Factor Analysis

1. WHAT IS FACTOR ANALYSIS & WHEN WE DO IT?
Purpose of factor analysis is to describe the covariance relationship
among many variables in terms of a few underlying but
UNOBSERVABLE RANDOM QUANTITIES called “FACTORS”.
Suppose variables can be grouped by their correlation i.e. all variables
within a particular group are highly correlated among themselves but
have relatively small correlation with variables of different group.
Then it is conceivable that each group of variables represent a single
factor that is responsible for the observed correlations.

2. MATHEMATICALLY ,
“ p” denotes the number of variables (X1, X2,…,Xp) and
“m” denotes the number of underlying factors (F1, F2,…,Fm).
“ Xj “ is the variable represented in latent factors.
Where j=1,2,3…p.
The factor loadings are aj1, aj2,…,ajm which denotes that aj1 is the factor
loading of j th variable on the 1st factor. The factor loadings give us an idea
about how much the variable has contributed to the factor.
The larger the factor loading the more the variable has contributed to that
factor.

3. THREE STAGES IN FACTOR ANALYSIS :
First, a correlation matrix is generated for all the variables.
Second, factors are extracted from the correlation matrix based on
the correlation coefficients of the variables. (i.e. EXTRACTION)
 Here extraction method is Principal Component Analysis.
Third, the factors are rotated in order to maximize the relationship
between the variables and some of the factors. (i.e. ROTATION)
The rotational method is Varimax .

4. Outputs & its Explanation:

5. CORRELATION MATRIX:

6. Kaiser-Meyer-Olkin (KMO) and Bartlett's Test
:
(measures strength of the relationship among variables)
The KMO measures the sampling adequacy which should be greater
than 0.5 for a satisfactory factor analysis to proceed.
The Bartlett’s test, test the null hypothesis that the correlation matrix is
Identity Matrix. Here the significance value is near about 0. This means the
correlation matrix is not an Identity Matrix.

7. COMMUNALITIES:
(MEASURE HOW MUCH OF THE VARIANCE IN THE VARIABLES HAS BEEN
ACCOUNTED FOR EXTRACTION OF FACTORS)
56.0% of the variance in
UN91 is accounted for
while
78.4% is accounted for
UT94

8. SCREE PLOT & TOTAL VARIANCE EXPLAINED BY THE FACTORS:
The scree plot is a graph
of the Eigen values
against all the factors. The
graph is useful for
determining how many
factors to retain. The point
of interest is where the
curve starts to flatten.
It begins to flatten after
the factor 3 but the Eigen
value of the factor 3 is
less than 1.
So here two factors are
retained.

9. Note that the first factor accounted for 59.9% & second 8.05% of the
Variance.

10. FACTOR MATRIX:
It shows the loading of 15
variables on two extracted
factors.

11. ROTATED FACTOR MATRIX:
Factors are rotated for better
interpretation since unrotated
factors are ambiguous.
The goal of rotation is to attain an
optimal simple structure which
attempts to have each variable
load on as few factors as possible
but
maximizes the number of high
loadings on each variable.
These factors can be used as
variables for further analysis. Factor 1
Factor 2

12. COMPONENT PLOT OF FACTOR 1 & FACTOR 2 :

13. Interpretation of Output :
Looking at the Rotated Factor Matrix , we see that the first
rotated factor is most highly correlated with the following
variables.
Factor 1
loading
I try always to be responsive to the needs of my family and friends. BC86 .694
Protecting society’s weak and vulnerable members is important to me .
UC87 .718
I think it is important that every person in the world have equal
opportunities in life.
UC88 .663
I want everyone to be treated justly, even people I doesn’t know . UC89 .676
One strongly believes that I should care for nature. UN90 .764
It is important to me to work against threats to the world of nature . UN91 .688
Protecting the natural environment from destruction or pollution is
important to me .
UN92 .797
I work to promote harmony and peace among diverse groups. UT93 .730
It is important to me to listen to people who are different from me . UT94 .837
Even when I disagree with people, it is important to me to understand
them.
UT95 .835

14. Factor 2 loadings
It is important to me to be loyal to those who are
close to me .
BD81 .747
I go out of my way to be a dependable and
trustworthy friend .
BD82 .719
I want those I spend time with to be able to rely on
me completely .
BD83 .728
It’s very important to me to help the people dear to
me .
BC84 .784
Caring for the well-being of people I am close to is
important to me.
BC85 .810
And the second rotated factor is most highly correlated with the
following variables.

15. The Equation:
As we have two factors ,we have two factor equations. Using the values from the
component/factor score coefficient matrix, we get the equations
One is
F1=(-.137)BD81 +(-.128)BD82+(-
.143)BD83……+(.228)UT95
Another is
F2=(.304)BD81+(.289)BD82+(.279)BD83+..+(-
.144)UT95

16. DIAGRAMMATIC PRESENTATION:
Based on
Section 3
FACTOR
1
BC86
UC87
UC88
UC89
UN90
UN91
UN92
UT93
UT94
UT95
FACTOR2
BD81
BD82
BD83
BC84
BC85
.694
.718
.663
.676
.764
.688
.797
.730
.837
.835
.747
.719
.728
.784
.810

17. THANKING YOU