Factor Analysis with a case study

1. Exploratory Factor Analysis
Dr. K.S.Harish, M.sc, MBA, Ph.D
Associate Professor

2. What is Factor Analysis (FA)?
• Patterns of correlations are identified
and either used as descriptive (PCA) or
as indicative of underlying theory (FA)
• Process of providing an operational
definition for latent construct (through
regression equation)

3. What is Factor Analysis (FA)?
• FA and PCA (principal components
analysis) are methods of data
reduction
– Take many variables and explain them
with a few “factors” or “components”
– Correlated variables are grouped
together and separated from other
variables with low or no correlation

4. Important Concepts
• Anti-image Correlation Matrix
– A matrix of partial correlation among the
variables used in the analysis highlights the
extent to which the variables are related.
– The diagonal terms gives the extent of
sampling adequacy for each variable.
– Off-diagonal terms give the partial correlations
among the variables

5. Important Concepts
• KMO Statistic:
–It is a measure of sampling adequacy
–The range of this statistic is 0 to 1
–Higher the KMO value, more suitable is
the number of observations for
conducting the factor analysis

6. Important Concepts
• Bartlett’s Test of Sphericity:
– It is a test of significance of the overall
correlation among the variables used in
the factor analysis
– It is given in the form of chi-square test
– If the obtained value is more than the
tabulated value, a factor analysis is required

7. Important Concepts
• Cronbach’s Alpha:
– This is a measure of reliability of the dimension of
the manifest variables
– After conducting the Factor Analysis, we use this
to find whether there is uni-dimensionality in the
variables considered for the factor
– The range of this coefficient is 0 – 1.
– Higher the value, more is the measure of
dimensionality of the variables
– This also suggests the extent to which a group of
manifest variables under a factor measure the
same construct

8. Important Concepts
• Manifest Variables:
– The variables on which data are collected
– Observed variables to be used in Factor
Analysis
Examples
– Occupation,
– Education,
– Income and
– Dwelling area related to group of persons

9. Important Concepts
• Factor
– A factor is underlying dimension that accounts for
several observed variables
– There can be one or more factors in a factor
analysis, which may dependent on the number of
variables used in it and the factor extraction option
exercised.
– It is also known as Latent Variable
• E.g.: Occupation, Education, Income having
close relation with socio economic class. This
can be referred to as a factor

10. “Good Factor”
• A good factor:
–Makes sense
–will be easy to interpret
–simple structure
–Lacks complex loadings

11. Important Concepts
• Component Matrix:
– It is a summary or of components or factors
carved out and presented in a tabular form with
columns indicating the factors.
– Number of columns indicate the number of
factors extracted
– A component matrix could be rotated or
unrotated depending on the option exercised to
carve out the number of factors

12. Important Concepts
• Factor Loading:
– The values of a factor are called factor
loadings
– These values represents the variable-factor
correlation
– High value indicates a high correlation of the
variable with the factor
– Absolute size of the loadings that are
important in interpretation and naming of the
factor

13. Important Concepts
• Communality (𝒉 𝟐
):
– It shows how much variance is shared by
each variable in the analysis with respect to
the number of factors extracted in the
Factor Analysis.
– A high value of communality means that not
much of the variables left are left over after
whatever the factors represent is taken into
consideration

14. Important Concepts
• Eigen Value:
– It is also known as latent root
– The sum of squared values of factor loadings under
a factor is referred to as eigenvalue
– Eigen value indicates the relative importance of
each factor in the analysis
– It gives the percent variance attribute to the factor
– This can be determined by the Principle of
component analysis
– The first factor is the highest importance and the
other factors are important the descending order

15. Important Concepts
• Scree Plot:
– It is a graphical representation of the relationship
between the two
– It highlights the relationship between the
eigenvalues and the components in the factor
analysis
– The change in the curvature in the line depicting
the relationship between eigen values and factors
indicate about the change in percent variation
– It can be used to decide on the number of
components or factors to be considered as it
gives change in percent variation from the first
factor to the last one

16. Important Concepts
• Cross Loading
• When we look into initial statistics of a factor
analysis output, we may find that there are
factor loadings that are relatively high on
more than one factor. Such situation is
referred as the output having cross loading of
variables.

17. Important Concepts
• Rotation:
– It is highly useful in identifying the relevant
variables important for a factor
– Unrotated factor matrix often does not show clear
factor loadings with respect to manifest variables
– The reference axes are turned around the origin
till some suitable position of axis is reached where
there is clear factor loading of manifest variables
on different factors with minimum cross loading

18. Important Concepts
• Types of factor rotation
– Orthogonal rotation
• Axes are maintained at 90 degrees
• Varimax, Quartrimax and Equimax rotations are
orthogonal rotations.
• Varimax rotation is the widely used in factor rotation
– Oblique rotation
• It is not necessary to maintain 90 degrees

19. Important Concepts
• Factor Scores:
• It is a composite score that represents the degree to
which each respondent gets scores relating to a
factor.
• Factor scores are calculated with specific procedure
giving weightages to the original score of the
respondent according to the factor weight or the
factor loadings

20. General Steps to FA
• Step 1: Selecting and Measuring a set of
variables in a given domain
• Step 2: Data screening in order to
prepare the correlation matrix
• Step 3: Factor Extraction
• Step 4: Factor Rotation to increase
interpretability
• Step 5: Interpretation
• Further Steps: Validation and Reliability
of the measures

21. Case study
Buying Behaviour of Consumers of Mobile Pohones
• A market researcher is trying to identify the
factors that affect the buying behaviour of
consumers for mobile handsets in the
market. He has identified a number of
attributes to which the consumers may
give different weightage while purchasing
them. The name of the attribute and the
abbreviation to be used while using these
in the statistical package to represent the
variables has been mentioned as follows

22. Variables Considered
• Service: Prompt-after
sales service
• Guarantee: Longer
Guareantee period
• Maintods: Maintenance
options by dealers
• Trendylo: Trendy looks
• Avpropar: Availability of
product parts
• GSM/CDMA: Deciding
factor for buying mobile
• Price: Price of the cell
phone
• Camnet: Inbuilt camera
or net connection options
• Recharge: Shorter
Recharge time
• Repofcom: Reputation of
the company and brand
• Range: Product range
with price and quality
• Weigsize: Weight and size

23. Case study
• The marketing manager is using these attributes
in a questionnaire and eliciting responses from a
sample of 190 respondents on a seven point scale
• Purpose is to find whether groups of attributes
have very close correlations among them or have
the same dimension.
• If they have same dimension the may collectively
point to a construct.(latent variables)

24. Output
• KMO and Bartlet’s test of sphericity
From the table KMO statistic is 0.62, which
suggests that a factor analysis can be performed
with a data set of the number of observations
and the variables
KMO measure of sampling adequacy 0.621
Bartlett’s test of
sphericity
Approx Chi-
square
401.193
Degrees of
freedom
66
Significance 0.000

25. Communalities
Variables Initial Extraction
Service 1.00 0.585
Guarantee 1.00 0.521
Maintods 1.00 0.528
Trendylo 1.00 0.639
Avpropar 1.00 0.696
Price 1.00 0.536
Camnet 1.00 0.573
Recharge 1.00 0.461
Repofcom 1.00 0.664
Ranges 1.00 0.511
GSM/CDMA 1.00 0.635
Weigsize 1.00 0.661

26. Interpretation
• The above table summarizes the
communalities for all the variables used in the
analysis
• The initial communalities refer to percent
variance accounted for by each of the
variables in the analysis
• The value of communality associated with a
factor is found by adding the squares of factor
loadings of the variables across the factors.

27. Total variance explained
Initial Eigen Values Extraction sums of
squared loadings
Rotation sums of
squared loadings
Compon
ent
Total % of
variance
Cumulative
% of
variance
Total % of
variance
Cumulati
ve % of
variance
Total % of
variance
Cumulati
ve % of
variance
1 2.627 21.896 21.896 2.627 21.896 21.896 2.152 17.930 17.930
2 1.926 16.048 37.944 1.926 16.048 37.944 1.861 15.507 33.438
3 1.314 10.949 48.893 1.314 10.949 48.893 1.745 14.542 47.980
4 1.142 9.519 58.412 1.142 9.519 58.412 1.252 10.431 58.411
5 0.937 7.807 66.219
6 0.799 6.655 72.874
7 0.754 6.285 79.159
8 0.705 5.878 85.037
9 0.567 4.723 89.76
10 0.470 3.917 93.677
11 0.437 3.641 97.318
12 0.322 2.681 99.999

28. Interpretation
• Table summarises total variance explained by
different factors.
• The first left hand side column lists the factors with
respective eigenvalues in the second column.
• The number of factors is equal to number of
variables used in the analysis.
• The eigen values of the first four factors are more
than 1, those four factors have been selected in the
analysis
• The eigenvalues and and the percent variation have
been reported in the second part of the table

29. Scree Plot
0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10 12 14
Eigenvalues
Component Number
Scree Plot
Eigen values

30. Output Unrotated Component Matrix
Variables Component
1 2 3 4
Service 0.551 -0.508 -0.120 -0.093
Guarantee 0.512 -0.506 -0.045 -0.045
Maintods 0.496 -0.380 -0.357 -0.099
Trendylo 0.364 0.639 0.273 0.150
Avpropar 0.665 0.237 -0.360 -0.262
Price 0.102 -0.530 0.347 0.354
Camnet 0.102 0.566 -0.152 0.468
Recharge 0.566 0.180 0.264 -0.198
Repofcom 0.205 -0.036 -0.264 0.742
Ranges 0.464 -0.130 0.450 0.276
GSM/CDMA 0.602 0.353 -0.377 -0.075
Weigsize 0.531 0.163 0.582 -0.119

31. Interpretation
• From the unrotated component matrix it may
be seen that the first factor has about 10
variables with a factor loading of more than
0.40, Where as the second and third factors,
there are only two variables in in the fourth
there is only one variable that has a factor
loading of more than 0.40

32. Rotated Component Matrix
Variables Component
1 2 3 4
Service 0.748 0.119 0.073 -0.075
Guarantee 0.704 0.151 -0.006 -0.056
Maintods 0.678 -0.056 0.179 0.182
Trendylo -0.324 0.596 0.303 0.293
Avpropar 0.317 0.207 0.743 0.032
Price 0.377 0.182 -0.587 0.130
Camnet -0.322 0.101 0.263 0.624
Recharge 0.145 0.592 0.283 0.097
Repofcom 0.221 -0.074 -0.069 0.778
Ranges 0.272 0.588 -0.230 0.197
GSM/CDMA 0.196 0.187 0.715 0.224
Weigsize 0.058 0.802 0.044 -0.114

33. Interpretation
• In the rotated factor matrix, the factor
loadings with respect to the different
factors change.
• These factors have variables with relatively
more factor loadings
• Which suggest that there is a clear picture
on factor variable correlations
• This situation is not observed in the
unrotated matrix where we may find cross
loading of one variable with more than one
factor

34. Summary of factors
Factor 1
(After Sales Services)
Factor – 2
(Looks and Ranges)
Factor – 3
(Availability of parts
and add-on technology)
Factor – 4
(Brand &
feature)
Service (0.75) Trendy Look
(0.60)
Availability (0.74) Reputation
(0.78)
Guarantee (0.70) Weight and
Size(0.8)
Add on technology Inbuilt
Camera(0.62)
Maintenance
(0.68)
Ranges (0.58)
Chronbach 0.62 Chronbach 0.6 Chronbach 0.66 Chronbach
0.18
The value closer to the index of one is more reliable. It may be
seen that value of Chrobach’s alpha has been mentioned