Other Multivariate Techniques

1 Hair, Babin, Money & Samouel, Essentials of Business Research, Wiley, 2003.

Learning Objectives:1. Explain the difference between dependence

and interdependence techniques.2. Understand how to use factor analysis to

simplify data analysis.3. Demonstrate the usefulness of cluster analysis.4. Understand when and how to use discriminant analysis.

Other Multivariate Other Multivariate TechniquesTechniquesChapter 13


Dependence vs. Dependence vs. Interdependence TechniquesInterdependence Techniques

Interdependence Techniques = instead of analyzing both sets of variables at the same time, we only examine one set. Thus, we do not compare independent and dependent variables.

Dependence Techniques = variables are divided into independent and dependent sets for analysis purposes.


Factor Analysis Factor Analysis

What is it?Why use it?

??


Factor Analysis Factor Analysis

. . . . an interdependence technique that combines many variables into a few factors to simplify our understanding of the data.


Exhibit 13-1 Ratings of Exhibit 13-1 Ratings of Fast Food RestaurantsFast Food Restaurants

Respondent Taste Portion Freshness Friendly Courteous Competent Size

#1 9 8 7 4 3 4 #2 8 7 8 4 5 3 #3 7 8 9 3 4 3 #4 8 9 7 4 4 3 #5 7 8 7 3 3 3 #6 9 7 8 5 4 5


Friendly

Courteous

Competent

Taste

Portion Size

Freshness

EMPLOYEES

FOOD

FACTORSVARIABLES

Exhibit 13-2 Factor Exhibit 13-2 Factor Analysis of Analysis of Selection FactorsSelection Factors

On Linehttp://www.burgerking.comhttp://www.mcdonalds.com

http://www.burgerking.com/

http://www.mcdonalds.com/




What can we do with factor What can we do with factor analysis?analysis?

1. Identify the structure of the relationships among either variables or respondents.

2. Identify representative variables from a much larger set of variables for use in subsequent analysis.

3. Create an entirely new set of variables for use in subsequent analysis.


Using Factor Using Factor AnalysisAnalysis

Extraction MethodsNumber of FactorsFactor Loadings/InterpretationUsing with Other Techniques


Extraction Methods:Extraction Methods:

Variance Considerations. Component Analysis Common Factor

Rotation Approaches. Orthogonal Oblique


Exhibit 13-3 Types of Variance Exhibit 13-3 Types of Variance in Factor Analysisin Factor Analysis

Error VarianceUnique

Variance

Common Variance

Common Factor Analysis

Principal Components Analysis


Component vs. Component vs. Common?Common?

Two Criteria:

1. Objectives of the factor analysis.

2. Amount of prior knowledge about the variance in the variables.


Exhibit 13-4 Orthogonal Exhibit 13-4 Orthogonal and Oblique Rotation of and Oblique Rotation of

FactorsFactors987897y98hojhkyuiyiuhbjk

0 .5 1.0

.5

F2 Oblique Rotation

F1 Oblique Rotation

F1 Orthogonal Rotation

X4

X5

X6

X3X2

X1

F2 Unrotated

F1

F2 Orthogonal Rotation


Comparison of Factor Analysis and Cluster Comparison of Factor Analysis and Cluster AnalysisAnalysis

Variables

1 2 3

Respondent

A 7 6 7

B 6 7 6

C 4 3 4

D 3 4 3

Scor

e

7654321

Respondent A Respondent BRespondent CRespondent D


Assumptions:Assumptions:

Multicollinearity. Measured by MSA (measure of

sampling adequacy).Homogeneity of sample.


Number of Factors?Number of Factors?

Latent Root CriterionPercentage of Variance


Which Factor LoadingsWhich Factor LoadingsAre Significant?Are Significant?

Customary Criteria = Practical Significance.Sample Size & Statistical Significance.Number of Factors and/or Variables.


Guidelines for Identifying Significant Factor Loadings Guidelines for Identifying Significant Factor Loadings Based on Sample SizeBased on Sample Size

Factor Loading Sample Size Needed for Significance*

.30.30 350350

.35.35 250250

.40.40 200200

.45.45 150150

.50.50 120120

.55.55 100100

.60.60 85 85

.65.65 70 70

.70.70 60 60

.75.75 50 50

*Significance is based on a .05 significance level , a power level of 80 percent, and standard errors assumed to be twice those of conventional correlation coefficients.


Exhibit 13-5 Example of Varimax-Exhibit 13-5 Example of Varimax-Rotated Principal Components Rotated Principal Components

Factor MatrixFactor Matrix

Variables Factor 1 Factor 2 Factor 3 Communality

X1 Friendly .93 .19 .09 .91

X2 Courteous .89 .27 .18 .90

X3 Competent .76 -.21 .27 .70

X4 Taste .11 .76 .31 .69

X5 Portion Size .03 .67 .44 .65

X6 Freshness .19 .81 .24 .75

Total

Sum of squares (eigenvalue) 2.32 1.83 .45 4.60

Percentage of trace* 38.7 30.7 7.5 76.97

Trace = 6.0 (number of variables analyzed)

Loadings


Exhibit 13-7 Descriptive Exhibit 13-7 Descriptive Statistics for Customer SurveyStatistics for Customer Survey

Variables Mean

X1 – Excellent Food Quality 5.53

X2 – Attractive Interior 4.70

X3 – Generous Portions 3.89

X4 – Excellent Food Taste 5.15

X5 – Good Value for the Money

4.33

X6 – Friendly Employees 3.66

Descriptive Statistics

X7 – Appears Clean and Neat 4.11

X8 – Fun Place to Go 3.39

X9 – Wide Variety of Menu Items 5.51

X10 – Reasonable Prices 4.06

X11 – Courteous Employees 2.40

X12 – Competent Employees 2.19


Exhibit 13-8 Rotated Factor Solution for Customer Survey Exhibit 13-8 Rotated Factor Solution for Customer Survey PerceptionsPerceptions

Components (Factors)

1 2 3 4

X4 – Excellent Food Taste .912

X9 – Wide Variety of Menu Items .901

X4 – Excellent Food Quality .883

X6 – Friendly Employees .892

X11 – Courteous Employees .850

X12 – Competent Employees .800

X8 – Fun Place to Go .869

X2 – Attractive Interior .854

X7 – Appears Clean and Neat .751

X3 – Generous Portions .896

X5 – Good Value for Money .775

X10 – Reasonable Prices .754


Exhibit 13-8 Rotated Factor Exhibit 13-8 Rotated Factor Solution for Customer Survey Solution for Customer Survey

Perceptions ContinuedPerceptions Continued

Component Rotation Sums of Squared Loadings % of Variance Cumulative %Total

1 2.543 21.188 21.188

2 2.251 18.758 39.946

3 2.100 17.498 57.444

4 2.060 17.170 74.614


Interpreting the Factor Interpreting the Factor MatrixMatrix

Steps:1. Examine the Factor Matrix of

Loadings.2. Identify the Highest Loading for Each

Variable.3. Assess Communalities of the

Variables.4. Label the Factors.


• Select Surrogate Select Surrogate Variables?Variables?

• Create Summated Scales?Create Summated Scales?

• Compute Factor Scores?Compute Factor Scores?

Using Factor Analysis Using Factor Analysis with with

Other Multivariate Other Multivariate TechniquesTechniques


Cluster Analysis Cluster Analysis OverviewOverview

What is it?

Why use it?


Cluster AnalysisCluster Analysis

. . . an interdependence technique that groups objects (respondents, products, firms, variables, etc.) so that each object is similar to the other objects in the cluster and different from objects in all the other clusters.


11

33

22

Low Frequency of Using Coupons HighLow Frequency of Using Coupons High

Low

Low

Fre

quen

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f Loo

king

for L

ow P

rices

Freq

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Loo

king

for L

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rices

Hig

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igh

Exhibit 13-9 Three Clusters Exhibit 13-9 Three Clusters of Shopper Typesof Shopper Types


HighHigh

LowLow LowLow High High

Scatter Diagram Scatter Diagram for Cluster for Cluster

ObservationsObservationsLe

vel o

f Edu

catio

nLe

vel o

f Edu

catio

n

Brand LoyaltyBrand Loyalty


HighHigh

LowLow LowLow High High



vel o

f Le

vel o

f Ed

ucat

ion

Educ

atio

n



HighHigh

LowLowLowLow HighHigh



vel o

f Ed

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Leve

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Educ

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Exhibit 13-10 Between Exhibit 13-10 Between and Within Cluster and Within Cluster

VariationVariation

Within Cluster Variation

Between Cluster Distances


““McDonald’s”McDonald’s”““Wendy’s”Wendy’s”

““Burger King”Burger King”

Low Preference for Tasty Burgers HighLow Preference for Tasty Burgers High

Low

Low

Inco

me

Hig

hIn

com

e

H

igh

Cluster AnalysisCluster Analysis


Three Phases of Cluster Three Phases of Cluster Analysis:Analysis:

Phase One: Divide the total sample into smaller subgroups.

Phase Two: Verify the subgroups identified are statistically different and theoretically meaningful.

Phase Three: Profile the clusters in terms of demographics, psychographics, and other relevant characteristics.


Questions to Answer in Questions to Answer in Phase One:Phase One:

1. How do we measure the distances between the objects we are clustering?

2. What procedure will be used to group similar objects into clusters?

3. How many clusters will we derive?


Research Design Research Design ConsiderationsConsiderations

in Using Cluster in Using Cluster Analysis:Analysis:

• Detecting Outliers• Similarity Measures

Distance• Standardizing the Data


Go On-Linewww.dssresearch.com

Nonhierarchical

Hierarchical

Cluster Grouping Cluster Grouping ApproachesApproaches

http://www.dssresearch.com/


Hierarchical vs. Hierarchical vs. Nonhierarchical Cluster Nonhierarchical Cluster

ApproachesApproaches

Nonhierarchical = referred to a K-means clustering, these procedures do not involve the tree-like process, but instead select one or more cluster seeds and then objects within a prespecified distance from the cluster seeds are considered to be in a particular cluster.

Hierarchical = develops a hierarchy or tree-like format using either a build-up or divisive approach.


Divisive = starts with all objects as a single cluster and then takes away one object at a time until each object is a separate cluster.

Build-up = also referred to as agglomerative, it starts with all the objects as separate clusters and combines them one at a time until there is a single cluster representing all the objects.

Build-up Build-up vs. Divisive vs. Divisive ApproachesApproaches


Exhibit 13-11 Dendogram of Exhibit 13-11 Dendogram of Hierarchical ClusteringHierarchical Clustering

Obj

ect N

umbe

r1

2

3

4

5

1 2 3 4 5 Steps


Phase Two – Cluster Phase Two – Cluster AnalysisAnalysis

. . . involves verifying that the identified groups are in fact statistically different and theoretically meaningful.


Phase Three – Cluster Phase Three – Cluster AnalysisAnalysis

. . . examines the demographic and other characteristics of the objects in each cluster and attempts to explain why the objects were grouped in the manner they were.


HOW MANY HOW MANY CLUSTERSCLUSTERS??

1. Run cluster; examine similarity or distance measure for two, three, four, etc. clusters?

2. Select number of clusters based on “a priori” criteria, practical judgement, common sense, and/or theoretical foundations.


Cluster Analysis Cluster Analysis ExampleExample

Variables Used:X6 – Friendly EmployeesX11 – Courteous Employees X12 – Competent Employees


Exhibit 13-12 Error Exhibit 13-12 Error Coefficients for Cluster Coefficients for Cluster

SolutionsSolutions

Error Coefficients Error Reduction

Four Clusters = 203.529 3 – 4 Clusters = 48.089 Three Clusters = 251.618 2 – 3 Clusters = 66.969 Two Clusters = 318.587 1 – 2 Clusters = 356.143 One Cluster = 674.730


Exhibit 13-13 Exhibit 13-13 Characteristics of Two-Characteristics of Two-Group Cluster SolutionGroup Cluster Solution

Variables Groups N Means

X6 – Friendly Employees 1 101 4.61

2 99 2.68

Total 200 3.66

X11 – Courteous Employees 1 101 3.04

2 99 1.75

Total 200 2.40

X12 – Competent Employees 1 101 2.83

2 99 1.53

Total 200 2.19

Descriptives


Exhibit 13-13 Characteristics Exhibit 13-13 Characteristics of Two-Group Cluster Solution of Two-Group Cluster Solution

ContinuedContinued

Variables F Sig.

X6 – Friendly Employees Between Groups 300.528 .000

X11 – Courteous Employees Between Groups 171.340 .000

X12 – Competent Employees Between Groups 170.960 .000

ANOVA


Exhibit 13-14 Exhibit 13-14 Demographic Profiles of Demographic Profiles of

Two Cluster SolutionTwo Cluster SolutionVariables Groups N Means

X22 – Gender 1 101 .47

2 99 .47

Total 200 .47

X23 – Age 1 101 2.37

2 99 3.30

Total 200 2.83

X24 – Income 1 101 3.17

2 99 3.80

Total 200 3.48

X25 – Competitor 1 101 .80

2 99 .19

Total 200 .50

Descriptives


Exhibit 13-14 Demographic Exhibit 13-14 Demographic Profiles of Two Cluster Solution Profiles of Two Cluster Solution

ContinuedContinued

Variables F Sig.

X22 – Gender Between Groups .018 .895

X23 – Age Between Groups 38.034 .000

X24 – Income Between Groups 13.913 .000

X25 – Competitor Between Groups 117.356 .000

ANOVA


Discriminant Analysis Discriminant Analysis

What is it?Why use it?

??


Discriminant Analysis Discriminant Analysis

. . . . a dependence technique that is used to predict which group an individual (object) is likely to belong to using two or more metric independent variables. The single dependent variable is non-metric.


““McDonald’s”McDonald’s”

““Burger King”Burger King”

Less Important Food Taste More ImportantLess Important Food Taste More ImportantLess

Impo

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ss Im

port

ant

F

un P

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for

Kid

s

Mor

e Im

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Fun

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ids

M

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Impo

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Exhibit 13-15 Two Dimensional Exhibit 13-15 Two Dimensional Discriminant Analysis Plot of Discriminant Analysis Plot of

Restaurant CustomersRestaurant Customers


What Can We Do WithWhat Can We Do WithDiscriminant AnalysisDiscriminant Analysis??

1. Determine whether statistically significant differences exist between the average score profiles on a set of variables for two (or more) a priori defined groups.

2. Establish procedures for classifying statistical units (individuals or objects) into groups on the basis of their composite Z scores computed from a set of independent variables.

3. Determine which of the independent variables account the most for the differences in the average score profiles of the two or more groups.


X2

X1

Z

B

A

DiscriminantDiscriminantFunctionFunction

A’

B’

Exhibit 13-16 Scatter Diagram and Exhibit 13-16 Scatter Diagram and Projection of Two-Group Projection of Two-Group Discriminant AnalysisDiscriminant Analysis


Z = W1X1 + W2X2 + . . . + WnXn

Potential Independent Variables:

X1 = incomeX2 = educationX3 = family sizeX4 = ? ?

Each respondent has a variate value (Z).

The Z value is a single composite Z score (linear combination) for each individual. It is computed from the entire set of independent variables so that it best achieves the statistical objective.


Using Discriminant Using Discriminant AnalysisAnalysis

Computational Method. Statistical Significance.

(Mahalanobis D2 ) Predictive Accuracy.

(Hit Ratio) Interpretation of Results.


Computational Computational Methods:Methods:

1. Simultaneous

2. Stepwise


Predictive Predictive Accuracy:Accuracy:

Group Centroids & Z Scores.

Classification Matrices. Cutting Score Determination. Hit Ratio. Costs of Misclassification.


Exhibit 13-17 Discriminant Exhibit 13-17 Discriminant Function Z Axis and Cutoff Function Z Axis and Cutoff

ScoresScores

DiscriminantDiscriminant FunctionFunction

DiscriminantDiscriminant FunctionFunction

Z

Z

A B

BA

Cutoff score

Cutoff score

(a)

(b)


Exhibit 13-18 Classification Exhibit 13-18 Classification Matrix for Burger King and Matrix for Burger King and

McDonald’s CustomersMcDonald’s Customers

Predicted Group Burger King McDonald’s Total

BK 160 40 200 (80%) (20%)

ActualGroup

McD 10 190 200 (5%) (95%)

Overall prediction accuracy (hit ratio) = 87.5% (160 + 190 = 350 / 400 = 87.5% )


Exhibit 13-19 Discriminant Analysis of Customer Exhibit 13-19 Discriminant Analysis of Customer SurveysSurveys

Test of Function(s) Wilks’ Lambda Sig.

1 .541 .000

Classification Results

*79% of original grouped cases correctly classified

Predicted Group Membership Total

X25 – Competitor Samouel’s Gino’s

OriginalGroup

Count Samouel’s 80 20 100Gino’s 14 86 100

% Samouel’s 80.0 20.0 100.0Gino’s 14.0 86.0 100.0


Exhibit 13-20 Tests of Equality of Exhibit 13-20 Tests of Equality of Group MeansGroup Means

Variables F Sig.

X1 – Excellent Food Quality 10.954 .001

X4 – Excellent Food Taste 11.951 .001

X6 – Friendly Employees 119.366 .000

X9 – Wide Variety of Menu Items .420 .518

X11 – Courteous Employees 54.821 .000

X12 – Competent Employees 105.073 .000


Exhibit 13-21 Structure Matrix for Exhibit 13-21 Structure Matrix for Restaurant Perceptions VariablesRestaurant Perceptions Variables

Variables Function 1

X6 – Friendly Employees .843

X12 – Competent Employees .791

X11 – Courteous Employees .571

X4 – Excellent Food Taste .267

X1 – Excellent Food Quality .255

X9 – Wide Variety of Menu Items .050

Correlations between discriminating variables and the discriminant function. Variables ordered by absolute size of correlation within function.


Exhibit 13-22 Means of Independent Exhibit 13-22 Means of Independent Variables for RestaurantsVariables for Restaurants

Variables Mean

Samouel’s Gino’s

X1 – Excellent Food Quality* 5.24 5.81

X4 – Excellent Food Taste* 5.16 5.73

X6 – Friendly Employees* 2.89 4.42

X9 – Wide Variety of Menu Items 5.45 5.56

X11 – Courteous Employees* 1.96 2.84

X12 – Competent Employees* 1.62 2.75

Function

X25 – Competitor 1

Samouel’s -.916

Gino’s .916

Functions at Group Centroids* Significant < .05 on a univariate basis.


Other Multivariate Other Multivariate TechniquesTechniques

Go On-Line www.psych.nmsu.edu

Explore this website and identify its value for business researchers.

http://www.psych.nmsu.edu/

Documents

Other Multivariate Techniques