Multivariate Analysis
Content
ARVIND BANGER
ASSISTANT PROFESSOR
DEPARTMENT OF MANAGEMENT
FACULTY OF SOCIAL SCIENCES
DEI
ARVIND BANGER
1
MULTIVARIATE ANALYSIS TECHNIQUE
It is used to simultaneously analyze more than two
variables on a sample of observations.
Objective : to represent a collection of massive data
in a simplified way.
i.e. transform a mass of observations into a smaller
number of composite scores in such a way that
they may reflect as much information as possible
contained in the raw data obtained concerning the
research study
ARVIND BANGER
2
All multivariate methods
Are some variables
dependent?
YES
NO
Dependence
methods
Interdependence
methods
How many variables
are dependent?
Are all inputs
Metric?
ONE
Several
Is it metric?
Are they metric?
YES
NO
Multiple
Regression
Multiple
discriminant
analysis
YES
YES
Factor
analysis
NO
Cluster
analysis
Metric
MDS
NO
Non-metric
MDS
Latent structure
analysis
Canonical analysis
MAV
ARVIND BANGER
3
VARIABLES IN MULTIVARIATE ANALYSIS
Explanatory Variable & Criterion Variable
If X may be considered to be the cause of Y, then X is
described as explanatory variable and Y is described as
criterion variable. In some cases both explanatory variable
& criterion variable may consist of a set of many variables
in which case set (X1, X2, X3, …..Xp) may be called a set of
explanatory variables and the set (Y1, Y2, Y3, …..Yp) may be
called a set of criterion variables if the variation of the
former may be supposed to cause the variation of the latter
as whole.
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4
OBSERVABLE VARIABLES & LATENT
VARIABLES
Explanatory variables described above are supposed to be
observable directly in some situations, and if this is so,
the same are termed as observable variables. However,
there are some unobservable variables which may
influence the criterion variables. We call such
unobservable variables as latent variables.
DISCRETE VARIABLE & CONTINUOUS
VARIABLE
Discrete variable is that variable which when measured
may take only the integer value whereas continuous
variable is one which, when measured, can assume any
real value.
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DUMMY VARIABLE
This term is being used in a technical sense and is useful in
algebraic manipulations in context of multivariate analysis.
We call Xi (i = 1,…..,m) a dummy variable, if only one of Xi is
1 and the others are all zero.
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6
IMPORTANT MULTIVARIATE
TECHNIQUES
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7
MULTIPLE DISCRIMINANT ANALYSIS
Through this method we classify individuals or objects
into two or more mutually exclusive & exhaustive
groups on the basis of a set of independent variables.
Used for single dependent variable which in nonmetric
E.g. brand preference, that depends on individual’s
age, education, income etc.
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8
Contd…
E.g. if an individual is 20 years old, has income of Rs 12,000
and 10 years of formal education
If b1, b2, b3 are the weights given to these independent
variables, then his score would be
Z=b1(20)+b2(12000)+b3(10)
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9
FACTOR ANALYSIS
APPLICABLE WHEN there is systematic
interdependence among a set of observed variables
and the researcher wants to find out something more
fundamental/latent which creates this commonality.
E.g. observed variables: income, education,
occupation, dwelling area
latent factor: social class
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10
i.e.
a large set of measured variables is resolved into
a few categories called
factors
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11
MATHEMATICAL BASIS OF FACTOR
ANALYSIS
SCORE MATRIX
Measures (variables)
objects
a
b
c
.
k
1
a1
b1
c1
.
k1
2
a2
b2
c2
.
k2
3
a3
b3
c3
.
k3
.
.
.
.
.
.
.
.
.
.
.
.
N
aN
bN
cN
.
kN
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FEW BASIC TERMS
FACTOR : It is an underlying dimension that account for
several observed variables.
FACTOR LOADING : They are the values which explain
how closely the variables are related to each one of the
factors discovered. It’s absolute size helps in interpreting
the factor.
COMMONALITY (h2) : It shows how much of each factor
is accounted for by the underlying factor taken together.
h2 of the variable=(ith factor loading of factor A)2+(ith
factor loading of factor B)2
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Contd…
EIGEN VALUE : It is the sum of squared values of the factor
loadings relating to a factor. It indicates relative
importance of each factor in accounting the particular set
of variables being analyzed.
TOTAL SUM OF SQUARES : It is the sum of squared
values of factor loadings related to a factor.
FACTOR SCORES : These represent the degree to which
each respondent gets high scores on the group of items that
load high on each factor.
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METHODS OF FACTOR ANALYSIS
Centroid method
Principal-components method
Maximum likelihood method
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CENTROID METHOD
This method tends to maximize the sum of loadings
ILLUSTRATION :
Given the following correlation matrix, R, relating to 8
variables with unities in the diagonal spaces. Work out
the first & second centroid factors:
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Variables
1
Variables
2
3
4
5
6
1
1
2
0.709
3
0.204 0.051
4
0.081 0.089 0.671
1
5
0.262 0.581 0.123
0.22
6
0.113 0.098 0.689 0.798 0.047
7
0.155 0.083 0.582 0.613 0.201 0.801
8
0.774 0.652 0.072 0.111 0.724
7
8
0.709 0.204 0.081 0.262 0.113 0.155 0.774
1
0.051 0.089 0.581 0.098 0.083 0.652
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1
0.671 0.123 0.689 0.582 0.072
0.022 0.798 0.613 0.111
1
0.047 0.201 0.724
1
0.12
0.891
0.12
1
0.152
0.152
1
17
SOLUTION:
As the matrix is positive manifold, the weights of
various variables are taken as (+1) i.e. variables are
simply summed.
a)
The sum of coefficients in each column of the
correlation matrix are worked out.
The sum of these columns (T) is obtained.
The sum of each column obtained as per (a) is divided
by the square root of T obtained in (b), to get centroid
loadings. The full set of loadings so obtained constitute
the first centroid factor.( say A)
b)
c)
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Variables
1
Variables
2
3
4
5
6
1
1
2
0.709
3
0.204 0.051
4
0.081 0.089 0.671
1
5
0.262 0.581 0.123
0.22
6
0.113 0.098 0.689 0.798 0.047
7
0.155 0.083 0.582 0.613 0.201 0.801
8
0.774 0.652 0.072 0.111 0.724
column sum(Si)
7
8
0.709 0.204 0.081 0.262 0.113 0.155 0.774
1
0.051 0.089 0.581 0.098 0.083 0.652
1
0.671 0.123 0.689 0.582 0.072
0.022 0.798 0.613 0.111
1
0.047 0.201 0.724
1
0.12
0.891
0.12
1
0.152
0.152
1
3.662 3.263 3.392 3.385 3.324 3.666 3.587 3.605
Sum of column sums (T) = 27.884, √T=5.281
First centroid
factor A, Si /√T
0.693 0.618 0.642 0.641 0.629 0.694 0.679 0.683
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We can also state the information as:Variables
Factor loadings concerning
first centroid factor A
1
0.693
2
0.618
3
0.642
4
0.641
5
0.629
6
0.694
7
0.679
8
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0.683
20
FINDING SECOND CENTROID FACTOR : The loadings for the
variables on the first centroid factor are multiplied. This is
done for all possible pairs of variables resulting matrix is
named as Q1
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First Matrix of Factor cross product (Q1)
→ 0.693 0.618 0.642 0.641 0.629 0.694 0.679 0.683
↓
0.693
0.48
0.428 0.445 0.444 0.436 0.481 0.471 0.437
First centroid 0.618 0.428 0.382 0.397 0.396 0.389 0.429
factor A
0.42
0.422
0.642 0.445 0.397 0.412 0.412 0.404 0.446 0.436 0.438
0.641 0.444 0.396 0.412 0.411 0.403 0.445 0.435 0.438
0.629 0.436 0.389 0.404 0.403 0.396 0.437 0.427
0.43
0.694 0.481 0.429 0.446 0.445 0.437 0.482 0.471 0.474
0.679 0.471
0.42
0.436 0.435 0.428 0.471 0.461 0.464
0.683 0.473 0.422 0.438 0.438
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0.43
0.474 0.464 0.466
22
Now, Q1 is subtracted element by element from the original matrix
R, resulting in matrix of residual coefficients R1.
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First matrix of residual coefficient(R1 )
1
1
2
0.52
3
0.281 -0.24
Variables 2 0.281 0.618 -0.35
4
5
6
-0.36
0.19
-0.37
-0.31 0.192 -0.33
8
-0.32 0.301
-0.34
0.23
3 -0.24
-0.35 0.588 0.259 -0.28
4 -0.36
-0.31 0.259 0.589 -0.38 0.353 0.178 -0.33
5
0.192 -0.28
0.19
0.43
7
-0.38 0.604 -0.39
0.146 -0.37
-0.22 0.294
6 -0.37
-0.33 0.243 0.353 -0.39 0.518
0.33
7 -0.32
-0.34 0.146 0.178 -0.23
0.33
0.539 -0.31
-0.33 0.294 -0.35
-0.31 0.534
8 0.301 -0.23
-0.37
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-0.35
24
Now reflecting the variables 3, 4, 6, 7 ,we obtain reflected matrix
of residual coefficients R1’ as given below. Again the same method
is repeated to get the centroid factor B
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First matrix of residual coefficient(R1)
1
Variables
2
3*
4*
0.281 0.241 0.363
5
0.19
6*
7*
1
0.52
2
0.281 0.618 0.346 0.307 0.192 0.331 0.337
3* 0.241 0.346 0.588 0.259 0.281
8
0.368 0.316 0.301
0.43
0.23
0.146 0.366
4* 0.363 0.307 0.259 0.589 0.381 0.353 0.178 0.327
5
0.19
0.192 0.281 0.381 0.604
0.39
0.217 0.294
0.39
0.518
0.33
7* 0.316 0.337 0.146 0.178 0.226
0.33
0.539 0.312
6* 0.368 0.331 0.243 0.353
0.354
8
0.301
0.23
0.366 0.327 0.294 0.354 0.312 0.534
∑
2.58
2.642
2.47
2.757 2.558 3.074 2.375 2.718
Sum of column sums(T)=20.987 , √T=4.581
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We can now write the matrix of factor loadings as under:
Variables
Factor loadings
Centroid factor A
Centroid factor B
1
0.693
0.563
2
0.618
0.577
3
0.642
-0.539
4
0.641
-0.602
5
0.629
0.558
6
0.694
-0.63
7
0.678
-0.518
8
0.683
0.593
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Hence centroid factor B, and commonality(h2) is as follows
Variables
Commonality(h2)
Factor loadings
Centroid factor A
Centroid factor B
A2+B2
1
0.693
0.563
0.797
2
0.618
0.577
0.715
3
0.642
-0.539
0.703
4
0.641
-0.602
0.773
5
0.629
0.558
0.707
6
0.694
-0.63
0.879
7
0.678
-0.518
0.729
8
0.683
0.593
0.818
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Proportion of variance
Variables
Factor loadings
Commonality(h2)
Centroid factor A
Centroid factor B
Eigen value
3.49
2.631
6.121
Proportion of
0.44
0.33
0.77
total variance
[44%]
[33%]
[77%]
Proportion of
0.57
0.43
1
common variance
[57%]
[43%]
[100%]
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PRINCIPAL-COMPONENTS METHOD
This method seeks to maximize the sum of squared loadings
of each factor .Hence the factors in this method explain more
variance than the loadings obtained from any other method of
factoring.
Principal components are constructed which are linear
combination of given set of variables.
p1 = a11X1+a12X2+….+a1nXn
p2= a21X1+a22X2+….+a2nXn
and so on till pn
The aij’s are called loadings and worked out in such a way that
PC are uncorrelated(orthogonal) and first PC has maximum
variance.
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ILLUSTRATION:
Take the correlation matrix R for 8 variables and compute:
(i) the first two principal component factors.
(ii) the communality for each variable on the basis of said
two component factors.
(iii) the proportion of total variance as well as the proportion
of common variance explained by each of the two
component factors.
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SOLUTION:
As the correlation matrix is positive manifold we work out the
1st principal component factor as follows:
The vector of column sums is referred to as Ua1 and when it
is normalized by, we call it Va1.
To normalize: square the column sums in Ua1 and then
divide each element in Ua1 by the square root of the sum of
squares.
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32
Variables
1
Variables
2
3
4
5
6
1
1
2
0.709
3
0.204 0.051
4
0.081 0.089 0.671
1
5
0.262 0.581 0.123
0.22
6
0.113 0.098 0.689 0.798 0.047
7
0.155 0.083 0.582 0.613 0.201 0.801
8
0.774 0.652 0.072 0.111 0.724
7
8
0.709 0.204 0.081 0.262 0.113 0.155 0.774
1
0.051 0.089 0.581 0.098 0.083 0.652
1
0.671 0.123 0.689 0.582 0.072
0.022 0.798 0.613 0.111
1
0.047 0.201 0.724
1
0.12
0.891
0.12
1
0.152
0.152
1
column sum Ua1
3.662 3.263 3.392 3.385 3.324 3.666 3.587 3.605
Va1
=
Ua1/normalizing
factor
0.371 0.331 0.334 0.343 0.337 0.372 0.363 0.365
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Normalizing factor:
=√{(3.622)2+(3.263)2+(3.392)2+(3.385)2+(3.324)2+(3.666)2+(3.587)2+
(3.605)2}
=9.868
We now obtain Ua2 by accumulatively multiplying Va1 row
by row into R resulting in:
Ua2 : [1.296, 1.143, 1.201, 1.201, 1.165, 1.308,1.280, 1.275]
Normalizing it we obtain:
Va2 : [0.371, 0.327, 0.344, 0.344, 0.344, 0.374,0.366,0.365]
Va1 and Va2 are almost equal i.e. convergence has occurred .
Finally we compute the loadings as follows:
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34
Variables
(Characteristic
*
vector Va1)
√normalizing
factor
Principal
of Ua2
=
component
1
0.371
*
1.868
=
0.69
2
0.331
*
1.868
=
0.62
3
0.334
*
1.868
=
0.64
4
0.343
*
1.868
=
0.64
5
0.337
*
1.868
=
0.63
6
0.372
*
1.868
=
0.70
7
0.363
*
1.868
=
0.68
8
0.365
*
1.868
=
0.68
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We now find principal component II (acc. to method followed to
obtain centroid factor B earlier) to get:
Variables
Principal component
II
1
0.57
2
0.59
3
-0.52
4
-0.59
5
0.57
6
-0.61
7
-0.49
8
-0.61
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Variables
Principal components
commonality(h2)
I
II
I2+II2
1
0.69
0.57
0.801
2
0.62
59
0.733
3
0.64
-0.52
0.68
4
0.64
-0.59
0.758
5
0.63
0.57
0.722
6
0.7
-61
0.862
7
0.68
-0.49
0.703
8
0.68
-0.61
0.835
Eigen value
3.4914
2.6007
6.0921
Proportion of
0.436
0.325
0.761
total variance
43.6%
32.5%
76%
Proportion of
0.537
0.427
1.00
57%
43%
100%
common varianceARVIND
BANGER
37
MAXIMUM LIKELIHOOD METHOD
If Rs→correlation matrix actually obtained from data in
sample
& Rp→correlation matrix obtained if entire population is
tested.
then ML method seeks to extrapolate what is known in Rs
in the best possible way to estimate Rp.
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38
CLUSTER ANALYSIS
Unlike techniques for analyzing relationship between
variables it attempts to reorganize a differentiated group of
people, events or objects into homogenous subgroups.
STEPS:
Selection of sample to be clustered (buyers, products etc.).
Definition of the variables on which to measure the
objects, events etc. (e.g. market segment characteristics,
product competition definitions etc.)
Computation of similarities among entities (through
correlation, euclidean distances, and other techniques)
Selection of mutually exclusive clusters(maximisation of
within cluster similarity and between cluster differences).
Cluster comparison & validation.
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CA used to segment car buying population into distinct markets
A→ minivan buyers
B →sports car buyers
Income
B
A
Age
Family size
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40
MULTIDIMENSIONAL SCALING (MDS)
This creates a special description of a respondent’s perception about
a product or service and helps business researcher to understand
difficult to measure construct like product quality.
Method:
We may take three type of attribute space, each representing a
multidimensional space
1. Objective space :object positioned in terms of measurable attributes
like object’s weight, flavor and nutritional value.
2. Subjective space : perceptions about object’s flavor, weight and
nutritional value.
3. Preference space: describes respondents preferences using object’s
attributes (ideal point).All objects close to this ideal point are
interpreted as preferred .
Ideal points from many people can be positioned in this preference
space to reveal the pattern and size of preference clusters. Thus
Cluster analysis and MDS can be combined to map market
segments and then examine product designed for those segments.
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41
CONJOINT ANALYSIS
Used in market research & product development
Takes non-metric & independent variables as input
E.g. considering purchase decision of a computer, if we have
3 prices, 3 brands, 3 speeds, 2 levels of educational values, 2
categories of games, & 2 categories of work assistance, then
model will have (3*3*3*2*2*2)=216 decision levels
Objective of Conjoint analysis is to secure Utility Scores, that
represent the importance of each aspect of the product in
buyer’s overall preference rating.
Utility Scores are computed from buyer’s ratings of set of
cards . Each card in the deck describes one possible
configuration of combined product attributes.
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42
Steps followed in Conjoint Analysis
Select the attribute most pertinent to the purchase
decision (called factor).
Find the possible values of attribute (called factor
levels).
After selecting the factors and their levels SPSS determines
the No. of product descriptions necessary to estimate the
utilities. It also builds a file structure for all possible
combinations , generate the subset required for testing ,
produce the card description and analysis the results.
ARVIND BANGER
43
THANK YOU
ARVIND BANGER
44
ASSISTANT PROFESSOR
DEPARTMENT OF MANAGEMENT
FACULTY OF SOCIAL SCIENCES
DEI
ARVIND BANGER
1
MULTIVARIATE ANALYSIS TECHNIQUE
It is used to simultaneously analyze more than two
variables on a sample of observations.
Objective : to represent a collection of massive data
in a simplified way.
i.e. transform a mass of observations into a smaller
number of composite scores in such a way that
they may reflect as much information as possible
contained in the raw data obtained concerning the
research study
ARVIND BANGER
2
All multivariate methods
Are some variables
dependent?
YES
NO
Dependence
methods
Interdependence
methods
How many variables
are dependent?
Are all inputs
Metric?
ONE
Several
Is it metric?
Are they metric?
YES
NO
Multiple
Regression
Multiple
discriminant
analysis
YES
YES
Factor
analysis
NO
Cluster
analysis
Metric
MDS
NO
Non-metric
MDS
Latent structure
analysis
Canonical analysis
MAV
ARVIND BANGER
3
VARIABLES IN MULTIVARIATE ANALYSIS
Explanatory Variable & Criterion Variable
If X may be considered to be the cause of Y, then X is
described as explanatory variable and Y is described as
criterion variable. In some cases both explanatory variable
& criterion variable may consist of a set of many variables
in which case set (X1, X2, X3, …..Xp) may be called a set of
explanatory variables and the set (Y1, Y2, Y3, …..Yp) may be
called a set of criterion variables if the variation of the
former may be supposed to cause the variation of the latter
as whole.
ARVIND BANGER
4
OBSERVABLE VARIABLES & LATENT
VARIABLES
Explanatory variables described above are supposed to be
observable directly in some situations, and if this is so,
the same are termed as observable variables. However,
there are some unobservable variables which may
influence the criterion variables. We call such
unobservable variables as latent variables.
DISCRETE VARIABLE & CONTINUOUS
VARIABLE
Discrete variable is that variable which when measured
may take only the integer value whereas continuous
variable is one which, when measured, can assume any
real value.
ARVIND BANGER
5
DUMMY VARIABLE
This term is being used in a technical sense and is useful in
algebraic manipulations in context of multivariate analysis.
We call Xi (i = 1,…..,m) a dummy variable, if only one of Xi is
1 and the others are all zero.
ARVIND BANGER
6
IMPORTANT MULTIVARIATE
TECHNIQUES
ARVIND BANGER
7
MULTIPLE DISCRIMINANT ANALYSIS
Through this method we classify individuals or objects
into two or more mutually exclusive & exhaustive
groups on the basis of a set of independent variables.
Used for single dependent variable which in nonmetric
E.g. brand preference, that depends on individual’s
age, education, income etc.
ARVIND BANGER
8
Contd…
E.g. if an individual is 20 years old, has income of Rs 12,000
and 10 years of formal education
If b1, b2, b3 are the weights given to these independent
variables, then his score would be
Z=b1(20)+b2(12000)+b3(10)
ARVIND BANGER
9
FACTOR ANALYSIS
APPLICABLE WHEN there is systematic
interdependence among a set of observed variables
and the researcher wants to find out something more
fundamental/latent which creates this commonality.
E.g. observed variables: income, education,
occupation, dwelling area
latent factor: social class
ARVIND BANGER
10
i.e.
a large set of measured variables is resolved into
a few categories called
factors
ARVIND BANGER
11
MATHEMATICAL BASIS OF FACTOR
ANALYSIS
SCORE MATRIX
Measures (variables)
objects
a
b
c
.
k
1
a1
b1
c1
.
k1
2
a2
b2
c2
.
k2
3
a3
b3
c3
.
k3
.
.
.
.
.
.
.
.
.
.
.
.
N
aN
bN
cN
.
kN
ARVIND BANGER
12
FEW BASIC TERMS
FACTOR : It is an underlying dimension that account for
several observed variables.
FACTOR LOADING : They are the values which explain
how closely the variables are related to each one of the
factors discovered. It’s absolute size helps in interpreting
the factor.
COMMONALITY (h2) : It shows how much of each factor
is accounted for by the underlying factor taken together.
h2 of the variable=(ith factor loading of factor A)2+(ith
factor loading of factor B)2
ARVIND BANGER
13
Contd…
EIGEN VALUE : It is the sum of squared values of the factor
loadings relating to a factor. It indicates relative
importance of each factor in accounting the particular set
of variables being analyzed.
TOTAL SUM OF SQUARES : It is the sum of squared
values of factor loadings related to a factor.
FACTOR SCORES : These represent the degree to which
each respondent gets high scores on the group of items that
load high on each factor.
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METHODS OF FACTOR ANALYSIS
Centroid method
Principal-components method
Maximum likelihood method
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CENTROID METHOD
This method tends to maximize the sum of loadings
ILLUSTRATION :
Given the following correlation matrix, R, relating to 8
variables with unities in the diagonal spaces. Work out
the first & second centroid factors:
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Variables
1
Variables
2
3
4
5
6
1
1
2
0.709
3
0.204 0.051
4
0.081 0.089 0.671
1
5
0.262 0.581 0.123
0.22
6
0.113 0.098 0.689 0.798 0.047
7
0.155 0.083 0.582 0.613 0.201 0.801
8
0.774 0.652 0.072 0.111 0.724
7
8
0.709 0.204 0.081 0.262 0.113 0.155 0.774
1
0.051 0.089 0.581 0.098 0.083 0.652
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0.671 0.123 0.689 0.582 0.072
0.022 0.798 0.613 0.111
1
0.047 0.201 0.724
1
0.12
0.891
0.12
1
0.152
0.152
1
17
SOLUTION:
As the matrix is positive manifold, the weights of
various variables are taken as (+1) i.e. variables are
simply summed.
a)
The sum of coefficients in each column of the
correlation matrix are worked out.
The sum of these columns (T) is obtained.
The sum of each column obtained as per (a) is divided
by the square root of T obtained in (b), to get centroid
loadings. The full set of loadings so obtained constitute
the first centroid factor.( say A)
b)
c)
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Variables
1
Variables
2
3
4
5
6
1
1
2
0.709
3
0.204 0.051
4
0.081 0.089 0.671
1
5
0.262 0.581 0.123
0.22
6
0.113 0.098 0.689 0.798 0.047
7
0.155 0.083 0.582 0.613 0.201 0.801
8
0.774 0.652 0.072 0.111 0.724
column sum(Si)
7
8
0.709 0.204 0.081 0.262 0.113 0.155 0.774
1
0.051 0.089 0.581 0.098 0.083 0.652
1
0.671 0.123 0.689 0.582 0.072
0.022 0.798 0.613 0.111
1
0.047 0.201 0.724
1
0.12
0.891
0.12
1
0.152
0.152
1
3.662 3.263 3.392 3.385 3.324 3.666 3.587 3.605
Sum of column sums (T) = 27.884, √T=5.281
First centroid
factor A, Si /√T
0.693 0.618 0.642 0.641 0.629 0.694 0.679 0.683
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We can also state the information as:Variables
Factor loadings concerning
first centroid factor A
1
0.693
2
0.618
3
0.642
4
0.641
5
0.629
6
0.694
7
0.679
8
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0.683
20
FINDING SECOND CENTROID FACTOR : The loadings for the
variables on the first centroid factor are multiplied. This is
done for all possible pairs of variables resulting matrix is
named as Q1
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First Matrix of Factor cross product (Q1)
→ 0.693 0.618 0.642 0.641 0.629 0.694 0.679 0.683
↓
0.693
0.48
0.428 0.445 0.444 0.436 0.481 0.471 0.437
First centroid 0.618 0.428 0.382 0.397 0.396 0.389 0.429
factor A
0.42
0.422
0.642 0.445 0.397 0.412 0.412 0.404 0.446 0.436 0.438
0.641 0.444 0.396 0.412 0.411 0.403 0.445 0.435 0.438
0.629 0.436 0.389 0.404 0.403 0.396 0.437 0.427
0.43
0.694 0.481 0.429 0.446 0.445 0.437 0.482 0.471 0.474
0.679 0.471
0.42
0.436 0.435 0.428 0.471 0.461 0.464
0.683 0.473 0.422 0.438 0.438
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0.43
0.474 0.464 0.466
22
Now, Q1 is subtracted element by element from the original matrix
R, resulting in matrix of residual coefficients R1.
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First matrix of residual coefficient(R1 )
1
1
2
0.52
3
0.281 -0.24
Variables 2 0.281 0.618 -0.35
4
5
6
-0.36
0.19
-0.37
-0.31 0.192 -0.33
8
-0.32 0.301
-0.34
0.23
3 -0.24
-0.35 0.588 0.259 -0.28
4 -0.36
-0.31 0.259 0.589 -0.38 0.353 0.178 -0.33
5
0.192 -0.28
0.19
0.43
7
-0.38 0.604 -0.39
0.146 -0.37
-0.22 0.294
6 -0.37
-0.33 0.243 0.353 -0.39 0.518
0.33
7 -0.32
-0.34 0.146 0.178 -0.23
0.33
0.539 -0.31
-0.33 0.294 -0.35
-0.31 0.534
8 0.301 -0.23
-0.37
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-0.35
24
Now reflecting the variables 3, 4, 6, 7 ,we obtain reflected matrix
of residual coefficients R1’ as given below. Again the same method
is repeated to get the centroid factor B
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First matrix of residual coefficient(R1)
1
Variables
2
3*
4*
0.281 0.241 0.363
5
0.19
6*
7*
1
0.52
2
0.281 0.618 0.346 0.307 0.192 0.331 0.337
3* 0.241 0.346 0.588 0.259 0.281
8
0.368 0.316 0.301
0.43
0.23
0.146 0.366
4* 0.363 0.307 0.259 0.589 0.381 0.353 0.178 0.327
5
0.19
0.192 0.281 0.381 0.604
0.39
0.217 0.294
0.39
0.518
0.33
7* 0.316 0.337 0.146 0.178 0.226
0.33
0.539 0.312
6* 0.368 0.331 0.243 0.353
0.354
8
0.301
0.23
0.366 0.327 0.294 0.354 0.312 0.534
∑
2.58
2.642
2.47
2.757 2.558 3.074 2.375 2.718
Sum of column sums(T)=20.987 , √T=4.581
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We can now write the matrix of factor loadings as under:
Variables
Factor loadings
Centroid factor A
Centroid factor B
1
0.693
0.563
2
0.618
0.577
3
0.642
-0.539
4
0.641
-0.602
5
0.629
0.558
6
0.694
-0.63
7
0.678
-0.518
8
0.683
0.593
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Hence centroid factor B, and commonality(h2) is as follows
Variables
Commonality(h2)
Factor loadings
Centroid factor A
Centroid factor B
A2+B2
1
0.693
0.563
0.797
2
0.618
0.577
0.715
3
0.642
-0.539
0.703
4
0.641
-0.602
0.773
5
0.629
0.558
0.707
6
0.694
-0.63
0.879
7
0.678
-0.518
0.729
8
0.683
0.593
0.818
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Proportion of variance
Variables
Factor loadings
Commonality(h2)
Centroid factor A
Centroid factor B
Eigen value
3.49
2.631
6.121
Proportion of
0.44
0.33
0.77
total variance
[44%]
[33%]
[77%]
Proportion of
0.57
0.43
1
common variance
[57%]
[43%]
[100%]
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PRINCIPAL-COMPONENTS METHOD
This method seeks to maximize the sum of squared loadings
of each factor .Hence the factors in this method explain more
variance than the loadings obtained from any other method of
factoring.
Principal components are constructed which are linear
combination of given set of variables.
p1 = a11X1+a12X2+….+a1nXn
p2= a21X1+a22X2+….+a2nXn
and so on till pn
The aij’s are called loadings and worked out in such a way that
PC are uncorrelated(orthogonal) and first PC has maximum
variance.
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ILLUSTRATION:
Take the correlation matrix R for 8 variables and compute:
(i) the first two principal component factors.
(ii) the communality for each variable on the basis of said
two component factors.
(iii) the proportion of total variance as well as the proportion
of common variance explained by each of the two
component factors.
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SOLUTION:
As the correlation matrix is positive manifold we work out the
1st principal component factor as follows:
The vector of column sums is referred to as Ua1 and when it
is normalized by, we call it Va1.
To normalize: square the column sums in Ua1 and then
divide each element in Ua1 by the square root of the sum of
squares.
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Variables
1
Variables
2
3
4
5
6
1
1
2
0.709
3
0.204 0.051
4
0.081 0.089 0.671
1
5
0.262 0.581 0.123
0.22
6
0.113 0.098 0.689 0.798 0.047
7
0.155 0.083 0.582 0.613 0.201 0.801
8
0.774 0.652 0.072 0.111 0.724
7
8
0.709 0.204 0.081 0.262 0.113 0.155 0.774
1
0.051 0.089 0.581 0.098 0.083 0.652
1
0.671 0.123 0.689 0.582 0.072
0.022 0.798 0.613 0.111
1
0.047 0.201 0.724
1
0.12
0.891
0.12
1
0.152
0.152
1
column sum Ua1
3.662 3.263 3.392 3.385 3.324 3.666 3.587 3.605
Va1
=
Ua1/normalizing
factor
0.371 0.331 0.334 0.343 0.337 0.372 0.363 0.365
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Normalizing factor:
=√{(3.622)2+(3.263)2+(3.392)2+(3.385)2+(3.324)2+(3.666)2+(3.587)2+
(3.605)2}
=9.868
We now obtain Ua2 by accumulatively multiplying Va1 row
by row into R resulting in:
Ua2 : [1.296, 1.143, 1.201, 1.201, 1.165, 1.308,1.280, 1.275]
Normalizing it we obtain:
Va2 : [0.371, 0.327, 0.344, 0.344, 0.344, 0.374,0.366,0.365]
Va1 and Va2 are almost equal i.e. convergence has occurred .
Finally we compute the loadings as follows:
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Variables
(Characteristic
*
vector Va1)
√normalizing
factor
Principal
of Ua2
=
component
1
0.371
*
1.868
=
0.69
2
0.331
*
1.868
=
0.62
3
0.334
*
1.868
=
0.64
4
0.343
*
1.868
=
0.64
5
0.337
*
1.868
=
0.63
6
0.372
*
1.868
=
0.70
7
0.363
*
1.868
=
0.68
8
0.365
*
1.868
=
0.68
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We now find principal component II (acc. to method followed to
obtain centroid factor B earlier) to get:
Variables
Principal component
II
1
0.57
2
0.59
3
-0.52
4
-0.59
5
0.57
6
-0.61
7
-0.49
8
-0.61
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Variables
Principal components
commonality(h2)
I
II
I2+II2
1
0.69
0.57
0.801
2
0.62
59
0.733
3
0.64
-0.52
0.68
4
0.64
-0.59
0.758
5
0.63
0.57
0.722
6
0.7
-61
0.862
7
0.68
-0.49
0.703
8
0.68
-0.61
0.835
Eigen value
3.4914
2.6007
6.0921
Proportion of
0.436
0.325
0.761
total variance
43.6%
32.5%
76%
Proportion of
0.537
0.427
1.00
57%
43%
100%
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MAXIMUM LIKELIHOOD METHOD
If Rs→correlation matrix actually obtained from data in
sample
& Rp→correlation matrix obtained if entire population is
tested.
then ML method seeks to extrapolate what is known in Rs
in the best possible way to estimate Rp.
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CLUSTER ANALYSIS
Unlike techniques for analyzing relationship between
variables it attempts to reorganize a differentiated group of
people, events or objects into homogenous subgroups.
STEPS:
Selection of sample to be clustered (buyers, products etc.).
Definition of the variables on which to measure the
objects, events etc. (e.g. market segment characteristics,
product competition definitions etc.)
Computation of similarities among entities (through
correlation, euclidean distances, and other techniques)
Selection of mutually exclusive clusters(maximisation of
within cluster similarity and between cluster differences).
Cluster comparison & validation.
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CA used to segment car buying population into distinct markets
A→ minivan buyers
B →sports car buyers
Income
B
A
Age
Family size
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MULTIDIMENSIONAL SCALING (MDS)
This creates a special description of a respondent’s perception about
a product or service and helps business researcher to understand
difficult to measure construct like product quality.
Method:
We may take three type of attribute space, each representing a
multidimensional space
1. Objective space :object positioned in terms of measurable attributes
like object’s weight, flavor and nutritional value.
2. Subjective space : perceptions about object’s flavor, weight and
nutritional value.
3. Preference space: describes respondents preferences using object’s
attributes (ideal point).All objects close to this ideal point are
interpreted as preferred .
Ideal points from many people can be positioned in this preference
space to reveal the pattern and size of preference clusters. Thus
Cluster analysis and MDS can be combined to map market
segments and then examine product designed for those segments.
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CONJOINT ANALYSIS
Used in market research & product development
Takes non-metric & independent variables as input
E.g. considering purchase decision of a computer, if we have
3 prices, 3 brands, 3 speeds, 2 levels of educational values, 2
categories of games, & 2 categories of work assistance, then
model will have (3*3*3*2*2*2)=216 decision levels
Objective of Conjoint analysis is to secure Utility Scores, that
represent the importance of each aspect of the product in
buyer’s overall preference rating.
Utility Scores are computed from buyer’s ratings of set of
cards . Each card in the deck describes one possible
configuration of combined product attributes.
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Steps followed in Conjoint Analysis
Select the attribute most pertinent to the purchase
decision (called factor).
Find the possible values of attribute (called factor
levels).
After selecting the factors and their levels SPSS determines
the No. of product descriptions necessary to estimate the
utilities. It also builds a file structure for all possible
combinations , generate the subset required for testing ,
produce the card description and analysis the results.
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THANK YOU
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