Marketing Research

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Marketing Research

• Aaker, Kumar, Day and Leone
• Tenth Edition
• Instructors Presentation Slides

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Chapter Twenty
Discriminant and Canonical Analysis
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Discriminant Analysis

• Used to classify individuals into one of two or
  more alternative groups on the basis of a set of
  measurements
• Used to identify variables that discriminate
  between naturally occurring groups

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Objectives of Discriminant Analysis

• Determining linear combinations of the predictor
  variables to separate groups by measuring
  between-group variation relative to within-group
  variation
• Developing procedures for assigning new objects,
  firms, or individuals, whose profiles, but not
  group identity are known, to one of the two
  groups
• Testing whether significant differences exist
  between the two groups based on the group
  centroids
• Determining which variables count most in
  explaining inter-group differences

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Basic Concept
If we can assume that two populations have the
same variance, then the usual value of C
is where X1 and XII are the mean values for the
two groups, respectively.
Distribution of two populations
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Discriminant Function
Zi b1 X1 b2 X2 b3 X3 ... bn Xn

• Where Z discriminant score
• b discriminant weights
• X predictor (independent)
  variables

In a particular group, each individual has a
discriminant score (zi) S zi centroid (group
mean) where i individual
Indicates most typical location of an individual
from a particular group
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Discriminant Function A Graphical Illustration
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Cut-off Score

• Criterion against which each individuals
  discriminant score is judged to determine into
  which group the individual should be classified

For equal group sizes
For unequal group sizes
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Determination of Significance

• Null Hypothesis In the population, the group
  means the discriminant function are equal
• Ho µA µB
• Generally, predictors with relatively large
  standardized coefficients contribute more to the
  discriminating power of the function
• Canonical or discriminant loadings show the
  variance that the predictor shares with the
  function

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Classification and Validation

• Holdout Method
• Uses part of sample to construct classification
  rule other subsample used for validation
• Uses classification matrix and hit ratio to
  evaluate groups classification
• Uses discriminant weights to generate
  discriminant scores for cases in subsample

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Classification and Validation (Contd.)

• U - method or Cross Validation
• Uses all available data without serious bias in
  estimating error rates
• Estimated classification error rates
• P1 m1/ n1 P2 m2 / n2
• where m1 and m2 number of sample observations
  mis-classified in groups G1 and G2

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Steps in Discriminant Analysis
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Export Data Set
Respid Will(y1) Govt(y2) Train(x5) Size(x1) Exp(x6
) Rev(x2) Years(x3) Prod(x4) 1 4 5 1 49 1 1000 5
.5 6 2 3 4 1 46 1 1000 6.5 4 3 5 4 1 54 1 1000 6
.0 7 4 2 3 1 31 0 3000 6.0 5 5 4 3 1 50 1 2000 6
.5 7 6 5 4 1 69 1 1000 5.5 9 . . . . . . . . . .
. . . . . . . . . . . . . . . . . 115 4 3 1 45
1 2000 6.0 6 116 5 4 1 44 1 2000
5.8 11 117 3 4 1 46 0 1000 7.0 3 118 3 4 1 54 1 10
00 7.0 4 119 4 3 1 49 1 1000 6.5 7 120 4 5 1 54 1
4000 6.5 7
Marketing Research 8th Edition Aaker, Kumar, Day
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Description of Variables
Variable Description Corresponding Name in Output Scale Values
Willingness to Export (Y1) Will 1(definitely not interested) to 5 (definitely interested)
Level of Interest in Seeking Govt Assistance (Y2) Govt 1(definitely not interested) to 5 (definitely interested)
Employee Size (X1) Size Greater than Zero
Firm Revenue (X2) Rev In millions of dollars
Years of Operation in the Domestic Market (X3) Years Actual number of years
Number of Products Currently Produced by the Firm (X4) Prod Actual number
Training of Employees (X5) Train 0 (no formal program) or 1 (existence of a formal program)
Management Experience in International Operation (X6) Exp 0 (no experience) or 1 (presence of experience)
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Export Data Set Discriminant Analysis Results
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Discriminant Analysis Results (Contd.)
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Discriminant Analysis Results (Contd.)
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Multiple Discriminant Analysis

• Number of possible discriminant functions
• Min (p, m-1)
• Where M number of groups
• P number of predictor variables

• Assumptions Underlying the Discriminant Function
• The p independent variables must have a
  multivariate normal
• distribution
• 2. The p x p variancecovariance matrix of the
  independent variables in each of the two groups
  must be the same

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Canonical Correlation Analysis

• Canonical correlation analysis is a multivariate
  statistical model that helps the study of
  interrelationships among sets of multiple
  dependent variables and multiple independent
  variables.
• Sets of variables on each side are combined to
  form linear composites such that the correlation
  between these linear composites (canonical
  variates) is maximized

Y1 Y2 Yn X1 X2 Xn
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Objectives of Canonical Correlation Analysis

• To determine whether two sets of variables are
  independent of one another and estimate the
  magnitude of the relationship between the two
  sets.
• Derive a set of weights for each set (dependent
  and independent) of variables so that the linear
  combinations of each set are maximally
  correlated.
• Explain nature of relationships among sets of
  variables by measuring the relative importance of
  each variable to the canonical functions
  (relationships).

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Canonical Loadings and Roots

• Canonical loadings or canonical structure
  coefficients measure the simple correlation
  between an original observed variable in the
  dependent or independent set and the sets
  canonical variate or the linear composite.
• reflects the variance that the original variable
  shares with the canonical variate or the relative
  contribution of each of the variable to the
  canonical function.
• Canonical roots or the eigenvalues are the
  squared canonical correlations (i.e. correlation
  between dependent and independent canonical
  variate)
• reflects the percentage of variance in the
  dependent canonical variate that can be explained
  by the independent canonical variate.

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Interpreting Canonical Functions

• Sign and magnitude of canonical weights
  (standardized coefficients) on each of the
  canonical functions help to identify the relative
  importance of each of the variables in deriving
  the canonical relationships.
• Maximum number of canonical function that can be
  extracted equals the number of variables in the
  smallest data set (independent set or dependent
  set).
• Redundancy index (the amount of variance in
  canonical variate explained by the other
  canonical variate in the canonical function
  obtained by multiplying the shared variance of
  the variate with the squared canonical
  correlation) helps to overcome the bias and
  uncertainty in using canonical roots as a measure
  of shared variance.

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Limitations of Canonical Correlation Analysis

• Procedures that maximize the correlation do not
  necessarily maximize interpretation of the pairs
  of canonical variates therefore canonical
  solutions are not easily interpretable.
• Rotation of canonical variate (like in factor
  analysis) to improve interpretability is not a
  common practice and not available in most
  computer programs.
• If a non-linear relationship between dimensions
  in a pair is suspected, use of canonical
  correlation may be inappropriate unless the
  variables are transformed or combined to capture
  the non-linear relationship.
• Only orthogonal solution is normally available.
• Changing variable in one set alters the
  composition of canonical variate in the other set
  significantly.
• There is no causal relationship but is only a
  correlational technique.

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Export Data Set Canonical Correlation Results
The CANCORR Procedure Attitude to
Exporting 2 Firm characteristics
6 Observations 120
Adjusted Approximate
Squared Canonical
Canonical Standard Canonical
Correlation
Correlation Error
Correlation 1 0.857700
0.850646 0.024233
0.735649 2 0.434392
0.405915 0.074372
0.188697
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Canonical Correlation Results (Contd.)
Raw Canonical Coefficients for the Attitude to
Exporting
attitude1 attitude2
y1 y1 0.663025751
-0.825828605 y2
y2 0.1747547312 1.1757781282
Raw Canonical Coefficients for the Firm
characteristics
demographics1 demographics2
x1 x1 0.0590789526
0.03138617 x2 x2
0.0001734106 0.0009537723
x3 x3 -0.372885396
0.1278689212 x4
x4 0.1427469498 -0.150119835
x5 x5 0.1194923096
0.4450507388 x6
x6 0.0015015543 -0.164606455
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Canonical Correlation Results (Contd.)
Standardized Canonical
Coefficients for the Attitude to Exporting
attitude1
attitude2 y1
y1 0.8531 -1.0625
y2 y2 0.2003
1.3478 Standardized Canonical
Coefficients for the Firm characteristics
demographics1
demographics2 x1
x1 0.6122 0.3252
x2 x2 0.1641
0.9028 x3
x3 -0.3222 0.1105
x4 x4 0.3605
-0.3791 x5
x5 0.0550 0.2048
x6 x6 0.0007
-0.0738
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Canonical Correlation Results (Contd.)

• Correlations Between the Attitude to Exporting
  and Their Canonical Variables
• attitude1 attitude2
• y1 y1 0.9891
  -0.1470
• y2 y2 0.7798
  0.6261
• Correlations Between the Firm characteristics and
  Their Canonical Variables
• demographics1 demographics2
• x1 x1 0.8771
  0.0208
• x2 x2 0.0223
  0.9038
• x3 x3 -0.4618
  0.4067
• x4 x4 0.7944
  -0.1369
• x5 x5 0.4331
  0.3525
• x6 x6 0.5672
  -0.1114

• Correlations Between the Attitude to Exporting
  and the Canonical Variables of the Firm
  characteristics
• demographics1 demographics2
• y1 y1 0.8484
  -0.0639
• y2 y2 0.6688
  0.2720
• Correlations Between the Firm characteristics and
  the Canonical Variables of the Attitude to
  Exporting
• attitude1 attitude2
• x1 x1 0.7523 0.0090
• x2 x2 0.0191 0.3926
• x3 x3 -0.3961 0.1767
• x4 x4 0.6814 -0.0595
• x5 x5 0.3714 0.1531
• x6 x6 0.4865 -0.04