Multidimensional Scaling and Correspondence Analysis

Hair, et al. 2006. Multivariate Data Analysis Sixth Edition, Chapter 9, US: Prentice Hall

Putu Eka Widya Shanti (29009020)M. Satya Oktamalandi (29009021)

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

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)

How MDS Works

Gathering Similarity Judgment

Creating a perceptual map

Interpreting the axes

THREE STEPS:

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

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

A - 2 13 4 3 8

B - 12 6 5 7

C - 9 10 11

D - 1 14

E - 15

F -

Dimension IBD AC

One Dimensional Perceptual Map of Six Observations

210-1-2

Dimension II

Dimension I

E

B

D

A

F

C

Two Dimensional Perceptual Map of Six Observations

A Decision Framework for Perceptual MappingObjectives 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

Stages 1 – 3 in the Multidimensional Scaling Diagram

Select an approach to Perceptual Mapping

Research Problem

Research Specification

Research Design Issue

Assumption

Type of Evaluation made

To Stage

4

STAGE 1

PreferencesSimilarities Both Similarities and Preferences Measure

Compositional Methods Decompositional Methods

STAGE 2

STAGE 3

Stage 1: OBJECTIVES OF MDS• MDS is an exploratory technique well suited for:

o Identifying unrecognized dimensions used by respondents in making comparisons between objects (brands, products, stores…)

o Providing an objective basis for comparison between objects based on these dimensionso 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 perceptionso Perceptual distances where they indicate how similar/dissimilar objects are to each other, oro “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:o Allows for construction of perceptual maps on a respondent-by-respondent basiso Assessment of variation among individuals. o Provides a basis for segmentation analysis.

• Aggregate (group) analysis:o Creates perceptual maps of one or more groupso Helps understand overall evaluations of objects and/or dimensions employed in those evaluations.o Should be found by using the average evaluations of all respondents within a group.

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.

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:

MDS Several Multivariate Techniques

Perceptual mapping with two basic objectives Portrayal among objects on a defined set of attributes

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

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 usableThe

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

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

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.

From Stage

3

Selecting the basis for the Perceptual Map

SimilarityPreference

Similarity-based Perceptual Maps

Estimation of the perceptual maps

Selecting the dimensionality of

the perceptual map

Identifying the dimensions

Validating the perceptual maps

External analysis

Internal analysis

STAGE 4

STAGE 5

STAGE 6

Preference-based Perceptual Maps

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.

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

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

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.

Correspondence Analysis

•interdependence technique for dimensional reduction and perceptual mapping. It also is known as optimal scaling or scoring, reciprocal averaging or homogeneity analysis.

Correspondence Analysis (CA) :

•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.

The distinguishing characteristics:

•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

Differences with other multivariate

technique :

Decision Framework for CA

Stage 1 : Objective of CA two basic objective :• A

ssociation 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 3 : Assumption in CA :• S

hares 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.

Decision Framework for CA (Cont’d)

Stage 5 : Interpretation the result :• I

nterpreting 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 :• S

ample – 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.

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.

Thank You