Transcript Document 7766711

Chapter 13
Other Multivariate Techniques
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.
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Business Research, Wiley, 2003.
1
Dependence vs. Interdependence
Techniques
Dependence Techniques = variables are divided into
independent and dependent sets for analysis purposes.
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.
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Business Research, Wiley, 2003.
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Factor Analysis
What is it?
?
Hair, Babin, Money & Samouel, Essentials of
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Why use it?
3
Factor Analysis
. . . . an interdependence technique
that combines many variables into
a few factors to simplify our
understanding of the data.
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Exhibit 13-1 Ratings of Fast Food
Restaurants
Respondent Taste Portion Freshness Friendly Courteous Competent
Size
#1
#2
#3
#4
#5
#6
9
8
7
8
7
9
8
7
8
9
8
7
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8
9
7
7
8
4
4
3
4
3
5
3
5
4
4
3
4
4
3
3
3
3
5
5
Exhibit 13-2 Factor Analysis of
Selection Factors
VARIABLES
FACTORS
Friendly
Courteous
EMPLOYEES
Competent
Taste
Portion Size
FOOD
Freshness
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On Line
http://www.burgerking.com
http://www.mcdonalds.com
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What can we do with factor 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.
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Using Factor Analysis
Extraction Methods
Number of Factors
Factor Loadings/Interpretation
Using with Other Techniques
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Extraction Methods:
 Variance Considerations.
 Component Analysis

Common Factor
 Rotation Approaches.
 Orthogonal
 Oblique
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Exhibit 13-3 Types of Variance in Factor
Analysis
Error
Variance
Unique
Variance
Principal Components
Analysis
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Common
Variance
Common Factor Analysis
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Component vs. Common?
Two Criteria:
1. Objectives of the factor analysis.
2. Amount of prior knowledge about
the variance in the variables.
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Exhibit 13-4 Orthogonal and
Oblique Rotation of Factors
9
8
7
8
9
7
y
9
8
h
o
j
h
k
y
u
i
y
i
u
h
b
j
k
F2
Unrotated
F2 Orthogonal Rotation
F2 Oblique Rotation
X4
.5
X5
X6
0
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1.0
.5
F1
X3
X2
X1
F1 Oblique Rotation
F1 Orthogonal Rotation
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Comparison of Factor Analysis and Cluster Analysis
Variables
1
2
3
A
7
6
7
B
6
7
6
C
4
3
4
D
3
4
3
Score
Respondent
7
6
5
4
3
2
1
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Respondent A
Respondent B
Respondent C
Respondent D
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Assumptions:
Multicollinearity.
 Measured by MSA (measure of
sampling adequacy).
Homogeneity of sample.
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Number of Factors?
Latent Root Criterion
Percentage of Variance
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Which Factor Loadings
Are Significant?
Customary Criteria = Practical Significance.
Sample Size & Statistical Significance.
Number of Factors and/or Variables.
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Guidelines for Identifying Significant Factor Loadings Based on Sample Size
Factor Loading
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
Sample Size Needed
for Significance*
350
250
200
150
120
100
85
70
60
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.
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Exhibit 13-5 Example of Varimax-Rotated
Principal Components Factor Matrix
Loadings
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)
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Exhibit 13-7 Descriptive Statistics for
Customer Survey
Descriptive Statistics
Variables
Mean
X1 – Excellent Food Quality
5.53
X2 – Attractive Interior
4.70
X3 – Generous Portions
3.89
X4 – Excellent Food Taste
5.15
X7 – Appears Clean and Neat
4.11
X5 – Good Value for the Money
4.33
X8 – Fun Place to Go
3.39
X6 – Friendly Employees
3.66
X9 – Wide Variety of Menu Items
5.51
X10 – Reasonable Prices
4.06
X11 – Courteous Employees
2.40
X12 – Competent Employees
2.19
19
Hair, Babin, Money & Samouel, Essentials of
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Exhibit 13-8 Rotated Factor Solution for Customer Survey Perceptions
Components (Factors)
1
X4 – Excellent Food Taste
.912
X9 – Wide Variety of Menu Items
.901
X4 – Excellent Food Quality
.883
2
X6 – Friendly Employees
.892
X11 – Courteous Employees
.850
X12 – Competent Employees
.800
3
X8 – Fun Place to Go
.869
X2 – Attractive Interior
.854
X7 – Appears Clean and Neat
.751
4
X3 – Generous Portions
.896
X5 – Good Value for Money
.775
X10 – Reasonable Prices
.754
Hair, Babin, Money & Samouel, Essentials of
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Exhibit 13-8 Rotated Factor Solution for
Customer Survey Perceptions 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
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Interpreting the Factor Matrix
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.
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Using Factor Analysis with
Other Multivariate Techniques
•
Select Surrogate Variables?
•
Create Summated Scales?
•
Compute Factor Scores?
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Cluster Analysis Overview
What is it?
Why use it?
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Cluster 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.
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3
1
2
Low
Frequency of Looking for Low Prices
High
Exhibit 13-9 Three Clusters of
Shopper Types
Low
Frequency of Using Coupons
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High
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Scatter Diagram for
Cluster Observations
Level of Education
High
Low
Low
High
Brand Loyalty
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Business Research, Wiley, 2003.
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Scatter Diagram for
Cluster Observations
Level of Education
High
Low
Low
Brand Loyalty
Hair, Babin, Money & Samouel, Essentials of
Business Research, Wiley, 2003.
High
28
Scatter Diagram for
Cluster Observations
Level of Education
High
Low
Low
Brand Loyalty
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High
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Exhibit 13-10 Between and
Within Cluster Variation
Within Cluster Variation
Between Cluster Distances
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High
Cluster Analysis
“Wendy’s”
Income
“McDonald’s”
Low
“Burger King”
Low
Preference for Tasty Burgers
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High
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Three Phases of Cluster 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.
Hair, Babin, Money & Samouel, Essentials of
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Questions to Answer in 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?
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Research Design Considerations
in Using Cluster Analysis:
•
•
•
Detecting Outliers
Similarity Measures
 Distance
Standardizing the Data
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Cluster Grouping Approaches
Hierarchical
Nonhierarchical
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Hierarchical vs. Nonhierarchical
Cluster Approaches
Hierarchical = develops a hierarchy or tree-like format using
either a build-up or divisive approach.
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.
Hair, Babin, Money & Samouel, Essentials of
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Build-up vs. Divisive Approaches
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.
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.
Hair, Babin, Money & Samouel, Essentials of
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Exhibit 13-11 Dendogram of Hierarchical Clustering
1
Object Number
2
3
4
5
1
2
3
4
5
Steps
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Phase Two – Cluster Analysis
. . . involves verifying that the
identified groups are in fact
statistically different and
theoretically meaningful.
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Phase Three – Cluster Analysis
. . . 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.
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HOW MANY CLUSTERS?
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.
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Cluster Analysis Example
Variables Used:
X6 – Friendly Employees
X11 – Courteous Employees
X12 – Competent Employees
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Exhibit 13-12 Error Coefficients
for Cluster Solutions
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
Hair, Babin, Money & Samouel, Essentials of
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Exhibit 13-13 Characteristics of
Two-Group Cluster Solution
Descriptives
Variables
Groups
N
Means
X6 – Friendly Employees
1
101
4.61
2
99
2.68
Total
200
3.66
1
101
3.04
2
99
1.75
Total
200
2.40
1
101
2.83
2
99
1.53
Total
200
2.19
X11 – Courteous Employees
X12 – Competent Employees
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Exhibit 13-13 Characteristics of TwoGroup Cluster Solution Continued
ANOVA
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
Hair, Babin, Money & Samouel, Essentials of
Business Research, Wiley, 2003.
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Exhibit 13-14 Demographic
Profiles of Two Cluster Solution
Descriptives
Variables
Groups
N
Means
X22 – Gender
1
101
.47
2
99
.47
Total
200
.47
1
101
2.37
2
99
3.30
Total
200
2.83
1
101
3.17
2
99
3.80
Total
200
3.48
1
101
.80
2
99
.19
Total
200
.50
X23 – Age
X24 – Income
X25 – Competitor
Hair, Babin, Money & Samouel, Essentials of
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Exhibit 13-14 Demographic Profiles of
Two Cluster Solution Continued
ANOVA
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
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Business Research, Wiley, 2003.
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Discriminant Analysis
?
What is it?
Why use it?
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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.
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“McDonald’s”
Less Important
Fun Place for Kids
More Important
Exhibit 13-15 Two Dimensional Discriminant
Analysis Plot of Restaurant Customers
“Burger King”
Less Important
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Food Taste
More Important
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What Can We Do With
Discriminant 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.
Hair, Babin, Money & Samouel, Essentials of
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Exhibit 13-16 Scatter Diagram and Projection
of Two-Group Discriminant Analysis
X2
A
B
A’
X1
B’
Discriminant
Function
Hair, Babin, Money & Samouel, Essentials of
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Z
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Z = W1X1 + W2X2 + . . . + WnXn
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.
Potential Independent Variables:
X1
X2
X3
X4
=
=
=
=
income
education
family size
??
Hair, Babin, Money & Samouel, Essentials of
Business Research, Wiley, 2003.
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Using Discriminant Analysis
Computational Method.
Statistical Significance.
(Mahalanobis D2 )
Predictive Accuracy.
(Hit Ratio)
Interpretation of Results.
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Business Research, Wiley, 2003.
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Computational Methods:
1.
Simultaneous
2.
Stepwise
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Predictive Accuracy:
Group Centroids & Z Scores.
Classification Matrices.
 Cutting Score Determination.
 Hit Ratio.
 Costs of Misclassification.
Hair, Babin, Money & Samouel, Essentials of
Business Research, Wiley, 2003.
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Exhibit 13-17 Discriminant Function Z
Axis and Cutoff Scores
A
B
(a)
Z
Discriminant Function
Cutoff score
A
B
(b)
Z
Cutoff score
Hair, Babin, Money & Samouel, Essentials of
Business Research, Wiley, 2003.
Discriminant Function
57
Exhibit 13-18 Classification Matrix for
Burger King and McDonald’s Customers
BK
Predicted Group
Burger King
McDonald’s
160
40
(80%)
(20%)
Total
200
Actual
Group
McD
10
(5%)
190
(95%)
200
Overall prediction accuracy (hit ratio) = 87.5% (160 + 190 = 350 / 400 = 87.5% )
Hair, Babin, Money & Samouel, Essentials of
Business Research, Wiley, 2003.
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Exhibit 13-19 Discriminant Analysis of Customer Surveys
Test of Function(s)
Wilks’ Lambda
Sig.
1
.541
.000
Classification Results
Predicted Group
Membership
Original
Group
Count
%
Total
X25 – Competitor
Samouel’s
Gino’s
Samouel’s
80
20
100
Gino’s
14
86
100
Samouel’s
80.0
20.0
100.0
Gino’s
14.0
86.0
100.0
*79% of original grouped cases correctly classified
Hair, Babin, Money & Samouel, Essentials of
Business Research, Wiley, 2003.
59
Exhibit 13-20 Tests of Equality of Group Means
Variables
F
Sig.
X1 – Excellent Food Quality
10.954
.001
X4 – Excellent Food Taste
11.951
.001
X6 – Friendly Employees
119.366
.000
.420
.518
X11 – Courteous Employees
54.821
.000
X12 – Competent Employees
105.073
.000
X9 – Wide Variety of Menu Items
Hair, Babin, Money & Samouel, Essentials of
Business Research, Wiley, 2003.
60
Exhibit 13-21 Structure Matrix for Restaurant
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.
Hair, Babin, Money & Samouel, Essentials of
Business Research, Wiley, 2003.
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Exhibit 13-22 Means of Independent Variables for Restaurants
Mean
Variables
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
* Significant < .05 on a
Functions at Group Centroids
univariate basis.
Function
X25 – Competitor
Hair, Babin, Money & Samouel, Essentials of
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1
Samouel’s
-.916
Gino’s
.916
62
Other Multivariate Techniques
Go On-Line
www.psych.nmsu.edu
Explore this website and identify its value for
business researchers.
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