# multivariate statistical analysis 2 by jpl7986

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```									Multivariate Analysis: Factor
Analysis, Clustering Methods,
Multidimensional Scaling, and
Conjoint Analysis
Chapter 19
Factor Analysis
• Factor analysis is a class of techniques
which reduce and summarize data

– For example, taking 14 variables, and finding
similarities and reducing those 14 variables to
4 factors

(These reduced variables are known as factors)
Factor Analysis
• Factor analysis is not about making
finding relationships between whole sets
of variables, and finding the strength of
those relationships
Factor Analysis: Key Terms
A set of techniques for finding the number and characteristics of variables underlying
Factor Analysis:         a large number of measurements made on individuals or objects.

A variable or construct that is not directly observable but is developed as a linear
Factor:                  combination of observed variables.

The correlation between a variable and a factor. It is computed by from correlating

A value for each factor that is assigned to each person. It is derived from a
Factor Score:            summation of the derived weights which are applied to the original data
variables.

The common variance of each variable summarized by the factors, or the amount
Communality (h2):        (percent) of each variable that is explained by the factors. The uniqueness
component of a variable’s variance is 1- h2.

The sum of squares of loadings of each factor. It is a measure of the variance of
Eigenvalue:              each factor, and if divided by the number of variables (i.e., the total variance), it
is the percent of variance summarized by the factor.
Factor Analysis—Example
• A grocery store administers a survey to customers, asking them to
rate stores in a variety of traits
–   Convenient / inconvenient location
–   Low-quality / high-quality products
–   Modern / old-fashioned
–   Unfriendly / friendly staff
–   Sophisticated / unsophisticated customers
–   Cluttered / spacious
–   Fast / slow check-out
–   Unorganized / organized layout
–   Enjoyable / unenjoyable shopping experience
–   Dull / exciting
Eigenvalues
Initial Eigenvalues
Factor                Total           % of Variance       Cumulative %
1                    5.448                38.917               38.917
2                    1.523                10.882               49.799
3                    1.245                8.890                58.689
4                    1.096                7.827                66.516
5                    .875                 6.247                72.763

Here are eigenvalues for the different numbers of factors.

Where the eigenvalue is less than one, that means that additional factors
explain less of the variance than variables by themselves. Here, since the
eignenvalues are greater than one up to four factors, there are four factors to be
extracted.

On the right-hand column, it displays the percentage of the variance explained
by the factors.
Factor Components
Factor

Variable          1               2                3                4             Communalities
(h2)

Location                 1.255E-02             .218       1.075E-02              .735                    .587
Quality of products           .789       3.318E-02              .237       -8.115E-02                    .687
Modern                       -.665             .216       -7.144E-02            -.221                    .543
Friendliness of cl…           .199            -.298             .606             .433                    .683
Customers                    -.235             .781            -.139       6.850E-02                     .689
Cluttered                7.027E-02            -.162             .894       5.166E-02                     .834
Check-out                     .170             .720       -3.992E-02            -.326                    .655
Layout                        .323       -5.814E-02             .742             .150                    .681
Shopping experie…            -.353             .448            -.183            -.552                    .664
Reputation                    .724            -.283       9.555E-02              .296                    .702
Service                      -.257             .339            -.393            -.588                    .680
Helpfulness of clerks         .281            -.338             .290             .597                    .634

Selection of prod…            .799       -6.586E-02             .184             .126                    .692

Dull                         -.284             .668            -.227       4.610E-02                     .581

The factor loading scores show the correlation between factors and individual variables on a scale from
-1 to 1. Where a factor loading is less than -0.5 or greater than 0.5 (more or less, depending on
researcher’s judgment), the variable is a component of that factor.
Factor Analysis—Example
• Factor analysis shows that the 14 variables fit into 4
factors

Factor 1           Factor 2    Factor 3                 Factor 4
Quality products   Customers   Friendliness of clerks   Location
Modern stores      Check-out   Cluttered                Shopping experience
Reputation         Dull        Layout                   Service

• For each respondent, a factor score to be used in future
analysis is generated for each respondent by taking the
sum of the products of the variable and a weighting for
that variable
–   Factor1= (Variable1 x Weight1) + (V2 x W2) + …
Cluster Analysis Defined
• Grouping instances into groups that show
as little difference between instances
within the group, and maximum
differences between the different groups
• Techniques designed to identify objects,
people, or variables that are similar with
respect to some criteria or characteristics
Cluster Analysis
• There are many approaches to finding
―clusters‖ or groups of data points with
similar values, always through use of
mathematical formulas
• Most statistical software packages have
tools to do all of the calculations
Applications of Cluster Analysis
• Segmentation
– Breaking consumers into different groups so
that they have similar preferences and
reactions to product configurations or
promotions
• Product positioning
– Allows marketers to see how various products
are positioned relative to competing brands
Multidimensional Scaling (MDS)
• Using distances, or differences between
data points to create a 2- or 3-dimensional
map to represent data
• ―psychological dissimilarity as geometric
distance‖
MDS—Example
MDS models can display data a few
different ways. In this example,
different breakfast breads are scored in
a variety of traits.
In this case, the arrows represent
different vectors, or traits. They begin
at the origin and head outward. Vectors
related—‖inexpensive‖ and ―not highly
filling‖ are related. Vectors heading off
at a 90 degree angle are unrelated, and
are negatively correlated.
The closer a point is to a vector, the
more it exemplifies the trait. Hard rolls
are ―eaten with other foods‖, somewhat
―hard to prepare‖ and not ―mainly for
kids‖
Applications of MDS
• Brand positioning—shows positioning of
all brands relative to product attributes
Conjoint Analysis
• Shows the economic trade-offs people
make when different product traits (brand,
configuration, packaging, etc.) are
combined
• Helps identify both important attributes
and ideal product configurations
Conjoint Methodologies
• Several methodologies exist, each with
drawbacks
–   Two-factor
–   Full profile
–   Choice-based conjoint
–   Self-explicated conjoint
–   Hybrid conjoint
–   Hierarchical bayes
• Ultimately, all methods measure how choices
change when several attributes are combined
Conjoint Example
• Product manager for dairy has the following ice cream
options:
– 3 ice cream formulations (gelato, premium, cheap)
– 12 different base flavors (vanilla, chocolate, strawberry, mocha,
etc.)
– 6 different flavor sworls (marshmallow, chocolate syrup, fudge,
strawberry, caramel)
– 12 different mix-ins (pralines, peanuts, brownie bits, cookie
dough, etc.)
• (3 x 12 x 6 x 12) = 2,592 different flavor combinations
• By giving test subjects the choice of different
combinations of attributes, the relative importance of
each category emerges, and the preferred level or trait in
each falls out
Points to Consider
• Different techniques exist for clustering,
MDS, and conjoint
– Different researchers have different
approaches
– Math behind these techniques is not
complicated, and not statistically validated
– Using these techniques is more an art than a
science

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