- Model . . . . . . . . . . . Indscal Maximum Dimensionality . . . . . 2 Minimum Dimensionality . . . . . 2 Negative Weights . . . . . . . Not Permitted Output Options- Job Option Header . . . . . . . Printed Data Matrices . . . . . . . . Not Printed Configurations and Transformations . Plotted Output Dataset . . . . . . . . Not Created Initial Stimulus Coordinates . . . Computed Initial Subject Weights . . . . . Computed Algorithmic Options- Maximum Iterations . . . . . . 30 Convergence Criterion . . . . . .00100 Minimum S-stress . . . . . . . .00500 Missing Data Estimated by . . . . Ulbounds _ 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 Stress RSQ 1 .243 .644 2 .197 .777 Averaged (rms) over matrices Stress = .22105 RSQ = .71037 _ Configuration derived in 2 dimensions Stimulus Coordinates Dimension Stimulus Stimulus 1 2 Number Name 1 MAGBULL -.3858 1.1261 2 HRTSURG -.6418 1.0086 3 DIETEXE -.3909 .9122 4 HRTCENT -1.0139 -1.8631 5 FAIRVW 1.0387 -.7798 6 PARMACOM -1.1670 -.3413 7 SWGENER 1.7995 .5537 8 METRO .7612 -.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 Dimension Subject Weird- 1 2 Number ness 1 .0759 .6155 .5146 2 .0714 .7347 .4871 Overall importance of each dimension: .4594 .2510 _ Flattened Subject Weights Variable Subject Plot 1 Number Symbol 1 1 -1.0000 2 2 1.0000
WMDS Euclidean Distance
Males and Females Group Space
Dimension 1
2.01.51.0.50.0-.5-1.0-1.5
Dim
ension 2
1.5
1.0
.5
0.0
-.5
-1.0
-1.5
-2.0
metro
swgener
parmacom
fairvw
hrtcent
dietexehrtsurg
magbull
WMDS
Subject Weights
Dimension 1
.74.72.70.68.66.64.62.60
Dim
ension 2
.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
Flattened Subject Weights
Individual differences (weighted) Euclidean distance model
One Dimensional Plot
.6.4.2-.0-.2-.4-.6
Variable 1
1.5
1.0
.5
0.0
-.5
-1.0
-1.5
2
1
Weighted Multidimensional Scaling: Three Dimensions
Alscal Procedure Options Data Options- Number of Rows (Observations/Matrix). 8 Number of Columns (Variables) . . . 8 Number of Matrices . . . . . . 2 Measurement Level . . . . . . . Interval Data Matrix Shape . . . . . . . Symmetric Type . . . . . . . . . . . Dissimilarity Approach to Ties . . . . . . . Leave Tied Conditionality . . . . . . . . Matrix Data Cutoff at . . . . . . . . .000000 Model Options- Model . . . . . . . . . . . Indscal Maximum Dimensionality . . . . . 3 Minimum Dimensionality . . . . . 3 Negative Weights . . . . . . . Not Permitted
Output Options- Job Option Header . . . . . . . Printed Data Matrices . . . . . . . . Not Printed Configurations and Transformations . Plotted Output Dataset . . . . . . . . Not Created Initial Stimulus Coordinates . . . Computed Initial Subject Weights . . . . . Computed Algorithmic Options- Maximum Iterations . . . . . . 30 Convergence Criterion . . . . . .00100 Minimum S-stress . . . . . . . .00500 Missing Data Estimated by . . . . Ulbounds _ Iteration history for the 3 dimensional solution (in squared distances) Young's S-stress formula 1 is used. Iteration S-stress Improvement 0 .18562 1 .18467 2 .17557 .00911 3 .17387 .00170 4 .17235 .00152 5 .17094 .00142 6 .16971 .00122 7 .16876 .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 Stress RSQ 1 .104 .853 2 .104 .884 Averaged (rms) over matrices Stress = .10424 RSQ = .86887 _ Configuration derived in 3 dimensions Stimulus Coordinates Dimension Stimulus Stimulus 1 2 3 Number Name 1 MAGBULL .0752 1.1438 1.1247 2 HRTSURG -.6683 1.0738 -.9630 3 DIETEXE -.2696 1.1741 -.0447 4 HRTCENT -1.2907 -1.6781 .3795 5 FAIRVW .7337 -1.0397 -1.1813 6 PARMACOM -1.2219 -.1611 .7880 7 SWGENER 1.7006 -.0265 -1.4006 8 METRO .9412 -.4863 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 Dimension Subject Weird- 1 2 3 Number ness 1 .1742 .5804 .5666 .4420 2 .2027 .7548 .5105 .2326 Overall importance of each dimension: .4533 .2908 .1247 _ Flattened Subject Weights Variable Subject Plot 1 2 Number Symbol 1 1 -1.0000 1.0000 2 2 1.0000 -1.0000
Derived Stimulus Configuration
Individual differences (weighted) Euclidean distance model
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 Weights
Individual differences (weighted) Euclidean distance model
Dimension 2
.8 .5
.51
.522
.53
.54
.55
1
.56
.7 .4
.57
Dimension 3Dimension 1.6 .3
Scatterplot of Linear Fit
Individual differences (weighted) 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
Flattened Subject Weights
Individual differences (weighted) Euclidean distance model