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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.
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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.
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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
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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(listwise)
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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 .
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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 .
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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 .
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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
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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 function.Largest absolute correlation between each variable and
any discriminant function
*.
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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
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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
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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.
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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.
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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).
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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
Dimension
Cadbury
Nestle
Amul
Price
Quality
Taste
Availability
Packaging
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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.