Discriminat Analysis

7/28/2019 Discriminat Analysis

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Attribute based Perceptual

Mapping Using Discriminant

Analysis


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An Example of Attribute Based MDS Using Discriminant Analysis

Problem : A chocolate company wants to draw a perceptualmap using an attribute based procedure, of its consumers

perceptions regarding its own brand and two competingbrands. Assume that it is Nestle against Cadburys and

Amul, for example.

Slide 1

DATA

Data was collected from 15 respondents (5 of each brand), on five attributes,namely Price, Quality, Availability, Packaging and Taste. The variables are

measured using different scales, but a higher value indicates a favourable ratingin each variables measurement.


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Input Data (Higher = Better)

BRAND PRICE QUALITY AVAILABILITY PACKAGING TASTE

1 12 34 500 5 18

1 11 35 234 4 15

1 10 36 250 4 14

1 13 22 345 5 12

1 12 23 432 3 13

2 10 14 234 2 15

2 11 17 231 3 11

2 15 23 45 4 10

2 13 14 35 3 12

2 12 15 25 2 10

3 10 22 75 4 8

3 12 24 80 4 7

3 13 28 90 5 10

3 11 17 96 2 12

3 11 18 59 2 6

Slide 2


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Means Output by Brand

Group Statistics

11.6000 1.14018 5 5.000

30.0000 6.89202 5 5.000

4.2000 .83666 5 5.000

14.4000 2.30217 5 5.000

352.2000 114.76149 5 5.000

12.2000 1.92354 5 5.000

16.6000 3.78153 5 5.000

2.8000 .83666 5 5.000

11.6000 2.07364 5 5.000

114.0000 108.41125 5 5.000

11.4000 1.14018 5 5.000

21.8000 4.49444 5 5.000

3.4000 1.34164 5 5.000

8.6000 2.40832 5 5.000

80.0000 14.33527 5 5.000

11.7333 1.38701 15 15.000

22.8000 7.48522 15 15.000

3.4667 1.12546 15 15.000

11.5333 3.22638 15 15.000

182.0667 151.30266 15 15.000

PRICE

QUALITY

PACKAG

TASTE

AVALBLTY

PRICE

QUALITY

PACKAG

TASTE

AVALBLTY

PRICE

QUALITY

PACKAG

TASTE

AVALBLTY

PRICE

QUALITY

PACKAG

TASTE

AVALBLTY

BRAND

1

2

3

Total

Mean Std. Deviation Unweighted Weighted

Valid N (l istwise)

Slide 3


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Univariate F tests

Tests of Equality of Group Means

.936 .413 2 12 .671

.418 8.349 2 12 .005

.722 2.313 2 12 .141

.423 8.195 2 12 .006

.314 13.131 2 12 .001

PRICE

QUALITY

PACKAG

TASTE

AVALBLTY

Wilks'

Lambda F df1 df2 Sig.

Slide 4


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Discrim Functions

Eigenvalues

4.749a 81.4 81.4 .909

1.083a 18.6 100.0 .721

Function

1

2

Eigenvalue % of Variance Cumulative %CanonicalCorrelation

First 2 canonical di scriminant functions were used in theanalysis.

a.

Slide 5


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Significance Test

Wilks' Lambda

.084 24.827 10 .006

.480 7.336 4 .119

Test of Function(s)

1 through 2

2

Wilks'

Lambda Chi-square df Sig.

Slide 6


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Standardised Coeffs.

tandardized Canonical Discriminant Function Coefficients

.207 .701

.988 -.454

-.398 -.293

-.136 .986.999 -.122

PRICE

QUALITY

PACKAG

TASTEAVAL BLT Y

1 2

Function

Slide 7


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Var. Loadings on Functions

Structure Matrix

.664* .294

.517* -.336

.268* -.203

.431 .668*

-.044 .235*

AVALBLTY

QUALITYPACKAG

TASTE

PRICE

1 2

Function

Pooled within-groups correlations between discriminating

variables and standardized canonical discriminant functions

Variables ordered by absolute size of correlation within functioLargest absolute correlation between each variable and

any discriminant function

*.

Slide 8


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Centroids of Brands on Functions

Functions at Group Centroids

2.745 .123

-1.596 1.073

-1.149 -1.196

BRAND1

2

3

1 2

Function

Unstandardized canonical discriminantfunctions evaluated at group means

Slide 9


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Plot of Brands on 2 Dimensions

Canonical Discriminant Functions

Function 1

6420-2-4

2

1

0

-1

-2

-3

BRAND

Group Centroids

3

2

1

3

2

1

Slide 10


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Putting Variables/Attribute Vectors on the Above Map

Vectors which represent the original attributes can be located on theabove map. If there are more than 3 brands, we may get more than2 dimensions, and may have to draw more than one plot of theabove type.

To plot the attributes on the map above, we can use thestandardized coefficients of the original variables in the discriminant

function. For example, for Taste, the standardized coefficients are -.136 and .986 on Dimensions 1 and 2 respectively. So we can locatethis point (-.136, .986) on the map, and draw an arrow from the

origin to that point. This will be labeled the Taste vector, and similarly, all other vectors

can be located, one for each of the five attributes - Price, Quality,

Availability, Packaging and Taste. The length of the arrowrepresents its effect in discriminating on each dimension. Longerarrows pointing more closely towards a given group centroidrepresent variables most strongly associated with the group (orBrand, in this case). Vectors pointing in the opposite direction from agiven group centroid represent lower association with a group.

Slide 11


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Variables with longer vectors in a given dimension, and those

closest to a given axis (dimension represented by the discriminantfunction) are contributing more to the interpretation of thatdimension. Looking at all variables that contribute to a given axis(dimension), we can label the dimension as a combination of thosevariables.

In this case, the interpretation in terms of the variables and theircorrelation to dimensions 1 and 2 can be found from the graphwhich follows (on next page).

Slide 11 Contd...


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Plot of Brands and Attribute Vectors

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1 2 3

Dimension 1

Dimension2

Cadbury

Nestle

Amul

Price

Quality

Taste

Availability

Packaging

Slide 12


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As seen from the graph, Nestle, Cadbury and Amul, the threebrands have their unique positions on the map. In addition, on thesame map, we now have plotted values of the attributes on the

same 2 dimensions (each discriAminant function represents adimension). As we can see, Dimension 1 seems to be a combinationof Availability (closest to the x-axis) and Quality. This is also evidentfrom the standardized discriminant coefficients for Availability (.999)and Quality (.988) on Dimension 1, from the earlier output table.

Dimension 2 seems to comprise of Taste and Price, the two vectors(arrows) that are closest to the vertical axis. This is also evident fromthe standardized coefficients, of .986 and .701 respectively, forTaste and Price on Dimension 2, from the earlier output table.

Packaging is not useful in defining any of the two dimensions, as itsarrow is not close to either of the two dimensions.

Brands and their Association with Attributes/Dimensions

Nestle seems to be stronger on Dimension 1 (Availability andQuality), and Cadbury on Dimension 2 (Taste and to a lesser extent,Price). Amul scores low on both dimensions compared to itscompetitors.

Slide 12 contd...

Documents

Discriminat Analysis