Other Multivariate Techniques

1
Other Multivariate Techniques
Chapter 13

• Learning Objectives
• Explain the difference between dependence
• and interdependence techniques.
• Understand how to use factor analysis to
• simplify data analysis.
• Demonstrate the usefulness of cluster analysis.
• Understand when and how to use discriminant
• analysis.

2
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.
3
Factor Analysis
What is it? Why use it?

?
4
Factor Analysis
. . . . an interdependence technique that
combines many variables into a few factors to
simplify our understanding of the data.
5
Exhibit 13-1 Ratings of Fast 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
6
Exhibit 13-2 Factor Analysis of Selection Factors
On Line http//www.burgerking.com http//www.mcdo
nalds.com
7
What can we do with factor analysis?

• Identify the structure of the relationships among
  either variables or respondents.
• Identify representative variables from a much
  larger set of variables for use in subsequent
  analysis.
• Create an entirely new set of variables for use
  in subsequent analysis.

8
Using Factor Analysis

• Extraction Methods
• Number of Factors
• Factor Loadings/Interpretation
• Using with Other Techniques

9
Extraction Methods

• Variance Considerations.
• Component Analysis
• Common Factor
• Rotation Approaches.
• Orthogonal
• Oblique

10
Exhibit 13-3 Types of Variance in Factor Analysis
Error Variance Unique Variance Common
Variance
Principal Components Analysis
Common Factor Analysis
11
Component vs. Common?

• Two Criteria
• 1. Objectives of the factor analysis.
• 2. Amount of prior knowledge about
• the variance in the variables.

12
Exhibit 13-4 Orthogonal and Oblique Rotation of
Factors
987897y98hojhkyuiyiuhbjk

F2 Unrotated
F2 Orthogonal Rotation
F2 Oblique Rotation
X4

.5
X5
X6
1.0
F1
0
.5
X3

X2
X1
F1 Oblique Rotation
F1 Orthogonal Rotation
13
Comparison of Factor Analysis and Cluster Analysis
Variables 1 2 3 Respondent
A 7 6 7 B 6 7 6 C 4 3 4
D 3 4 3
7 6 5 4 3 2 1
Respondent A Respondent B
Score
Respondent C Respondent D
14
Assumptions

• Multicollinearity.
• Measured by MSA (measure of sampling
  adequacy).
• Homogeneity of sample.

15
Number of Factors?

• Latent Root Criterion
• Percentage of Variance

16
Which Factor LoadingsAre Significant?

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

17
Guidelines for Identifying Significant Factor
Loadings Based on Sample Size
Factor Loading Sample Size Needed for
Significance
.30 350 .35 250 .40 200 .45 150 .
50 120 .55 100 .60 85 .65
70 .70 60 .75 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.
18
Exhibit 13-5 Example of Varimax-Rotated
Principal Components Factor Matrix
19
Exhibit 13-7 Descriptive Statistics for Customer
Survey
Descriptive Statistics
20
Exhibit 13-8 Rotated Factor Solution for
Customer Survey Perceptions
21
Exhibit 13-8 Rotated Factor Solution for
Customer Survey Perceptions Continued
22
Interpreting the Factor Matrix

• Steps
• Examine the Factor Matrix of Loadings.
• Identify the Highest Loading for Each Variable.
• Assess Communalities of the Variables.
• Label the Factors.

23
Using Factor Analysis with Other Multivariate
Techniques

• Select Surrogate Variables?
• Create Summated Scales?
• Compute Factor Scores?

24
Cluster Analysis Overview

• What is it?
• Why use it?

25
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.

26
Exhibit 13-9 Three Clusters of Shopper Types
Low Frequency of Looking for Low Prices
High
3
1
2
Low Frequency of Using Coupons
High
27
Scatter Diagram for Cluster Observations
Level of Education
High Low Low High
Brand Loyalty
28
Scatter Diagram for Cluster Observations
High Low Low
High
Level of Education
Brand Loyalty
29
Scatter Diagram for Cluster Observations
Level of Education
High Low Low High
Brand Loyalty
30
Exhibit 13-10 Between and Within Cluster
Variation

Within Cluster Variation
Between Cluster Distances
31
Cluster Analysis
Low Income
High
Wendys
McDonalds
Burger King
Low Preference for Tasty Burgers
High
32
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.

33
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?

34
Research Design Considerations in Using Cluster
Analysis

• Detecting Outliers
• Similarity Measures
• Distance
• Standardizing the Data

35
Cluster Grouping Approaches
Hierarchical
Nonhierarchical
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36
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.
37
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.
38
Exhibit 13-11 Dendogram of Hierarchical
Clustering
1
2
Object Number
3
4
5
1 2 3 4 5
Steps
39
Phase Two Cluster Analysis

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

40
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.

41
HOW MANY CLUSTERS?

• Run cluster examine similarity or distance
  measure for two, three, four, etc. clusters?
• Select number of clusters based on a priori
  criteria, practical judgement, common sense,
  and/or theoretical foundations.

42
Cluster Analysis Example
Variables Used X6 Friendly Employees X11
Courteous Employees X12 Competent Employees
43
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

44
Exhibit 13-13 Characteristics of Two-Group
Cluster Solution
Descriptives
45
Exhibit 13-13 Characteristics of Two-Group
Cluster Solution Continued
ANOVA
46
Exhibit 13-14 Demographic Profiles of Two
Cluster Solution
Descriptives
47
Exhibit 13-14 Demographic Profiles of Two
Cluster Solution Continued
ANOVA
48
Discriminant Analysis

?
What is it? Why use it?
49
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.
50
Exhibit 13-15 Two Dimensional Discriminant
Analysis Plot of Restaurant Customers
Less Important Fun Place for Kids
More Important
McDonalds
Burger King
Less Important Food Taste
More Important
51
What Can We Do With Discriminant Analysis?

• Determine whether statistically significant
  differences exist between the average score
  profiles on a set of variables for two (or more)
  a priori defined groups.
• 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.
• Determine which of the independent variables
  account the most for the differences in the
  average score profiles of the two or more groups.

52
Exhibit 13-16 Scatter Diagram and Projection of
Two-Group Discriminant Analysis
53
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
income X2 education X3 family size X4
? ?
54
Using Discriminant Analysis

• Computational Method.
• Statistical Significance. (Mahalanobis D2 )
• Predictive Accuracy.
• (Hit Ratio)
• Interpretation of Results.

55
Computational Methods

• Simultaneous
• Stepwise

56
Predictive Accuracy

• Group Centroids Z Scores.
• Classification Matrices.
• Cutting Score Determination.
• Hit Ratio.
• Costs of Misclassification.

57
Exhibit 13-17 Discriminant Function Z Axis and
Cutoff Scores
58
Exhibit 13-18 Classification Matrix for Burger
King and McDonalds Customers
Predicted Group Burger King McDonalds Total
BK 160 40 200 (80)
(20) Actual Group McD 10 190 200
(5) (95)
Overall prediction accuracy (hit ratio) 87.5
(160 190 350 / 400 87.5 )
59
Exhibit 13-19 Discriminant Analysis of Customer
Surveys
Classification Results
79 of original grouped cases correctly
classified
60
Exhibit 13-20 Tests of Equality of Group Means
61
Exhibit 13-21 Structure Matrix for Restaurant
Perceptions Variables
Correlations between discriminating variables
and the discriminant function. Variables
ordered by absolute size of correlation within
function.
62
Exhibit 13-22 Means of Independent Variables
for Restaurants
Functions at Group Centroids
Significant 63
Other Multivariate Techniques
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value for business researchers.