1. Multidimensional Scaling and Correspondence Analysis
Hair, et al Multivariate Data Analysis Sixth Edition, Chapter 9, US: Prentice Hall
Putu Eka Widya Shanti ( )
M. Satya Oktamalandi ( )

2. Learning Objectives
Define multidimensional scaling and describe how it is performed
Understand how to create a perceptual map
Select between a decompositional or compositional approach
Determine the comparability and number of objects
Understand the differences between similarity data and preference data
Explain correspondence analysis as a method of perceptual mapping

3. What is Multidimensional Scaling (MDS)?
A procedure that enables a researcher to determine the perceived relative image of a set of objects (firms, products, ideas, or other items associated with commonly held perceptions)
Purpose: transform consumer judgments of overall similarity or preference into distances represented in multidimensional space
Based on the comparison of objects (e.q., product, service, person, aroma)

4. How MDS Works Three steps: Gathering Similarity Judgment
Creating a perceptual map
Interpreting the axes
Three steps:

5. Gathering Similarity Judgment  simple global responses
The statements could be:
Rate the similarity of product A and B on a 10 point scale
Product A is more similar to B than to C
I like product A better than product B

6. Creating a perceptual map
Comparing 6 candy bars  (6x5/2 = 15 pairs)
1 = the most similar pair, 15= the least similar pair
Table 9-1
Similarity Data (Rank Order) for Candy Bar Pairs
Candy Bar A B C D E F - 2 13 4 3 8 12 6 5 7 9 10 11 1 14 15

7. One Dimensional Perceptual Map of Six Observations
Dimension I -2 -1 1 2
One Dimensional Perceptual Map of Six Observations

8. Two Dimensional Perceptual Map of Six Observations
Dimension II F B A
Dimension I E D C
Two Dimensional Perceptual Map of Six Observations

9. A Decision Framework for Perceptual Mapping
Objectives of MDS
Research Design of MDS
Assumptions of MDS Analysis
Deriving the MDS Solution and Assessing Overall Fit
Interpreting the MDS Results
Validating The MDS Results
Six stages:
Stages 1 – 3
Stages 4 - 6

10. Stages 1 – 3 in the Multidimensional Scaling Diagram
Research Problem
STAGE 1
Research Specification
STAGE 2
Select an approach to Perceptual Mapping
Compositional Methods
Decompositional Methods
Research Design Issue
Type of Evaluation made
Similarities
Preferences
Both Similarities and Preferences Measure
STAGE 3
Assumption
To Stage 4

11. Stage 1: OBJECTIVES OF MDS
MDS is an exploratory technique well suited for:
Identifying unrecognized dimensions used by respondents in making comparisons between objects (brands, products, stores…)
Providing an objective basis for comparison between objects based on these dimensions
Identifying specific attributes that may correspond to these dimensions
An MDS solution requires Identification of all relevant objects (e.g., all competing brands within a product category) which set the boundaries for the research question.
Respondents’ provide one or both types of perceptions
Perceptual distances where they indicate how similar/dissimilar objects are to each other, or
“Good-bad” assessments of competing objects (preference comparisons) which assist in identifying combinations of attributes that are valued highly.
MDS can be performed at the individual or group level:
Disaggregate (individual) analysis:
Allows for construction of perceptual maps on a respondent-by-respondent basis
Assessment of variation among individuals.
Provides a basis for segmentation analysis.
Aggregate (group) analysis:
Creates perceptual maps of one or more groups
Helps understand overall evaluations of objects and/or dimensions employed in those evaluations.
Should be found by using the average evaluations of all respondents within a group.

12. Stage 2: Research Design of MDS
Perceptual maps can be generated through decompositional or compositional approaches:
Decompositional approaches are the “traditional” and most common MDS method requiring only overall comparisons of similarity between objects.
Compositional approaches are used when the research objectives involve comparing objects on a defined set of attributes.
The number of objects to be evaluated is a tradeoff between:
A small number of objects to facilitate the respondents’ task.
Four times as many objects as dimensions desired (i.e., 5 objects for one dimension, 9 objects for two dimensions ) to obtain a stable solution.

13. Selection of Either a Decompositional (Attribute – Free) or Compositional (Attribute – Based) Approach
Decompositional
Rely on global/overall measures of similarity from which the perceptual maps and relative positioning of objects are formed
[+] respondents do not have to detail the attributes or the importance of each attribute they evaluated
[-] little guidance for specific action
Compositional
The assessment of similarity in which a defined set of attributes is considered in developing the similarity between objects
[+] Respondent provides detailed evaluation of the attributes  evaluative criteria represented by the dimension is easier to ascertain  offers unique managerial insight into the competitive marketplace
[-] The similarity between objects is limited to only attributes rated by respondents  the researcher must assume some method of combining these attributes to represent overall similarity  data collection efforts is substantial
[-]`Result are not typically available for the individual respondents
Three basic groups:
Perceptual mapping with two basic objectives
Portrayal among objects on a defined set of attributes
MDS
Several Multivariate Techniques

14. Three Basic Groups… Compositional approach
Graphical or pos hoc approaches  semantic differential plots
Conventional multivariate statistical techniques  factor analysis, discriminant analysis
Specialized perceptual mapping methods  correspondence analysis

15. Objects: Their Number and Selection
Selecting Objects
The objects being compared should have some set of underlying attributes that characterize each object and form the basis for comparison by the respondents
Non comparable objects is not usable
The Number of Objects
Four times as many objects as dimensions desired (i.e., 5 objects for one dimension, 9 objects for two dimensions ) to obtain a stable solution.
Objects

16. Similarities versus Preferences Data
Similarities data
Respondent does not apply any ‘good – bad’ aspects of evaluation in the comparison
Represent attribute similarities and perceptual dimensions of comparison but do not reflect any direct insight into determinants of choice
Preference data
Respondent apply any ‘good – bad’ assessment
Reflect preferred choices but may not correspond in any way to the similarity- based position, because respondent may base their choices on entirely different dimensions or criteria

17. Stage 3: Assumption of MDS Analysis
Variation in dimensionality
respondents may vary in the dimensionality they use to form their perceptions of an object (although it is thought that most people judge in terms of a limited number of characteristics or dimensions).
Variation in importance
respondents need not attach the same level of importance to a dimension, even if all respondents perceive this dimension.
Variation over time
judgments of a stimulus in terms of either dimensions or levels of importance are likely to change over time.

18. Selecting the basis for the Perceptual Map STAGE 4
From Stage 3
Selecting the basis for the Perceptual Map
STAGE 4
Preference
Similarity
Preference-based Perceptual Maps
Similarity-based Perceptual Maps
Internal analysis
External analysis
Estimation of the perceptual maps
Selecting the dimensionality of the perceptual map
Identifying the dimensions
STAGE 5
Validating the perceptual maps
STAGE 6

19. Stage 4: Deriving The MDS Solution and Assessing Overall Fit
Selecting the basis for the perceptual map : does the map represent perception of similarity of preference?
Similiarity-based perceptual map : relative positions of objects reflect similarity on perceived dimension
Preference-based perceptual map : preference reflect by position of objects to a ideal points.
Determining an Object’s position in the perceptual map
Select an initial configuration of stimuli at a desired initial dimensionality.
Compute the distance between the stimuli and compare the relationship.
If the measure of fit does not meet a selected predefined stopping value, find a new configuration for which the measure of fit is further minimized.

20. Stage 4: Deriving The MDS Solution and Assessing Overall Fit Cont’d
Selecting the Dimensionality of the Perceptual Map :
Subjective evaluation : using researcher’s evaluation of the perceptual map and determines whether the configuration looks reasonable
Stress measurement : indicates the proportion of the variance of the disparities not accounted for by MDS model. The data analyzed using Kruskal’s stress. Stress is minimized when the objects are placed in a configuration so htat the distances between the objects best match the original distances. Stress always improves with increased dimension
Index of fit (R2 ): the measurement of how well the raw data fit the MDS model. Measures .6 or higher is considered acceptable
Incorporating Preference into MDS : To determine the preferred mix of characteristic for a set of stimuli that predict preference.
Ideal point : the point that represents the most preferred combination of perceived attributes  farther from the ideal should be less preferred
Determining ideal point :
Explicit estimation : from direct response of the subject
Implicit estimation : measured to maximally consistent with individual respondence preference

21. Stage 5: Interpreting the MDS Results
The differences in interpretation of compositional & decompositional methods : based on the amount of information directly provided in the analysis and the generalizability of the results to the actual decision-making process.
For compositional methods, the perceptual map can be directly interpreted with the attributes incorporated in the analysis. Validation using other measures of perception
For decompositional methods, the most important issues is the description of the perceptual dimension and their correspondence to attributes. Evaluation of similarity or preference are done without regard to attributes (avoiding specification error issues) but also can incorrectly translated in the dimension of the evaluation.
Identifying the Dimension :
Subjective procedure : labeling the dimensions of the perceptual map by the respondent
Objective procedure : collects attribute rating for each object and then finds the best correspondence of each attribute to derived perceptual space  PROFIT (Property Fitting) method
The researcher must select the type of procedure that best suits both the objectives of the research and the available information

22. Stage 6: Validating the MDS Results
Issues in validating MDS results :
The only output that can be used for comparative purposes involves the relative positions of the objects, even the positions can be compared, the underlying dimensions have no basis for comparison.
Systematic methods of comparison have not been developed into the statistical programs.
Approaches to Validation :
Split-sample analysis : multiple solutions are generated by either splitting the original sample or collecting new data. Validity is indicated if the results are match.
Comparison of Decompositional vs Compositional solutions : applying both methods to the same sample, decompositional method applied first then compositional method used to confirm the result.
The lack of internal methods of direct comparison between solutions, difficulties of comparing perceptual solution made the approach to validation none of them are completely satisfactory.

23. Correspondence Analysis
Correspondence Analysis (CA) :
interdependence technique for dimensional reduction and perceptual mapping. It also is known as optimal scaling or scoring, reciprocal averaging or homogeneity analysis.
The distinguishing characteristics:
It is a compositional technique, rather than a decompositional approach.
Its most direct application is portraying the “correspondence” of categories of variables.
The unique benefits of CA lie in its abilities for representing rows and columns, in joint space.
Differences with other multivariate technique :
CA can be used with nominal data rater than metric rating of each object on each object.
CA created preceptual maps in a single step, where variables and objects are simultaneously plotted in the perceptual map based directly on the association of variables and objects

24. Decision Framework for CA
Stage 1 : Objective of CA  two basic objective :
Association among only row or column categories: CA can be used to examine the association among the categories of just a row or column. examining the categories of a scale The categories can be compared to see if two can be combined or if they provide
Association between both row and column categories: This portrays the association between categories of the rows and columns, such as product sales by age group.
Stage 2 : Research Design of CA :
Requires only a rectangular data matrix (cross-tabulation) of nonnegative entries. The most common type of input matrix is a contingency table with specific categories defining the rows and columns.
Stage 3 : Assumption in CA :
Shares with the more traditional MDS techniques a relative freedom from assumptions. The use of strictly nonmetric data in its simplest form (cross-tabulated data) represents linear and nonlinear relationships equally well.

25. Decision Framework for CA (Cont’d)
Stage 5 : Interpretation the result :
Interpreting the dimensions to understand the basis for the association among categories.
Assessing the degree of association between categories, either within a row/column or between rows and columns.
Stage 6 : Validation of the result : two key questions concerning generalizability :
Sample – as with all MDS techniques, an emphasis must be made to ensure generalizability through split- or multisample analyses.
Objects – the generalizability of the objects (represented individually and as a set by the categories) must also be established. The sensitivity of the results to the addition or deletion of a category can be evaluated. The goal is to assess whether the analysis is dependent on only a few objects and/or attributes.

26. Rule of Thumb 9.4 CORRESPONDENCE ANALYSIS
Correspondence analysis (CA) is best suited for exploratory research and is not appropriate for hypothesis testing.
CA is a form of compositional technique which requires specification of both objects and attributes to be compared.
Correspondence analysis is very sensitive to outliers which should be eliminated prior to using the technique.
The number of dimensions to be retained in the solution is based on:
Dimensions with inertia (eigenvalues) greater than .2.
Enough dimensions to meet the research objectives (usually two or three).
Dimensions can be “named” based on the decomposition of inertia measures across a dimension:
These values show the extent of association for each category individually with each dimension.
They can be used for description much like loadings in factor analysis.

27. Thank You