chapter16 slides

7/31/2019 Chapter16 Slides

1/17

An Example of Attribute Based MDS Using

Discriminant Analysis

Problem : A chocolate companywants to draw a perceptual mapusing an attribute based

procedure, of its consumersperceptions regarding its ownbrand and two competing brands.

Assume that it is Nestle against

Cadburys and Amul, for example.


2/17

DATA

Data was collected from 15respondents (5 of each brand), onfive attributes, namely Price,Quality, Availability, Packagingand Taste. The variables aremeasured using different scales,

but a higher value indicates afavourable rating in eachvariables measurement.


3/17

Input Data (High = 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


4/17

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(listwise)


5/17

Univariate F tests

T e s ts o f E q u ality o f G r o u p M e an s

.9 3 6 .4 1 3 2 1 2 .6 7 1

.4 1 8 8 .3 4 9 2 1 2 .0 0 5

.7 2 2 2 .3 1 3 2 1 2 .1 4 1

.4 2 3 8 .1 9 5 2 1 2 .0 0 6

.3 1 4 1 3 .1 3 1 2 1 2 .0 0 1

P R I C E

Q U A LITY

P A C K A G

TA S TE

A V A LB LTY

W ilks '

La m b d a F d f1 d f2 S ig .


6/17

Discrim Functions

E i g e n v a l u e s

4 .7 4 9a 8 1 .4 8 1 .4 .9 0 9

1 .083a 1 8 .6 1 0 0 .0 .7 2 1

Fu n ctio n1

2

Eigenva lue

% o f V a r ian ce

C um ula tive %

C an o n ica lC o rr e la tio n

F ir s t 2 ca no n ica l d is c r im inan t func tio ns w ere us ed in thea na lys is .

a .


7/17

Significance Test

W ilk s ' Lam b d a

.0 8 4 2 4 .8 2 7 1 0 .0 0 6

.4 8 0 7 .3 3 6 4 .1 1 9

Te s t o f Fu n c tio n (s )

1 th ro u g h 2

2

W ilks '

La m b d a C h i-s q u a r e d f S ig .


8/17

Standardised Coeffs.

Standardized Canonical Discriminant Function Coefficients

.207 .7 01

.988 -.454

-.398 -.293

-.136 .986.999 -.122

PRICE

QUALITY

PACKAG

TASTEAVALBLTY

1 2

Function


9/17

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 function.Largest absolute correlation between each variable and

any discriminant function

*.


10/17

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


11/17

Plot of Brands on 2 Dimensions

Canonical Discriminant Functions

Function 1

6420-2-4

Function2

2

1

0

-1

-2

-3

BRAND

Group Centroids

3

2

1

3

2

1


12/17

Putting Variables/Attribute Vectors on theAbove Map

Vectors which represent the original attributescan be located on the above map. If there aremore than 3 brands, we may get more than 2dimensions, and may have to draw more than

one plot of the above type. To plot the attributes on the map above, we can

use the standardized coefficients of the originalvariables in the discriminant function. Forexample, for Taste, the standardized coefficientsare -.136 and .986 on Dimensions 1 and 2respectively. So we can locate this point (-.136, .986) on the map, and draw an arrow from theorigin to that point.


13/17

This will be labeled the Taste vector, andsimilarly, all other vectors can be located,

one for each of the five attributes - Price,Quality, Availability, Packaging and Taste.The length of the arrow represents itseffect in discriminating on each dimension.

Longer arrows pointing more closelytowards a given group centroid representvariables most strongly associated withthe group (or Brand, in this case). Vectorspointing in the opposite direction from agiven group centroid represent lowerassociation with a group.


14/17

Variables with longer vectors in a givendimension, and those closest to a given

axis (dimension represented by thediscriminant function) are contributingmore to the interpretation of thatdimension. Looking at all variables that

contribute to a given axis (dimension), wecan label the dimension as a combinationof those variables.

In this case, the interpretation in terms ofthe variables and their correlation todimensions 1 and 2 can be found from thegraph which follows (on next page).


15/17

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

Dimension

Cadbury

Nestle

Amul

Price

Quality

Taste

Availability

Packaging


16/17

As seen from the graph, Nestle, Cadburyand Amul, the three brands have their

unique positions on the map. In addition,on the same map, we now have plottedvalues of the attributes on the same 2dimensions (each discriminant function

represents a dimension). As we can see,Dimension 1 seems to be a combination of

Availability (closest to the x-axis) andQuality. This is also evident from thestandardized discriminant coefficients for

Availability (.999) and Quality (.988) onDimension 1, from the earlier output table.


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Dimension 2 seems to comprise of Taste andPrice, the two vectors (arrows) that are closest to

the vertical axis. This is also evident from thestandardized coefficients, of .986 and .701respectively, for Taste and Price on Dimension2, from the earlier output table.

Packaging is not useful in defining any of the two

dimensions, as its arrow is not close to either ofthe two dimensions.

Brands and their Association withAttributes/Dimensions

Nestle seems to be stronger on Dimension 1(Availability and Quality), and Cadbury onDimension 2 (Taste and to a lesser extent,Price). Amul scores low on both dimensionscompared to its competitors.

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