Perceptual mapping by multidimensional scaling MDS

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 These analyses address issues of concern to marketing and advertising professionals and to academic researchers investigating consumer behavior. The reports present original research and cutting edge analyses conducted by faculty and graduate students in the ConsumerIndustrial Research Program at Cleveland State University.

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AVAILABLE PUBLICATIONS: Professional Series

Lyttle, B. & Weizenecker, M. Focus groups: A basic introduction, February, 2005. Arab, F., Blake, B.F., & Neuendorf, K.A. Attracting Internet shoppers in the Iranian market, February, 2003. Liu, C., Blake, B.F., & Neuendorf , K.A. Internet shopping in Taiwan and U.S., February, 2003. Jurik, R., Blake, B.F., & Neuendorf, K.A. Attracting Internet shoppers in the Austrian market, January, 2003. Blake, B.F., & Smith, L. Marketers, Get More Actionable Results for Your Research Dollar!, October, 2002.

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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. Blake, B.F., Dostal, J., & Neuendorf, K.A. Identifying constellations of website features: Documentation of a proposed methodology, February, 2005. Saaka, A., Sidon , C., & Blake, B.F. Laddering: A “How to do it” manual – with a note of caution, February, 2004. Blake, B.F., Schulze, S., & Hughes, J.M. Perceptual mapping by multidimensional scaling: A step by step primer, July, 2003.

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Shamatta, C., Blake, B.F., Neuendorf, K.A, Dostal, J., & Guo, F. Comparing website attribute preferences across nationalities: The case of China, Poland, and the USA, October, 2005. Blake, B.F., Dostal, J., & Neuendorf, K.A. Website feature preference constellations: Conceptualization and measurement, February, 2005. Blake, B.F., Dostal, J., Neuendorf, K.A., Salamon, C., & Cambria, N.A. Attribute preference nets: An approach to specifying desired characteristics of an innovation, February, 2005. Blake, B.F., Neuendorf, K.A., Valdiserri, C.M., & Valdiserri, J. The Online Shopping Profile in the cross-national context: The roles of innovativeness and perceived newness, February, 2005. Blake, B.F., & Neuendorf , K.A. Cross-national differences in website appeal: A framework for assessment, July, 2003. Blake, B.F., Neuendorf , K.A., & Valdiserri , C.M. Appealing to those most likely to shop new websites, June, 2003. Blake, B.F., Neuendorf , K.A., & Valdiserri , C.M. Innovativeness and variety of information shopping, April, 2003.

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

=


.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


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




. . . .

. . . .

. . . .

. . . .

. . . .

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 _

=


.95017

Configuration derived in 3 dimensions


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


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-


. . . .

. . . .

. . . .

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




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. . . .

. . . .

. . . .

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


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. . . .

. . . .

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 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

Perceptual mapping by multidimensional scaling MDS

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